88:
1741:
222:
24:
3459:
24021:
16260:
this process eventually erases all the
Fermionic lines: this is the Euler algorithm to 2-color a graph, which works whenever each vertex has even degree. The number of steps in the Euler algorithm is only equal to the number of independent Fermionic homology cycles in the common special case that all terms in the Lagrangian are exactly quadratic in the Fermi fields, so that each vertex has exactly two Fermionic lines. When there are four-Fermi interactions (like in the Fermi effective theory of the
1859:
9242:
1452:. The Feynman diagrams are much easier to keep track of than "old-fashioned" terms, because the old-fashioned way treats the particle and antiparticle contributions as separate. Each Feynman diagram is the sum of exponentially many old-fashioned terms, because each internal line can separately represent either a particle or an antiparticle. In a non-relativistic theory, there are no antiparticles and there is no doubling, so each Feynman diagram includes only one term.
14948:
18765:
16183:
8817:
1357:
1556:, who devised a similar notation many years earlier. Stueckelberg was motivated by the need for a manifestly covariant formalism for quantum field theory, but did not provide as automated a way to handle symmetry factors and loops, although he was first to find the correct physical interpretation in terms of forward and backward in time particle paths, all without the path-integral.
9651:
14626:
2485:
2732:
18470:
18100:
3444:
12680:
15874:
4988:
16477:
10571:
9237:{\displaystyle \left\langle \phi (k_{1})\phi (k_{2})\phi (k_{3})\phi (k_{4})\right\rangle ={\frac {\delta (k_{1}-k_{2})}{k_{1}^{2}}}{\frac {\delta (k_{3}-k_{4})}{k_{3}^{2}}}+{\frac {\delta (k_{1}-k_{3})}{k_{3}^{2}}}{\frac {\delta (k_{2}-k_{4})}{k_{2}^{2}}}+{\frac {\delta (k_{1}-k_{4})}{k_{1}^{2}}}{\frac {\delta (k_{2}-k_{3})}{k_{2}^{2}}}}
20603:
22287:
15118:
22839:, nonperturbative effects like tunneling do not show up, because any effect that goes to zero faster than any polynomial does not affect the Taylor series. Even bound states are absent, since at any finite order particles are only exchanged a finite number of times, and to make a bound state, the binding force must last forever.
22827:), is one whose Fourier transform is constant—a delta-function potential. The lowest order scattering of the theory reveals the non-relativistic interpretation of this theory—it describes a collection of particles with a delta-function repulsion. Two such particles have an aversion to occupying the same point at the same time.
9850:
10966:
7244:
7578:
6812:
15853:
22003:
2923:
17269:
16268:-integrals than Fermi loops. In this case, the counting rule should apply the Euler algorithm by pairing up the Fermi lines at each vertex into pairs that together form a bosonic factor of the term in the Lagrangian, and when entering a vertex by one line, the algorithm should always leave with the partner line.
4224:
16272:
loops, and when traversing the loop, one can reorder the vertex terms one after the other as one goes around without any sign cost. The exception is when you return to the starting point, and the final half-line must be joined with the unlinked first half-line. This requires one permutation to move the last
5625:
9410:
21600:
15443:
1571:'s derivation from old-fashioned perturbation theory borrowed from the perturbative expansions in statistical mechanics was easier to follow for physicists trained in earlier methods. Feynman had to lobby hard for the diagrams, which confused the establishment physicists trained in equations and graphs.
14229:
14943:{\displaystyle Z=\int e^{{\bar {\psi }}M\psi +{\bar {\eta }}\psi +{\bar {\psi }}\eta }\,D{\bar {\psi }}\,D\psi =\int e^{\left({\bar {\psi }}+{\bar {\eta }}M^{-1}\right)M\left(\psi +M^{-1}\eta \right)-{\bar {\eta }}M^{-1}\eta }\,D{\bar {\psi }}\,D\psi =\mathrm {Det} (M)e^{-{\bar {\eta }}M^{-1}\eta }\,,}
22855:
superposition of particles whose local interactions are described by
Feynman diagrams. When the coupling is small, these become collective processes that involve large numbers of particles, but where the interactions between each of the particles is simple. (The perturbation series of any interacting
15535:
formally gives the equations of motion and the anticommutation relations of the Dirac field, just as the Klein Gordon
Lagrangian in an ordinary path integral gives the equations of motion and commutation relations of the scalar field. By using the spatial Fourier transform of the Dirac field as a new
22842:
But this point of view is misleading, because the diagrams not only describe scattering, but they also are a representation of the short-distance field theory correlations. They encode not only asymptotic processes like particle scattering, they also describe the multiplication rules for fields, the
8061:
1711:
Feynman diagrams are a pictorial representation of a contribution to the total amplitude for a process that can happen in several different ways. When a group of incoming particles are to scatter off each other, the process can be thought of as one where the particles travel over all possible paths,
22879:
The number of
Feynman diagrams at high orders of perturbation theory is very large, because there are as many diagrams as there are graphs with a given number of nodes. Nonperturbative effects leave a signature on the way in which the number of diagrams and resummations diverge at high order. It is
16909:
The one difference is that the sign of one propagator is wrong in the
Lorentz case: the timelike component has an opposite sign propagator. This means that these particle states have negative norm—they are not physical states. In the case of photons, it is easy to show by diagram methods that these
16259:
The rule follows from the observation that the number of Fermi lines at a vertex is always even. Each term in the
Lagrangian must always be Bosonic. A Fermi loop is counted by following Fermionic lines until one comes back to the starting point, then removing those lines from the diagram. Repeating
12867:
diagrams includes only those diagrams each of whose connected components end on at least one external line. The vacuum bubbles are the same whatever the external lines, and give an overall multiplicative factor. The denominator is the sum over all vacuum bubbles, and dividing gets rid of the second
7593:
is the number of configurations, and the sum is of the product of the field values on each configuration. The
Euclidean correlation function is just the same as the correlation function in statistics or statistical mechanics. The quantum mechanical correlation functions are an analytic continuation
16285:
This rule is the only visible effect of the exclusion principle in internal lines. When there are external lines, the amplitudes are antisymmetric when two Fermi insertions for identical particles are interchanged. This is automatic in the source formalism, because the sources for Fermi fields are
11494:
The number of ways to form a given
Feynman diagram by joining together half-lines is large, and by Wick's theorem, each way of pairing up the half-lines contributes equally. Often, this completely cancels the factorials in the denominator of each term, but the cancellation is sometimes incomplete.
11406:
A forest diagram is one where all the internal lines have momentum that is completely determined by the external lines and the condition that the incoming and outgoing momentum are equal at each vertex. The contribution of these diagrams is a product of propagators, without any integration. A tree
1727:
that represent the interaction of particles rather than the physical position of the particle during a scattering process. Unlike a bubble chamber picture, only the sum of all the
Feynman diagrams represent any given particle interaction; particles do not choose a particular diagram each time they
22863:
This means that nonperturbative effects show up asymptotically in resummations of infinite classes of diagrams, and these diagrams can be locally simple. The graphs determine the local equations of motion, while the allowed large-scale configurations describe non-perturbative physics. But because
12245:
9247:
An interpretation of Wick's theorem is that each field insertion can be thought of as a dangling line, and the expectation value is calculated by linking up the lines in pairs, putting a delta function factor that ensures that the momentum of each partner in the pair is equal, and dividing by the
4800:
and in the limit that A and B recede to the infinite past and the infinite future, the only contribution that matters is from the ground state (this is only rigorously true if the path-integral is defined slightly rotated into imaginary time). The path integral can be thought of as analogous to a
20984:
6162:
is equivalent to integrating over all
Fourier modes, because taking a Fourier transform is a unitary linear transformation of field coordinates. When you change coordinates in a multidimensional integral by a linear transformation, the value of the new integral is given by the determinant of the
1478:
In addition to their value as a mathematical tool, Feynman diagrams provide deep physical insight into the nature of particle interactions. Particles interact in every way available; in fact, intermediate virtual particles are allowed to propagate faster than light. The probability of each final
16271:
To clarify and prove the rule, consider a Feynman diagram formed from vertices, terms in the Lagrangian, with Fermion fields. The full term is Bosonic, it is a commuting element of the Grassmann algebra, so the order in which the vertices appear is not important. The Fermi lines are linked into
3817:
22050:
produces a superposition of Wigner particle states. In the free field theory, the field produces one particle states only. But when there are interactions, the field operator can also produce 3-particle, 5-particle (if there is no +/− symmetry also 2, 4, 6 particle) states too. To compute the
22036:
neutrons, electrons and photons, but it excludes quarks, which are permanently confined, so the modern point of view is more accommodating: a particle is anything whose interaction can be described in terms of Feynman diagrams, which have an interpretation as a sum over particle trajectories.
18770:
The diagrams are derived from this action. The propagator for the spin-1 fields has the usual Feynman form. There are vertices of degree 3 with momentum factors whose couplings are the structure constants, and vertices of degree 4 whose couplings are products of structure constants. There are
4795:
2245:
22035:
gave a mathematical definition for single-particle states: they are a collection of states that form an irreducible representation of the Poincaré group. Single particle states describe an object with a finite mass, a well defined momentum, and a spin. This definition is fine for protons and
23146:
In this solution, the 'negative energy states' appear in a form which may be pictured (as by Stückelberg) in space-time as waves traveling away from the external potential backwards in time. Experimentally, such a wave corresponds to a positron approaching the potential and annihilating the
19262:
18760:{\displaystyle S=\int \operatorname {Tr} \partial _{\mu }A_{\nu }\partial ^{\mu }A^{\nu }+f_{jk}^{i}\partial ^{\nu }A_{i}^{\mu }A_{\mu }^{j}A_{\nu }^{k}+f_{jr}^{i}f_{kl}^{r}A_{i}A_{j}A^{k}A^{l}+\operatorname {Tr} \partial _{\mu }{\bar {\eta }}\partial ^{\mu }\eta +{\bar {\eta }}A_{j}\eta \,}
17089:
11466:
valued), the value associated with each line is the momentum. The boundary operator takes each line to the sum of the end-vertices with a positive sign at the head and a negative sign at the tail. The condition that the momentum is conserved is exactly the condition that the boundary of the
2503:
16255:
Dirac particles are as follows: The propagator is the inverse of the Dirac operator, the lines have arrows just as for a complex scalar field, and the diagram acquires an overall factor of −1 for each closed Fermi loop. If there are an odd number of Fermi loops, the diagram changes sign.
12858:
4020:
20187:
8806:
13157:
The sum of the connected diagrams in the presence of sources includes a term for each connected diagram in the absence of sources, except now the diagrams can end on the source. Traditionally, a source is represented by a little "×" with one line extending out, exactly as an insertion.
12990:, because the momentum conservation is over-enforced. In finite volume, this factor can be identified as the total volume of space time. Dividing by the volume, the remaining integral for the vacuum bubble has an interpretation: it is a contribution to the energy density of the vacuum.
4406:
17907:
13613:
11486:(i.e. up to boundaries of 2-cells) define the first homology of a graph. The number of independent momenta that are not determined is then equal to the number of independent homology loops. For many graphs, this is equal to the number of loops as counted in the most intuitive way.
3631:
12424:
16178:{\displaystyle \left\langle {\bar {\psi }}(k_{1}){\bar {\psi }}(k_{2})\cdots {\bar {\psi }}(k_{n})\psi (k'_{1})\cdots \psi (k_{n})\right\rangle =\sum _{\mathrm {pairings} }(-1)^{S}\prod _{\mathrm {pairs} \;i,j}\delta \left(k_{i}-k_{j}\right){\frac {1}{\gamma \cdot k_{i}-m}}}
10175:
16904:
118:, "Since the middle of the 20th century, theoretical physicists have increasingly turned to this tool to help them undertake critical calculations. Feynman diagrams have revolutionized nearly every aspect of theoretical physics." While the diagrams are applied primarily to
12487:
7072:
4807:
23008:"It was Dyson's contribution to indicate how Feynman's visual insights could be used He realized that Feynman diagrams can also be viewed as a representation of the logical content of field theories (as stated in their perturbative expansions)". Schweber, op.cit (1994)
15298:
19074:
16300:
10283:
22693:
20786:
202:
of quantum field theory represents the transition amplitude as a weighted sum of all possible histories of the system from the initial to the final state, in terms of either particles or fields. The transition amplitude is then given as the matrix element of the
22091:
8531:
If the equations of motion are linear, the propagator will always be the reciprocal of the quadratic-form matrix that defines the free Lagrangian, since this gives the equations of motion. This is also easy to see directly from the path integral. The factor of
14977:
3191:
11145:
1792:
The particles in the initial state are depicted by lines sticking out in the direction of the initial state (e.g., to the left), the particles in the final state are represented by lines sticking out in the direction of the final state (e.g., to the right).
21366:
22028:, the uncertainty in energy is large enough to produce more particles and antiparticles of the same type from the vacuum. This means that the notion of a single-particle state is to some extent incompatible with the notion of an object localized in space.
22023:
The correlation functions of a quantum field theory describe the scattering of particles. The definition of "particle" in relativistic field theory is not self-evident, because if you try to determine the position so that the uncertainty is less than the
9666:
17660:
17522:
10758:
7083:
20355:
16236:, the integral is zero, and it is easy to check that the sum over pairings gives zero in this case (there are always an even number of them). This is the Grassmann analog of the higher Gaussian moments that completed the Bosonic Wick's theorem earlier.
7400:
6616:
11328:
which multiplies the contribution. This means that the two half-lines that make a line are forced to have equal and opposite momentum. The line itself should be labelled by an arrow, drawn parallel to the line, and labeled by the momentum in the line
1412:
to describe how the particles deflect one another. The amplitude for scattering is the sum of each possible interaction history over all possible intermediate particle states. The number of times the interaction Hamiltonian acts is the order of the
19579:
13326:
17790:
14418:
5176:
21777:
18919:
13756:
7866:
12685:
The top is the sum over all Feynman diagrams, including disconnected diagrams that do not link up to external lines at all. In terms of the connected diagrams, the numerator includes the same contributions of vacuum bubbles as the denominator:
11444:
are joined to external lines, while the remaining two lines are joined to each other. The two lines joined to each other can have any momentum at all, since they both enter and leave the same vertex. A more complicated example is one where two
8222:
2765:
2109:
5368:
14054:
10272:
18447:
16607:
114:, who introduced the diagrams in 1948. The interaction of subatomic particles can be complex and difficult to understand; Feynman diagrams give a simple visualization of what would otherwise be an arcane and abstract formula. According to
19741:
11478:-values can be arbitrarily redefined whenever there is a closed loop. A closed loop is a cyclical path of adjacent vertices that never revisits the same vertex. Such a cycle can be thought of as the boundary of a hypothetical 2-cell. The
6357:
14554:
10721:
9646:{\displaystyle {\frac {\partial ^{n}}{\partial a^{n}}}I=\int {\frac {x^{2n}}{2^{n}}}e^{-ax^{2}/2}dx={\frac {1\cdot 3\cdot 5\ldots \cdot (2n-1)}{2\cdot 2\cdot 2\ldots \;\;\;\;\;\cdot 2\;\;\;\;\;\;}}{\sqrt {2\pi }}\,a^{-{\frac {2n+1}{2}}}}
1764:• at each vertex there is 4-momentum conservation using delta functions, 4-momenta entering the vertex are positive while those leaving are negative, the factors at each vertex and internal line are multiplied in the amplitude integral,
8388:
15671:
6049:
5416:
20315:
17372:
12998:
Correlation functions are the sum of the connected Feynman diagrams, but the formalism treats the connected and disconnected diagrams differently. Internal lines end on vertices, while external lines go off to insertions. Introducing
4526:
21377:
17100:
15304:
18188:
14083:
15628:
11752:
This theorem has an interpretation in terms of particle-paths: when identical particles are present, the integral over all intermediate particles must not double-count states that differ only by interchanging identical particles.
8642:
22864:
Feynman propagators are nonlocal in time, translating a field process to a coherent particle language is not completely intuitive, and has only been explicitly worked out in certain special cases. In the case of nonrelativistic
15530:
2211:
12480:
An immediate consequence of the linked-cluster theorem is that all vacuum bubbles, diagrams without external lines, cancel when calculating correlation functions. A correlation function is given by a ratio of path-integrals:
12084:
20804:
2480:{\displaystyle S=\sum _{n=0}^{\infty }{\frac {(-i)^{n}}{n!}}\left(\prod _{j=1}^{n}\int d^{4}x_{j}\right){\mathcal {T}}\left\{\prod _{j=1}^{n}{\mathcal {H}}_{V}\left(x_{j}\right)\right\}\equiv \sum _{n=0}^{\infty }S^{(n)}\;.}
19400:
15455:, the path integral is a Pfaffian times a source quadratic form, and the formulas give the square root of the determinant, just as they do for real Bosonic fields. The propagator is still the inverse of the quadratic part.
7753:
4050:
16692:
4633:
2727:{\displaystyle S=\sum _{n=0}^{\infty }{\frac {i^{n}}{n!}}\left(\prod _{j=1}^{n}\int d^{4}x_{j}\right){\mathcal {T}}\left\{\prod _{j=1}^{n}{\mathcal {L}}_{V}\left(x_{j}\right)\right\}\equiv \sum _{n=0}^{\infty }S^{(n)}\;.}
16963:
4423:, the field Lagrangian, integrated over all possible field histories, defines the probability amplitude to go from one field configuration to another. In order to make sense, the field theory should have a well-defined
21685:
7877:
5723:
1974:
9276:
s have the same momentum? If it's an odd number, the integral is zero; negative values cancel with the positive values. But if the number is even, the integral is positive. The previous demonstration assumed that the
1676:
interactions. The particles are represented by the lines of the diagram, which can be squiggly or straight, with an arrow or without, depending on the type of particle. A point where lines connect to other lines is a
12692:
11618:
The symmetry factor theorem gives the symmetry factor for a general diagram: the contribution of each Feynman diagram must be divided by the order of its group of automorphisms, the number of symmetries that it has.
22880:
only because non-perturbative effects appear in hidden form in diagrams that it was possible to analyze nonperturbative effects in string theory, where in many cases a Feynman description is the only one available.
19982:
13829:
8661:
6139:
19119:
16764:
9989:
7669:
1847:. A bosonic propagator is represented by a wavy line connecting two vertices (•~•). A fermionic propagator is represented by a solid line (with an arrow in one or another direction) connecting two vertices, (•←•).
22783:
18095:{\displaystyle \int DA\int \delta {\big (}F(A){\big )}\det \left({\frac {\partial F}{\partial G}}\right)\,DGe^{iS}=\int DG\int \delta {\big (}F(A){\big )}\det \left({\frac {\partial F}{\partial G}}\right)e^{iS}\,}
13114:
13464:
9404:
7290:, which is formally infinite, but that just means that the fluctuations are unbounded in infinite volume. In any finite volume, the integral is replaced by a discrete sum, and the variance of the integral is
3650:
3032:
5943:
12256:
9899:
4623:
at each point of the final hypersurface defines the final field value, which is allowed to vary, giving a different amplitude to end up at different values. This is the field-to-field transition amplitude.
3828:
18354:
12675:{\displaystyle \left\langle \phi _{1}(x_{1})\cdots \phi _{n}(x_{n})\right\rangle ={\frac {\displaystyle \int e^{-S}\phi _{1}(x_{1})\cdots \phi _{n}(x_{n})\,D\phi }{\displaystyle \int e^{-S}\,D\phi }}\,.}
10054:
4983:{\displaystyle {\frac {\displaystyle \int e^{iS}\phi (x_{1})\cdots \phi (x_{n})\,D\phi }{\displaystyle \int e^{iS}\,D\phi }}=\left\langle 0\left|\phi (x_{1})\cdots \phi (x_{n})\right|0\right\rangle \,.}
4235:
12053:
11323:
8305:
6917:
6888:
15145:
12959:
16472:{\displaystyle S=\int {\tfrac {1}{4}}F^{\mu \nu }F_{\mu \nu }=\int -{\tfrac {1}{2}}\left(\partial ^{\mu }A_{\nu }\partial _{\mu }A^{\nu }-\partial ^{\mu }A_{\mu }\partial _{\nu }A^{\nu }\right)\,.}
10566:{\displaystyle S=\int _{k}k^{2}\left|\phi (k)\right|^{2}+{\frac {\lambda }{4!}}\int _{k_{1}k_{2}k_{3}k_{4}}\phi (k_{1})\phi (k_{2})\phi (k_{3})\phi (k_{4})\delta (k_{1}+k_{2}+k_{3}+k_{4})=S_{F}+X.}
19896:
18930:
17896:
11398:
The number of ways of making a diagram by joining half-lines into lines almost completely cancels the factorial factors coming from the Taylor series of the exponential and the 4! at each vertex.
8471:
1708:, there is no past and future and all the lines are internal. The particles then begin and end on little x's, which represent the positions of the operators whose correlation is being calculated.
18292:
13443:
3465:
19966:
11756:
Proof: To prove this theorem, label all the internal and external lines of a diagram with a unique name. Then form the diagram by linking a half-line to a name and then to the other half line.
4025:
is the electromagnetic contraction (propagator) in the Feynman gauge. This term is represented by the Feynman diagram at the right. This diagram gives contributions to the following processes:
13911:
22533:
22433:
22282:{\displaystyle \left\langle \phi (k_{1})\phi (k_{2})\phi (k_{3})\phi (k_{4})\right\rangle ={\frac {i}{k_{1}^{2}}}{\frac {i}{k_{2}^{2}}}{\frac {i}{k_{3}^{2}}}{\frac {i}{k_{4}^{2}}}i\lambda \,}
21160:
16779:
21762:
20627:
15113:{\displaystyle \left\langle {\bar {\psi }}\psi \right\rangle ={\frac {1}{Z}}{\frac {\partial }{\partial \eta }}{\frac {\partial }{\partial {\bar {\eta }}}}Z|_{\eta ={\bar {\eta }}=0}=M^{-1}}
4588:
3439:{\displaystyle S^{(2)}={\frac {(ie)^{2}}{2!}}\int d^{4}x\,d^{4}x'\,T{\bar {\psi }}(x)\,\gamma ^{\mu }\,\psi (x)\,A_{\mu }(x)\,{\bar {\psi }}(x')\,\gamma ^{\nu }\,\psi (x')\,A_{\nu }(x').\;}
18241:
17279:
To find the Feynman rules for non-Abelian gauge fields, the procedure that performs the gauge fixing must be carefully corrected to account for a change of variables in the path-integral.
14063:. This expression is useful for formally changing field coordinates in the path integral, much as a delta function is used to change coordinates in an ordinary multi-dimensional integral.
8526:
16256:
Historically, the −1 rule was very difficult for Feynman to discover. He discovered it after a long process of trial and error, since he lacked a proper theory of Grassmann integration.
11038:
10188:
Interactions are represented by higher order contributions, since quadratic contributions are always Gaussian. The simplest interaction is the quartic self-interaction, with an action:
9845:{\displaystyle \left\langle x^{2n}\right\rangle ={\frac {\displaystyle \int x^{2n}e^{-ax^{2}/2}}{\displaystyle \int e^{-ax^{2}/2}}}=1\cdot 3\cdot 5\ldots \cdot (2n-1){\frac {1}{a^{n}}}}
21231:
11924:
10961:{\displaystyle \left\langle \phi (k_{1})\phi (k_{2})\phi (k_{3})\phi (k_{4})\right\rangle ={\frac {\displaystyle \int e^{-S}\phi (k_{1})\phi (k_{2})\phi (k_{3})\phi (k_{4})D\phi }{Z}}}
7239:{\displaystyle \left\langle \phi (x_{1})\cdots \phi (x_{n})\right\rangle ={\frac {\displaystyle \int e^{-S}\phi (x_{1})\cdots \phi (x_{n})\,D\phi }{\displaystyle \int e^{-S}\,D\phi }}}
5836:
22008:
Since the numerator of the integrand is not involved, the same prescription works for any loop, no matter what the spins are carried by the legs. The interpretation of the parameters
20598:{\displaystyle \int _{0}^{1}\int {\frac {1}{\left(k^{2}+m^{2}+2vp\cdot k+vp^{2}\right)^{2}}}\,dk\,dv=\int _{0}^{1}\int {\frac {1}{\left(k'^{2}+m^{2}+v(1-v)p^{2}\right)^{2}}}\,dk'\,dv}
7573:{\displaystyle \left\langle \phi (x_{1})\cdots \phi (x_{n})\right\rangle =\lim _{|C|\rightarrow \infty }{\frac {\displaystyle \sum _{C}\phi _{C}(x_{1})\cdots \phi _{C}(x_{n})}{|C|}}}
7320:
The path integral defines a probabilistic algorithm to generate a Euclidean scalar field configuration. Randomly pick the real and imaginary parts of each Fourier mode at wavenumber
6807:{\displaystyle \int \exp \left({\frac {i}{2}}\sum _{k}k^{2}\phi ^{*}(k)\phi (k)\right)\,D\phi =\prod _{k}\int _{\phi _{k}}e^{{\frac {i}{2}}k^{2}\left|\phi _{k}\right|^{2}\,d^{d}k}\,}
1591:
gave good arguments for taking the original, non-regularized Feynman diagrams as the most succinct representation of our present knowledge about the physics of quantum scattering of
1526:
compensates for this effect and eliminates the troublesome infinities. After renormalization, calculations using Feynman diagrams match experimental results with very high accuracy.
23563:
22051:
scattering amplitude for single particle states only requires a careful limit, sending the fields to infinity and integrating over space to get rid of the higher-order corrections.
17537:
17399:
11544:, because it does not link up to any external lines. There are 4! ways to form this diagram, but the denominator includes a 2! (from the expansion of the exponential, there are two
11433:
It is easy to verify that in all these cases, the momenta on all the internal lines is determined by the external momenta and the condition of momentum conservation in each vertex.
6561:
18790:
Feynman diagrams were originally discovered by Feynman, by trial and error, as a way to represent the contribution to the S-matrix from different classes of particle trajectories.
22510:
6221:
5004:
right from the beginning, because the fluctuations in the field can become unbounded. So the path-integral can be thought of as on a discrete square lattice, with lattice spacing
21220:
8405:
coincide, and the operator order matters. The form of the singularity can be understood from the canonical commutation relations to be a delta-function. Defining the (Euclidean)
6405:
1840:
In QED a vertex always has three lines attached to it: one bosonic line, one fermionic line with arrow toward the vertex, and one fermionic line with arrow away from the vertex.
21998:{\displaystyle {\frac {1}{D_{0}\cdots D_{n}}}=n!\int _{\mathrm {simplex} }{\frac {1}{\left(v_{0}D_{0}+v_{1}D_{1}\cdots +v_{n}D_{n}\right)^{n+1}}}\,dv_{1}\,dv_{2}\cdots dv_{n}}
13164:
6494:
22364:
over all possible final states is the probability for the scattering event. The normalization of the single-particle states must be chosen carefully, however, to ensure that
17675:
14304:
11864:
5037:
3113:
2918:{\displaystyle {\mathcal {T}}\prod _{j=1}^{n}{\mathcal {L}}_{V}\left(x_{j}\right)=\sum _{\text{A}}(\pm ){\mathcal {N}}\prod _{j=1}^{n}{\mathcal {L}}_{V}\left(x_{j}\right)\;,}
18301:
is the adjoint valued element of the Lie algebra at every point that performs the infinitesimal gauge transformation. This adds the Faddeev Popov determinant to the action:
11426:
run off to external lines. These are all also forest diagrams (as every tree is a forest); an example of a forest that is not a tree is when eight external lines end on two
22349:
18804:
13635:
11456:-integrals that are left undetermined by momentum conservation is equal to the number of independent closed loops in the diagram, where independent loops are counted as in
23279:
8094:
7597:
For free fields with a quadratic action, the probability distribution is a high-dimensional Gaussian, and the statistical average is given by an explicit formula. But the
2003:
5264:
1881:
In the initial state (at the bottom; early time) there is one electron (e) and one positron (e) and in the final state (at the top; late time) there are two photons (γ).
13930:
6601:
10194:
18365:
16955:
16515:
22876:, the Shifman–Vainshtein–Zakharov sum rules describe non-perturbatively excited long-wavelength field modes in particle language, but only in a phenomenological way.
15848:{\displaystyle \left\langle {\bar {\psi }}(k')\psi (k)\right\rangle =\delta (k+k'){\frac {1}{\gamma \cdot k-m}}=\delta (k+k'){\frac {\gamma \cdot k+m}{k^{2}-m^{2}}}}
6232:
5620:{\displaystyle S=\int _{k}{\Big (}{\big (}1-\cos(k_{1}){\big )}+{\big (}1-\cos(k_{2}){\big )}+\cdots +{\big (}1-\cos(k_{d}){\big )}{\Big )}\phi _{k}^{*}\phi ^{k}\,.}
18105:
The factor in front is the volume of the gauge group, and it contributes a constant, which can be discarded. The remaining integral is over the gauge fixed action.
14449:
10597:
9318:
9295:
9274:
1426:
21595:{\displaystyle \int _{0}^{\infty }\int _{\mathrm {simplex} }u^{n}e^{-u\left(v_{0}D_{0}+v_{1}D_{1}+v_{2}D_{2}\cdots +v_{n}D_{n}\right)}\,dv_{1}\cdots dv_{n}\,du\,,}
17264:{\displaystyle \left\langle A_{\mu }(k)A_{\nu }(k')\right\rangle =\delta \left(k+k'\right){\frac {g_{\mu \nu }-\lambda {\frac {k_{\mu }k_{\nu }}{k^{2}}}}{k^{2}}}.}
8316:
1700:). Traditionally, the bottom of the diagram is the past and the top the future; other times, the past is to the left and the future to the right. When calculating
15438:{\displaystyle \left\langle \phi ^{*}\phi \right\rangle ={\frac {1}{Z}}{\frac {\partial }{\partial h}}{\frac {\partial }{\partial h^{*}}}Z|_{h=h^{*}=0}=M^{-1}\,.}
7772:
5969:
20206:
17288:
14224:{\displaystyle \left\langle \phi (x)\right\rangle ={\frac {1}{Z}}{\frac {\partial }{\partial h(x)}}Z={\frac {\partial }{\partial h(x)}}\log {\big (}Z{\big )}\,.}
4449:
1471:; each vertex where lines meet gives a factor derived from an interaction term in the Lagrangian, and incoming and outgoing lines carry an energy, momentum, and
19425:
11387:, the propagator, and either goes from vertex to vertex, or ends at an insertion. If it is internal, it is integrated over. At each vertex, the total incoming
1780:
A Feynman diagram represents a perturbative contribution to the amplitude of a quantum transition from some initial quantum state to some final quantum state.
18111:
24266:
15542:
1786:
The initial state is often assumed to be at the left of the diagram and the final state at the right (although other conventions are also used quite often).
11158:
to the external half-lines. Each of these different ways of matching the half-lines together in pairs contributes exactly once, regardless of the values of
8560:
19274:
The Schwinger representation is both useful for making manifest the particle aspect of the propagator, and for symmetrizing denominators of loop diagrams.
16910:
states are not physical—their contribution cancels with longitudinal photons to only leave two physical photon polarization contributions for any value of
15464:
12240:{\displaystyle \left({\frac {1}{0!}}+{\frac {C_{1}}{1!}}+{\frac {C_{1}^{2}}{2!}}+\cdots \right)\left(1+C_{2}+{\frac {1}{2}}C_{2}^{2}+\cdots \right)\cdots }
20979:{\displaystyle {\frac {1}{D_{0}D_{1}\cdots D_{n}}}=\int _{0}^{\infty }\cdots \int _{0}^{\infty }e^{-u_{0}D_{0}\cdots -u_{n}D_{n}}\,du_{0}\cdots du_{n}\,.}
2124:
1822:
Positron in the initial state is represented by a solid line, with an arrow indicating the spin of the particle e.g. pointing away from the vertex: (←•).
4219:{\displaystyle N{\bar {\psi }}(x)\,\gamma ^{\mu }\,{\underline {\psi (x)\,{\bar {\psi }}(x')}}\,\gamma ^{\nu }\,\psi (x')\,A_{\mu }(x)\,A_{\nu }(x')\;,}
6897:
is the side-length of the box. Each separate factor is an oscillatory Gaussian, and the width of the Gaussian diverges as the volume goes to infinity.
4790:{\displaystyle \int _{A}^{B}e^{iS}\phi (x_{1})\cdots \phi (x_{n})\,D\phi =\left\langle A\left|\phi (x_{1})\cdots \phi (x_{n})\right|B\right\rangle \,,}
19298:
7691:
134:"would have been literally unthinkable without Feynman diagrams, as would calculations that established a route to production and observation of the
24327:
23796:
17084:{\displaystyle S=\int {\tfrac {1}{2}}\left(\partial ^{\mu }A^{\nu }\partial _{\mu }A_{\nu }-\lambda \left(\partial _{\mu }A^{\mu }\right)^{2}\right)}
13452:
at every point. One way to interpret this expression is that it is taking the Fourier transform in field space. If there is a probability density on
471:
16622:
8056:{\displaystyle \left\langle \phi (k)\phi (k')\right\rangle =\delta (k-k'){\frac {1}{2{\big (}d-\cos(k_{1})+\cos(k_{2})\cdots +\cos(k_{d}){\big )}}}}
1611:
in a form in close contact with the experimental numbers one wants to understand. Although the statement of the theory in terms of graphs may imply
8257:
The form of the propagator can be more easily found by using the equation of motion for the field. From the Lagrangian, the equation of motion is:
2938:
of the operators and (±) takes care of the possible sign change when commuting the fermionic operators to bring them together for a contraction (a
23564:
https://www.marinabaysands.com/content/dam/singapore/marinabaysands/master/main/home/museum/Feynman/Richard%20Feynman%20by%20Harald%20Fritzsch.pdf
13006:
Sources are external fields, fields that contribute to the action, but are not dynamical variables. A scalar field source is another scalar field
12853:{\displaystyle \int e^{-S}\phi _{1}(x_{1})\cdots \phi _{n}(x_{n})\,D\phi =\left(\sum E_{i}\right)\left(\exp \left(\sum _{i}C_{i}\right)\right)\,.}
21611:
20182:{\displaystyle \int _{u,v}ue^{-u\left(k^{2}+m^{2}+v2p\cdot k+vp^{2}\right)}=\int {\frac {1}{\left(k^{2}+m^{2}+v2p\cdot k-vp^{2}\right)^{2}}}\,dv}
19601:
8801:{\displaystyle \left\langle \phi (k_{1})\cdots \phi (k_{2n})\right\rangle =\sum \prod _{i,j}{\frac {\delta \left(k_{i}-k_{j}\right)}{k_{i}^{2}}}}
5640:
1911:
1819:
Electron in the final state is represented by a line, with an arrow indicating the spin of the particle e.g. pointing away from the vertex: (•→).
22860:, complicating the limit of the infinite series of diagrams needed (in the limit of vanishing coupling) to describe such field configurations.)
19257:{\displaystyle \Delta (x)=\int _{0}^{\infty }d\tau \int DX\,e^{-\int \limits _{0}^{\tau }\left({\frac {{\dot {x}}^{2}}{2}}+m^{2}\right)d\tau '}}
4612:
on the starting hypersurface give the initial value of the field, analogous to the starting position for a point particle, and the field values
1783:
For example, in the process of electron-positron annihilation the initial state is one electron and one positron, the final state: two photons.
1338:
13767:
13119:
In the Feynman expansion, this contributes H terms with one half-line ending on a vertex. Lines in a Feynman diagram can now end either on an
6060:
1825:
Positron in the final state is represented by a line, with an arrow indicating the spin of the particle e.g. pointing toward the vertex: (•←).
22988:
16703:
9921:
8241:
is the ordinary one-dimensional Dirac delta function. This convention for delta-functions is not universal—some authors keep the factors of 2
22712:
15123:
again, in a schematic matrix notation. The meaning of the formula above is that the derivative with respect to the appropriate component of
13608:{\displaystyle \int \rho (y)e^{iky}\,d^{n}y=\left\langle e^{iky}\right\rangle =\left\langle \prod _{i=1}^{n}e^{ih_{i}y_{i}}\right\rangle \,}
7077:
The expectation value of the field is the statistical expectation value of the field when chosen according to the probability distribution:
13016:
22354:
which is a constant, independent of the incoming and outgoing momentum. The interpretation of the scattering amplitude is that the sum of
3812:{\displaystyle N{\bar {\psi }}(x)\gamma ^{\mu }\psi (x){\bar {\psi }}(x')\gamma ^{\nu }\psi (x'){\underline {A_{\mu }(x)A_{\nu }(x')}}\;,}
24200:
22792:
is then the probability amplitude for relativistically normalized incoming states to become relativistically normalized outgoing states.
4997:
for the field, and it coincides with the statistical mechanical partition function at zero temperature when rotated into imaginary time.
391:
12419:{\displaystyle Z\propto \prod _{i}{\sum _{n_{i}=0}^{\infty }{\frac {C_{i}^{n_{i}}}{n_{i}!}}}=\exp {\sum _{i}{C_{i}}}\propto \exp {W}\,.}
10277:
The reason for the combinatorial factor 4! will be clear soon. Writing the action in terms of the lattice (or continuum) Fourier modes:
7615:
4015:{\displaystyle {\underline {A_{\mu }(x)A_{\nu }(x')}}=\int {\frac {d^{4}k}{(2\pi )^{4}}}{\frac {-ig_{\mu \nu }}{k^{2}+i0}}e^{-ik(x-x')}}
24316:
24222:
11794:
if all vertices and propagator lines are linked by a sequence of vertices and propagators of the diagram itself. If one views it as an
9326:
2966:
1681:, and this is where the particles meet and interact: by emitting or absorbing new particles, deflecting one another, or changing type.
19812:
10180:
and this is in fact the correct answer. So Wick's theorem holds no matter how many of the momenta of the internal variables coincide.
5866:
24058:
11799:
9856:
431:
18464:
as in the abelian case. The full gauge fixed action is then the Yang Mills action in Feynman gauge with an additional ghost action:
18307:
11777:
But the number of unnamed diagrams is smaller than the number of named diagram by the order of the automorphism group of the graph.
10170:{\displaystyle \left\langle x^{2n}\right\rangle =(2n-1)\cdot (2n-3)\ldots \cdot 5\cdot 3\cdot 1\left\langle x^{2}\right\rangle ^{n}}
4401:{\displaystyle {\underline {\psi (x){\bar {\psi }}(x')}}=\int {\frac {d^{4}p}{(2\pi )^{4}}}{\frac {i}{\gamma p-m+i0}}e^{-ip(x-x')}}
19292:
s together in two half-lines, and making the remaining lines external, the integral over the internal propagators in the loop is:
7067:{\displaystyle e^{\int _{k}-{\tfrac {1}{2}}k^{2}\phi _{k}^{*}\phi _{k}}=\prod _{k}e^{-k^{2}\left|\phi _{k}\right|^{2}\,d^{d}k}\,.}
24168:
1612:
1414:
176:
23481:
23276:
15293:{\displaystyle \int e^{\phi ^{*}M\phi +h^{*}\phi +\phi ^{*}h}\,D\phi ^{*}\,D\phi ={\frac {e^{h^{*}M^{-1}h}}{\mathrm {Det} (M)}}}
11798:
it is connected. The remarkable relevance of such diagrams in QFTs is due to the fact that they are sufficient to determine the
11367:
s with the external half-lines, representing insertions, are the Feynman diagrams of this theory. Each line carries a factor of
24422:
11990:
11258:
8263:
6831:
1619:
shows that this formalism is flexible enough to deal with phenomena of nonperturbative characters ... Some modification of the
19069:{\displaystyle \Delta (x)=\int _{0}^{\infty }d\tau e^{-m^{2}\tau }{\frac {1}{({4\pi \tau })^{d/2}}}e^{\frac {-x^{2}}{4\tau }}}
10745:. The half-lines meet at a vertex, which contributes a delta-function that ensures that the sum of the momenta are all equal.
24211:
23789:
23351:
23324:
16697:
The additional contribution from gauge-fixing cancels the second half of the free Lagrangian, giving the Feynman Lagrangian:
12881:
11774:
gives a factor of 4!. So a named diagram can be formed in exactly as many ways as the denominator of the Feynman expansion.
3626:{\displaystyle N{\bar {\psi }}(x)ie\gamma ^{\mu }\psi (x){\bar {\psi }}(x')ie\gamma ^{\nu }\psi (x')A_{\mu }(x)A_{\nu }(x')}
22934:
Maxivan in 1975, and had it painted with Feynman diagrams. The van is currently owned by video game designer and physicist
17821:
8418:
23055:
22062:
particles in a scattering event is the given by the sum of the Feynman diagrams that go into the correlation function for
19282:
The Schwinger representation has an immediate practical application to loop diagrams. For example, for the diagram in the
18252:
14953:
where the last equality is a consequence of the translation invariance of the Grassmann integral. The Grassmann variables
13377:
6904:
becomes positive definite, and can be interpreted as a probability distribution. The probability of a field having values
1196:
24091:
22688:{\displaystyle \int \delta (E^{2}-k^{2}-m^{2})|E,k\rangle \langle E,k|\,dE\,dk=\int {dk \over 2E}|k\rangle \langle k|\,.}
18924:
The meaning of this identity (which is an elementary integration) is made clearer by Fourier transforming to real space.
16899:{\displaystyle \left\langle A_{\mu }(k)A_{\nu }(k')\right\rangle =\delta \left(k+k'\right){\frac {g_{\mu \nu }}{k^{2}}}.}
15448:
So that the propagator is the inverse of the matrix in the quadratic part of the action in both the Bose and Fermi case.
23674:
20781:{\displaystyle \int dk\,{\frac {1}{k^{2}+m^{2}}}{\frac {1}{(k+p_{1})^{2}+m^{2}}}\cdots {\frac {1}{(k+p_{n})^{2}+m^{2}}}}
15139:. This is exactly analogous to the bosonic path integration formula for a Gaussian integral of a complex bosonic field:
7394:
To find any correlation function, generate a field again and again by this procedure, and find the statistical average:
4801:
probability distribution, and it is convenient to define it so that multiplying by a constant does not change anything:
24255:
23708:
23098:
16505:, since it only determines the choice of gauge. This procedure inserts the following factor into the path integral for
16220:
pair is a commuting element of the Grassmann algebra, it does not matter what order the pairs are in. If more than one
13840:
1867:
1331:
22385:
21090:
13618:
The Fourier transform is the expectation of an oscillatory exponential. The path integral in the presence of a source
10997:
s in the expansion. The insertions should also be thought of as half-lines, four in this case, which carry a momentum
24024:
23662:
23639:
23303:
21696:
11502:
of the diagram. The contribution of each diagram to the correlation function must be divided by its symmetry factor.
8550:
Because each field mode is an independent Gaussian, the expectation values for the product of many field modes obeys
1850:
The number of vertices gives the order of the term in the perturbation series expansion of the transition amplitude.
8811:
where the sum is over each partition of the field modes into pairs, and the product is over the pairs. For example,
7601:
also works well for bosonic interacting field theories where there is no closed form for the correlation functions.
4537:
23896:
23782:
19911:
18199:
11140:{\displaystyle \lambda {\frac {1}{k_{1}^{2}}}{\frac {1}{k_{2}^{2}}}{\frac {1}{k_{3}^{2}}}{\frac {1}{k_{4}^{2}}}\,.}
8482:
961:
21361:{\displaystyle u^{n}\,du\wedge dv_{1}\wedge dv_{2}\cdots \wedge dv_{n}=du_{0}\wedge du_{1}\cdots \wedge du_{n}\,.}
24340:
24178:
23819:
23072:
19821:
11936:
constructed from some (arbitrary) Feynman diagram that can be thought to consist of several connected components
1409:
234:
14283:
The field path integral can be extended to the Fermi case, but only if the notion of integration is expanded. A
5247:
modes are also a lattice. This is not strictly as necessary as the space-lattice limit, because interactions in
1623:
of calculation may well outlive the elaborate mathematical structure of local canonical quantum field theory ...
17655:{\displaystyle \int f(r)\,dx\,dy=\int d\theta \,\int f(r)\delta (y)\left|{\frac {dy}{d\theta }}\right|\,dx\,dy}
17517:{\displaystyle \int f(r)\,dx\,dy=\int f(r)\int d\theta \,\delta (y)\left|{\frac {dy}{d\theta }}\right|\,dx\,dy}
13834:
The Fourier transform of a delta-function is a constant, which gives a formal expression for a delta function:
11875:
11529:
half lines to make the diagram is only 4 × 3, so the contribution of this diagram is divided by two.
5783:
87:
22039:
A field operator can act to produce a one-particle state from the vacuum, which means that the field operator
1432:
The Dyson series can be alternatively rewritten as a sum over Feynman diagrams, where at each vertex both the
24402:
24189:
24051:
20614:
is more negative than four times the mass of the particle in the loop, which happens in a physical region of
13371:
is just a parameter in the Lagrangian, which varies from point to point. The path integral for the field is:
11449:
s are joined to each other by matching the legs one to the other. This diagram has no external lines at all.
8412:
as the Fourier transform of the time-ordered two-point function (the one that comes from the path-integral):
2956:
The diagrams are drawn according to the Feynman rules, which depend upon the interaction Lagrangian. For the
1498:, because the short-distance particle interactions require a careful limiting procedure, to include particle
1324:
509:
348:
308:
6505:
5860:, while the imaginary part is odd. The Fourier transform avoids double-counting, so that it can be written:
24392:
24101:
23384:
22799:, the relativistic normalization is the same as the nonrelativistic normalization (up to a constant factor
22054:
The relation between scattering and correlation functions is the LSZ-theorem: The scattering amplitude for
16493:
To fix this problem, one needs to fix a gauge. The most convenient way is to demand that the divergence of
11795:
1724:
1422:
1066:
991:
338:
22458:
11032:, should match up with the momenta of the external half-lines in pairs. The new contribution is equal to:
6169:
5015:
should be taken carefully. If the final results do not depend on the shape of the lattice or the value of
24417:
23946:
23829:
23437:
22958:
22869:
22519:
point equally, the measure is not Lorentz-invariant. The invariant measure integrates over all values of
21166:
13321:{\displaystyle \log {\big (}Z{\big )}=\sum _{n,C}h(k_{1})h(k_{2})\cdots h(k_{n})C(k_{1},\cdots ,k_{n})\,}
9998:
are all the same variable, the index is just to keep track of the number of ways to pair them. The first
6372:
504:
441:
106:
is a pictorial representation of the mathematical expressions describing the behavior and interaction of
23461:
17785:{\displaystyle \int f(r)\,dx\,dy=\int d\theta _{0}\,\int f(x)\left|{\frac {dy}{d\theta }}\right|\,dx\,.}
14413:{\displaystyle \int e^{M_{ij}{\bar {\psi }}^{i}\psi ^{j}}\,D{\bar {\psi }}\,D\psi =\mathrm {Det} (M)\,,}
11603:
could link up to either pair, giving an additional factor of 2. The remaining two half-lines in the two
9904:
If Wick's theorem were correct, the higher moments would be given by all possible pairings of a list of
5171:{\displaystyle \phi (x)=\int {\frac {dk}{(2\pi )^{d}}}\phi (k)e^{ik\cdot x}=\int _{k}\phi (k)e^{ikx}\,.}
24397:
24136:
22844:
16612:
The first factor, the delta function, fixes the gauge. The second factor sums over different values of
1638:
343:
23729:
18914:{\displaystyle {\frac {1}{p^{2}+m^{2}}}=\int _{0}^{\infty }e^{-\tau \left(p^{2}+m^{2}\right)}\,d\tau }
13751:{\displaystyle Z=\int e^{iS}e^{i\int _{x}h(x)\phi (x)}\,D\phi =\left\langle e^{ih\phi }\right\rangle }
7609:
Each mode is independently Gaussian distributed. The expectation of field modes is easy to calculate:
6449:
1176:
24233:
24121:
24116:
24096:
23967:
23906:
23834:
22973:
11813:
8217:{\displaystyle \delta (k)=(2\pi )^{d}\delta _{D}(k_{1})\delta _{D}(k_{2})\cdots \delta _{D}(k_{d})\,}
4420:
3075:
2104:{\displaystyle S=\lim _{t_{2}\rightarrow +\infty }\lim _{t_{1}\rightarrow -\infty }U(t_{2},t_{1})\;.}
1696:
extend from a vertex to "the future" and represent the final state (the latter two are also known as
1632:
1488:
1460:
494:
436:
199:
22847:. Nonperturbative tunneling processes involve field configurations that on average get big when the
20618:, the integral has a cut. This is exactly when the external momentum can create physical particles.
17377:
To find the form of the determinant, consider first a simple two-dimensional integral of a function
11482:-labellings of a graph that conserve momentum (i.e. which has zero boundary) up to redefinitions of
5363:{\displaystyle S=\sum _{\langle x,y\rangle }{\tfrac {1}{2}}{\big (}\phi (x)-\phi (y){\big )}^{2}\,,}
5243:
It is also convenient from time to time to consider the space-time volume to be finite, so that the
24412:
24141:
24044:
23767:
22983:
22953:
22324:
16261:
14049:{\displaystyle \delta (\phi -\eta )=\int e^{ih(x){\big (}\phi (x)-\eta (x){\big )}\,d^{d}x}\,Dh\,,}
11607:
s can be linked to each other in two ways, so that the total number of ways to form the diagram is
5728:
Now we have the continuum Fourier transform of the original action. In finite volume, the quantity
4431:. The path integral formalism is completely equivalent to the canonical operator formalism above.
1642:
1580:
1397:
996:
621:
514:
451:
411:
115:
15536:
basis for the Grassmann algebra, the quadratic part of the Dirac action becomes simple to invert:
13356:
external lines carrying momentum as indicated. The sum is over all connected diagrams, as before.
10267:{\displaystyle S=\int \partial ^{\mu }\phi \partial _{\mu }\phi +{\frac {\lambda }{4!}}\phi ^{4}.}
8647:
is zero unless the field modes coincide in pairs. This means that it is zero for an odd number of
986:
24366:
24111:
24086:
23860:
23805:
22924:
18442:{\displaystyle \int e^{{\bar {\eta }}\partial _{\mu }\,D^{\mu }\eta }\,D{\bar {\eta }}\,D\eta \,}
16602:{\displaystyle \int \delta \left(\partial _{\mu }A^{\mu }-f\right)e^{-{\frac {f^{2}}{2}}}\,Df\,.}
2957:
1991:
1893:
for a transition of a quantum system (between asymptotically free states) from the initial state
1797:
1752:
for intermediate particles and processes, which has a propagator factor ("prop"), external lines
1276:
736:
544:
539:
381:
23159:
Penco, R.; Mauro, D. (2006). "Perturbation theory via Feynman diagrams in classical mechanics".
22823:
The nonrelativistic potential, which scatters in all directions with an equal amplitude (in the
13916:
This tells you what a field delta function looks like in a path-integral. For two scalar fields
10583:
is the free action, whose correlation functions are given by Wick's theorem. The exponential of
6576:
4627:
The path integral gives the expectation value of operators between the initial and final state:
1559:
Historically, as a book-keeping device of covariant perturbation theory, the graphs were called
24106:
23476:
22873:
17815:, so this exercise reproduces the standard formula for polar integration of a radial function:
16937:, the two terms do not cancel completely. This gives a covariant Lagrangian with a coefficient
6352:{\displaystyle \det(A)\int dx_{1}\,dx_{2}\cdots \,dx_{n}=\int dy_{1}\,dy_{2}\cdots \,dy_{n}\,.}
4035:
e e scattering (initial state at the bottom/top, final state at the top/bottom of the diagram).
1628:
1449:
1146:
1126:
616:
549:
446:
406:
184:
169:
131:
23982:
21023:
and an integral over the fraction of the proper time in all but the first segment of the loop
16940:
14549:{\displaystyle \int e^{{\frac {1}{2}}A_{ij}\psi ^{i}\psi ^{j}}\,D\psi =\mathrm {Pfaff} (A)\,,}
10748:
To compute a correlation function in the interacting theory, there is a contribution from the
10716:{\displaystyle e^{-S}=e^{-S_{F}}\left(1+X+{\frac {1}{2!}}XX+{\frac {1}{3!}}XXX+\cdots \right)}
6570:. On a finite volume lattice, the determinant is nonzero and independent of the field values.
24244:
24131:
23412:
22857:
16501:, whose value is random from point to point. It does no harm to integrate over the values of
10042:, and so on. This means that Wick's theorem, uncorrected, says that the expectation value of
8383:{\displaystyle \partial _{\mu }\partial ^{\mu }\left\langle \phi (x)\phi (y)\right\rangle =0}
1890:
1729:
1701:
1592:
1530:
1031:
886:
582:
519:
426:
288:
161:
16198:
to put the ones that are paired up to make the delta-functions next to each other, with the
11513:
joined to each other. There are 4 × 3 ways to join the external half-lines to the
7861:{\displaystyle \left\langle \phi (k)\phi (k')\right\rangle =\delta (k-k'){\frac {1}{k^{2}}}}
7766:-values coincide, since this is the variance of the Gaussian. In the infinite volume limit,
6044:{\displaystyle S=\int {\tfrac {1}{2}}\partial _{\mu }\phi ^{*}\partial ^{\mu }\phi \,d^{d}x}
5732:
is not infinitesimal, but becomes the volume of a box made by neighboring Fourier modes, or
4601:
are space-like hypersurfaces that define the boundary conditions. The collection of all the
24345:
24146:
24126:
23178:
23110:
22836:
20310:{\displaystyle {\frac {1}{AB}}=\int _{0}^{1}{\frac {1}{{\big (}vA+(1-v)B{\big )}^{2}}}\,dv}
20196:-integral. This method, invented by Schwinger but usually attributed to Feynman, is called
17367:{\displaystyle \delta \left(\partial _{\mu }A_{\mu }-f\right)e^{-{\frac {f^{2}}{2}}}\det M}
14077:
The correlation functions are derivatives of the path integral with respect to the source:
9303:
9280:
9259:
4521:{\displaystyle S=\int {\tfrac {1}{2}}\partial _{\mu }\phi \partial ^{\mu }\phi \,d^{d}x\,.}
2228:
1812:
Electron in the initial state is represented by a solid line, with an arrow indicating the
1789:
A Feynman diagram consists of points, called vertices, and lines attached to the vertices.
1705:
1649:
1608:
1596:
1479:
state is then obtained by summing over all such possibilities. This is closely tied to the
1281:
1056:
587:
421:
213:
180:
119:
23992:
23724:
23577:
23456:
19574:{\displaystyle \int _{t,t'}e^{-t(k^{2}+m^{2})-t'\left((k+p)^{2}+m^{2}\right)}\,dt\,dt'\,.}
9256:
There is a subtle point left before Wick's theorem is proved—what if more than two of the
1800:
there are two types of particles: matter particles such as electrons or positrons (called
333:
8:
23936:
23213:
22899:
18193:
To get a covariant gauge, the gauge fixing condition is the same as in the Abelian case:
14234:
In Euclidean space, source contributions to the action can still appear with a factor of
5251:
are not localized, but it is convenient for keeping track of the factors in front of the
2115:
1673:
1534:
1480:
1291:
611:
123:
99:
23182:
23114:
18460:
can give the Feynman propagator (or a covariant propagator) by choosing the measure for
18183:{\displaystyle \int \det \left({\frac {\partial F}{\partial G}}\right)e^{iS_{GF}}\,DA\,}
13761:
which, on a lattice, is the product of an oscillatory exponential for each field value:
11615:. The total symmetry factor is 2, and the contribution of this diagram is divided by 2.
7685:, since then the two Gaussian random variables are independent and both have zero mean.
4427:, and the integral should be performed a little bit rotated into imaginary time, i.e. a
1776:
axes are not always shown, directions of external lines correspond to passage of time.
1494:
The naïve application of such calculations often produces diagrams whose amplitudes are
1001:
726:
183:
or correlation function of a quantum mechanical or statistical field theory. Within the
24321:
24002:
23977:
23901:
23697:
23194:
23168:
23126:
23038:
22824:
22025:
19808:
parametrizes the fraction of the proper time on the top of the loop versus the bottom.
16294:
The naive propagator for photons is infinite, since the Lagrangian for the A-field is:
15623:{\displaystyle S=\int _{k}{\bar {\psi }}\left(i\gamma ^{\mu }k_{\mu }-m\right)\psi \,.}
11505:
For example, consider the Feynman diagram formed from two external lines joined to one
11013:
in the Taylor expansion of the action. Wick's theorem requires that the momenta in the
8073:, for field fluctuations long compared to the lattice spacing, the two forms coincide.
7598:
2236:
1584:
1553:
1523:
1507:
1377:
1301:
1226:
1156:
1036:
1021:
951:
911:
901:
861:
776:
666:
626:
396:
358:
313:
301:
278:
273:
142:
107:
23540:"Fermilab | TUFTE Exhibit | April 12-June 26, 2014 | About the Exhibit"
11532:
For another example, consider the diagram formed by joining all the half-lines of one
11344:. If one of the two half-lines is external, this kills the integral over the internal
8637:{\displaystyle \left\langle \phi (k_{1})\phi (k_{2})\cdots \phi (k_{n})\right\rangle }
5236:. Sometimes, instead of a lattice, the field modes are just cut off at high values of
24407:
24296:
24151:
23962:
23931:
23704:
23658:
23651:
23635:
23525:
23408:
23347:
23320:
23299:
23190:
23130:
22848:
22816:
invariant scattering amplitude is still constant. The particles created by the field
15525:{\displaystyle \int {\bar {\psi }}\left(\gamma ^{\mu }\partial _{\mu }-m\right)\psi }
12469:
8545:
5229:
are not allowed to contribute right away, they only start to contribute in the limit
5218:
3641:
2738:
1740:
1716:
1616:
1484:
1256:
1186:
1071:
1061:
906:
841:
811:
781:
641:
636:
466:
386:
376:
268:
188:
19419:. The asymmetry can be fixed by putting everything in the Schwinger representation.
11763:
s gives a different pattern of linking names to half-lines, and this is a factor of
11517:, and then there is only one way to join the two remaining lines to each other. The
11410:
An example of a tree diagram is the one where each of four external lines end on an
4032:
e e scattering (initial state at the left, final state at the right of the diagram);
4029:
e e scattering (initial state at the right, final state at the left of the diagram);
2206:{\displaystyle S={\mathcal {T}}e^{-i\int _{-\infty }^{+\infty }d\tau H_{V}(\tau )}.}
1672:
A Feynman diagram is a representation of quantum field theory processes in terms of
1448:
is not necessarily equal to the mass, i.e. the intermediate particles are so-called
971:
23997:
23880:
23618:
23198:
23186:
23118:
23047:
19902:
16483:
15452:
14295:, which defines the new type of Gaussian integration appropriate for Fermi fields.
14284:
1657:
1588:
1545:
1519:
1499:
1464:
1441:
1401:
1373:
1296:
1211:
1191:
1161:
1111:
931:
921:
916:
801:
786:
716:
401:
328:
318:
253:
23:
22379:. This is because the nonrelativistic unit operator on single particle states is:
17901:
In the path-integral for a nonabelian gauge field, the analogous manipulation is:
17282:
The gauge fixing factor has an extra determinant from popping the delta function:
16769:
which is just like four independent free scalar fields, one for each component of
8655:, it is equal to a contribution from each pair separately, with a delta function.
24067:
23941:
23926:
23755:
23558:
23341:
23283:
22968:
22935:
22919:
22915:, i.e. the paths followed by Ph.D. students when interacting with their advisors.
12465:
11457:
5001:
2239:
into a perturbation series in the powers of the interaction Hamiltonian density,
1828:
Virtual Photon in the initial and the final state is represented by a wavy line (
1503:
1408:
that describes the incoming and outgoing particles, and including an interaction
1241:
1236:
1206:
1201:
1171:
1131:
1101:
1081:
981:
871:
866:
831:
816:
791:
771:
751:
681:
671:
592:
554:
456:
248:
221:
164:
in theoretical particle physics requires the use of rather large and complicated
111:
91:
22872:
describes the class of diagrams to include to describe a relativistic atom. For
19395:{\displaystyle \int _{k}{\frac {1}{k^{2}+m^{2}}}{\frac {1}{(k+p)^{2}+m^{2}}}\,.}
16482:
The quadratic form defining the propagator is non-invertible. The reason is the
12983:
at each vertex, and when there are no external lines, this contains a factor of
11759:
Now count the number of ways to form the named diagram. Each permutation of the
7748:{\displaystyle \left\langle \phi _{k}\phi _{k}\right\rangle ={\frac {V}{k^{2}}}}
4039:
3458:
187:
formulation of quantum field theory, a Feynman diagram represents a term in the
23972:
23030:
22852:
5028:
1813:
1720:
1472:
1385:
1266:
1216:
1136:
1121:
1106:
1086:
1051:
1046:
1041:
1026:
1011:
976:
966:
891:
826:
796:
731:
696:
686:
661:
532:
489:
353:
23495:
19815:
for this transformation of variables is easy to work out from the identities:
16687:{\displaystyle e^{-{\frac {\left(\partial _{\mu }A_{\mu }\right)^{2}}{2}}}\,.}
1016:
24386:
24334:
23725:
AMS article: "What's New in Mathematics: Finite-dimensional Feynman Diagrams"
23680:
23632:
Drawing Theories Apart: The Dispersion of Feynman Diagrams in Postwar Physics
23370:
22931:
22032:
20615:
18778:
In the Abelian case, the determinant for covariant gauges does not depend on
18771:
additional ghost loops, which cancel out timelike and longitudinal states in
12968:
is the energy density in the vacuum. Each vacuum bubble contains a factor of
11248:
By Wick's theorem, each pair of half-lines must be paired together to make a
6418:, and the sign depends on whether the rotation includes a reflection or not.
5853:
4428:
1620:
1568:
1515:
1445:
1369:
1271:
1261:
1251:
1246:
1166:
1151:
946:
941:
766:
761:
721:
691:
656:
646:
577:
572:
127:
23539:
23134:
21084:
The Jacobian for the coordinate transformation can be worked out as before:
18359:
which can be rewritten as a Grassmann integral by introducing ghost fields:
7372:. For real scalar fields, the algorithm must generate only one of each pair
3069:
A fermionic propagator is represented by a solid line connecting two points;
157:
are represented as moving backward along the time axis in Feynman diagrams.
24287:
23614:
23122:
22963:
21680:{\displaystyle v_{i}>0\quad {\mbox{and}}\quad \sum _{i=1}^{n}v_{i}<1}
19736:{\displaystyle \int _{t,t'}e^{-(t+t')(k^{2}+m^{2})-t'2p\cdot k-t'p^{2}}\,,}
11623:
11452:
The reason loop diagrams are called loop diagrams is because the number of
10727:
5718:{\displaystyle S=\int _{k}{\tfrac {1}{2}}k^{2}\left|\phi (k)\right|^{2}\,.}
4424:
2935:
2752:
2737:
A Feynman diagram is a graphical representation of a single summand in the
2232:
1969:{\displaystyle S_{\rm {fi}}=\langle \mathrm {f} |S|\mathrm {i} \rangle \;,}
1805:
1723:
images because they all describe particle scattering. Feynman diagrams are
1600:
1418:
1306:
1231:
1116:
1091:
1076:
1006:
926:
881:
876:
756:
746:
741:
706:
701:
676:
631:
499:
461:
283:
263:
154:
44:
23774:
23622:
22927:, features the titular vacuum diagram, a specific type of Feynman diagram.
22015:
is that they are the fraction of the total proper time spent on each leg.
5395:. The discretization should be thought of as defining what the derivative
1858:
1648:
in the evaluation of Feynman diagrams; it assigns values to them that are
1567:, because the path integral was unfamiliar when they were introduced, and
23921:
23911:
23875:
23850:
23763:
22865:
22292:
Stripping off the external propagators, that is, removing the factors of
20621:
When the loop has more vertices, there are more denominators to combine:
18246:
Whose variation under an infinitesimal gauge transformation is given by:
14288:
14241:
11591:
links up to two external lines, and the remaining two half-lines of each
11154:
is canceled because there are exactly 4! ways to match the half-lines in
10752:
terms now. For example, the path-integral for the four-field correlator:
1878:
has a contribution from the second order Feynman diagram shown adjacent:
1425:(collections of particles with a definite momentum) the series is called
1286:
1221:
1181:
1141:
1096:
956:
856:
851:
836:
821:
806:
135:
23669:(expanded, updated version of 't Hooft & Veltman, 1973, cited above)
23051:
22527:, restricting to the hyperbola with a Lorentz-invariant delta function:
13824:{\displaystyle \left\langle \prod _{x}e^{ih_{x}\phi _{x}}\right\rangle }
7871:
Strictly speaking, this is an approximation: the lattice propagator is:
6134:{\displaystyle S=\int _{k}{\tfrac {1}{2}}k^{2}\left|\phi (k)\right|^{2}}
5255:-integrals and the momentum-conserving delta functions that will arise.
1417:, and the time-dependent perturbation theory for fields is known as the
1404:, the interaction between particles can be described by starting from a
24355:
23987:
23916:
23591:
23366:
23173:
22978:
22907:
20320:
But this form does not provide the physical motivation for introducing
16759:{\displaystyle S=\int \partial ^{\mu }A^{\nu }\partial _{\mu }A_{\nu }}
9984:{\displaystyle \left\langle x_{1}x_{2}x_{3}\cdots x_{2n}\right\rangle }
7267:
is independently Gaussian distributed. The variance of the Gaussian is
3063:
2939:
1844:
1692:
extend from "the past" to a vertex and represent an initial state, and
1511:
1468:
1405:
936:
846:
711:
651:
323:
258:
22778:{\displaystyle {\sqrt {E}}=\left(k^{2}-m^{2}\right)^{\frac {1}{4}}\,.}
20996:
can be split up as before into an integral over the total proper time
13003:
unifies the formalism, by making new vertices where one line can end.
9251:
3048:, the Feynman rules can be formulated in coordinate space as follows:
23462:
https://www.flickr.com/photos/jkannenberg/14287569353/in/photostream/
22894:
22072:
field insertions, leaving out the propagators for the external legs.
16188:
where S is the sign of the permutation that reorders the sequence of
13109:{\displaystyle \int h(x)\phi (x)\,d^{d}x=\int h(k)\phi (k)\,d^{d}k\,}
5000:
The initial-to-final amplitudes are ill-defined if one thinks of the
60:
23699:
QED and the men who made it: Dyson, Feynman, Schwinger, and Tomonaga
22371:
Non-relativistic single particle states are labeled by the momentum
21771:
integral gives the general prescription for combining denominators:
12875:
itself, which from the definition of the path integral is equal to:
11683:. If the line is undirected, as it is for a real scalar field, then
207:-matrix between the initial and final states of the quantum system.
23870:
23855:
23747:
14292:
11226:
s. These all end on a vertex, and are integrated over all possible
11004:
The lowest-order contribution comes from the first nontrivial term
9320:
are equal, and this is a notable property of Gaussian integration:
1984:
1801:
1732:—every diagram contributes to the total amplitude for the process.
1645:
1495:
1437:
1392:, represented by the blue sine wave and green spiral, respectively.
1365:
896:
192:
165:
150:
146:
36:
28:
24036:
13363:
is not dynamical, which means that there is no path integral over
12062:
labels the (infinitely) many connected Feynman diagrams possible.
7664:{\displaystyle \left\langle \phi _{k}\phi _{k'}\right\rangle =0\,}
5410:
In terms of the lattice Fourier modes, the action can be written:
22889:
The use of the above diagram of the virtual particle producing a
20328:
is the proportion of proper time on one of the legs of the loop.
18798:
The Euclidean scalar propagator has a suggestive representation:
17527:
The derivative-factor ensures that popping the delta function in
9399:{\displaystyle I=\int e^{-ax^{2}/2}dx={\sqrt {\frac {2\pi }{a}}}}
3027:{\displaystyle L_{v}=-g{\bar {\psi }}\gamma ^{\mu }\psi A_{\mu }}
23744:
A graphical application for creating publication ready diagrams.
22442:-states for a particle of mass m integrates over a hyperbola in
10734:
should be thought of as four half-lines, one for each factor of
5938:{\displaystyle S=\int _{k}{\tfrac {1}{2}}k^{2}\phi (k)\phi (-k)}
5225:-integrals in QFT. The lattice means that fluctuations at large
1421:. When the intermediate states at intermediate times are energy
23865:
9894:{\displaystyle \left\langle x^{2}\right\rangle ={\frac {1}{a}}}
1433:
48:
16:
Pictorial representation of the behavior of subatomic particles
23741:
22375:, and they are chosen to have the same norm at every value of
19746:
except for the asymmetrical little bit. Defining the variable
18349:{\displaystyle \det \left(\partial _{\mu }\,D_{\mu }\right)\,}
11363:
The diagrams that are formed by linking the half-lines in the
11333:. The half-line at the tail end of the arrow carries momentum
172:. Feynman diagrams can represent these integrals graphically.
22890:
20791:
The general rule follows from the Schwinger prescription for
19113:
The path-integral representation for the propagator is then:
18782:, so the ghosts do not contribute to the connected diagrams.
14059:
which integrates over the Fourier transform coordinate, over
12464:
The linked-cluster theorem was first proved to order four by
11977:. However, in the end each contribution of a Feynman diagram
11782:
9300:
But the theorem is correct even when arbitrarily many of the
1656:, called the dimension. Dimensional regularization writes a
1389:
1356:
72:
56:
24234:
Feynman's Lost Lecture: The Motion of Planets Around the Sun
23477:
https://www.nst.com.my/cbt/2019/04/481648/original-space-van
12065:
A scheme to successively create such contributions from the
11548:
s) and two factors of 4!. The contribution is multiplied by
11179:
gives a series of terms with progressively higher number of
6606:
and the path integral is a separate factor at each value of
6566:
which is the complex conjugate-transpose, up to factors of 2
6443:
can be read off from the definition of a Fourier transform.
22702:-states are different from the relativistically normalized
17531:
removes the integral. Exchanging the order of integration,
14298:
The two fundamental formulas of Grassmann integration are:
10730:
of corrections to the free action. The term represented by
4040:
Compton scattering and annihilation/generation of e e pairs
1595:. Their motivations are consistent with the convictions of
1529:
Feynman diagram and path integral methods are also used in
1455:
Feynman gave a prescription for calculating the amplitude (
1381:
1361:
122:, they can also be used in other areas of physics, such as
23735:
21605:
where the simplex is the region defined by the conditions
19102:
of a normalized Gaussian, the probability of ending up at
10971:
which in the free field was only nonzero when the momenta
7391:, and make the second the complex conjugate of the first.
5948:
over an integration domain that integrates over each pair
4439:
A simple example is the free relativistic scalar field in
1843:
The vertices might be connected by a bosonic or fermionic
1744:
General features of the scattering process A + B → C + D:
12048:{\displaystyle \prod _{i}{\frac {C_{i}^{n_{i}}}{n_{i}!}}}
11318:{\displaystyle {\frac {\delta (k_{1}+k_{2})}{k_{1}^{2}}}}
8300:{\displaystyle \partial _{\mu }\partial ^{\mu }\phi =0\,}
6883:{\displaystyle d^{d}k=\left({\frac {1}{L}}\right)^{d}\,,}
6499:
and the Fourier inversion theorem tells you the inverse:
3644:
of the integrand gives (among others) the following term
3176:
1853:
14070:, and the physical partition function is the value when
12871:
The vacuum bubbles then are only useful for determining
11595:
are joined to each other. The number of ways to link an
5258:
On a lattice, (ii), the action needs to be discretized:
1808:). They are represented in Feynman diagrams as follows:
23457:
https://www.flickr.com/photos/122759998@N03/14430191165
23340:
Fetter, Alexander L.; Walecka, John Dirk (2003-06-20).
22085:
contribution to the (Lorentz) correlation function is:
13458:, the Fourier transform of the probability density is:
12954:{\displaystyle Z=\int e^{-S}D\phi =e^{-HT}=e^{-\rho V}}
11807:. More precisely, connected Feynman diagrams determine
11583:
Another example is the Feynman diagram formed from two
11337:, while the half-line at the head-end carries momentum
10726:
The path integral for the interacting action is then a
5841:
In terms of real and imaginary parts, the real part of
21633:
16977:
16361:
16314:
14066:
The partition function is now a function of the field
10975:
were equal in pairs, is now nonzero for all values of
6940:
6081:
5983:
5887:
5661:
5297:
4463:
3115:
is represented by a wiggly line attached to the point
3066:
is represented by a wiggly line connecting two points;
3059:
is represented by a point (sometimes called a vertex);
1607:
The Feynman graphs and rules of calculation summarize
1429:(or time-dependent/time-ordered perturbation theory).
24267:
Perfectly Reasonable Deviations from the Beaten Track
23271:, CERN Yellow Report 1973, reprinted in G. 't Hooft,
22715:
22536:
22461:
22388:
22327:
22094:
21780:
21699:
21614:
21380:
21234:
21169:
21093:
20807:
20630:
20358:
20209:
19985:
19914:
19824:
19604:
19428:
19301:
19122:
18933:
18807:
18473:
18368:
18310:
18255:
18202:
18114:
17910:
17891:{\displaystyle \int f(r)\,dx\,dy=2\pi \int f(x)x\,dx}
17824:
17678:
17540:
17402:
17291:
17103:
16966:
16943:
16782:
16706:
16625:
16518:
16303:
15877:
15674:
15545:
15467:
15307:
15148:
14980:
14629:
14452:
14307:
14086:
13933:
13843:
13770:
13638:
13467:
13380:
13167:
13019:
13010:
that contributes a term to the (Lorentz) Lagrangian:
12884:
12695:
12641:
12561:
12490:
12259:
12087:
11993:
11878:
11816:
11599:
to two external lines is 4 × 3, and either
11440:
diagram, and an example is one where two lines of an
11261:
11041:
10853:
10761:
10600:
10591:, giving a series of corrections to the free action.
10286:
10197:
10057:
9924:
9859:
9745:
9696:
9669:
9413:
9329:
9306:
9283:
9262:
8820:
8664:
8563:
8485:
8466:{\displaystyle \partial ^{2}\Delta (x)=i\delta (x)\,}
8421:
8319:
8266:
8097:
7880:
7775:
7694:
7618:
7486:
7403:
7209:
7143:
7086:
6920:
6834:
6619:
6579:
6508:
6452:
6375:
6235:
6172:
6063:
5972:
5869:
5786:
5643:
5419:
5267:
5040:
4993:
The normalization factor on the bottom is called the
4879:
4813:
4810:
4636:
4540:
4452:
4238:
4053:
3831:
3653:
3468:
3194:
3165:
is represented by a solid line attached to the point
3139:
is represented by a solid line attached to the point
3078:
2969:
2768:
2506:
2248:
2127:
2006:
1914:
1884:
1816:
of the particle e.g. pointing toward the vertex (→•).
1728:
interact. The law of summation is in accord with the
1574:
175:
A Feynman diagram is a graphical representation of a
23482:
mom, wife, Feynman, daughter, son, van, Ensenada, MX
22913:
visualize and describe quantum academic interactions
21073:
are positive and add up to less than 1, so that the
18287:{\displaystyle \partial _{\mu }\,D_{\mu }\alpha \,,}
16616:
that are inequivalent gauge fixings. This is simply
13438:{\displaystyle Z=\int e^{iS+i\int h\phi }\,D\phi \,}
11418:, and the remaining half-line joins up with another
1660:
as an integral depending on the spacetime dimension
1463:. Each internal line corresponds to a factor of the
55:, represented by the blue sine wave) that becomes a
23612:
14439:are independent Grassmann variables for each index
11436:A diagram that is not a forest diagram is called a
9252:
Higher Gaussian moments — completing Wick's theorem
23753:
23696:
23650:
23526:https://www.researchgate.net/publication/319224539
23510:Pathways for Theoretical Advances in Visualization
23275:(World Scientific, Singapore, 1994), Introduction
22777:
22687:
22504:
22427:
22343:
22281:
21997:
21756:
21679:
21594:
21360:
21225:Wedging all these equations together, one obtains
21214:
21154:
20978:
20780:
20597:
20309:
20181:
19960:
19890:
19735:
19573:
19394:
19256:
19068:
18913:
18759:
18441:
18348:
18286:
18235:
18182:
18094:
17890:
17784:
17654:
17516:
17366:
17263:
17083:
16949:
16898:
16758:
16686:
16601:
16471:
16177:
15847:
15622:
15524:
15437:
15292:
15112:
14942:
14548:
14412:
14262:
14223:
14048:
13905:
13823:
13750:
13607:
13437:
13320:
13108:
12953:
12852:
12674:
12418:
12239:
12047:
11918:
11858:
11317:
11139:
10960:
10715:
10587:in the path integral can be expanded in powers of
10565:
10266:
10169:
9983:
9893:
9844:
9645:
9398:
9312:
9289:
9268:
9236:
8800:
8636:
8520:
8465:
8397:, and the identity is true everywhere except when
8382:
8299:
8216:
8055:
7860:
7747:
7663:
7572:
7238:
7066:
6882:
6821:is the infinitesimal volume of a discrete cell in
6806:
6595:
6555:
6488:
6399:
6351:
6215:
6133:
6043:
5937:
5830:
5717:
5619:
5362:
5209:are not allowed. It is important to note that the
5170:
4982:
4789:
4582:
4520:
4400:
4218:
4014:
3811:
3625:
3438:
3107:
3026:
2917:
2726:
2479:
2205:
2103:
1968:
130:wrote that the calculations that won him the 2004
22911:of January 11, 2012, shows Feynman diagrams that
13906:{\displaystyle \delta (x-y)=\int e^{ik(x-y)}\,dk}
11770:. Each permutation of the half-lines in a single
11414:. Another is when three external lines end on an
11233:external half-lines, which are the come from the
5583:
5438:
1862:Feynman diagram of electron/positron annihilation
1758:for incoming/outgoing particles to/from vertices
1583:, written from the particle physics perspective,
24384:
22930:Feynman and his wife, Gweneth Howarth, bought a
22820:scatter in all directions with equal amplitude.
22428:{\displaystyle \int dk\,|k\rangle \langle k|\,.}
21155:{\displaystyle du=du_{0}+du_{1}\cdots +du_{n}\,}
18311:
18118:
18047:
17952:
17358:
14567:is a collection of Grassmann variables, and the
7459:
6580:
6236:
5777:is real-valued, so the Fourier transform obeys:
5027:On a lattice, (i), the field can be expanded in
3037:describing the interaction of a fermionic field
2040:
2014:
22835:Thinking of Feynman diagrams as a perturbation
22452:space defined by the energy–momentum relation:
22081:interaction of the previous section, the order
21757:{\displaystyle v_{0}=1-\sum _{i=1}^{n}v_{i}\,.}
20331:Once the denominators are combined, a shift in
18785:
11984:to the partition function has the generic form
11634:of the vertices with the following properties:
11183:s. The contribution from the term with exactly
7324:to be a Gaussian random variable with variance
1667:
110:. The scheme is named after American physicist
24328:Quantum Man: Richard Feynman's Life in Science
23732:explained by Flip Tanedo at Quantumdiaries.com
23613:'t Hooft, Gerardus; Veltman, Martinus (1973).
23498:. 21 October 2021 – via www.youtube.com.
23237:
23211:
11790:Roughly speaking, a Feynman diagram is called
11211:internal half-lines, which are the factors of
4583:{\displaystyle \int _{A}^{B}e^{iS}\,D\phi \,,}
2490:Equivalently, with the interaction Lagrangian
24052:
23790:
23339:
22989:Angular momentum diagrams (quantum mechanics)
20286:
20251:
20200:. Abstractly, it is the elementary identity:
19961:{\displaystyle u\,du\wedge dv=dt\wedge dt'\,}
19804:is the total proper time for the loop, while
18236:{\displaystyle \partial _{\mu }A^{\mu }=f\,,}
18042:
18023:
17947:
17928:
17274:
14238:, so that they still do a Fourier transform.
14212:
14193:
14014:
13980:
13195:
13176:
8521:{\displaystyle \Delta (k)={\frac {i}{k^{2}}}}
8045:
7957:
5576:
5541:
5525:
5490:
5480:
5445:
5345:
5310:
4044:Another interesting term in the expansion is
1332:
23750:A Java program for drawing Feynman diagrams.
23738:C++ library that produces PostScript output.
23634:. Chicago, IL: University of Chicago Press.
23401:
23314:
22897:pair was featured in the television sit-com
22670:
22667:
22602:
22599:
22410:
22407:
17665:but now the delta-function can be popped in
15633:The propagator is the inverse of the matrix
14287:of a free Fermi field is a high-dimensional
5291:
5279:
4531:The probability amplitude for a process is:
4414:
1959:
1933:
24201:QED: The Strange Theory of Light and Matter
23804:
23653:Diagrammatica: The Path to Feynman Diagrams
23296:Diagrammatica: The Path to Feynman Diagrams
23090:
22314:, gives the invariant scattering amplitude
20989:The integral over the Schwinger parameters
19891:{\displaystyle d(uv)=dt'\quad du=dt+dt'\,,}
19094:. The total propagation function from 0 to
18793:
7358:at random, and the Fourier transform gives
5221:, this is the best standard convention for
4411:is the fermionic contraction (propagator).
24059:
24045:
23797:
23783:
23238:Gribbin, John; Gribbin, Mary (1997). "5".
23158:
23022:
22903:, in the episode "The Bat Jar Conjecture".
22830:
16921:is done with a coefficient different from
16096:
11509:, and the remaining two half-lines in the
11498:The uncancelled denominator is called the
11356:. If both are internal, the integral over
9597:
9596:
9595:
9594:
9593:
9592:
9585:
9584:
9583:
9582:
9581:
4434:
4212:
3805:
3435:
3181:The second order perturbation term in the
2911:
2720:
2473:
2097:
1962:
1712:including paths that go backward in time.
1684:There are three different types of lines:
1339:
1325:
220:
24223:What Do You Care What Other People Think?
23736:Drawing Feynman diagrams with FeynDiagram
23672:
23172:
22771:
22681:
22626:
22619:
22498:
22421:
22398:
22340:
22278:
21965:
21951:
21750:
21588:
21581:
21551:
21354:
21245:
21208:
21151:
20972:
20942:
20640:
20588:
20576:
20465:
20458:
20300:
20172:
19957:
19918:
19884:
19729:
19567:
19555:
19548:
19388:
19277:
19168:
19083:to the propagator is a Gaussian of width
18904:
18756:
18438:
18431:
18415:
18399:
18345:
18329:
18280:
18266:
18229:
18179:
18172:
18091:
17983:
17881:
17847:
17840:
17778:
17771:
17727:
17701:
17694:
17645:
17638:
17582:
17563:
17556:
17507:
17500:
17459:
17425:
17418:
17094:and the covariant propagator for QED is:
16680:
16595:
16588:
16465:
15616:
15431:
15221:
15207:
14936:
14868:
14852:
14716:
14700:
14616:are Grassmann-valued column vectors, and
14542:
14506:
14406:
14376:
14360:
14217:
14042:
14035:
14019:
13896:
13714:
13604:
13499:
13434:
13427:
13317:
13105:
13091:
13047:
12846:
12767:
12668:
12658:
12633:
12412:
11919:{\displaystyle Z\propto \sum _{k}{D_{k}}}
11175:The expansion of the action in powers of
11133:
10993:can now match up with the momenta of the
9611:
8462:
8296:
8213:
7660:
7226:
7201:
7060:
7044:
6876:
6803:
6787:
6700:
6592:
6552:
6485:
6421:The matrix that changes coordinates from
6345:
6331:
6314:
6281:
6264:
6209:
6151:Integrating over all different values of
6027:
5831:{\displaystyle \phi (k)^{*}=\phi (-k)\,.}
5824:
5711:
5613:
5356:
5164:
4976:
4896:
4871:
4783:
4706:
4576:
4569:
4514:
4500:
4187:
4167:
4149:
4138:
4105:
4089:
4078:
3449:
3407:
3389:
3378:
3351:
3331:
3318:
3307:
3282:
3263:
1715:Feynman diagrams are often confused with
1631:the Feynman diagrams are obtained from a
1351:
23694:
23252:
22851:gets small, but each configuration is a
19267:which is a path-integral rewrite of the
19098:is a weighted sum over all proper times
11525:, but the number of ways to link up the
11460:. The homology is real-valued (actually
8310:and in an expectation value, this says:
8076:The delta functions contain factors of 2
7594:of the Euclidean correlation functions.
6054:the Fourier transform is unconstrained:
3457:
1857:
1739:
1355:
86:
71:), after which the antiquark radiates a
22:
23648:
23516:IEEE Computer Graphics and Applications
23371:Academic Interaction – Feynman Diagrams
23343:Quantum Theory of Many-particle Systems
23319:. New York: McGraw-Hill. p. viii.
23096:
23073:"Why Feynman Diagrams Are So Important"
11737:If there are different types of lines,
11422:, and the remaining half-lines of this
11407:diagram is a connected forest diagram.
5963:For a complex scalar field with action
5387:is a pair of nearest lattice neighbors
1548:always referred to Feynman diagrams as
24385:
23657:. Cambridge Lecture Notes in Physics.
23629:
23407:
23382:
23298:, Cambridge Lecture Notes in Physics,
23028:
22959:Julian Schwinger#Schwinger and Feynman
14961:, and differentiating with respect to
13448:and it is a function of the values of
6556:{\displaystyle A_{kx}^{-1}=e^{-ikx}\,}
4443:dimensions, whose action integral is:
3177:Example: second order processes in QED
2948:represents all possible contractions.
1854:Electron–positron annihilation example
1384:, with intermediate steps involving a
24040:
23778:
23315:Bjorken, J. D.; Drell, S. D. (1965).
23267:Gerardus 't Hooft, Martinus Veltman,
22883:
22438:In relativity, the integral over the
20608:This form shows that the moment that
19976:integral to be evaluated explicitly:
19079:The contribution at any one value of
15858:The analog of Wick's theorem matches
14583:is to prevent double-counting (since
8252:
24169:There's Plenty of Room at the Bottom
23754:Bowley, Roger; Copeland, Ed (2010).
23570:
22505:{\displaystyle E^{2}-k^{2}=m^{2}\,.}
13127:vertex, and only one line enters an
12468:in 1955, and for infinite orders by
12446:one simply calculates all connected
11869:To see this, one should recall that
11626:of a Feynman graph is a permutation
8536:disappears in the Euclidean theory.
7604:
6216:{\displaystyle y_{i}=A_{ij}x_{j}\,,}
5185:restricted to a cube of side length
5181:Here the integration domain is over
1540:
24066:
22964:Stueckelberg–Feynman interpretation
21215:{\displaystyle d(uv_{i})=du_{i}\,.}
19584:Now the exponent mostly depends on
16486:of the field; adding a gradient to
11489:
11170:
8245:in the delta functions (and in the
6400:{\displaystyle A^{\mathrm {T} }A=I}
5019:, then the continuum limit exists.
2741:of the time-ordered product in the
2223:is the interaction Hamiltonian and
13:
24256:The Pleasure of Finding Things Out
24212:Surely You're Joking, Mr. Feynman!
23273:Under the Spell of Gauge Principle
23240:Richard Feynman: A Life in Science
21846:
21843:
21840:
21837:
21834:
21831:
21828:
21420:
21417:
21414:
21411:
21408:
21405:
21402:
21391:
20881:
20863:
19148:
19123:
18959:
18934:
18853:
18716:
18694:
18551:
18510:
18490:
18452:The determinant is independent of
18390:
18320:
18257:
18204:
18136:
18128:
18065:
18057:
17970:
17962:
17301:
17046:
17014:
16994:
16957:, which does not affect anything:
16737:
16717:
16643:
16531:
16441:
16421:
16398:
16378:
16289:
16092:
16089:
16086:
16083:
16080:
16049:
16046:
16043:
16040:
16037:
16034:
16031:
16028:
15499:
15365:
15361:
15350:
15346:
15274:
15271:
15268:
15040:
15036:
15025:
15021:
14885:
14882:
14879:
14608:are Grassmann-valued row vectors,
14529:
14526:
14523:
14520:
14517:
14393:
14390:
14387:
14167:
14163:
14128:
14124:
12309:
11950:(identical) copies of a component
11252:, and this line gives a factor of
11001:, but one that is not integrated.
10221:
10208:
9427:
9417:
8486:
8432:
8423:
8331:
8321:
8278:
8268:
7479:
6382:
6015:
5995:
5213:-measure contains the factors of 2
4488:
4475:
3172:with an arrow away from the point;
2882:
2853:
2800:
2771:
2699:
2646:
2612:
2529:
2452:
2399:
2365:
2271:
2168:
2160:
2136:
2060:
2034:
1955:
1937:
1924:
1921:
1885:Canonical quantization formulation
1652:of an auxiliary complex parameter
1615:, use of graphical methods in the
1575:Representation of physical reality
1456:
79:, represented by the green helix).
14:
24434:
24025:Template:Quantum mechanics topics
23718:
17799:just gives an overall factor of 2
12475:
12450:, i.e., the diagrams without any
11536:to all the half-lines of another
7344:. This generates a configuration
1804:) and exchange particles (called
1427:old-fashioned perturbation theory
24020:
24019:
23897:Anomalous magnetic dipole moment
23649:Veltman, Martinus (1994-06-16).
23061:from the original on 2012-05-27.
23031:"Physics and Feynman's Diagrams"
13131:vertex. The Feynman rule for an
11471:-valued weighted graph is zero.
10979:. The momenta of the insertions
9297:s would only match up in pairs.
6825:-space, in a square lattice box
6489:{\displaystyle A_{kx}=e^{ikx}\,}
3462:The Feynman diagram of the term
2951:
24341:Feynman Prize in Nanotechnology
24179:The Feynman Lectures on Physics
24092:Wheeler–Feynman absorber theory
23584:
23550:
23532:
23502:
23488:
23468:
23448:
23430:
23376:
23360:
23333:
23308:
23288:
21639:
21631:
19854:
19800:goes from 0 to 1. The variable
19405:Here one line carries momentum
15451:For real Grassmann fields, for
11859:{\displaystyle iW\equiv \ln Z.}
11734:ends on the same external line.
11630:of the lines and a permutation
11391:is equal to the total outgoing
11348:, since it forces the internal
8539:
8080:, so that they cancel out the 2
5022:
3146:with an arrow toward the point;
3108:{\displaystyle A_{\mu }(x_{i})}
1905:is given by the matrix element
1487:, also invented by Feynman—see
1459:) for any given diagram from a
153:moving backward in time. Thus,
23703:. Princeton University Press.
23617:(Report). CERN Yellow Report.
23383:Jepsen, Kathryn (2014-08-05).
23261:
23246:
23231:
23205:
23152:
23065:
23002:
22856:quantum field theory has zero
22795:For nonrelativistic values of
22677:
22660:
22615:
22586:
22582:
22543:
22417:
22400:
22173:
22160:
22154:
22141:
22135:
22122:
22116:
22103:
21189:
21173:
20753:
20733:
20699:
20679:
20550:
20538:
20277:
20265:
19837:
19828:
19680:
19654:
19651:
19634:
19519:
19506:
19487:
19461:
19363:
19350:
19132:
19126:
19014:
18999:
18943:
18937:
18737:
18709:
18425:
18383:
18037:
18031:
17942:
17936:
17875:
17869:
17837:
17831:
17803:, while the rate of change of
17740:
17734:
17691:
17685:
17607:
17601:
17595:
17589:
17553:
17547:
17469:
17463:
17447:
17441:
17415:
17409:
17149:
17138:
17125:
17119:
16828:
16817:
16804:
16798:
16065:
16055:
16011:
15998:
15989:
15973:
15967:
15954:
15948:
15936:
15923:
15917:
15908:
15895:
15889:
15796:
15779:
15746:
15729:
15715:
15709:
15703:
15692:
15686:
15568:
15477:
15386:
15284:
15278:
15080:
15063:
15049:
14992:
14912:
14895:
14889:
14862:
14828:
14760:
14745:
14710:
14689:
14671:
14650:
14539:
14533:
14403:
14397:
14370:
14336:
14207:
14201:
14179:
14173:
14155:
14149:
14140:
14134:
14101:
14095:
14009:
14003:
13994:
13988:
13975:
13969:
13949:
13937:
13891:
13879:
13859:
13847:
13709:
13703:
13697:
13691:
13648:
13642:
13480:
13474:
13390:
13384:
13352:is the connected diagram with
13314:
13282:
13276:
13263:
13254:
13241:
13235:
13222:
13190:
13184:
13135:vertex is that a line from an
13088:
13082:
13076:
13070:
13044:
13038:
13032:
13026:
12764:
12751:
12735:
12722:
12630:
12617:
12601:
12588:
12548:
12535:
12519:
12506:
12408:
12402:
12269:
12263:
11888:
11882:
11850:
11844:
11829:
11823:
11294:
11268:
10943:
10930:
10924:
10911:
10905:
10892:
10886:
10873:
10840:
10827:
10821:
10808:
10802:
10789:
10783:
10770:
10538:
10486:
10480:
10467:
10461:
10448:
10442:
10429:
10423:
10410:
10328:
10322:
10183:
10118:
10103:
10097:
10082:
9822:
9807:
9558:
9543:
9213:
9187:
9160:
9134:
9104:
9078:
9051:
9025:
8995:
8969:
8942:
8916:
8899:
8886:
8880:
8867:
8861:
8848:
8842:
8829:
8711:
8695:
8686:
8673:
8626:
8613:
8604:
8591:
8585:
8572:
8495:
8489:
8459:
8453:
8441:
8435:
8366:
8360:
8354:
8348:
8210:
8197:
8181:
8168:
8155:
8142:
8123:
8113:
8107:
8101:
8040:
8027:
8012:
7999:
7987:
7974:
7943:
7926:
7912:
7901:
7895:
7889:
7838:
7821:
7807:
7796:
7790:
7784:
7563:
7555:
7549:
7536:
7520:
7507:
7476:
7472:
7464:
7447:
7434:
7425:
7412:
7315:
7198:
7185:
7176:
7163:
7130:
7117:
7108:
7095:
6692:
6686:
6680:
6674:
6245:
6239:
6117:
6111:
5932:
5923:
5917:
5911:
5821:
5812:
5797:
5790:
5697:
5691:
5571:
5558:
5520:
5507:
5475:
5462:
5339:
5333:
5324:
5318:
5145:
5139:
5101:
5095:
5080:
5070:
5050:
5044:
4960:
4947:
4938:
4925:
4868:
4855:
4846:
4833:
4767:
4754:
4745:
4732:
4703:
4690:
4681:
4668:
4393:
4376:
4320:
4310:
4277:
4266:
4260:
4251:
4245:
4209:
4198:
4184:
4178:
4164:
4153:
4129:
4118:
4112:
4102:
4096:
4075:
4069:
4063:
4007:
3990:
3918:
3908:
3875:
3864:
3851:
3845:
3796:
3785:
3772:
3766:
3750:
3739:
3723:
3712:
3706:
3697:
3691:
3675:
3669:
3663:
3620:
3609:
3596:
3590:
3577:
3566:
3544:
3533:
3527:
3518:
3512:
3490:
3484:
3478:
3453:
3429:
3418:
3404:
3393:
3375:
3364:
3358:
3348:
3342:
3328:
3322:
3304:
3298:
3292:
3227:
3217:
3206:
3200:
3102:
3089:
2995:
2848:
2842:
2715:
2709:
2468:
2462:
2289:
2279:
2195:
2189:
2094:
2068:
2054:
2028:
1950:
1942:
1868:electron–positron annihilation
1735:
1444:, but where the length of the
520:Renormalization group equation
1:
24423:Eponymous theorems of physics
24190:The Character of Physical Law
23015:
22923:, a science fiction story by
22368:is a relativistic invariant.
22344:{\displaystyle M=i\lambda \,}
22018:
19288:theory formed by joining two
17389:. Inserting an integral over
16773:. The Feynman propagator is:
16490:does not change the physics.
16286:themselves Grassmann valued.
11401:
8393:Where the derivatives act on
6144:and the integral is over all
349:Spontaneous symmetry breaking
309:Symmetry in quantum mechanics
23606:
23522:DOI:10.1109/MCG.2017.3271463
23212:George Johnson (July 2000).
22515:If the integral weighs each
19106:after a random walk of time
18786:Particle-path representation
18456:, so the path-integral over
16278:to go in front of the first
15661:, since different values of
15133:gives the matrix element of
14563:is an antisymmetric matrix,
11352:to be equal to the external
8651:, and for an even number of
3052:Each integration coordinate
1668:Particle-path interpretation
27:In this Feynman diagram, an
7:
23820:Euler–Heisenberg Lagrangian
23413:"The Brilliant Mr. Feynman"
23317:Relativistic Quantum Fields
23161:European Journal of Physics
22946:
16282:, and this gives the sign.
15458:The free Dirac Lagrangian:
14427:is an arbitrary matrix and
11957:within the Feynman diagram
11611:, while the denominator is
11540:. This diagram is called a
11244:insertions in the integral.
8084:factors in the measure for
7256:is a product, the value of
3041:with a bosonic gauge field
1552:, after a Swiss physicist,
1533:and can even be applied to
1446:energy-momentum four-vector
10:
24439:
24137:Quantum cellular automaton
23253:Mlodinow, Leonard (2011).
23191:10.1088/0143-0807/27/5/023
22845:operator product expansion
17275:Spin 1: non-Abelian ghosts
14596:In matrix notation, where
12993:
12457:(sometimes referred to as
11800:quantum partition function
11723:ends on an external line,
8543:
7263:at each separate value of
6596:{\displaystyle \det A=1\,}
6163:transformation matrix. If
5205:, so that large values of
1730:principle of superposition
1639:Dimensional regularization
1579:In their presentations of
344:Explicit symmetry breaking
24309:
24280:
24160:
24117:Path integral formulation
24074:
24011:
23955:
23889:
23843:
23835:Path integral formulation
23812:
23099:"The Theory of Positrons"
23097:Feynman, Richard (1949).
22974:Path integral formulation
22942:was the license plate ID.
21371:This gives the integral:
16262:weak nuclear interactions
14957:are external sources for
14620:is a real-valued matrix:
11748:should preserve the type.
7249:Since the probability of
4415:Path integral formulation
2235:expands the time-ordered
1489:path integral formulation
1398:scattering cross-sections
510:Bargmann–Wigner equations
437:Path Integral Formulation
200:path integral formulation
145:'s interpretation of the
24142:Rogers Commission Report
24102:Hellmann–Feynman theorem
24003:Photon-photon scattering
23768:University of Nottingham
23695:Schweber, S. S. (1994).
23679:. Script. Archived from
23673:Srednicki, Mark (2006).
23385:"Saving the Feynman van"
23214:"The Jaguar and the Fox"
22995:
22984:List of Feynman diagrams
22954:One-loop Feynman diagram
22788:The invariant amplitude
20349:symmetrizes everything:
19269:Schwinger representation
18794:Schwinger representation
16950:{\displaystyle \lambda }
16204:coming right before the
14258:; "photons" and "ghosts"
8249:-integration) explicit.
3149:An anti-fermionic field
1581:fundamental interactions
1457:the Feynman rules, below
515:Schwinger-Dyson equation
24368:The Challenger Disaster
24112:Feynman parametrization
23947:Ward–Takahashi identity
23830:Gupta–Bleuler formalism
23806:Quantum electrodynamics
23578:"Dr. Feynman's Doodles"
23346:. Courier Corporation.
22870:Bethe–Salpeter equation
22831:Nonperturbative effects
22706:-states by a factor of
12461:of a Feynman diagram).
6900:In imaginary time, the
4435:Scalar field Lagrangian
2960:interaction Lagrangian
1992:time-evolution operator
1461:field theory Lagrangian
545:Electroweak interaction
540:Quantum electrodynamics
505:Wheeler–DeWitt equation
382:Background field method
168:over a large number of
24107:Feynman slash notation
23630:Kaiser, David (2005).
23257:. Vintage. p. 29.
23123:10.1103/PhysRev.76.749
23029:Kaiser, David (2005).
22874:quantum chromodynamics
22810:). In this limit, the
22779:
22689:
22506:
22429:
22345:
22283:
21999:
21758:
21739:
21681:
21660:
21596:
21362:
21216:
21156:
21081:-dimensional simplex.
20980:
20782:
20599:
20311:
20183:
19962:
19892:
19737:
19575:
19396:
19278:Combining denominators
19258:
19191:
19070:
18915:
18761:
18443:
18350:
18288:
18237:
18184:
18096:
17892:
17786:
17656:
17518:
17368:
17265:
17085:
16951:
16917:If the averaging over
16900:
16760:
16688:
16603:
16473:
16179:
15849:
15624:
15526:
15439:
15294:
15114:
14965:pulls down factors of
14944:
14550:
14414:
14225:
14074:is the zero function:
14050:
13907:
13825:
13752:
13609:
13568:
13439:
13322:
13110:
12955:
12854:
12676:
12420:
12313:
12241:
12049:
11920:
11860:
11785:linked-cluster theorem
11319:
11141:
10962:
10717:
10567:
10268:
10171:
9985:
9895:
9846:
9647:
9400:
9314:
9291:
9270:
9238:
8802:
8638:
8522:
8467:
8384:
8301:
8218:
8057:
7862:
7749:
7665:
7574:
7240:
7068:
6884:
6808:
6597:
6557:
6490:
6401:
6353:
6217:
6135:
6045:
5939:
5832:
5719:
5621:
5364:
5172:
4984:
4791:
4584:
4522:
4402:
4220:
4016:
3813:
3633:
3627:
3450:Scattering of fermions
3440:
3109:
3028:
2936:normal-ordered product
2919:
2878:
2796:
2728:
2703:
2642:
2581:
2533:
2481:
2456:
2395:
2334:
2275:
2207:
2105:
1970:
1863:
1777:
1688:connect two vertices,
1664:and spacetime points.
1629:quantum field theories
1625:
1561:Feynman–Dyson diagrams
1415:perturbation expansion
1393:
1352:Motivation and history
550:Quantum chromodynamics
407:Effective field theory
162:probability amplitudes
132:Nobel Prize in Physics
95:
80:
24245:The Meaning of It All
24132:One-electron universe
24097:Bethe–Feynman formula
23968:Breit–Wheeler process
23907:Klein–Nishina formula
23730:Draw Feynman diagrams
23623:10.5170/CERN-1973-009
22858:radius of convergence
22780:
22690:
22507:
22430:
22346:
22284:
22075:For example, for the
22000:
21759:
21719:
21682:
21640:
21597:
21363:
21217:
21157:
20981:
20783:
20600:
20312:
20198:combining denominator
20184:
19963:
19893:
19738:
19576:
19397:
19259:
19177:
19071:
18916:
18762:
18444:
18351:
18289:
18238:
18185:
18097:
17893:
17787:
17657:
17519:
17381:that depends only on
17369:
17266:
17086:
16952:
16901:
16761:
16689:
16604:
16474:
16180:
15850:
15665:do not mix together.
15625:
15527:
15440:
15295:
15115:
14945:
14551:
14415:
14279:: Grassmann integrals
14226:
14051:
13908:
13826:
13753:
13610:
13548:
13440:
13323:
13111:
12956:
12855:
12677:
12421:
12286:
12250:and therefore yields
12242:
12050:
11964:one has to include a
11921:
11861:
11609:4 × 3 × 4 × 3 × 2 × 2
11320:
11167:, by Wick's theorem.
11142:
10963:
10718:
10568:
10269:
10172:
9986:
9896:
9847:
9648:
9401:
9315:
9313:{\displaystyle \phi }
9292:
9290:{\displaystyle \phi }
9271:
9269:{\displaystyle \phi }
9239:
8803:
8639:
8523:
8468:
8385:
8302:
8219:
8058:
7863:
7750:
7666:
7575:
7241:
7069:
6885:
6809:
6598:
6558:
6491:
6402:
6354:
6218:
6136:
6046:
5940:
5833:
5720:
5622:
5365:
5173:
4985:
4792:
4585:
4523:
4403:
4221:
4017:
3814:
3628:
3461:
3441:
3110:
3029:
2920:
2858:
2776:
2729:
2683:
2622:
2561:
2513:
2482:
2436:
2375:
2314:
2255:
2208:
2106:
1971:
1891:probability amplitude
1861:
1743:
1706:scattering amplitudes
1702:correlation functions
1650:meromorphic functions
1605:
1593:fundamental particles
1550:Stueckelberg diagrams
1531:statistical mechanics
1359:
495:Klein–Gordon equation
427:LSZ reduction formula
198:. Alternatively, the
90:
26:
24403:Quantum field theory
24147:Feynman checkerboard
24127:Sticky bead argument
23676:Quantum Field Theory
22713:
22534:
22459:
22386:
22325:
22092:
21778:
21697:
21612:
21378:
21232:
21167:
21091:
21077:integral is over an
20805:
20628:
20356:
20207:
19983:
19912:
19822:
19602:
19426:
19299:
19120:
18931:
18805:
18471:
18366:
18308:
18253:
18200:
18112:
17908:
17822:
17676:
17538:
17400:
17289:
17101:
16964:
16941:
16780:
16704:
16623:
16516:
16301:
15875:
15672:
15543:
15465:
15305:
15146:
14978:
14627:
14450:
14305:
14084:
13931:
13841:
13768:
13636:
13465:
13378:
13165:
13017:
12882:
12693:
12488:
12257:
12085:
11991:
11943:. If one encounters
11876:
11814:
11783:Connected diagrams:
11259:
11201:th order terms has:
11039:
10759:
10598:
10284:
10195:
10055:
10018:. The next unpaired
9922:
9857:
9667:
9411:
9327:
9304:
9281:
9260:
8818:
8662:
8561:
8483:
8419:
8317:
8264:
8095:
7878:
7773:
7692:
7616:
7401:
7084:
6918:
6832:
6617:
6577:
6506:
6450:
6373:
6366:is a rotation, then
6233:
6170:
6061:
5970:
5867:
5784:
5641:
5417:
5265:
5038:
4808:
4634:
4538:
4450:
4236:
4051:
3829:
3651:
3466:
3192:
3076:
2967:
2766:
2504:
2246:
2229:time-ordered product
2125:
2004:
1912:
1609:quantum field theory
1597:James Daniel Bjorken
588:Theory of everything
422:Lattice field theory
392:Correlation function
214:Quantum field theory
191:of the perturbative
181:transition amplitude
179:contribution to the
120:quantum field theory
24393:Concepts in physics
24087:Feynman–Kac formula
23983:Delbrück scattering
23937:Vacuum polarization
23861:Faddeev–Popov ghost
23742:Online Diagram Tool
23373:, January 11, 2012.
23183:2006EJPh...27.1241P
23115:1949PhRv...76..749F
23052:10.1511/2005.52.957
22900:The Big Bang Theory
22269:
22247:
22225:
22203:
22058:particles to go to
21395:
20885:
20867:
20489:
20373:
20242:
19152:
18963:
18857:
18643:
18625:
18604:
18589:
18574:
18549:
17385:, not on the angle
16239:The rules for spin-
16232:pair have the same
15988:
12863:Where the sum over
12337:
12222:
12152:
12027:
11312:
11130:
11108:
11086:
11064:
10022:can be paired with
10002:can be paired with
9231:
9178:
9122:
9069:
9013:
8960:
8795:
6975:
6529:
5634:near zero this is:
5602:
4651:
4555:
2172:
2116:interaction picture
1990:. In terms of the
1899:to the final state
1613:perturbation theory
1535:classical mechanics
1514:and implemented by
1502:. The technique of
1481:functional integral
1360:In this diagram, a
567:Incomplete theories
160:The calculation of
108:subatomic particles
100:theoretical physics
24418:1948 introductions
24322:Cargo cult science
23978:Compton scattering
23756:"Feynman Diagrams"
23409:Dubner, Stephen J.
23294:Martinus Veltman,
23282:2005-03-19 at the
23039:American Scientist
22884:In popular culture
22825:Born approximation
22775:
22698:So the normalized
22685:
22502:
22425:
22341:
22279:
22255:
22233:
22211:
22189:
22026:compton wavelength
21995:
21754:
21677:
21637:
21592:
21381:
21358:
21212:
21152:
20976:
20871:
20853:
20778:
20595:
20475:
20359:
20307:
20228:
20179:
19958:
19888:
19733:
19571:
19392:
19254:
19138:
19066:
18949:
18911:
18843:
18757:
18626:
18608:
18590:
18575:
18560:
18532:
18439:
18346:
18284:
18233:
18180:
18092:
17888:
17795:The integral over
17782:
17652:
17514:
17364:
17261:
17081:
16986:
16947:
16896:
16756:
16684:
16599:
16469:
16370:
16323:
16175:
16107:
16054:
15976:
15845:
15620:
15522:
15435:
15290:
15110:
14940:
14546:
14410:
14285:Grassmann integral
14221:
14046:
13903:
13821:
13785:
13748:
13605:
13435:
13318:
13218:
13106:
12979:zeroing the total
12951:
12850:
12825:
12672:
12665:
12640:
12416:
12375:
12316:
12284:
12237:
12208:
12138:
12045:
12006:
12003:
11916:
11903:
11856:
11315:
11298:
11137:
11116:
11094:
11072:
11050:
10958:
10952:
10713:
10563:
10264:
10167:
9981:
9891:
9842:
9780:
9744:
9643:
9396:
9310:
9287:
9266:
9234:
9217:
9164:
9108:
9055:
8999:
8946:
8798:
8781:
8740:
8634:
8518:
8463:
8407:Feynman propagator
8380:
8297:
8253:Equation of motion
8214:
8053:
7858:
7745:
7661:
7599:Monte Carlo method
7570:
7552:
7496:
7483:
7236:
7233:
7208:
7064:
7000:
6961:
6949:
6880:
6804:
6719:
6653:
6593:
6553:
6509:
6486:
6397:
6349:
6213:
6131:
6090:
6041:
5992:
5935:
5896:
5828:
5715:
5670:
5617:
5588:
5360:
5306:
5295:
5219:Fourier transforms
5168:
4995:partition function
4980:
4903:
4878:
4787:
4637:
4580:
4541:
4518:
4472:
4398:
4284:
4216:
4136:
4012:
3882:
3809:
3803:
3634:
3623:
3436:
3125:A fermionic field
3105:
3024:
2915:
2841:
2724:
2477:
2237:matrix exponential
2203:
2152:
2118:, this expands to
2101:
2064:
2038:
1966:
1864:
1778:
1717:spacetime diagrams
1635:by Feynman rules.
1554:Ernst Stueckelberg
1508:Ernst Stueckelberg
1394:
432:Partition function
359:Topological charge
279:General relativity
274:Special relativity
143:Ernst Stueckelberg
124:solid-state theory
96:
81:
24398:Scattering theory
24380:
24379:
24372:
24351:
24297:Charles Hirshberg
24152:Feynman sprinkler
24034:
24033:
23993:Møller scattering
23963:Bhabha scattering
23932:Uehling potential
23881:Virtual particles
23496:"The Feynman Van"
23389:Symmetry Magazine
23353:978-0-486-42827-7
23326:978-0-07-005494-3
23255:Feynman's Rainbow
23242:. Penguin-Putnam.
22849:coupling constant
22768:
22721:
22657:
22270:
22248:
22226:
22204:
21949:
21811:
21636:
20848:
20776:
20722:
20671:
20574:
20456:
20298:
20223:
20192:leaving only the
20170:
19386:
19342:
19221:
19209:
19063:
19032:
18838:
18740:
18712:
18428:
18386:
18143:
18072:
17977:
17807:with a change in
17765:
17632:
17494:
17354:
17256:
17243:
16985:
16891:
16676:
16584:
16497:is some function
16369:
16322:
16264:) there are more
16173:
16074:
16022:
15951:
15920:
15892:
15843:
15771:
15689:
15571:
15480:
15453:Majorana fermions
15379:
15357:
15342:
15288:
15083:
15056:
15052:
15032:
15017:
14995:
14915:
14865:
14831:
14763:
14748:
14713:
14692:
14674:
14653:
14469:
14373:
14339:
14183:
14144:
14120:
13776:
13203:
13143:gets a factor of
13123:vertex, or on an
12816:
12666:
12470:Jeffrey Goldstone
12429:To establish the
12366:
12353:
12275:
12206:
12161:
12131:
12106:
12043:
11994:
11894:
11642:goes from vertex
11521:comes divided by
11313:
11131:
11109:
11087:
11065:
10956:
10691:
10667:
10358:
10249:
9889:
9840:
9781:
9639:
9609:
9599:
9477:
9441:
9394:
9393:
9232:
9179:
9123:
9070:
9014:
8961:
8796:
8725:
8516:
8051:
7856:
7758:in finite volume
7743:
7605:Scalar propagator
7568:
7487:
7458:
7234:
6991:
6948:
6864:
6750:
6710:
6644:
6642:
6089:
5991:
5895:
5669:
5305:
5274:
5090:
4904:
4471:
4360:
4330:
4263:
4240:
4115:
4091:
4066:
3974:
3928:
3833:
3754:
3709:
3666:
3638:
3637:
3530:
3481:
3361:
3295:
3245:
2998:
2839:
2832:
2554:
2307:
2039:
2013:
1902:| f ⟩
1746:• internal lines
1617:many-body problem
1541:Alternative names
1500:self-interactions
1485:quantum mechanics
1396:When calculating
1370:strange antiquark
1349:
1348:
412:Expectation value
387:BRST quantization
334:Poincaré symmetry
289:Yang–Mills theory
269:Quantum mechanics
149:as if it were an
85:
84:
24430:
24370:
24362:
24349:
24302:
24293:
24273:
24262:
24251:
24240:
24229:
24218:
24207:
24196:
24185:
24174:
24061:
24054:
24047:
24038:
24037:
24023:
24022:
23998:Schwinger effect
23799:
23792:
23785:
23776:
23775:
23771:
23714:
23702:
23691:
23689:
23688:
23668:
23656:
23645:
23626:
23600:
23599:
23588:
23582:
23581:
23580:. July 12, 2005.
23574:
23568:
23554:
23548:
23547:
23536:
23530:
23506:
23500:
23499:
23492:
23486:
23472:
23466:
23452:
23446:
23445:
23438:"Fermilab Today"
23434:
23428:
23427:
23425:
23423:
23405:
23399:
23398:
23396:
23395:
23380:
23374:
23364:
23358:
23357:
23337:
23331:
23330:
23312:
23306:
23292:
23286:
23265:
23259:
23258:
23250:
23244:
23243:
23235:
23229:
23228:
23226:
23224:
23209:
23203:
23202:
23176:
23167:(5): 1241–1250.
23156:
23150:
23149:
23143:
23142:
23133:. Archived from
23094:
23088:
23087:
23085:
23084:
23069:
23063:
23062:
23060:
23035:
23026:
23009:
23006:
22819:
22815:
22809:
22808:
22807:
22798:
22791:
22784:
22782:
22781:
22776:
22770:
22769:
22761:
22759:
22755:
22754:
22753:
22741:
22740:
22722:
22717:
22705:
22701:
22694:
22692:
22691:
22686:
22680:
22663:
22658:
22656:
22648:
22640:
22618:
22589:
22581:
22580:
22568:
22567:
22555:
22554:
22526:
22522:
22518:
22511:
22509:
22508:
22503:
22497:
22496:
22484:
22483:
22471:
22470:
22451:
22441:
22434:
22432:
22431:
22426:
22420:
22403:
22378:
22374:
22367:
22363:
22361:
22350:
22348:
22347:
22342:
22317:
22313:
22312:
22310:
22309:
22304:
22301:
22288:
22286:
22285:
22280:
22271:
22268:
22263:
22251:
22249:
22246:
22241:
22229:
22227:
22224:
22219:
22207:
22205:
22202:
22197:
22185:
22180:
22176:
22172:
22171:
22153:
22152:
22134:
22133:
22115:
22114:
22084:
22080:
22071:
22061:
22057:
22049:
22014:
22004:
22002:
22001:
21996:
21994:
21993:
21978:
21977:
21964:
21963:
21950:
21948:
21947:
21936:
21932:
21931:
21930:
21921:
21920:
21905:
21904:
21895:
21894:
21882:
21881:
21872:
21871:
21853:
21851:
21850:
21849:
21812:
21810:
21809:
21808:
21796:
21795:
21782:
21770:
21763:
21761:
21760:
21755:
21749:
21748:
21738:
21733:
21709:
21708:
21686:
21684:
21683:
21678:
21670:
21669:
21659:
21654:
21638:
21634:
21624:
21623:
21601:
21599:
21598:
21593:
21580:
21579:
21564:
21563:
21550:
21549:
21548:
21544:
21543:
21542:
21533:
21532:
21517:
21516:
21507:
21506:
21494:
21493:
21484:
21483:
21471:
21470:
21461:
21460:
21435:
21434:
21425:
21424:
21423:
21394:
21389:
21367:
21365:
21364:
21359:
21353:
21352:
21334:
21333:
21318:
21317:
21302:
21301:
21283:
21282:
21267:
21266:
21244:
21243:
21221:
21219:
21218:
21213:
21207:
21206:
21188:
21187:
21161:
21159:
21158:
21153:
21150:
21149:
21131:
21130:
21115:
21114:
21080:
21076:
21072:
21065:
21054:
21053:
21051:
21050:
21045:
21042:
21022:
20995:
20985:
20983:
20982:
20977:
20971:
20970:
20955:
20954:
20941:
20940:
20939:
20938:
20929:
20928:
20913:
20912:
20903:
20902:
20884:
20879:
20866:
20861:
20849:
20847:
20846:
20845:
20833:
20832:
20823:
20822:
20809:
20797:
20787:
20785:
20784:
20779:
20777:
20775:
20774:
20773:
20761:
20760:
20751:
20750:
20728:
20723:
20721:
20720:
20719:
20707:
20706:
20697:
20696:
20674:
20672:
20670:
20669:
20668:
20656:
20655:
20642:
20613:
20604:
20602:
20601:
20596:
20587:
20575:
20573:
20572:
20567:
20563:
20562:
20561:
20531:
20530:
20518:
20517:
20516:
20494:
20488:
20483:
20457:
20455:
20454:
20449:
20445:
20444:
20443:
20410:
20409:
20397:
20396:
20378:
20372:
20367:
20348:
20334:
20327:
20323:
20316:
20314:
20313:
20308:
20299:
20297:
20296:
20295:
20290:
20289:
20255:
20254:
20244:
20241:
20236:
20224:
20222:
20211:
20195:
20188:
20186:
20185:
20180:
20171:
20169:
20168:
20163:
20159:
20158:
20157:
20124:
20123:
20111:
20110:
20092:
20084:
20083:
20082:
20078:
20077:
20076:
20043:
20042:
20030:
20029:
20001:
20000:
19975:
19972:This allows the
19967:
19965:
19964:
19959:
19956:
19897:
19895:
19894:
19889:
19883:
19853:
19807:
19803:
19799:
19795:
19791:
19787:
19786:
19784:
19783:
19778:
19775:
19760:
19742:
19740:
19739:
19734:
19728:
19727:
19726:
19725:
19716:
19693:
19679:
19678:
19666:
19665:
19650:
19625:
19624:
19623:
19594:
19580:
19578:
19577:
19572:
19566:
19547:
19546:
19545:
19541:
19540:
19539:
19527:
19526:
19500:
19486:
19485:
19473:
19472:
19449:
19448:
19447:
19418:
19408:
19401:
19399:
19398:
19393:
19387:
19385:
19384:
19383:
19371:
19370:
19345:
19343:
19341:
19340:
19339:
19327:
19326:
19313:
19311:
19310:
19291:
19287:
19263:
19261:
19260:
19255:
19253:
19252:
19251:
19240:
19236:
19235:
19234:
19222:
19217:
19216:
19211:
19210:
19202:
19198:
19190:
19185:
19151:
19146:
19109:
19105:
19101:
19097:
19093:
19092:
19091:
19082:
19075:
19073:
19072:
19067:
19065:
19064:
19062:
19054:
19053:
19052:
19039:
19033:
19031:
19030:
19029:
19025:
19012:
18994:
18992:
18991:
18987:
18986:
18962:
18957:
18920:
18918:
18917:
18912:
18903:
18902:
18901:
18897:
18896:
18895:
18883:
18882:
18856:
18851:
18839:
18837:
18836:
18835:
18823:
18822:
18809:
18781:
18774:
18766:
18764:
18763:
18758:
18752:
18751:
18742:
18741:
18733:
18724:
18723:
18714:
18713:
18705:
18702:
18701:
18683:
18682:
18673:
18672:
18663:
18662:
18653:
18652:
18642:
18637:
18624:
18619:
18603:
18598:
18588:
18583:
18573:
18568:
18559:
18558:
18548:
18543:
18528:
18527:
18518:
18517:
18508:
18507:
18498:
18497:
18463:
18459:
18455:
18448:
18446:
18445:
18440:
18430:
18429:
18421:
18414:
18413:
18409:
18408:
18398:
18397:
18388:
18387:
18379:
18355:
18353:
18352:
18347:
18344:
18340:
18339:
18338:
18328:
18327:
18300:
18293:
18291:
18290:
18285:
18276:
18275:
18265:
18264:
18242:
18240:
18239:
18234:
18222:
18221:
18212:
18211:
18189:
18187:
18186:
18181:
18171:
18170:
18169:
18168:
18148:
18144:
18142:
18134:
18126:
18101:
18099:
18098:
18093:
18090:
18089:
18077:
18073:
18071:
18063:
18055:
18046:
18045:
18027:
18026:
18002:
18001:
17982:
17978:
17976:
17968:
17960:
17951:
17950:
17932:
17931:
17897:
17895:
17894:
17889:
17814:
17810:
17806:
17802:
17798:
17791:
17789:
17788:
17783:
17770:
17766:
17764:
17756:
17748:
17726:
17725:
17668:
17661:
17659:
17658:
17653:
17637:
17633:
17631:
17623:
17615:
17530:
17523:
17521:
17520:
17515:
17499:
17495:
17493:
17485:
17477:
17392:
17388:
17384:
17380:
17373:
17371:
17370:
17365:
17357:
17356:
17355:
17350:
17349:
17340:
17330:
17326:
17319:
17318:
17309:
17308:
17270:
17268:
17267:
17262:
17257:
17255:
17254:
17245:
17244:
17242:
17241:
17232:
17231:
17230:
17221:
17220:
17210:
17202:
17201:
17188:
17186:
17182:
17181:
17156:
17152:
17148:
17137:
17136:
17118:
17117:
17090:
17088:
17087:
17082:
17080:
17076:
17075:
17074:
17069:
17065:
17064:
17063:
17054:
17053:
17032:
17031:
17022:
17021:
17012:
17011:
17002:
17001:
16987:
16978:
16956:
16954:
16953:
16948:
16936:
16934:
16933:
16930:
16927:
16920:
16913:
16905:
16903:
16902:
16897:
16892:
16890:
16889:
16880:
16879:
16867:
16865:
16861:
16860:
16835:
16831:
16827:
16816:
16815:
16797:
16796:
16772:
16765:
16763:
16762:
16757:
16755:
16754:
16745:
16744:
16735:
16734:
16725:
16724:
16693:
16691:
16690:
16685:
16679:
16678:
16677:
16672:
16671:
16666:
16662:
16661:
16660:
16651:
16650:
16635:
16615:
16608:
16606:
16605:
16600:
16587:
16586:
16585:
16580:
16579:
16570:
16560:
16556:
16549:
16548:
16539:
16538:
16508:
16504:
16500:
16496:
16489:
16484:gauge invariance
16478:
16476:
16475:
16470:
16464:
16460:
16459:
16458:
16449:
16448:
16439:
16438:
16429:
16428:
16416:
16415:
16406:
16405:
16396:
16395:
16386:
16385:
16371:
16362:
16350:
16349:
16337:
16336:
16324:
16315:
16281:
16277:
16276:
16267:
16254:
16252:
16251:
16248:
16245:
16235:
16231:
16229:
16219:
16217:
16207:
16203:
16202:
16197:
16193:
16192:
16184:
16182:
16181:
16176:
16174:
16172:
16165:
16164:
16145:
16143:
16139:
16138:
16137:
16125:
16124:
16106:
16095:
16073:
16072:
16053:
16052:
16018:
16014:
16010:
16009:
15984:
15966:
15965:
15953:
15952:
15944:
15935:
15934:
15922:
15921:
15913:
15907:
15906:
15894:
15893:
15885:
15867:
15866:
15861:
15854:
15852:
15851:
15846:
15844:
15842:
15841:
15840:
15828:
15827:
15817:
15800:
15795:
15772:
15770:
15750:
15745:
15722:
15718:
15702:
15691:
15690:
15682:
15664:
15660:
15653:
15647:
15636:
15629:
15627:
15626:
15621:
15612:
15608:
15601:
15600:
15591:
15590:
15573:
15572:
15564:
15561:
15560:
15531:
15529:
15528:
15523:
15518:
15514:
15507:
15506:
15497:
15496:
15482:
15481:
15473:
15444:
15442:
15441:
15436:
15430:
15429:
15414:
15413:
15406:
15405:
15389:
15380:
15378:
15377:
15376:
15360:
15358:
15356:
15345:
15343:
15335:
15330:
15326:
15322:
15321:
15299:
15297:
15296:
15291:
15289:
15287:
15277:
15265:
15264:
15260:
15259:
15247:
15246:
15232:
15220:
15219:
15206:
15205:
15201:
15200:
15185:
15184:
15166:
15165:
15138:
15132:
15131:
15126:
15119:
15117:
15116:
15111:
15109:
15108:
15093:
15092:
15085:
15084:
15076:
15066:
15057:
15055:
15054:
15053:
15045:
15035:
15033:
15031:
15020:
15018:
15010:
15005:
15001:
14997:
14996:
14988:
14970:
14969:
14964:
14960:
14956:
14949:
14947:
14946:
14941:
14935:
14934:
14930:
14929:
14917:
14916:
14908:
14888:
14867:
14866:
14858:
14851:
14850:
14846:
14845:
14833:
14832:
14824:
14818:
14814:
14810:
14809:
14783:
14779:
14778:
14777:
14765:
14764:
14756:
14750:
14749:
14741:
14715:
14714:
14706:
14699:
14698:
14694:
14693:
14685:
14676:
14675:
14667:
14655:
14654:
14646:
14619:
14615:
14611:
14607:
14606:
14601:
14600:
14592:
14582:
14580:
14579:
14576:
14573:
14566:
14562:
14555:
14553:
14552:
14547:
14532:
14505:
14504:
14503:
14502:
14493:
14492:
14483:
14482:
14470:
14462:
14442:
14438:
14437:
14426:
14419:
14417:
14416:
14411:
14396:
14375:
14374:
14366:
14359:
14358:
14357:
14356:
14347:
14346:
14341:
14340:
14332:
14328:
14327:
14278:
14276:
14275:
14272:
14269:
14257:
14255:
14254:
14251:
14248:
14237:
14230:
14228:
14227:
14222:
14216:
14215:
14197:
14196:
14184:
14182:
14162:
14145:
14143:
14123:
14121:
14113:
14108:
14104:
14073:
14069:
14062:
14055:
14053:
14052:
14047:
14034:
14033:
14029:
14028:
14018:
14017:
13984:
13983:
13923:
13919:
13912:
13910:
13909:
13904:
13895:
13894:
13830:
13828:
13827:
13822:
13820:
13816:
13815:
13814:
13813:
13812:
13803:
13802:
13784:
13757:
13755:
13754:
13749:
13747:
13743:
13742:
13713:
13712:
13687:
13686:
13669:
13668:
13628:
13614:
13612:
13611:
13606:
13603:
13599:
13598:
13597:
13596:
13595:
13586:
13585:
13567:
13562:
13539:
13535:
13534:
13509:
13508:
13498:
13497:
13457:
13451:
13444:
13442:
13441:
13436:
13426:
13425:
13370:
13366:
13362:
13355:
13351:
13327:
13325:
13324:
13319:
13313:
13312:
13294:
13293:
13275:
13274:
13253:
13252:
13234:
13233:
13217:
13199:
13198:
13180:
13179:
13153:
13142:
13138:
13134:
13130:
13126:
13122:
13115:
13113:
13112:
13107:
13101:
13100:
13057:
13056:
13009:
12989:
12982:
12978:
12967:
12960:
12958:
12957:
12952:
12950:
12949:
12931:
12930:
12906:
12905:
12874:
12866:
12859:
12857:
12856:
12851:
12845:
12841:
12840:
12836:
12835:
12834:
12824:
12799:
12795:
12794:
12793:
12763:
12762:
12750:
12749:
12734:
12733:
12721:
12720:
12711:
12710:
12681:
12679:
12678:
12673:
12667:
12657:
12656:
12629:
12628:
12616:
12615:
12600:
12599:
12587:
12586:
12577:
12576:
12560:
12555:
12551:
12547:
12546:
12534:
12533:
12518:
12517:
12505:
12504:
12456:
12445:
12425:
12423:
12422:
12417:
12411:
12388:
12387:
12386:
12385:
12374:
12355:
12354:
12352:
12348:
12347:
12336:
12335:
12334:
12324:
12315:
12312:
12307:
12300:
12299:
12283:
12246:
12244:
12243:
12238:
12233:
12229:
12221:
12216:
12207:
12199:
12194:
12193:
12173:
12169:
12162:
12160:
12151:
12146:
12137:
12132:
12130:
12122:
12121:
12112:
12107:
12105:
12094:
12077:
12071:
12061:
12054:
12052:
12051:
12046:
12044:
12042:
12038:
12037:
12026:
12025:
12024:
12014:
12005:
12002:
11983:
11976:
11963:
11956:
11949:
11942:
11935:
11925:
11923:
11922:
11917:
11915:
11914:
11913:
11902:
11865:
11863:
11862:
11857:
11806:
11796:undirected graph
11773:
11769:
11762:
11747:
11733:
11722:
11715:
11704:
11693:
11682:
11671:
11660:
11649:
11645:
11641:
11633:
11629:
11614:
11610:
11606:
11602:
11598:
11594:
11590:
11586:
11579:
11577:
11576:
11573:
11570:
11563:
11561:
11560:
11557:
11554:
11547:
11539:
11535:
11528:
11524:
11520:
11516:
11512:
11508:
11490:Symmetry factors
11485:
11481:
11477:
11470:
11465:
11455:
11448:
11443:
11429:
11425:
11421:
11417:
11413:
11394:
11390:
11386:
11385:
11383:
11382:
11377:
11374:
11366:
11359:
11355:
11351:
11347:
11343:
11336:
11332:
11324:
11322:
11321:
11316:
11314:
11311:
11306:
11297:
11293:
11292:
11280:
11279:
11263:
11243:
11229:
11225:
11221:
11210:
11200:
11193:
11189:
11186:
11182:
11178:
11171:Feynman diagrams
11166:
11157:
11153:
11146:
11144:
11143:
11138:
11132:
11129:
11124:
11112:
11110:
11107:
11102:
11090:
11088:
11085:
11080:
11068:
11066:
11063:
11058:
11046:
11031:
11027:
11017:half-lines, the
11016:
11012:
11000:
10996:
10992:
10978:
10974:
10967:
10965:
10964:
10959:
10957:
10942:
10941:
10923:
10922:
10904:
10903:
10885:
10884:
10869:
10868:
10852:
10847:
10843:
10839:
10838:
10820:
10819:
10801:
10800:
10782:
10781:
10751:
10744:
10733:
10722:
10720:
10719:
10714:
10712:
10708:
10692:
10690:
10679:
10668:
10666:
10655:
10636:
10635:
10634:
10633:
10613:
10612:
10590:
10586:
10582:
10572:
10570:
10569:
10564:
10553:
10552:
10537:
10536:
10524:
10523:
10511:
10510:
10498:
10497:
10479:
10478:
10460:
10459:
10441:
10440:
10422:
10421:
10406:
10405:
10404:
10403:
10394:
10393:
10384:
10383:
10374:
10373:
10359:
10357:
10346:
10341:
10340:
10335:
10331:
10312:
10311:
10302:
10301:
10273:
10271:
10270:
10265:
10260:
10259:
10250:
10248:
10237:
10229:
10228:
10216:
10215:
10176:
10174:
10173:
10168:
10166:
10165:
10160:
10156:
10155:
10078:
10074:
10073:
10047:
10041:
10033:
10029:
10021:
10017:
10010:others, leaving
10009:
10001:
9997:
9990:
9988:
9987:
9982:
9980:
9976:
9975:
9974:
9959:
9958:
9949:
9948:
9939:
9938:
9914:
9910:
9900:
9898:
9897:
9892:
9890:
9882:
9877:
9873:
9872:
9851:
9849:
9848:
9843:
9841:
9839:
9838:
9826:
9782:
9779:
9778:
9774:
9769:
9768:
9743:
9742:
9738:
9733:
9732:
9712:
9711:
9695:
9690:
9686:
9685:
9659:
9652:
9650:
9649:
9644:
9642:
9641:
9640:
9635:
9621:
9610:
9602:
9600:
9598:
9561:
9520:
9509:
9508:
9504:
9499:
9498:
9478:
9476:
9475:
9466:
9465:
9453:
9442:
9440:
9439:
9438:
9425:
9424:
9415:
9405:
9403:
9402:
9397:
9395:
9389:
9381:
9380:
9369:
9368:
9364:
9359:
9358:
9319:
9317:
9316:
9311:
9296:
9294:
9293:
9288:
9275:
9273:
9272:
9267:
9243:
9241:
9240:
9235:
9233:
9230:
9225:
9216:
9212:
9211:
9199:
9198:
9182:
9180:
9177:
9172:
9163:
9159:
9158:
9146:
9145:
9129:
9124:
9121:
9116:
9107:
9103:
9102:
9090:
9089:
9073:
9071:
9068:
9063:
9054:
9050:
9049:
9037:
9036:
9020:
9015:
9012:
9007:
8998:
8994:
8993:
8981:
8980:
8964:
8962:
8959:
8954:
8945:
8941:
8940:
8928:
8927:
8911:
8906:
8902:
8898:
8897:
8879:
8878:
8860:
8859:
8841:
8840:
8807:
8805:
8804:
8799:
8797:
8794:
8789:
8780:
8779:
8775:
8774:
8773:
8761:
8760:
8742:
8739:
8718:
8714:
8710:
8709:
8685:
8684:
8654:
8650:
8643:
8641:
8640:
8635:
8633:
8629:
8625:
8624:
8603:
8602:
8584:
8583:
8535:
8527:
8525:
8524:
8519:
8517:
8515:
8514:
8502:
8472:
8470:
8469:
8464:
8431:
8430:
8411:
8404:
8400:
8396:
8389:
8387:
8386:
8381:
8373:
8369:
8339:
8338:
8329:
8328:
8306:
8304:
8303:
8298:
8286:
8285:
8276:
8275:
8248:
8244:
8240:
8223:
8221:
8220:
8215:
8209:
8208:
8196:
8195:
8180:
8179:
8167:
8166:
8154:
8153:
8141:
8140:
8131:
8130:
8087:
8083:
8079:
8072:
8062:
8060:
8059:
8054:
8052:
8050:
8049:
8048:
8039:
8038:
8011:
8010:
7986:
7985:
7961:
7960:
7947:
7942:
7919:
7915:
7911:
7867:
7865:
7864:
7859:
7857:
7855:
7854:
7842:
7837:
7814:
7810:
7806:
7765:
7761:
7754:
7752:
7751:
7746:
7744:
7742:
7741:
7729:
7724:
7720:
7719:
7718:
7709:
7708:
7684:
7670:
7668:
7667:
7662:
7653:
7649:
7648:
7647:
7646:
7633:
7632:
7592:
7590:
7579:
7577:
7576:
7571:
7569:
7567:
7566:
7558:
7548:
7547:
7535:
7534:
7519:
7518:
7506:
7505:
7495:
7485:
7482:
7475:
7467:
7454:
7450:
7446:
7445:
7424:
7423:
7390:
7371:
7357:
7343:
7342:
7340:
7339:
7334:
7331:
7323:
7311:
7310:
7308:
7307:
7302:
7299:
7289:
7288:
7286:
7285:
7277:
7274:
7266:
7262:
7255:
7245:
7243:
7242:
7237:
7235:
7225:
7224:
7197:
7196:
7175:
7174:
7159:
7158:
7142:
7137:
7133:
7129:
7128:
7107:
7106:
7073:
7071:
7070:
7065:
7059:
7058:
7054:
7053:
7043:
7042:
7037:
7033:
7032:
7018:
7017:
6999:
6987:
6986:
6985:
6984:
6974:
6969:
6960:
6959:
6950:
6941:
6935:
6934:
6910:
6902:Euclidean action
6896:
6889:
6887:
6886:
6881:
6875:
6874:
6869:
6865:
6857:
6844:
6843:
6824:
6820:
6813:
6811:
6810:
6805:
6802:
6801:
6797:
6796:
6786:
6785:
6780:
6776:
6775:
6761:
6760:
6751:
6743:
6736:
6735:
6734:
6733:
6718:
6699:
6695:
6673:
6672:
6663:
6662:
6652:
6643:
6635:
6609:
6602:
6600:
6599:
6594:
6569:
6562:
6560:
6559:
6554:
6551:
6550:
6528:
6520:
6495:
6493:
6492:
6487:
6484:
6483:
6465:
6464:
6442:
6431:
6417:
6406:
6404:
6403:
6398:
6387:
6386:
6385:
6365:
6358:
6356:
6355:
6350:
6344:
6343:
6327:
6326:
6313:
6312:
6294:
6293:
6277:
6276:
6263:
6262:
6222:
6220:
6219:
6214:
6208:
6207:
6198:
6197:
6182:
6181:
6161:
6147:
6140:
6138:
6137:
6132:
6130:
6129:
6124:
6120:
6101:
6100:
6091:
6082:
6079:
6078:
6050:
6048:
6047:
6042:
6037:
6036:
6023:
6022:
6013:
6012:
6003:
6002:
5993:
5984:
5959:
5944:
5942:
5941:
5936:
5907:
5906:
5897:
5888:
5885:
5884:
5859:
5851:
5837:
5835:
5834:
5829:
5805:
5804:
5776:
5769:
5768:
5767:
5755:
5753:
5752:
5747:
5744:
5731:
5724:
5722:
5721:
5716:
5710:
5709:
5704:
5700:
5681:
5680:
5671:
5662:
5659:
5658:
5633:
5626:
5624:
5623:
5618:
5612:
5611:
5601:
5596:
5587:
5586:
5580:
5579:
5570:
5569:
5545:
5544:
5529:
5528:
5519:
5518:
5494:
5493:
5484:
5483:
5474:
5473:
5449:
5448:
5442:
5441:
5435:
5434:
5406:
5394:
5390:
5386:
5385:
5369:
5367:
5366:
5361:
5355:
5354:
5349:
5348:
5314:
5313:
5307:
5298:
5294:
5254:
5250:
5246:
5239:
5235:
5228:
5224:
5216:
5212:
5208:
5204:
5203:
5201:
5200:
5195:
5192:
5184:
5177:
5175:
5174:
5169:
5163:
5162:
5135:
5134:
5122:
5121:
5091:
5089:
5088:
5087:
5068:
5060:
5018:
5014:
5007:
4989:
4987:
4986:
4981:
4975:
4971:
4967:
4963:
4959:
4958:
4937:
4936:
4905:
4895:
4894:
4867:
4866:
4845:
4844:
4829:
4828:
4812:
4796:
4794:
4793:
4788:
4782:
4778:
4774:
4770:
4766:
4765:
4744:
4743:
4702:
4701:
4680:
4679:
4664:
4663:
4650:
4645:
4622:
4611:
4600:
4596:
4589:
4587:
4586:
4581:
4568:
4567:
4554:
4549:
4527:
4525:
4524:
4519:
4510:
4509:
4496:
4495:
4483:
4482:
4473:
4464:
4442:
4407:
4405:
4404:
4399:
4397:
4396:
4392:
4361:
4359:
4333:
4331:
4329:
4328:
4327:
4308:
4304:
4303:
4293:
4285:
4280:
4276:
4265:
4264:
4256:
4225:
4223:
4222:
4217:
4208:
4197:
4196:
4177:
4176:
4163:
4148:
4147:
4137:
4132:
4128:
4117:
4116:
4108:
4088:
4087:
4068:
4067:
4059:
4021:
4019:
4018:
4013:
4011:
4010:
4006:
3975:
3973:
3963:
3962:
3952:
3951:
3950:
3931:
3929:
3927:
3926:
3925:
3906:
3902:
3901:
3891:
3883:
3878:
3874:
3863:
3862:
3844:
3843:
3818:
3816:
3815:
3810:
3804:
3799:
3795:
3784:
3783:
3765:
3764:
3749:
3735:
3734:
3722:
3711:
3710:
3702:
3687:
3686:
3668:
3667:
3659:
3642:Wick's expansion
3632:
3630:
3629:
3624:
3619:
3608:
3607:
3589:
3588:
3576:
3562:
3561:
3543:
3532:
3531:
3523:
3508:
3507:
3483:
3482:
3474:
3454:
3445:
3443:
3442:
3437:
3428:
3417:
3416:
3403:
3388:
3387:
3374:
3363:
3362:
3354:
3341:
3340:
3317:
3316:
3297:
3296:
3288:
3281:
3273:
3272:
3259:
3258:
3246:
3244:
3236:
3235:
3234:
3215:
3210:
3209:
3184:
3171:
3164:
3155:
3145:
3138:
3121:
3114:
3112:
3111:
3106:
3101:
3100:
3088:
3087:
3072:A bosonic field
3058:
3047:
3040:
3033:
3031:
3030:
3025:
3023:
3022:
3010:
3009:
3000:
2999:
2991:
2979:
2978:
2947:
2933:
2924:
2922:
2921:
2916:
2910:
2906:
2905:
2892:
2891:
2886:
2885:
2877:
2872:
2857:
2856:
2840:
2837:
2828:
2824:
2823:
2810:
2809:
2804:
2803:
2795:
2790:
2775:
2774:
2758:
2750:
2744:
2739:Wick's expansion
2733:
2731:
2730:
2725:
2719:
2718:
2702:
2697:
2679:
2675:
2674:
2670:
2669:
2656:
2655:
2650:
2649:
2641:
2636:
2616:
2615:
2609:
2605:
2604:
2603:
2594:
2593:
2580:
2575:
2555:
2553:
2545:
2544:
2535:
2532:
2527:
2496:
2486:
2484:
2483:
2478:
2472:
2471:
2455:
2450:
2432:
2428:
2427:
2423:
2422:
2409:
2408:
2403:
2402:
2394:
2389:
2369:
2368:
2362:
2358:
2357:
2356:
2347:
2346:
2333:
2328:
2308:
2306:
2298:
2297:
2296:
2277:
2274:
2269:
2226:
2222:
2212:
2210:
2209:
2204:
2199:
2198:
2188:
2187:
2171:
2163:
2140:
2139:
2110:
2108:
2107:
2102:
2093:
2092:
2080:
2079:
2063:
2053:
2052:
2037:
2027:
2026:
1996:
1987:
1982:
1975:
1973:
1972:
1967:
1958:
1953:
1945:
1940:
1929:
1928:
1927:
1904:
1903:
1898:
1897:
1775:
1771:
1757:
1751:
1663:
1658:Feynman integral
1655:
1641:is a method for
1589:Martinus Veltman
1546:Murray Gell-Mann
1465:virtual particle
1402:particle physics
1341:
1334:
1327:
402:Effective action
329:Lorentz symmetry
254:Electromagnetism
224:
210:
209:
206:
195:
189:Wick's expansion
19:
18:
24438:
24437:
24433:
24432:
24431:
24429:
24428:
24427:
24413:Richard Feynman
24383:
24382:
24381:
24376:
24360:
24305:
24300:
24291:
24276:
24271:
24260:
24249:
24238:
24227:
24216:
24205:
24194:
24183:
24172:
24156:
24082:Feynman diagram
24070:
24068:Richard Feynman
24065:
24035:
24030:
24029:
24007:
23951:
23942:Vertex function
23927:Schwinger limit
23902:Furry's theorem
23885:
23839:
23825:Feynman diagram
23808:
23803:
23721:
23711:
23686:
23684:
23665:
23642:
23609:
23604:
23603:
23590:
23589:
23585:
23576:
23575:
23571:
23559:Harald Fritzsch
23555:
23551:
23538:
23537:
23533:
23507:
23503:
23494:
23493:
23489:
23473:
23469:
23453:
23449:
23436:
23435:
23431:
23421:
23419:
23411:(Feb 7, 2024).
23406:
23402:
23393:
23391:
23381:
23377:
23365:
23361:
23354:
23338:
23334:
23327:
23313:
23309:
23293:
23289:
23284:Wayback Machine
23266:
23262:
23251:
23247:
23236:
23232:
23222:
23220:
23210:
23206:
23157:
23153:
23140:
23138:
23103:Physical Review
23095:
23091:
23082:
23080:
23077:Quanta Magazine
23071:
23070:
23066:
23058:
23033:
23027:
23023:
23018:
23013:
23012:
23007:
23003:
22998:
22993:
22969:Penguin diagram
22949:
22936:Seamus Blackley
22932:Dodge Tradesman
22920:Vacuum Diagrams
22886:
22833:
22817:
22811:
22803:
22801:
22800:
22796:
22789:
22760:
22749:
22745:
22736:
22732:
22731:
22727:
22726:
22716:
22714:
22711:
22710:
22703:
22699:
22676:
22659:
22649:
22641:
22639:
22614:
22585:
22576:
22572:
22563:
22559:
22550:
22546:
22535:
22532:
22531:
22524:
22520:
22516:
22492:
22488:
22479:
22475:
22466:
22462:
22460:
22457:
22456:
22443:
22439:
22416:
22399:
22387:
22384:
22383:
22376:
22372:
22365:
22357:
22355:
22326:
22323:
22322:
22315:
22305:
22302:
22297:
22296:
22294:
22293:
22264:
22259:
22250:
22242:
22237:
22228:
22220:
22215:
22206:
22198:
22193:
22184:
22167:
22163:
22148:
22144:
22129:
22125:
22110:
22106:
22099:
22095:
22093:
22090:
22089:
22082:
22076:
22063:
22059:
22055:
22040:
22021:
22013:
22009:
21989:
21985:
21973:
21969:
21959:
21955:
21937:
21926:
21922:
21916:
21912:
21900:
21896:
21890:
21886:
21877:
21873:
21867:
21863:
21862:
21858:
21857:
21852:
21827:
21826:
21822:
21804:
21800:
21791:
21787:
21786:
21781:
21779:
21776:
21775:
21768:
21767:Performing the
21744:
21740:
21734:
21723:
21704:
21700:
21698:
21695:
21694:
21665:
21661:
21655:
21644:
21632:
21619:
21615:
21613:
21610:
21609:
21575:
21571:
21559:
21555:
21538:
21534:
21528:
21524:
21512:
21508:
21502:
21498:
21489:
21485:
21479:
21475:
21466:
21462:
21456:
21452:
21451:
21447:
21440:
21436:
21430:
21426:
21401:
21400:
21396:
21390:
21385:
21379:
21376:
21375:
21348:
21344:
21329:
21325:
21313:
21309:
21297:
21293:
21278:
21274:
21262:
21258:
21239:
21235:
21233:
21230:
21229:
21202:
21198:
21183:
21179:
21168:
21165:
21164:
21145:
21141:
21126:
21122:
21110:
21106:
21092:
21089:
21088:
21078:
21074:
21071:
21067:
21056:
21046:
21043:
21040:
21035:
21034:
21032:
21029:
21024:
21020:
21014:
21007:
20997:
20994:
20990:
20966:
20962:
20950:
20946:
20934:
20930:
20924:
20920:
20908:
20904:
20898:
20894:
20890:
20886:
20880:
20875:
20862:
20857:
20841:
20837:
20828:
20824:
20818:
20814:
20813:
20808:
20806:
20803:
20802:
20792:
20769:
20765:
20756:
20752:
20746:
20742:
20732:
20727:
20715:
20711:
20702:
20698:
20692:
20688:
20678:
20673:
20664:
20660:
20651:
20647:
20646:
20641:
20629:
20626:
20625:
20609:
20580:
20568:
20557:
20553:
20526:
20522:
20512:
20508:
20504:
20503:
20499:
20498:
20493:
20484:
20479:
20450:
20439:
20435:
20405:
20401:
20392:
20388:
20387:
20383:
20382:
20377:
20368:
20363:
20357:
20354:
20353:
20336:
20332:
20325:
20321:
20291:
20285:
20284:
20283:
20250:
20249:
20248:
20243:
20237:
20232:
20215:
20210:
20208:
20205:
20204:
20193:
20164:
20153:
20149:
20119:
20115:
20106:
20102:
20101:
20097:
20096:
20091:
20072:
20068:
20038:
20034:
20025:
20021:
20020:
20016:
20009:
20005:
19990:
19986:
19984:
19981:
19980:
19973:
19949:
19913:
19910:
19909:
19876:
19846:
19823:
19820:
19819:
19805:
19801:
19797:
19793:
19792:goes from 0 to
19789:
19788:, the variable
19779:
19776:
19770:
19769:
19767:
19762:
19747:
19721:
19717:
19709:
19686:
19674:
19670:
19661:
19657:
19643:
19630:
19626:
19616:
19609:
19605:
19603:
19600:
19599:
19585:
19559:
19535:
19531:
19522:
19518:
19505:
19501:
19493:
19481:
19477:
19468:
19464:
19454:
19450:
19440:
19433:
19429:
19427:
19424:
19423:
19410:
19406:
19379:
19375:
19366:
19362:
19349:
19344:
19335:
19331:
19322:
19318:
19317:
19312:
19306:
19302:
19300:
19297:
19296:
19289:
19283:
19280:
19244:
19230:
19226:
19212:
19201:
19200:
19199:
19197:
19196:
19192:
19186:
19181:
19173:
19169:
19147:
19142:
19121:
19118:
19117:
19107:
19103:
19099:
19095:
19087:
19085:
19084:
19080:
19055:
19048:
19044:
19040:
19038:
19034:
19021:
19017:
19013:
19002:
18998:
18993:
18982:
18978:
18974:
18970:
18958:
18953:
18932:
18929:
18928:
18891:
18887:
18878:
18874:
18873:
18869:
18862:
18858:
18852:
18847:
18831:
18827:
18818:
18814:
18813:
18808:
18806:
18803:
18802:
18796:
18788:
18779:
18772:
18747:
18743:
18732:
18731:
18719:
18715:
18704:
18703:
18697:
18693:
18678:
18674:
18668:
18664:
18658:
18654:
18648:
18644:
18638:
18630:
18620:
18612:
18599:
18594:
18584:
18579:
18569:
18564:
18554:
18550:
18544:
18536:
18523:
18519:
18513:
18509:
18503:
18499:
18493:
18489:
18472:
18469:
18468:
18461:
18457:
18453:
18420:
18419:
18404:
18400:
18393:
18389:
18378:
18377:
18376:
18372:
18367:
18364:
18363:
18334:
18330:
18323:
18319:
18318:
18314:
18309:
18306:
18305:
18298:
18271:
18267:
18260:
18256:
18254:
18251:
18250:
18217:
18213:
18207:
18203:
18201:
18198:
18197:
18161:
18157:
18153:
18149:
18135:
18127:
18125:
18121:
18113:
18110:
18109:
18082:
18078:
18064:
18056:
18054:
18050:
18041:
18040:
18022:
18021:
17994:
17990:
17969:
17961:
17959:
17955:
17946:
17945:
17927:
17926:
17909:
17906:
17905:
17823:
17820:
17819:
17812:
17808:
17804:
17800:
17796:
17757:
17749:
17747:
17743:
17721:
17717:
17677:
17674:
17673:
17666:
17624:
17616:
17614:
17610:
17539:
17536:
17535:
17528:
17486:
17478:
17476:
17472:
17401:
17398:
17397:
17390:
17386:
17382:
17378:
17345:
17341:
17339:
17335:
17331:
17314:
17310:
17304:
17300:
17299:
17295:
17290:
17287:
17286:
17277:
17250:
17246:
17237:
17233:
17226:
17222:
17216:
17212:
17211:
17209:
17194:
17190:
17189:
17187:
17174:
17167:
17163:
17141:
17132:
17128:
17113:
17109:
17108:
17104:
17102:
17099:
17098:
17070:
17059:
17055:
17049:
17045:
17044:
17040:
17039:
17027:
17023:
17017:
17013:
17007:
17003:
16997:
16993:
16992:
16988:
16976:
16965:
16962:
16961:
16942:
16939:
16938:
16931:
16928:
16925:
16924:
16922:
16918:
16911:
16885:
16881:
16872:
16868:
16866:
16853:
16846:
16842:
16820:
16811:
16807:
16792:
16788:
16787:
16783:
16781:
16778:
16777:
16770:
16750:
16746:
16740:
16736:
16730:
16726:
16720:
16716:
16705:
16702:
16701:
16667:
16656:
16652:
16646:
16642:
16641:
16637:
16636:
16634:
16630:
16626:
16624:
16621:
16620:
16613:
16575:
16571:
16569:
16565:
16561:
16544:
16540:
16534:
16530:
16529:
16525:
16517:
16514:
16513:
16506:
16502:
16498:
16494:
16487:
16454:
16450:
16444:
16440:
16434:
16430:
16424:
16420:
16411:
16407:
16401:
16397:
16391:
16387:
16381:
16377:
16376:
16372:
16360:
16342:
16338:
16329:
16325:
16313:
16302:
16299:
16298:
16292:
16290:Spin 1: photons
16279:
16274:
16273:
16265:
16249:
16246:
16243:
16242:
16240:
16233:
16227:
16221:
16215:
16209:
16205:
16200:
16199:
16195:
16190:
16189:
16160:
16156:
16149:
16144:
16133:
16129:
16120:
16116:
16115:
16111:
16079:
16078:
16068:
16064:
16027:
16026:
16005:
16001:
15980:
15961:
15957:
15943:
15942:
15930:
15926:
15912:
15911:
15902:
15898:
15884:
15883:
15882:
15878:
15876:
15873:
15872:
15864:
15863:
15859:
15836:
15832:
15823:
15819:
15818:
15801:
15799:
15788:
15754:
15749:
15738:
15695:
15681:
15680:
15679:
15675:
15673:
15670:
15669:
15662:
15651:
15649:
15638:
15634:
15596:
15592:
15586:
15582:
15578:
15574:
15563:
15562:
15556:
15552:
15544:
15541:
15540:
15502:
15498:
15492:
15488:
15487:
15483:
15472:
15471:
15466:
15463:
15462:
15422:
15418:
15401:
15397:
15390:
15385:
15384:
15372:
15368:
15364:
15359:
15349:
15344:
15334:
15317:
15313:
15312:
15308:
15306:
15303:
15302:
15267:
15266:
15252:
15248:
15242:
15238:
15237:
15233:
15231:
15215:
15211:
15196:
15192:
15180:
15176:
15161:
15157:
15156:
15152:
15147:
15144:
15143:
15134:
15129:
15128:
15124:
15101:
15097:
15075:
15074:
15067:
15062:
15061:
15044:
15043:
15039:
15034:
15024:
15019:
15009:
14987:
14986:
14985:
14981:
14979:
14976:
14975:
14967:
14966:
14962:
14958:
14954:
14922:
14918:
14907:
14906:
14902:
14898:
14878:
14857:
14856:
14838:
14834:
14823:
14822:
14802:
14798:
14791:
14787:
14770:
14766:
14755:
14754:
14740:
14739:
14738:
14734:
14733:
14729:
14705:
14704:
14684:
14683:
14666:
14665:
14645:
14644:
14643:
14639:
14628:
14625:
14624:
14617:
14613:
14609:
14604:
14603:
14598:
14597:
14584:
14577:
14574:
14571:
14570:
14568:
14564:
14560:
14516:
14498:
14494:
14488:
14484:
14475:
14471:
14461:
14460:
14456:
14451:
14448:
14447:
14440:
14433:
14428:
14424:
14386:
14365:
14364:
14352:
14348:
14342:
14331:
14330:
14329:
14320:
14316:
14315:
14311:
14306:
14303:
14302:
14281:
14273:
14270:
14267:
14266:
14264:
14260:
14252:
14249:
14246:
14245:
14243:
14235:
14211:
14210:
14192:
14191:
14166:
14161:
14127:
14122:
14112:
14091:
14087:
14085:
14082:
14081:
14071:
14067:
14060:
14024:
14020:
14013:
14012:
13979:
13978:
13962:
13958:
13932:
13929:
13928:
13921:
13917:
13872:
13868:
13842:
13839:
13838:
13808:
13804:
13798:
13794:
13790:
13786:
13780:
13775:
13771:
13769:
13766:
13765:
13732:
13728:
13724:
13682:
13678:
13674:
13670:
13661:
13657:
13637:
13634:
13633:
13619:
13591:
13587:
13581:
13577:
13573:
13569:
13563:
13552:
13547:
13543:
13524:
13520:
13516:
13504:
13500:
13487:
13483:
13466:
13463:
13462:
13453:
13449:
13403:
13399:
13379:
13376:
13375:
13368:
13364:
13360:
13353:
13348:
13342:
13332:
13308:
13304:
13289:
13285:
13270:
13266:
13248:
13244:
13229:
13225:
13207:
13194:
13193:
13175:
13174:
13166:
13163:
13162:
13144:
13140:
13136:
13132:
13128:
13124:
13120:
13096:
13092:
13052:
13048:
13018:
13015:
13014:
13007:
12996:
12984:
12980:
12969:
12965:
12939:
12935:
12920:
12916:
12898:
12894:
12883:
12880:
12879:
12872:
12864:
12830:
12826:
12820:
12815:
12811:
12804:
12800:
12789:
12785:
12781:
12777:
12758:
12754:
12745:
12741:
12729:
12725:
12716:
12712:
12703:
12699:
12694:
12691:
12690:
12649:
12645:
12624:
12620:
12611:
12607:
12595:
12591:
12582:
12578:
12569:
12565:
12559:
12542:
12538:
12529:
12525:
12513:
12509:
12500:
12496:
12495:
12491:
12489:
12486:
12485:
12478:
12466:Keith Brueckner
12454:
12448:vacuum diagrams
12439:
12433:
12398:
12381:
12377:
12376:
12370:
12365:
12343:
12339:
12338:
12330:
12326:
12325:
12320:
12314:
12308:
12295:
12291:
12290:
12285:
12279:
12258:
12255:
12254:
12217:
12212:
12198:
12189:
12185:
12178:
12174:
12153:
12147:
12142:
12136:
12123:
12117:
12113:
12111:
12098:
12093:
12092:
12088:
12086:
12083:
12082:
12078:is obtained by
12073:
12070:
12066:
12059:
12033:
12029:
12028:
12020:
12016:
12015:
12010:
12004:
11998:
11992:
11989:
11988:
11982:
11978:
11973:
11968:
11966:symmetry factor
11962:
11958:
11955:
11951:
11948:
11944:
11941:
11937:
11934:
11930:
11909:
11905:
11904:
11898:
11877:
11874:
11873:
11815:
11812:
11811:
11802:
11788:
11771:
11764:
11760:
11738:
11724:
11720:
11706:
11695:
11684:
11673:
11662:
11651:
11647:
11643:
11639:
11631:
11627:
11612:
11608:
11604:
11600:
11596:
11592:
11588:
11584:
11574:
11571:
11568:
11567:
11565:
11558:
11555:
11552:
11551:
11549:
11545:
11537:
11533:
11526:
11522:
11518:
11514:
11510:
11506:
11500:symmetry factor
11492:
11483:
11479:
11475:
11474:A set of valid
11468:
11461:
11458:homology theory
11453:
11446:
11441:
11427:
11423:
11419:
11415:
11411:
11404:
11392:
11388:
11378:
11375:
11372:
11371:
11369:
11368:
11364:
11357:
11353:
11349:
11345:
11338:
11334:
11330:
11307:
11302:
11288:
11284:
11275:
11271:
11264:
11262:
11260:
11257:
11256:
11234:
11227:
11223:
11212:
11205:
11198:
11191:
11187:
11184:
11180:
11176:
11173:
11165:
11159:
11155:
11151:
11125:
11120:
11111:
11103:
11098:
11089:
11081:
11076:
11067:
11059:
11054:
11045:
11040:
11037:
11036:
11029:
11018:
11014:
11005:
10998:
10994:
10989:
10980:
10976:
10972:
10937:
10933:
10918:
10914:
10899:
10895:
10880:
10876:
10861:
10857:
10851:
10834:
10830:
10815:
10811:
10796:
10792:
10777:
10773:
10766:
10762:
10760:
10757:
10756:
10749:
10735:
10731:
10683:
10678:
10659:
10654:
10641:
10637:
10629:
10625:
10621:
10617:
10605:
10601:
10599:
10596:
10595:
10588:
10584:
10581:
10577:
10548:
10544:
10532:
10528:
10519:
10515:
10506:
10502:
10493:
10489:
10474:
10470:
10455:
10451:
10436:
10432:
10417:
10413:
10399:
10395:
10389:
10385:
10379:
10375:
10369:
10365:
10364:
10360:
10350:
10345:
10336:
10318:
10314:
10313:
10307:
10303:
10297:
10293:
10285:
10282:
10281:
10255:
10251:
10241:
10236:
10224:
10220:
10211:
10207:
10196:
10193:
10192:
10186:
10161:
10151:
10147:
10143:
10142:
10066:
10062:
10058:
10056:
10053:
10052:
10043:
10035:
10031:
10023:
10019:
10011:
10003:
9999:
9995:
9967:
9963:
9954:
9950:
9944:
9940:
9934:
9930:
9929:
9925:
9923:
9920:
9919:
9912:
9905:
9881:
9868:
9864:
9860:
9858:
9855:
9854:
9834:
9830:
9825:
9770:
9764:
9760:
9753:
9749:
9734:
9728:
9724:
9717:
9713:
9704:
9700:
9694:
9678:
9674:
9670:
9668:
9665:
9664:
9657:
9622:
9620:
9616:
9612:
9601:
9562:
9521:
9519:
9500:
9494:
9490:
9483:
9479:
9471:
9467:
9458:
9454:
9452:
9434:
9430:
9426:
9420:
9416:
9414:
9412:
9409:
9408:
9382:
9379:
9360:
9354:
9350:
9343:
9339:
9328:
9325:
9324:
9305:
9302:
9301:
9282:
9279:
9278:
9261:
9258:
9257:
9254:
9226:
9221:
9207:
9203:
9194:
9190:
9183:
9181:
9173:
9168:
9154:
9150:
9141:
9137:
9130:
9128:
9117:
9112:
9098:
9094:
9085:
9081:
9074:
9072:
9064:
9059:
9045:
9041:
9032:
9028:
9021:
9019:
9008:
9003:
8989:
8985:
8976:
8972:
8965:
8963:
8955:
8950:
8936:
8932:
8923:
8919:
8912:
8910:
8893:
8889:
8874:
8870:
8855:
8851:
8836:
8832:
8825:
8821:
8819:
8816:
8815:
8790:
8785:
8769:
8765:
8756:
8752:
8751:
8747:
8743:
8741:
8729:
8702:
8698:
8680:
8676:
8669:
8665:
8663:
8660:
8659:
8652:
8648:
8620:
8616:
8598:
8594:
8579:
8575:
8568:
8564:
8562:
8559:
8558:
8548:
8542:
8533:
8510:
8506:
8501:
8484:
8481:
8480:
8426:
8422:
8420:
8417:
8416:
8409:
8402:
8398:
8394:
8344:
8340:
8334:
8330:
8324:
8320:
8318:
8315:
8314:
8281:
8277:
8271:
8267:
8265:
8262:
8261:
8255:
8246:
8242:
8233:
8228:
8204:
8200:
8191:
8187:
8175:
8171:
8162:
8158:
8149:
8145:
8136:
8132:
8126:
8122:
8096:
8093:
8092:
8085:
8081:
8077:
8067:
8044:
8043:
8034:
8030:
8006:
8002:
7981:
7977:
7956:
7955:
7951:
7946:
7935:
7904:
7885:
7881:
7879:
7876:
7875:
7850:
7846:
7841:
7830:
7799:
7780:
7776:
7774:
7771:
7770:
7763:
7762:, when the two
7759:
7737:
7733:
7728:
7714:
7710:
7704:
7700:
7699:
7695:
7693:
7690:
7689:
7675:
7639:
7638:
7634:
7628:
7624:
7623:
7619:
7617:
7614:
7613:
7607:
7586:
7584:
7562:
7554:
7553:
7543:
7539:
7530:
7526:
7514:
7510:
7501:
7497:
7491:
7484:
7471:
7463:
7462:
7441:
7437:
7419:
7415:
7408:
7404:
7402:
7399:
7398:
7373:
7364:
7359:
7350:
7345:
7335:
7332:
7329:
7328:
7326:
7325:
7321:
7318:
7303:
7300:
7295:
7294:
7292:
7291:
7278:
7275:
7272:
7271:
7269:
7268:
7264:
7261:
7257:
7254:
7250:
7217:
7213:
7192:
7188:
7170:
7166:
7151:
7147:
7141:
7124:
7120:
7102:
7098:
7091:
7087:
7085:
7082:
7081:
7049:
7045:
7038:
7028:
7024:
7020:
7019:
7013:
7009:
7005:
7001:
6995:
6980:
6976:
6970:
6965:
6955:
6951:
6939:
6930:
6926:
6925:
6921:
6919:
6916:
6915:
6909:
6905:
6894:
6870:
6856:
6852:
6851:
6839:
6835:
6833:
6830:
6829:
6822:
6818:
6792:
6788:
6781:
6771:
6767:
6763:
6762:
6756:
6752:
6742:
6741:
6737:
6729:
6725:
6724:
6720:
6714:
6668:
6664:
6658:
6654:
6648:
6634:
6633:
6629:
6618:
6615:
6614:
6607:
6578:
6575:
6574:
6567:
6537:
6533:
6521:
6513:
6507:
6504:
6503:
6473:
6469:
6457:
6453:
6451:
6448:
6447:
6433:
6422:
6411:
6381:
6380:
6376:
6374:
6371:
6370:
6363:
6339:
6335:
6322:
6318:
6308:
6304:
6289:
6285:
6272:
6268:
6258:
6254:
6234:
6231:
6230:
6203:
6199:
6190:
6186:
6177:
6173:
6171:
6168:
6167:
6152:
6145:
6125:
6107:
6103:
6102:
6096:
6092:
6080:
6074:
6070:
6062:
6059:
6058:
6032:
6028:
6018:
6014:
6008:
6004:
5998:
5994:
5982:
5971:
5968:
5967:
5949:
5902:
5898:
5886:
5880:
5876:
5868:
5865:
5864:
5857:
5842:
5800:
5796:
5785:
5782:
5781:
5774:
5766:
5763:
5762:
5761:
5760:
5759:
5748:
5745:
5742:
5741:
5739:
5738:
5737:
5733:
5729:
5705:
5687:
5683:
5682:
5676:
5672:
5660:
5654:
5650:
5642:
5639:
5638:
5631:
5607:
5603:
5597:
5592:
5582:
5581:
5575:
5574:
5565:
5561:
5540:
5539:
5524:
5523:
5514:
5510:
5489:
5488:
5479:
5478:
5469:
5465:
5444:
5443:
5437:
5436:
5430:
5426:
5418:
5415:
5414:
5402:
5396:
5392:
5388:
5375:
5374:
5350:
5344:
5343:
5342:
5309:
5308:
5296:
5278:
5266:
5263:
5262:
5252:
5248:
5244:
5237:
5230:
5226:
5222:
5214:
5210:
5206:
5196:
5193:
5190:
5189:
5187:
5186:
5182:
5152:
5148:
5130:
5126:
5108:
5104:
5083:
5079:
5069:
5061:
5059:
5039:
5036:
5035:
5025:
5016:
5009:
5005:
5002:continuum limit
4954:
4950:
4932:
4928:
4921:
4917:
4913:
4909:
4887:
4883:
4862:
4858:
4840:
4836:
4821:
4817:
4811:
4809:
4806:
4805:
4761:
4757:
4739:
4735:
4728:
4724:
4720:
4716:
4697:
4693:
4675:
4671:
4656:
4652:
4646:
4641:
4635:
4632:
4631:
4613:
4602:
4598:
4594:
4560:
4556:
4550:
4545:
4539:
4536:
4535:
4505:
4501:
4491:
4487:
4478:
4474:
4462:
4451:
4448:
4447:
4440:
4437:
4417:
4385:
4366:
4362:
4337:
4332:
4323:
4319:
4309:
4299:
4295:
4294:
4292:
4269:
4255:
4254:
4241:
4239:
4237:
4234:
4233:
4201:
4192:
4188:
4172:
4168:
4156:
4143:
4139:
4121:
4107:
4106:
4092:
4090:
4083:
4079:
4058:
4057:
4052:
4049:
4048:
4042:
3999:
3980:
3976:
3958:
3954:
3953:
3943:
3939:
3932:
3930:
3921:
3917:
3907:
3897:
3893:
3892:
3890:
3867:
3858:
3854:
3839:
3835:
3834:
3832:
3830:
3827:
3826:
3788:
3779:
3775:
3760:
3756:
3755:
3753:
3742:
3730:
3726:
3715:
3701:
3700:
3682:
3678:
3658:
3657:
3652:
3649:
3648:
3612:
3603:
3599:
3584:
3580:
3569:
3557:
3553:
3536:
3522:
3521:
3503:
3499:
3473:
3472:
3467:
3464:
3463:
3452:
3421:
3412:
3408:
3396:
3383:
3379:
3367:
3353:
3352:
3336:
3332:
3312:
3308:
3287:
3286:
3274:
3268:
3264:
3254:
3250:
3237:
3230:
3226:
3216:
3214:
3199:
3195:
3193:
3190:
3189:
3182:
3179:
3170:
3166:
3161:
3151:
3150:
3144:
3140:
3135:
3126:
3120:
3116:
3096:
3092:
3083:
3079:
3077:
3074:
3073:
3057:
3053:
3046:
3042:
3038:
3018:
3014:
3005:
3001:
2990:
2989:
2974:
2970:
2968:
2965:
2964:
2954:
2943:
2929:
2901:
2897:
2893:
2887:
2881:
2880:
2879:
2873:
2862:
2852:
2851:
2836:
2819:
2815:
2811:
2805:
2799:
2798:
2797:
2791:
2780:
2770:
2769:
2767:
2764:
2763:
2756:
2746:
2742:
2708:
2704:
2698:
2687:
2665:
2661:
2657:
2651:
2645:
2644:
2643:
2637:
2626:
2621:
2617:
2611:
2610:
2599:
2595:
2589:
2585:
2576:
2565:
2560:
2556:
2546:
2540:
2536:
2534:
2528:
2517:
2505:
2502:
2501:
2495:
2491:
2461:
2457:
2451:
2440:
2418:
2414:
2410:
2404:
2398:
2397:
2396:
2390:
2379:
2374:
2370:
2364:
2363:
2352:
2348:
2342:
2338:
2329:
2318:
2313:
2309:
2299:
2292:
2288:
2278:
2276:
2270:
2259:
2247:
2244:
2243:
2233:Dyson's formula
2224:
2221:
2217:
2183:
2179:
2164:
2156:
2145:
2141:
2135:
2134:
2126:
2123:
2122:
2088:
2084:
2075:
2071:
2048:
2044:
2043:
2022:
2018:
2017:
2005:
2002:
2001:
1997:, it is simply
1994:
1985:
1980:
1954:
1949:
1941:
1936:
1920:
1919:
1915:
1913:
1910:
1909:
1901:
1900:
1896:|i⟩
1895:
1894:
1887:
1856:
1835:
1831:
1773:
1767:
1765:
1763:
1753:
1747:
1745:
1738:
1670:
1661:
1653:
1585:Gerard 't Hooft
1577:
1543:
1506:, suggested by
1504:renormalization
1483:formulation of
1354:
1345:
1316:
1315:
1314:
1312:
606:
598:
597:
593:Quantum gravity
568:
560:
559:
555:Higgs mechanism
535:
525:
524:
500:Proca equations
485:
477:
476:
472:Wightman Axioms
457:Renormalization
417:Feynman diagram
372:
364:
363:
304:
294:
293:
244:
229:
227:Feynman diagram
204:
193:
112:Richard Feynman
104:Feynman diagram
92:Richard Feynman
17:
12:
11:
5:
24436:
24426:
24425:
24420:
24415:
24410:
24405:
24400:
24395:
24378:
24377:
24375:
24374:
24364:
24353:
24343:
24338:
24331:
24324:
24319:
24313:
24311:
24307:
24306:
24304:
24303:
24294:
24284:
24282:
24278:
24277:
24275:
24274:
24263:
24252:
24241:
24230:
24219:
24208:
24197:
24186:
24175:
24164:
24162:
24158:
24157:
24155:
24154:
24149:
24144:
24139:
24134:
24129:
24124:
24119:
24114:
24109:
24104:
24099:
24094:
24089:
24084:
24078:
24076:
24072:
24071:
24064:
24063:
24056:
24049:
24041:
24032:
24031:
24028:
24027:
24013:
24012:
24009:
24008:
24006:
24005:
24000:
23995:
23990:
23985:
23980:
23975:
23973:Bremsstrahlung
23970:
23965:
23959:
23957:
23953:
23952:
23950:
23949:
23944:
23939:
23934:
23929:
23924:
23919:
23914:
23909:
23904:
23899:
23893:
23891:
23887:
23886:
23884:
23883:
23878:
23873:
23868:
23863:
23858:
23853:
23847:
23845:
23841:
23840:
23838:
23837:
23832:
23827:
23822:
23816:
23814:
23810:
23809:
23802:
23801:
23794:
23787:
23779:
23773:
23772:
23751:
23745:
23739:
23733:
23727:
23720:
23719:External links
23717:
23716:
23715:
23710:978-0691033273
23709:
23692:
23670:
23663:
23646:
23640:
23627:
23608:
23605:
23602:
23601:
23596:Liz Alzona Art
23583:
23569:
23567:
23566:
23561:
23549:
23531:
23529:
23528:
23523:
23520:
23517:
23514:
23511:
23501:
23487:
23485:
23484:
23479:
23467:
23465:
23464:
23459:
23447:
23429:
23400:
23375:
23359:
23352:
23332:
23325:
23307:
23287:
23260:
23245:
23230:
23204:
23174:hep-th/0605061
23151:
23109:(6): 749–759.
23089:
23064:
23020:
23019:
23017:
23014:
23011:
23010:
23000:
22999:
22997:
22994:
22992:
22991:
22986:
22981:
22976:
22971:
22966:
22961:
22956:
22950:
22948:
22945:
22944:
22943:
22928:
22925:Stephen Baxter
22916:
22904:
22885:
22882:
22832:
22829:
22786:
22785:
22774:
22767:
22764:
22758:
22752:
22748:
22744:
22739:
22735:
22730:
22725:
22720:
22696:
22695:
22684:
22679:
22675:
22672:
22669:
22666:
22662:
22655:
22652:
22647:
22644:
22638:
22635:
22632:
22629:
22625:
22622:
22617:
22613:
22610:
22607:
22604:
22601:
22598:
22595:
22592:
22588:
22584:
22579:
22575:
22571:
22566:
22562:
22558:
22553:
22549:
22545:
22542:
22539:
22513:
22512:
22501:
22495:
22491:
22487:
22482:
22478:
22474:
22469:
22465:
22436:
22435:
22424:
22419:
22415:
22412:
22409:
22406:
22402:
22397:
22394:
22391:
22352:
22351:
22339:
22336:
22333:
22330:
22290:
22289:
22277:
22274:
22267:
22262:
22258:
22254:
22245:
22240:
22236:
22232:
22223:
22218:
22214:
22210:
22201:
22196:
22192:
22188:
22183:
22179:
22175:
22170:
22166:
22162:
22159:
22156:
22151:
22147:
22143:
22140:
22137:
22132:
22128:
22124:
22121:
22118:
22113:
22109:
22105:
22102:
22098:
22031:In the 1930s,
22020:
22017:
22011:
22006:
22005:
21992:
21988:
21984:
21981:
21976:
21972:
21968:
21962:
21958:
21954:
21946:
21943:
21940:
21935:
21929:
21925:
21919:
21915:
21911:
21908:
21903:
21899:
21893:
21889:
21885:
21880:
21876:
21870:
21866:
21861:
21856:
21848:
21845:
21842:
21839:
21836:
21833:
21830:
21825:
21821:
21818:
21815:
21807:
21803:
21799:
21794:
21790:
21785:
21765:
21764:
21753:
21747:
21743:
21737:
21732:
21729:
21726:
21722:
21718:
21715:
21712:
21707:
21703:
21688:
21687:
21676:
21673:
21668:
21664:
21658:
21653:
21650:
21647:
21643:
21630:
21627:
21622:
21618:
21603:
21602:
21591:
21587:
21584:
21578:
21574:
21570:
21567:
21562:
21558:
21554:
21547:
21541:
21537:
21531:
21527:
21523:
21520:
21515:
21511:
21505:
21501:
21497:
21492:
21488:
21482:
21478:
21474:
21469:
21465:
21459:
21455:
21450:
21446:
21443:
21439:
21433:
21429:
21422:
21419:
21416:
21413:
21410:
21407:
21404:
21399:
21393:
21388:
21384:
21369:
21368:
21357:
21351:
21347:
21343:
21340:
21337:
21332:
21328:
21324:
21321:
21316:
21312:
21308:
21305:
21300:
21296:
21292:
21289:
21286:
21281:
21277:
21273:
21270:
21265:
21261:
21257:
21254:
21251:
21248:
21242:
21238:
21223:
21222:
21211:
21205:
21201:
21197:
21194:
21191:
21186:
21182:
21178:
21175:
21172:
21162:
21148:
21144:
21140:
21137:
21134:
21129:
21125:
21121:
21118:
21113:
21109:
21105:
21102:
21099:
21096:
21069:
21038:
21027:
21018:
21012:
21005:
20992:
20987:
20986:
20975:
20969:
20965:
20961:
20958:
20953:
20949:
20945:
20937:
20933:
20927:
20923:
20919:
20916:
20911:
20907:
20901:
20897:
20893:
20889:
20883:
20878:
20874:
20870:
20865:
20860:
20856:
20852:
20844:
20840:
20836:
20831:
20827:
20821:
20817:
20812:
20798:denominators:
20789:
20788:
20772:
20768:
20764:
20759:
20755:
20749:
20745:
20741:
20738:
20735:
20731:
20726:
20718:
20714:
20710:
20705:
20701:
20695:
20691:
20687:
20684:
20681:
20677:
20667:
20663:
20659:
20654:
20650:
20645:
20639:
20636:
20633:
20606:
20605:
20594:
20591:
20586:
20583:
20579:
20571:
20566:
20560:
20556:
20552:
20549:
20546:
20543:
20540:
20537:
20534:
20529:
20525:
20521:
20515:
20511:
20507:
20502:
20497:
20492:
20487:
20482:
20478:
20474:
20471:
20468:
20464:
20461:
20453:
20448:
20442:
20438:
20434:
20431:
20428:
20425:
20422:
20419:
20416:
20413:
20408:
20404:
20400:
20395:
20391:
20386:
20381:
20376:
20371:
20366:
20362:
20318:
20317:
20306:
20303:
20294:
20288:
20282:
20279:
20276:
20273:
20270:
20267:
20264:
20261:
20258:
20253:
20247:
20240:
20235:
20231:
20227:
20221:
20218:
20214:
20190:
20189:
20178:
20175:
20167:
20162:
20156:
20152:
20148:
20145:
20142:
20139:
20136:
20133:
20130:
20127:
20122:
20118:
20114:
20109:
20105:
20100:
20095:
20090:
20087:
20081:
20075:
20071:
20067:
20064:
20061:
20058:
20055:
20052:
20049:
20046:
20041:
20037:
20033:
20028:
20024:
20019:
20015:
20012:
20008:
20004:
19999:
19996:
19993:
19989:
19970:
19969:
19955:
19952:
19948:
19945:
19942:
19939:
19936:
19933:
19930:
19927:
19924:
19921:
19917:
19899:
19898:
19887:
19882:
19879:
19875:
19872:
19869:
19866:
19863:
19860:
19857:
19852:
19849:
19845:
19842:
19839:
19836:
19833:
19830:
19827:
19744:
19743:
19732:
19724:
19720:
19715:
19712:
19708:
19705:
19702:
19699:
19696:
19692:
19689:
19685:
19682:
19677:
19673:
19669:
19664:
19660:
19656:
19653:
19649:
19646:
19642:
19639:
19636:
19633:
19629:
19622:
19619:
19615:
19612:
19608:
19582:
19581:
19570:
19565:
19562:
19558:
19554:
19551:
19544:
19538:
19534:
19530:
19525:
19521:
19517:
19514:
19511:
19508:
19504:
19499:
19496:
19492:
19489:
19484:
19480:
19476:
19471:
19467:
19463:
19460:
19457:
19453:
19446:
19443:
19439:
19436:
19432:
19409:and the other
19403:
19402:
19391:
19382:
19378:
19374:
19369:
19365:
19361:
19358:
19355:
19352:
19348:
19338:
19334:
19330:
19325:
19321:
19316:
19309:
19305:
19279:
19276:
19265:
19264:
19250:
19247:
19243:
19239:
19233:
19229:
19225:
19220:
19215:
19208:
19205:
19195:
19189:
19184:
19180:
19176:
19172:
19167:
19164:
19161:
19158:
19155:
19150:
19145:
19141:
19137:
19134:
19131:
19128:
19125:
19077:
19076:
19061:
19058:
19051:
19047:
19043:
19037:
19028:
19024:
19020:
19016:
19011:
19008:
19005:
19001:
18997:
18990:
18985:
18981:
18977:
18973:
18969:
18966:
18961:
18956:
18952:
18948:
18945:
18942:
18939:
18936:
18922:
18921:
18910:
18907:
18900:
18894:
18890:
18886:
18881:
18877:
18872:
18868:
18865:
18861:
18855:
18850:
18846:
18842:
18834:
18830:
18826:
18821:
18817:
18812:
18795:
18792:
18787:
18784:
18768:
18767:
18755:
18750:
18746:
18739:
18736:
18730:
18727:
18722:
18718:
18711:
18708:
18700:
18696:
18692:
18689:
18686:
18681:
18677:
18671:
18667:
18661:
18657:
18651:
18647:
18641:
18636:
18633:
18629:
18623:
18618:
18615:
18611:
18607:
18602:
18597:
18593:
18587:
18582:
18578:
18572:
18567:
18563:
18557:
18553:
18547:
18542:
18539:
18535:
18531:
18526:
18522:
18516:
18512:
18506:
18502:
18496:
18492:
18488:
18485:
18482:
18479:
18476:
18450:
18449:
18437:
18434:
18427:
18424:
18418:
18412:
18407:
18403:
18396:
18392:
18385:
18382:
18375:
18371:
18357:
18356:
18343:
18337:
18333:
18326:
18322:
18317:
18313:
18295:
18294:
18283:
18279:
18274:
18270:
18263:
18259:
18244:
18243:
18232:
18228:
18225:
18220:
18216:
18210:
18206:
18191:
18190:
18178:
18175:
18167:
18164:
18160:
18156:
18152:
18147:
18141:
18138:
18133:
18130:
18124:
18120:
18117:
18103:
18102:
18088:
18085:
18081:
18076:
18070:
18067:
18062:
18059:
18053:
18049:
18044:
18039:
18036:
18033:
18030:
18025:
18020:
18017:
18014:
18011:
18008:
18005:
18000:
17997:
17993:
17989:
17986:
17981:
17975:
17972:
17967:
17964:
17958:
17954:
17949:
17944:
17941:
17938:
17935:
17930:
17925:
17922:
17919:
17916:
17913:
17899:
17898:
17887:
17884:
17880:
17877:
17874:
17871:
17868:
17865:
17862:
17859:
17856:
17853:
17850:
17846:
17843:
17839:
17836:
17833:
17830:
17827:
17793:
17792:
17781:
17777:
17774:
17769:
17763:
17760:
17755:
17752:
17746:
17742:
17739:
17736:
17733:
17730:
17724:
17720:
17716:
17713:
17710:
17707:
17704:
17700:
17697:
17693:
17690:
17687:
17684:
17681:
17663:
17662:
17651:
17648:
17644:
17641:
17636:
17630:
17627:
17622:
17619:
17613:
17609:
17606:
17603:
17600:
17597:
17594:
17591:
17588:
17585:
17581:
17578:
17575:
17572:
17569:
17566:
17562:
17559:
17555:
17552:
17549:
17546:
17543:
17525:
17524:
17513:
17510:
17506:
17503:
17498:
17492:
17489:
17484:
17481:
17475:
17471:
17468:
17465:
17462:
17458:
17455:
17452:
17449:
17446:
17443:
17440:
17437:
17434:
17431:
17428:
17424:
17421:
17417:
17414:
17411:
17408:
17405:
17375:
17374:
17363:
17360:
17353:
17348:
17344:
17338:
17334:
17329:
17325:
17322:
17317:
17313:
17307:
17303:
17298:
17294:
17276:
17273:
17272:
17271:
17260:
17253:
17249:
17240:
17236:
17229:
17225:
17219:
17215:
17208:
17205:
17200:
17197:
17193:
17185:
17180:
17177:
17173:
17170:
17166:
17162:
17159:
17155:
17151:
17147:
17144:
17140:
17135:
17131:
17127:
17124:
17121:
17116:
17112:
17107:
17092:
17091:
17079:
17073:
17068:
17062:
17058:
17052:
17048:
17043:
17038:
17035:
17030:
17026:
17020:
17016:
17010:
17006:
17000:
16996:
16991:
16984:
16981:
16975:
16972:
16969:
16946:
16907:
16906:
16895:
16888:
16884:
16878:
16875:
16871:
16864:
16859:
16856:
16852:
16849:
16845:
16841:
16838:
16834:
16830:
16826:
16823:
16819:
16814:
16810:
16806:
16803:
16800:
16795:
16791:
16786:
16767:
16766:
16753:
16749:
16743:
16739:
16733:
16729:
16723:
16719:
16715:
16712:
16709:
16695:
16694:
16683:
16675:
16670:
16665:
16659:
16655:
16649:
16645:
16640:
16633:
16629:
16610:
16609:
16598:
16594:
16591:
16583:
16578:
16574:
16568:
16564:
16559:
16555:
16552:
16547:
16543:
16537:
16533:
16528:
16524:
16521:
16480:
16479:
16468:
16463:
16457:
16453:
16447:
16443:
16437:
16433:
16427:
16423:
16419:
16414:
16410:
16404:
16400:
16394:
16390:
16384:
16380:
16375:
16368:
16365:
16359:
16356:
16353:
16348:
16345:
16341:
16335:
16332:
16328:
16321:
16318:
16312:
16309:
16306:
16291:
16288:
16186:
16185:
16171:
16168:
16163:
16159:
16155:
16152:
16148:
16142:
16136:
16132:
16128:
16123:
16119:
16114:
16110:
16105:
16102:
16099:
16094:
16091:
16088:
16085:
16082:
16077:
16071:
16067:
16063:
16060:
16057:
16051:
16048:
16045:
16042:
16039:
16036:
16033:
16030:
16025:
16021:
16017:
16013:
16008:
16004:
16000:
15997:
15994:
15991:
15987:
15983:
15979:
15975:
15972:
15969:
15964:
15960:
15956:
15950:
15947:
15941:
15938:
15933:
15929:
15925:
15919:
15916:
15910:
15905:
15901:
15897:
15891:
15888:
15881:
15856:
15855:
15839:
15835:
15831:
15826:
15822:
15816:
15813:
15810:
15807:
15804:
15798:
15794:
15791:
15787:
15784:
15781:
15778:
15775:
15769:
15766:
15763:
15760:
15757:
15753:
15748:
15744:
15741:
15737:
15734:
15731:
15728:
15725:
15721:
15717:
15714:
15711:
15708:
15705:
15701:
15698:
15694:
15688:
15685:
15678:
15631:
15630:
15619:
15615:
15611:
15607:
15604:
15599:
15595:
15589:
15585:
15581:
15577:
15570:
15567:
15559:
15555:
15551:
15548:
15533:
15532:
15521:
15517:
15513:
15510:
15505:
15501:
15495:
15491:
15486:
15479:
15476:
15470:
15446:
15445:
15434:
15428:
15425:
15421:
15417:
15412:
15409:
15404:
15400:
15396:
15393:
15388:
15383:
15375:
15371:
15367:
15363:
15355:
15352:
15348:
15341:
15338:
15333:
15329:
15325:
15320:
15316:
15311:
15300:
15286:
15283:
15280:
15276:
15273:
15270:
15263:
15258:
15255:
15251:
15245:
15241:
15236:
15230:
15227:
15224:
15218:
15214:
15210:
15204:
15199:
15195:
15191:
15188:
15183:
15179:
15175:
15172:
15169:
15164:
15160:
15155:
15151:
15121:
15120:
15107:
15104:
15100:
15096:
15091:
15088:
15082:
15079:
15073:
15070:
15065:
15060:
15051:
15048:
15042:
15038:
15030:
15027:
15023:
15016:
15013:
15008:
15004:
15000:
14994:
14991:
14984:
14951:
14950:
14939:
14933:
14928:
14925:
14921:
14914:
14911:
14905:
14901:
14897:
14894:
14891:
14887:
14884:
14881:
14877:
14874:
14871:
14864:
14861:
14855:
14849:
14844:
14841:
14837:
14830:
14827:
14821:
14817:
14813:
14808:
14805:
14801:
14797:
14794:
14790:
14786:
14782:
14776:
14773:
14769:
14762:
14759:
14753:
14747:
14744:
14737:
14732:
14728:
14725:
14722:
14719:
14712:
14709:
14703:
14697:
14691:
14688:
14682:
14679:
14673:
14670:
14664:
14661:
14658:
14652:
14649:
14642:
14638:
14635:
14632:
14557:
14556:
14545:
14541:
14538:
14535:
14531:
14528:
14525:
14522:
14519:
14515:
14512:
14509:
14501:
14497:
14491:
14487:
14481:
14478:
14474:
14468:
14465:
14459:
14455:
14421:
14420:
14409:
14405:
14402:
14399:
14395:
14392:
14389:
14385:
14382:
14379:
14372:
14369:
14363:
14355:
14351:
14345:
14338:
14335:
14326:
14323:
14319:
14314:
14310:
14280:
14261:
14259:
14240:
14232:
14231:
14220:
14214:
14209:
14206:
14203:
14200:
14195:
14190:
14187:
14181:
14178:
14175:
14172:
14169:
14165:
14160:
14157:
14154:
14151:
14148:
14142:
14139:
14136:
14133:
14130:
14126:
14119:
14116:
14111:
14107:
14103:
14100:
14097:
14094:
14090:
14057:
14056:
14045:
14041:
14038:
14032:
14027:
14023:
14016:
14011:
14008:
14005:
14002:
13999:
13996:
13993:
13990:
13987:
13982:
13977:
13974:
13971:
13968:
13965:
13961:
13957:
13954:
13951:
13948:
13945:
13942:
13939:
13936:
13914:
13913:
13902:
13899:
13893:
13890:
13887:
13884:
13881:
13878:
13875:
13871:
13867:
13864:
13861:
13858:
13855:
13852:
13849:
13846:
13832:
13831:
13819:
13811:
13807:
13801:
13797:
13793:
13789:
13783:
13779:
13774:
13759:
13758:
13746:
13741:
13738:
13735:
13731:
13727:
13723:
13720:
13717:
13711:
13708:
13705:
13702:
13699:
13696:
13693:
13690:
13685:
13681:
13677:
13673:
13667:
13664:
13660:
13656:
13653:
13650:
13647:
13644:
13641:
13616:
13615:
13602:
13594:
13590:
13584:
13580:
13576:
13572:
13566:
13561:
13558:
13555:
13551:
13546:
13542:
13538:
13533:
13530:
13527:
13523:
13519:
13515:
13512:
13507:
13503:
13496:
13493:
13490:
13486:
13482:
13479:
13476:
13473:
13470:
13446:
13445:
13433:
13430:
13424:
13421:
13418:
13415:
13412:
13409:
13406:
13402:
13398:
13395:
13392:
13389:
13386:
13383:
13346:
13340:
13329:
13328:
13316:
13311:
13307:
13303:
13300:
13297:
13292:
13288:
13284:
13281:
13278:
13273:
13269:
13265:
13262:
13259:
13256:
13251:
13247:
13243:
13240:
13237:
13232:
13228:
13224:
13221:
13216:
13213:
13210:
13206:
13202:
13197:
13192:
13189:
13186:
13183:
13178:
13173:
13170:
13139:with momentum
13117:
13116:
13104:
13099:
13095:
13090:
13087:
13084:
13081:
13078:
13075:
13072:
13069:
13066:
13063:
13060:
13055:
13051:
13046:
13043:
13040:
13037:
13034:
13031:
13028:
13025:
13022:
12995:
12992:
12962:
12961:
12948:
12945:
12942:
12938:
12934:
12929:
12926:
12923:
12919:
12915:
12912:
12909:
12904:
12901:
12897:
12893:
12890:
12887:
12861:
12860:
12849:
12844:
12839:
12833:
12829:
12823:
12819:
12814:
12810:
12807:
12803:
12798:
12792:
12788:
12784:
12780:
12776:
12773:
12770:
12766:
12761:
12757:
12753:
12748:
12744:
12740:
12737:
12732:
12728:
12724:
12719:
12715:
12709:
12706:
12702:
12698:
12683:
12682:
12671:
12664:
12661:
12655:
12652:
12648:
12644:
12639:
12636:
12632:
12627:
12623:
12619:
12614:
12610:
12606:
12603:
12598:
12594:
12590:
12585:
12581:
12575:
12572:
12568:
12564:
12558:
12554:
12550:
12545:
12541:
12537:
12532:
12528:
12524:
12521:
12516:
12512:
12508:
12503:
12499:
12494:
12477:
12476:Vacuum bubbles
12474:
12437:
12427:
12426:
12415:
12410:
12407:
12404:
12401:
12397:
12394:
12391:
12384:
12380:
12373:
12369:
12364:
12361:
12358:
12351:
12346:
12342:
12333:
12329:
12323:
12319:
12311:
12306:
12303:
12298:
12294:
12289:
12282:
12278:
12274:
12271:
12268:
12265:
12262:
12248:
12247:
12236:
12232:
12228:
12225:
12220:
12215:
12211:
12205:
12202:
12197:
12192:
12188:
12184:
12181:
12177:
12172:
12168:
12165:
12159:
12156:
12150:
12145:
12141:
12135:
12129:
12126:
12120:
12116:
12110:
12104:
12101:
12097:
12091:
12068:
12056:
12055:
12041:
12036:
12032:
12023:
12019:
12013:
12009:
12001:
11997:
11980:
11971:
11960:
11953:
11946:
11939:
11932:
11927:
11926:
11912:
11908:
11901:
11897:
11893:
11890:
11887:
11884:
11881:
11867:
11866:
11855:
11852:
11849:
11846:
11843:
11840:
11837:
11834:
11831:
11828:
11825:
11822:
11819:
11787:
11781:
11750:
11749:
11735:
11717:
11523:4! = 4 × 3 × 2
11491:
11488:
11403:
11400:
11326:
11325:
11310:
11305:
11301:
11296:
11291:
11287:
11283:
11278:
11274:
11270:
11267:
11246:
11245:
11231:
11172:
11169:
11163:
11150:The 4! inside
11148:
11147:
11136:
11128:
11123:
11119:
11115:
11106:
11101:
11097:
11093:
11084:
11079:
11075:
11071:
11062:
11057:
11053:
11049:
11044:
10987:
10969:
10968:
10955:
10951:
10948:
10945:
10940:
10936:
10932:
10929:
10926:
10921:
10917:
10913:
10910:
10907:
10902:
10898:
10894:
10891:
10888:
10883:
10879:
10875:
10872:
10867:
10864:
10860:
10856:
10850:
10846:
10842:
10837:
10833:
10829:
10826:
10823:
10818:
10814:
10810:
10807:
10804:
10799:
10795:
10791:
10788:
10785:
10780:
10776:
10772:
10769:
10765:
10724:
10723:
10711:
10707:
10704:
10701:
10698:
10695:
10689:
10686:
10682:
10677:
10674:
10671:
10665:
10662:
10658:
10653:
10650:
10647:
10644:
10640:
10632:
10628:
10624:
10620:
10616:
10611:
10608:
10604:
10579:
10574:
10573:
10562:
10559:
10556:
10551:
10547:
10543:
10540:
10535:
10531:
10527:
10522:
10518:
10514:
10509:
10505:
10501:
10496:
10492:
10488:
10485:
10482:
10477:
10473:
10469:
10466:
10463:
10458:
10454:
10450:
10447:
10444:
10439:
10435:
10431:
10428:
10425:
10420:
10416:
10412:
10409:
10402:
10398:
10392:
10388:
10382:
10378:
10372:
10368:
10363:
10356:
10353:
10349:
10344:
10339:
10334:
10330:
10327:
10324:
10321:
10317:
10310:
10306:
10300:
10296:
10292:
10289:
10275:
10274:
10263:
10258:
10254:
10247:
10244:
10240:
10235:
10232:
10227:
10223:
10219:
10214:
10210:
10206:
10203:
10200:
10185:
10182:
10178:
10177:
10164:
10159:
10154:
10150:
10146:
10141:
10138:
10135:
10132:
10129:
10126:
10123:
10120:
10117:
10114:
10111:
10108:
10105:
10102:
10099:
10096:
10093:
10090:
10087:
10084:
10081:
10077:
10072:
10069:
10065:
10061:
9992:
9991:
9979:
9973:
9970:
9966:
9962:
9957:
9953:
9947:
9943:
9937:
9933:
9928:
9902:
9901:
9888:
9885:
9880:
9876:
9871:
9867:
9863:
9852:
9837:
9833:
9829:
9824:
9821:
9818:
9815:
9812:
9809:
9806:
9803:
9800:
9797:
9794:
9791:
9788:
9785:
9777:
9773:
9767:
9763:
9759:
9756:
9752:
9748:
9741:
9737:
9731:
9727:
9723:
9720:
9716:
9710:
9707:
9703:
9699:
9693:
9689:
9684:
9681:
9677:
9673:
9654:
9653:
9638:
9634:
9631:
9628:
9625:
9619:
9615:
9608:
9605:
9591:
9588:
9580:
9577:
9574:
9571:
9568:
9565:
9560:
9557:
9554:
9551:
9548:
9545:
9542:
9539:
9536:
9533:
9530:
9527:
9524:
9518:
9515:
9512:
9507:
9503:
9497:
9493:
9489:
9486:
9482:
9474:
9470:
9464:
9461:
9457:
9451:
9448:
9445:
9437:
9433:
9429:
9423:
9419:
9406:
9392:
9388:
9385:
9378:
9375:
9372:
9367:
9363:
9357:
9353:
9349:
9346:
9342:
9338:
9335:
9332:
9309:
9286:
9265:
9253:
9250:
9245:
9244:
9229:
9224:
9220:
9215:
9210:
9206:
9202:
9197:
9193:
9189:
9186:
9176:
9171:
9167:
9162:
9157:
9153:
9149:
9144:
9140:
9136:
9133:
9127:
9120:
9115:
9111:
9106:
9101:
9097:
9093:
9088:
9084:
9080:
9077:
9067:
9062:
9058:
9053:
9048:
9044:
9040:
9035:
9031:
9027:
9024:
9018:
9011:
9006:
9002:
8997:
8992:
8988:
8984:
8979:
8975:
8971:
8968:
8958:
8953:
8949:
8944:
8939:
8935:
8931:
8926:
8922:
8918:
8915:
8909:
8905:
8901:
8896:
8892:
8888:
8885:
8882:
8877:
8873:
8869:
8866:
8863:
8858:
8854:
8850:
8847:
8844:
8839:
8835:
8831:
8828:
8824:
8809:
8808:
8793:
8788:
8784:
8778:
8772:
8768:
8764:
8759:
8755:
8750:
8746:
8738:
8735:
8732:
8728:
8724:
8721:
8717:
8713:
8708:
8705:
8701:
8697:
8694:
8691:
8688:
8683:
8679:
8675:
8672:
8668:
8645:
8644:
8632:
8628:
8623:
8619:
8615:
8612:
8609:
8606:
8601:
8597:
8593:
8590:
8587:
8582:
8578:
8574:
8571:
8567:
8552:Wick's theorem
8546:Wick's theorem
8544:Main article:
8541:
8538:
8529:
8528:
8513:
8509:
8505:
8500:
8497:
8494:
8491:
8488:
8474:
8473:
8461:
8458:
8455:
8452:
8449:
8446:
8443:
8440:
8437:
8434:
8429:
8425:
8391:
8390:
8379:
8376:
8372:
8368:
8365:
8362:
8359:
8356:
8353:
8350:
8347:
8343:
8337:
8333:
8327:
8323:
8308:
8307:
8295:
8292:
8289:
8284:
8280:
8274:
8270:
8254:
8251:
8231:
8225:
8224:
8212:
8207:
8203:
8199:
8194:
8190:
8186:
8183:
8178:
8174:
8170:
8165:
8161:
8157:
8152:
8148:
8144:
8139:
8135:
8129:
8125:
8121:
8118:
8115:
8112:
8109:
8106:
8103:
8100:
8064:
8063:
8047:
8042:
8037:
8033:
8029:
8026:
8023:
8020:
8017:
8014:
8009:
8005:
8001:
7998:
7995:
7992:
7989:
7984:
7980:
7976:
7973:
7970:
7967:
7964:
7959:
7954:
7950:
7945:
7941:
7938:
7934:
7931:
7928:
7925:
7922:
7918:
7914:
7910:
7907:
7903:
7900:
7897:
7894:
7891:
7888:
7884:
7869:
7868:
7853:
7849:
7845:
7840:
7836:
7833:
7829:
7826:
7823:
7820:
7817:
7813:
7809:
7805:
7802:
7798:
7795:
7792:
7789:
7786:
7783:
7779:
7756:
7755:
7740:
7736:
7732:
7727:
7723:
7717:
7713:
7707:
7703:
7698:
7672:
7671:
7659:
7656:
7652:
7645:
7642:
7637:
7631:
7627:
7622:
7606:
7603:
7581:
7580:
7565:
7561:
7557:
7551:
7546:
7542:
7538:
7533:
7529:
7525:
7522:
7517:
7513:
7509:
7504:
7500:
7494:
7490:
7481:
7478:
7474:
7470:
7466:
7461:
7457:
7453:
7449:
7444:
7440:
7436:
7433:
7430:
7427:
7422:
7418:
7414:
7411:
7407:
7362:
7348:
7317:
7314:
7259:
7252:
7247:
7246:
7232:
7229:
7223:
7220:
7216:
7212:
7207:
7204:
7200:
7195:
7191:
7187:
7184:
7181:
7178:
7173:
7169:
7165:
7162:
7157:
7154:
7150:
7146:
7140:
7136:
7132:
7127:
7123:
7119:
7116:
7113:
7110:
7105:
7101:
7097:
7094:
7090:
7075:
7074:
7063:
7057:
7052:
7048:
7041:
7036:
7031:
7027:
7023:
7016:
7012:
7008:
7004:
6998:
6994:
6990:
6983:
6979:
6973:
6968:
6964:
6958:
6954:
6947:
6944:
6938:
6933:
6929:
6924:
6907:
6891:
6890:
6879:
6873:
6868:
6863:
6860:
6855:
6850:
6847:
6842:
6838:
6815:
6814:
6800:
6795:
6791:
6784:
6779:
6774:
6770:
6766:
6759:
6755:
6749:
6746:
6740:
6732:
6728:
6723:
6717:
6713:
6709:
6706:
6703:
6698:
6694:
6691:
6688:
6685:
6682:
6679:
6676:
6671:
6667:
6661:
6657:
6651:
6647:
6641:
6638:
6632:
6628:
6625:
6622:
6604:
6603:
6591:
6588:
6585:
6582:
6564:
6563:
6549:
6546:
6543:
6540:
6536:
6532:
6527:
6524:
6519:
6516:
6512:
6497:
6496:
6482:
6479:
6476:
6472:
6468:
6463:
6460:
6456:
6408:
6407:
6396:
6393:
6390:
6384:
6379:
6360:
6359:
6348:
6342:
6338:
6334:
6330:
6325:
6321:
6317:
6311:
6307:
6303:
6300:
6297:
6292:
6288:
6284:
6280:
6275:
6271:
6267:
6261:
6257:
6253:
6250:
6247:
6244:
6241:
6238:
6224:
6223:
6212:
6206:
6202:
6196:
6193:
6189:
6185:
6180:
6176:
6142:
6141:
6128:
6123:
6119:
6116:
6113:
6110:
6106:
6099:
6095:
6088:
6085:
6077:
6073:
6069:
6066:
6052:
6051:
6040:
6035:
6031:
6026:
6021:
6017:
6011:
6007:
6001:
5997:
5990:
5987:
5981:
5978:
5975:
5960:exactly once.
5946:
5945:
5934:
5931:
5928:
5925:
5922:
5919:
5916:
5913:
5910:
5905:
5901:
5894:
5891:
5883:
5879:
5875:
5872:
5839:
5838:
5827:
5823:
5820:
5817:
5814:
5811:
5808:
5803:
5799:
5795:
5792:
5789:
5764:
5757:
5756:
5735:
5734:
5726:
5725:
5714:
5708:
5703:
5699:
5696:
5693:
5690:
5686:
5679:
5675:
5668:
5665:
5657:
5653:
5649:
5646:
5628:
5627:
5616:
5610:
5606:
5600:
5595:
5591:
5585:
5578:
5573:
5568:
5564:
5560:
5557:
5554:
5551:
5548:
5543:
5538:
5535:
5532:
5527:
5522:
5517:
5513:
5509:
5506:
5503:
5500:
5497:
5492:
5487:
5482:
5477:
5472:
5468:
5464:
5461:
5458:
5455:
5452:
5447:
5440:
5433:
5429:
5425:
5422:
5398:
5371:
5370:
5359:
5353:
5347:
5341:
5338:
5335:
5332:
5329:
5326:
5323:
5320:
5317:
5312:
5304:
5301:
5293:
5290:
5287:
5284:
5281:
5277:
5273:
5270:
5179:
5178:
5167:
5161:
5158:
5155:
5151:
5147:
5144:
5141:
5138:
5133:
5129:
5125:
5120:
5117:
5114:
5111:
5107:
5103:
5100:
5097:
5094:
5086:
5082:
5078:
5075:
5072:
5067:
5064:
5058:
5055:
5052:
5049:
5046:
5043:
5024:
5021:
5008:and the limit
4991:
4990:
4979:
4974:
4970:
4966:
4962:
4957:
4953:
4949:
4946:
4943:
4940:
4935:
4931:
4927:
4924:
4920:
4916:
4912:
4908:
4902:
4899:
4893:
4890:
4886:
4882:
4877:
4874:
4870:
4865:
4861:
4857:
4854:
4851:
4848:
4843:
4839:
4835:
4832:
4827:
4824:
4820:
4816:
4798:
4797:
4786:
4781:
4777:
4773:
4769:
4764:
4760:
4756:
4753:
4750:
4747:
4742:
4738:
4734:
4731:
4727:
4723:
4719:
4715:
4712:
4709:
4705:
4700:
4696:
4692:
4689:
4686:
4683:
4678:
4674:
4670:
4667:
4662:
4659:
4655:
4649:
4644:
4640:
4591:
4590:
4579:
4575:
4572:
4566:
4563:
4559:
4553:
4548:
4544:
4529:
4528:
4517:
4513:
4508:
4504:
4499:
4494:
4490:
4486:
4481:
4477:
4470:
4467:
4461:
4458:
4455:
4436:
4433:
4416:
4413:
4409:
4408:
4395:
4391:
4388:
4384:
4381:
4378:
4375:
4372:
4369:
4365:
4358:
4355:
4352:
4349:
4346:
4343:
4340:
4336:
4326:
4322:
4318:
4315:
4312:
4307:
4302:
4298:
4291:
4288:
4283:
4279:
4275:
4272:
4268:
4262:
4259:
4253:
4250:
4247:
4244:
4227:
4226:
4215:
4211:
4207:
4204:
4200:
4195:
4191:
4186:
4183:
4180:
4175:
4171:
4166:
4162:
4159:
4155:
4152:
4146:
4142:
4135:
4131:
4127:
4124:
4120:
4114:
4111:
4104:
4101:
4098:
4095:
4086:
4082:
4077:
4074:
4071:
4065:
4062:
4056:
4041:
4038:
4037:
4036:
4033:
4030:
4023:
4022:
4009:
4005:
4002:
3998:
3995:
3992:
3989:
3986:
3983:
3979:
3972:
3969:
3966:
3961:
3957:
3949:
3946:
3942:
3938:
3935:
3924:
3920:
3916:
3913:
3910:
3905:
3900:
3896:
3889:
3886:
3881:
3877:
3873:
3870:
3866:
3861:
3857:
3853:
3850:
3847:
3842:
3838:
3820:
3819:
3808:
3802:
3798:
3794:
3791:
3787:
3782:
3778:
3774:
3771:
3768:
3763:
3759:
3752:
3748:
3745:
3741:
3738:
3733:
3729:
3725:
3721:
3718:
3714:
3708:
3705:
3699:
3696:
3693:
3690:
3685:
3681:
3677:
3674:
3671:
3665:
3662:
3656:
3636:
3635:
3622:
3618:
3615:
3611:
3606:
3602:
3598:
3595:
3592:
3587:
3583:
3579:
3575:
3572:
3568:
3565:
3560:
3556:
3552:
3549:
3546:
3542:
3539:
3535:
3529:
3526:
3520:
3517:
3514:
3511:
3506:
3502:
3498:
3495:
3492:
3489:
3486:
3480:
3477:
3471:
3451:
3448:
3447:
3446:
3434:
3431:
3427:
3424:
3420:
3415:
3411:
3406:
3402:
3399:
3395:
3392:
3386:
3382:
3377:
3373:
3370:
3366:
3360:
3357:
3350:
3347:
3344:
3339:
3335:
3330:
3327:
3324:
3321:
3315:
3311:
3306:
3303:
3300:
3294:
3291:
3285:
3280:
3277:
3271:
3267:
3262:
3257:
3253:
3249:
3243:
3240:
3233:
3229:
3225:
3222:
3219:
3213:
3208:
3205:
3202:
3198:
3178:
3175:
3174:
3173:
3168:
3159:
3147:
3142:
3133:
3123:
3118:
3104:
3099:
3095:
3091:
3086:
3082:
3070:
3067:
3060:
3055:
3044:
3035:
3034:
3021:
3017:
3013:
3008:
3004:
2997:
2994:
2988:
2985:
2982:
2977:
2973:
2953:
2950:
2934:signifies the
2926:
2925:
2914:
2909:
2904:
2900:
2896:
2890:
2884:
2876:
2871:
2868:
2865:
2861:
2855:
2850:
2847:
2844:
2835:
2831:
2827:
2822:
2818:
2814:
2808:
2802:
2794:
2789:
2786:
2783:
2779:
2773:
2745:th-order term
2735:
2734:
2723:
2717:
2714:
2711:
2707:
2701:
2696:
2693:
2690:
2686:
2682:
2678:
2673:
2668:
2664:
2660:
2654:
2648:
2640:
2635:
2632:
2629:
2625:
2620:
2614:
2608:
2602:
2598:
2592:
2588:
2584:
2579:
2574:
2571:
2568:
2564:
2559:
2552:
2549:
2543:
2539:
2531:
2526:
2523:
2520:
2516:
2512:
2509:
2493:
2488:
2487:
2476:
2470:
2467:
2464:
2460:
2454:
2449:
2446:
2443:
2439:
2435:
2431:
2426:
2421:
2417:
2413:
2407:
2401:
2393:
2388:
2385:
2382:
2378:
2373:
2367:
2361:
2355:
2351:
2345:
2341:
2337:
2332:
2327:
2324:
2321:
2317:
2312:
2305:
2302:
2295:
2291:
2287:
2284:
2281:
2273:
2268:
2265:
2262:
2258:
2254:
2251:
2231:of operators.
2227:signifies the
2219:
2214:
2213:
2202:
2197:
2194:
2191:
2186:
2182:
2178:
2175:
2170:
2167:
2162:
2159:
2155:
2151:
2148:
2144:
2138:
2133:
2130:
2112:
2111:
2100:
2096:
2091:
2087:
2083:
2078:
2074:
2070:
2067:
2062:
2059:
2056:
2051:
2047:
2042:
2036:
2033:
2030:
2025:
2021:
2016:
2012:
2009:
1977:
1976:
1965:
1961:
1957:
1952:
1948:
1944:
1939:
1935:
1932:
1926:
1923:
1918:
1886:
1883:
1876:
1875:
1855:
1852:
1838:
1837:
1833:
1829:
1826:
1823:
1820:
1817:
1737:
1734:
1721:bubble chamber
1698:external lines
1694:outgoing lines
1690:incoming lines
1686:internal lines
1669:
1666:
1576:
1573:
1542:
1539:
1372:, decays both
1353:
1350:
1347:
1346:
1344:
1343:
1336:
1329:
1321:
1318:
1317:
1310:
1309:
1304:
1299:
1294:
1289:
1284:
1279:
1274:
1269:
1264:
1259:
1254:
1249:
1244:
1239:
1234:
1229:
1224:
1219:
1214:
1209:
1204:
1199:
1194:
1189:
1184:
1179:
1174:
1169:
1164:
1159:
1154:
1149:
1144:
1139:
1134:
1129:
1124:
1119:
1114:
1109:
1104:
1099:
1094:
1089:
1084:
1079:
1074:
1069:
1064:
1059:
1054:
1049:
1044:
1039:
1034:
1029:
1024:
1019:
1014:
1009:
1004:
999:
994:
989:
984:
979:
974:
969:
964:
959:
954:
949:
944:
939:
934:
929:
924:
919:
914:
909:
904:
899:
894:
889:
884:
879:
874:
869:
864:
859:
854:
849:
844:
839:
834:
829:
824:
819:
814:
809:
804:
799:
794:
789:
784:
779:
774:
769:
764:
759:
754:
749:
744:
739:
734:
729:
724:
719:
714:
709:
704:
699:
694:
689:
684:
679:
674:
669:
664:
659:
654:
649:
644:
639:
634:
629:
624:
619:
614:
608:
607:
604:
603:
600:
599:
596:
595:
590:
585:
580:
575:
569:
566:
565:
562:
561:
558:
557:
552:
547:
542:
536:
533:Standard Model
531:
530:
527:
526:
523:
522:
517:
512:
507:
502:
497:
492:
490:Dirac equation
486:
483:
482:
479:
478:
475:
474:
469:
467:Wick's theorem
464:
459:
454:
452:Regularization
449:
444:
439:
434:
429:
424:
419:
414:
409:
404:
399:
394:
389:
384:
379:
373:
370:
369:
366:
365:
362:
361:
356:
354:Noether charge
351:
346:
341:
339:Gauge symmetry
336:
331:
326:
321:
316:
311:
305:
300:
299:
296:
295:
292:
291:
286:
281:
276:
271:
266:
261:
256:
251:
245:
242:
241:
238:
237:
231:
230:
225:
217:
216:
136:Higgs particle
83:
82:
47:, producing a
15:
9:
6:
4:
3:
2:
24435:
24424:
24421:
24419:
24416:
24414:
24411:
24409:
24406:
24404:
24401:
24399:
24396:
24394:
24391:
24390:
24388:
24373:
24369:
24365:
24363:
24358:
24354:
24352:
24348:
24344:
24342:
24339:
24337:
24336:
24335:Tuva or Bust!
24332:
24330:
24329:
24325:
24323:
24320:
24318:
24315:
24314:
24312:
24308:
24298:
24295:
24289:
24286:
24285:
24283:
24279:
24269:
24268:
24264:
24258:
24257:
24253:
24247:
24246:
24242:
24236:
24235:
24231:
24225:
24224:
24220:
24214:
24213:
24209:
24203:
24202:
24198:
24192:
24191:
24187:
24181:
24180:
24176:
24170:
24166:
24165:
24163:
24159:
24153:
24150:
24148:
24145:
24143:
24140:
24138:
24135:
24133:
24130:
24128:
24125:
24123:
24120:
24118:
24115:
24113:
24110:
24108:
24105:
24103:
24100:
24098:
24095:
24093:
24090:
24088:
24085:
24083:
24080:
24079:
24077:
24073:
24069:
24062:
24057:
24055:
24050:
24048:
24043:
24042:
24039:
24026:
24018:
24015:
24014:
24010:
24004:
24001:
23999:
23996:
23994:
23991:
23989:
23986:
23984:
23981:
23979:
23976:
23974:
23971:
23969:
23966:
23964:
23961:
23960:
23958:
23954:
23948:
23945:
23943:
23940:
23938:
23935:
23933:
23930:
23928:
23925:
23923:
23920:
23918:
23915:
23913:
23910:
23908:
23905:
23903:
23900:
23898:
23895:
23894:
23892:
23888:
23882:
23879:
23877:
23874:
23872:
23869:
23867:
23864:
23862:
23859:
23857:
23854:
23852:
23849:
23848:
23846:
23842:
23836:
23833:
23831:
23828:
23826:
23823:
23821:
23818:
23817:
23815:
23811:
23807:
23800:
23795:
23793:
23788:
23786:
23781:
23780:
23777:
23769:
23765:
23761:
23760:Sixty Symbols
23757:
23752:
23749:
23746:
23743:
23740:
23737:
23734:
23731:
23728:
23726:
23723:
23722:
23712:
23706:
23701:
23700:
23693:
23683:on 2011-07-25
23682:
23678:
23677:
23671:
23666:
23664:0-521-45692-4
23660:
23655:
23654:
23647:
23643:
23641:0-226-42266-6
23637:
23633:
23628:
23624:
23620:
23616:
23611:
23610:
23597:
23593:
23587:
23579:
23573:
23565:
23562:
23560:
23557:
23556:
23553:
23545:
23541:
23535:
23527:
23524:
23521:
23519:37(4):103-112
23518:
23515:
23512:
23509:
23508:
23505:
23497:
23491:
23483:
23480:
23478:
23475:
23474:
23471:
23463:
23460:
23458:
23455:
23454:
23451:
23443:
23439:
23433:
23418:
23414:
23410:
23404:
23390:
23386:
23379:
23372:
23368:
23363:
23355:
23349:
23345:
23344:
23336:
23328:
23322:
23318:
23311:
23305:
23304:0-521-45692-4
23301:
23297:
23291:
23285:
23281:
23278:
23274:
23270:
23264:
23256:
23249:
23241:
23234:
23219:
23215:
23208:
23200:
23196:
23192:
23188:
23184:
23180:
23175:
23170:
23166:
23162:
23155:
23148:
23137:on 2022-08-09
23136:
23132:
23128:
23124:
23120:
23116:
23112:
23108:
23104:
23100:
23093:
23079:. 5 July 2016
23078:
23074:
23068:
23057:
23053:
23049:
23045:
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23032:
23025:
23021:
23005:
23001:
22990:
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22965:
22962:
22960:
22957:
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22951:
22941:
22937:
22933:
22929:
22926:
22922:
22921:
22917:
22914:
22910:
22909:
22905:
22902:
22901:
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22888:
22887:
22881:
22877:
22875:
22871:
22867:
22861:
22859:
22854:
22850:
22846:
22840:
22838:
22828:
22826:
22821:
22814:
22806:
22793:
22772:
22765:
22762:
22756:
22750:
22746:
22742:
22737:
22733:
22728:
22723:
22718:
22709:
22708:
22707:
22682:
22673:
22664:
22653:
22650:
22645:
22642:
22636:
22633:
22630:
22627:
22623:
22620:
22611:
22608:
22605:
22596:
22593:
22590:
22577:
22573:
22569:
22564:
22560:
22556:
22551:
22547:
22540:
22537:
22530:
22529:
22528:
22499:
22493:
22489:
22485:
22480:
22476:
22472:
22467:
22463:
22455:
22454:
22453:
22450:
22446:
22422:
22413:
22404:
22395:
22392:
22389:
22382:
22381:
22380:
22369:
22360:
22337:
22334:
22331:
22328:
22321:
22320:
22319:
22308:
22300:
22275:
22272:
22265:
22260:
22256:
22252:
22243:
22238:
22234:
22230:
22221:
22216:
22212:
22208:
22199:
22194:
22190:
22186:
22181:
22177:
22168:
22164:
22157:
22149:
22145:
22138:
22130:
22126:
22119:
22111:
22107:
22100:
22096:
22088:
22087:
22086:
22079:
22073:
22070:
22066:
22052:
22047:
22043:
22037:
22034:
22029:
22027:
22016:
21990:
21986:
21982:
21979:
21974:
21970:
21966:
21960:
21956:
21952:
21944:
21941:
21938:
21933:
21927:
21923:
21917:
21913:
21909:
21906:
21901:
21897:
21891:
21887:
21883:
21878:
21874:
21868:
21864:
21859:
21854:
21823:
21819:
21816:
21813:
21805:
21801:
21797:
21792:
21788:
21783:
21774:
21773:
21772:
21751:
21745:
21741:
21735:
21730:
21727:
21724:
21720:
21716:
21713:
21710:
21705:
21701:
21693:
21692:
21691:
21674:
21671:
21666:
21662:
21656:
21651:
21648:
21645:
21641:
21628:
21625:
21620:
21616:
21608:
21607:
21606:
21589:
21585:
21582:
21576:
21572:
21568:
21565:
21560:
21556:
21552:
21545:
21539:
21535:
21529:
21525:
21521:
21518:
21513:
21509:
21503:
21499:
21495:
21490:
21486:
21480:
21476:
21472:
21467:
21463:
21457:
21453:
21448:
21444:
21441:
21437:
21431:
21427:
21397:
21386:
21382:
21374:
21373:
21372:
21355:
21349:
21345:
21341:
21338:
21335:
21330:
21326:
21322:
21319:
21314:
21310:
21306:
21303:
21298:
21294:
21290:
21287:
21284:
21279:
21275:
21271:
21268:
21263:
21259:
21255:
21252:
21249:
21246:
21240:
21236:
21228:
21227:
21226:
21209:
21203:
21199:
21195:
21192:
21184:
21180:
21176:
21170:
21163:
21146:
21142:
21138:
21135:
21132:
21127:
21123:
21119:
21116:
21111:
21107:
21103:
21100:
21097:
21094:
21087:
21086:
21085:
21082:
21063:
21059:
21049:
21041:
21030:
21021:
21011:
21004:
21000:
20973:
20967:
20963:
20959:
20956:
20951:
20947:
20943:
20935:
20931:
20925:
20921:
20917:
20914:
20909:
20905:
20899:
20895:
20891:
20887:
20876:
20872:
20868:
20858:
20854:
20850:
20842:
20838:
20834:
20829:
20825:
20819:
20815:
20810:
20801:
20800:
20799:
20795:
20770:
20766:
20762:
20757:
20747:
20743:
20739:
20736:
20729:
20724:
20716:
20712:
20708:
20703:
20693:
20689:
20685:
20682:
20675:
20665:
20661:
20657:
20652:
20648:
20643:
20637:
20634:
20631:
20624:
20623:
20622:
20619:
20617:
20616:Lorentz space
20612:
20592:
20589:
20584:
20581:
20577:
20569:
20564:
20558:
20554:
20547:
20544:
20541:
20535:
20532:
20527:
20523:
20519:
20513:
20509:
20505:
20500:
20495:
20490:
20485:
20480:
20476:
20472:
20469:
20466:
20462:
20459:
20451:
20446:
20440:
20436:
20432:
20429:
20426:
20423:
20420:
20417:
20414:
20411:
20406:
20402:
20398:
20393:
20389:
20384:
20379:
20374:
20369:
20364:
20360:
20352:
20351:
20350:
20347:
20343:
20339:
20329:
20304:
20301:
20292:
20280:
20274:
20271:
20268:
20262:
20259:
20256:
20245:
20238:
20233:
20229:
20225:
20219:
20216:
20212:
20203:
20202:
20201:
20199:
20176:
20173:
20165:
20160:
20154:
20150:
20146:
20143:
20140:
20137:
20134:
20131:
20128:
20125:
20120:
20116:
20112:
20107:
20103:
20098:
20093:
20088:
20085:
20079:
20073:
20069:
20065:
20062:
20059:
20056:
20053:
20050:
20047:
20044:
20039:
20035:
20031:
20026:
20022:
20017:
20013:
20010:
20006:
20002:
19997:
19994:
19991:
19987:
19979:
19978:
19977:
19953:
19950:
19946:
19943:
19940:
19937:
19934:
19931:
19928:
19925:
19922:
19919:
19915:
19908:
19907:
19906:
19904:
19885:
19880:
19877:
19873:
19870:
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19864:
19861:
19858:
19855:
19850:
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19843:
19840:
19834:
19831:
19825:
19818:
19817:
19816:
19814:
19809:
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19773:
19765:
19758:
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19722:
19718:
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19710:
19706:
19703:
19700:
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19690:
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19671:
19667:
19662:
19658:
19647:
19644:
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19627:
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19613:
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19556:
19552:
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19512:
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19502:
19497:
19494:
19490:
19482:
19478:
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19441:
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19303:
19295:
19294:
19293:
19286:
19275:
19272:
19270:
19248:
19245:
19241:
19237:
19231:
19227:
19223:
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19213:
19206:
19203:
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19143:
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19135:
19129:
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19115:
19114:
19111:
19090:
19059:
19056:
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19009:
19006:
19003:
18995:
18988:
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18967:
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18954:
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18926:
18925:
18908:
18905:
18898:
18892:
18888:
18884:
18879:
18875:
18870:
18866:
18863:
18859:
18848:
18844:
18840:
18832:
18828:
18824:
18819:
18815:
18810:
18801:
18800:
18799:
18791:
18783:
18776:
18753:
18748:
18744:
18734:
18728:
18725:
18720:
18706:
18698:
18690:
18687:
18684:
18679:
18675:
18669:
18665:
18659:
18655:
18649:
18645:
18639:
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18631:
18627:
18621:
18616:
18613:
18609:
18605:
18600:
18595:
18591:
18585:
18580:
18576:
18570:
18565:
18561:
18555:
18545:
18540:
18537:
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18529:
18524:
18520:
18514:
18504:
18500:
18494:
18486:
18483:
18480:
18477:
18474:
18467:
18466:
18465:
18435:
18432:
18422:
18416:
18410:
18405:
18401:
18394:
18380:
18373:
18369:
18362:
18361:
18360:
18341:
18335:
18331:
18324:
18315:
18304:
18303:
18302:
18281:
18277:
18272:
18268:
18261:
18249:
18248:
18247:
18230:
18226:
18223:
18218:
18214:
18208:
18196:
18195:
18194:
18176:
18173:
18165:
18162:
18158:
18154:
18150:
18145:
18139:
18131:
18122:
18115:
18108:
18107:
18106:
18086:
18083:
18079:
18074:
18068:
18060:
18051:
18034:
18028:
18018:
18015:
18012:
18009:
18006:
18003:
17998:
17995:
17991:
17987:
17984:
17979:
17973:
17965:
17956:
17939:
17933:
17923:
17920:
17917:
17914:
17911:
17904:
17903:
17902:
17885:
17882:
17878:
17872:
17866:
17863:
17860:
17857:
17854:
17851:
17848:
17844:
17841:
17834:
17828:
17825:
17818:
17817:
17816:
17779:
17775:
17772:
17767:
17761:
17758:
17753:
17750:
17744:
17737:
17731:
17728:
17722:
17718:
17714:
17711:
17708:
17705:
17702:
17698:
17695:
17688:
17682:
17679:
17672:
17671:
17670:
17649:
17646:
17642:
17639:
17634:
17628:
17625:
17620:
17617:
17611:
17604:
17598:
17592:
17586:
17583:
17579:
17576:
17573:
17570:
17567:
17564:
17560:
17557:
17550:
17544:
17541:
17534:
17533:
17532:
17511:
17508:
17504:
17501:
17496:
17490:
17487:
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17479:
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17466:
17460:
17456:
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17438:
17435:
17432:
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17426:
17422:
17419:
17412:
17406:
17403:
17396:
17395:
17394:
17361:
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17346:
17342:
17336:
17332:
17327:
17323:
17320:
17315:
17311:
17305:
17296:
17292:
17285:
17284:
17283:
17280:
17258:
17251:
17247:
17238:
17234:
17227:
17223:
17217:
17213:
17206:
17203:
17198:
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17183:
17178:
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17164:
17160:
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17145:
17142:
17133:
17129:
17122:
17114:
17110:
17105:
17097:
17096:
17095:
17077:
17071:
17066:
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17056:
17050:
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17036:
17033:
17028:
17024:
17018:
17008:
17004:
16998:
16989:
16982:
16979:
16973:
16970:
16967:
16960:
16959:
16958:
16944:
16915:
16893:
16886:
16882:
16876:
16873:
16869:
16862:
16857:
16854:
16850:
16847:
16843:
16839:
16836:
16832:
16824:
16821:
16812:
16808:
16801:
16793:
16789:
16784:
16776:
16775:
16774:
16751:
16747:
16741:
16731:
16727:
16721:
16713:
16710:
16707:
16700:
16699:
16698:
16681:
16673:
16668:
16663:
16657:
16653:
16647:
16638:
16631:
16627:
16619:
16618:
16617:
16596:
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16589:
16581:
16576:
16572:
16566:
16562:
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16553:
16550:
16545:
16541:
16535:
16526:
16522:
16519:
16512:
16511:
16510:
16491:
16485:
16466:
16461:
16455:
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16445:
16435:
16431:
16425:
16417:
16412:
16408:
16402:
16392:
16388:
16382:
16373:
16366:
16363:
16357:
16354:
16351:
16346:
16343:
16339:
16333:
16330:
16326:
16319:
16316:
16310:
16307:
16304:
16297:
16296:
16295:
16287:
16283:
16269:
16263:
16257:
16237:
16230:
16224:
16218:
16212:
16169:
16166:
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16157:
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16150:
16146:
16140:
16134:
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16103:
16100:
16097:
16075:
16069:
16061:
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16023:
16019:
16015:
16006:
16002:
15995:
15992:
15985:
15981:
15977:
15970:
15962:
15958:
15945:
15939:
15931:
15927:
15914:
15903:
15899:
15886:
15879:
15871:
15870:
15869:
15837:
15833:
15829:
15824:
15820:
15814:
15811:
15808:
15805:
15802:
15792:
15789:
15785:
15782:
15776:
15773:
15767:
15764:
15761:
15758:
15755:
15751:
15742:
15739:
15735:
15732:
15726:
15723:
15719:
15712:
15706:
15699:
15696:
15683:
15676:
15668:
15667:
15666:
15658:
15654:
15645:
15641:
15617:
15613:
15609:
15605:
15602:
15597:
15593:
15587:
15583:
15579:
15575:
15565:
15557:
15553:
15549:
15546:
15539:
15538:
15537:
15519:
15515:
15511:
15508:
15503:
15493:
15489:
15484:
15474:
15468:
15461:
15460:
15459:
15456:
15454:
15449:
15432:
15426:
15423:
15419:
15415:
15410:
15407:
15402:
15398:
15394:
15391:
15381:
15373:
15369:
15353:
15339:
15336:
15331:
15327:
15323:
15318:
15314:
15309:
15301:
15281:
15261:
15256:
15253:
15249:
15243:
15239:
15234:
15228:
15225:
15222:
15216:
15212:
15208:
15202:
15197:
15193:
15189:
15186:
15181:
15177:
15173:
15170:
15167:
15162:
15158:
15153:
15149:
15142:
15141:
15140:
15137:
15105:
15102:
15098:
15094:
15089:
15086:
15077:
15071:
15068:
15058:
15046:
15028:
15014:
15011:
15006:
15002:
14998:
14989:
14982:
14974:
14973:
14972:
14937:
14931:
14926:
14923:
14919:
14909:
14903:
14899:
14892:
14875:
14872:
14869:
14859:
14853:
14847:
14842:
14839:
14835:
14825:
14819:
14815:
14811:
14806:
14803:
14799:
14795:
14792:
14788:
14784:
14780:
14774:
14771:
14767:
14757:
14751:
14742:
14735:
14730:
14726:
14723:
14720:
14717:
14707:
14701:
14695:
14686:
14680:
14677:
14668:
14662:
14659:
14656:
14647:
14640:
14636:
14633:
14630:
14623:
14622:
14621:
14594:
14591:
14587:
14543:
14536:
14513:
14510:
14507:
14499:
14495:
14489:
14485:
14479:
14476:
14472:
14466:
14463:
14457:
14453:
14446:
14445:
14444:
14436:
14431:
14407:
14400:
14383:
14380:
14377:
14367:
14361:
14353:
14349:
14343:
14333:
14324:
14321:
14317:
14312:
14308:
14301:
14300:
14299:
14296:
14294:
14290:
14286:
14239:
14218:
14204:
14198:
14188:
14185:
14176:
14170:
14158:
14152:
14146:
14137:
14131:
14117:
14114:
14109:
14105:
14098:
14092:
14088:
14080:
14079:
14078:
14075:
14064:
14043:
14039:
14036:
14030:
14025:
14021:
14006:
14000:
13997:
13991:
13985:
13972:
13966:
13963:
13959:
13955:
13952:
13946:
13943:
13940:
13934:
13927:
13926:
13925:
13900:
13897:
13888:
13885:
13882:
13876:
13873:
13869:
13865:
13862:
13856:
13853:
13850:
13844:
13837:
13836:
13835:
13817:
13809:
13805:
13799:
13795:
13791:
13787:
13781:
13777:
13772:
13764:
13763:
13762:
13744:
13739:
13736:
13733:
13729:
13725:
13721:
13718:
13715:
13706:
13700:
13694:
13688:
13683:
13679:
13675:
13671:
13665:
13662:
13658:
13654:
13651:
13645:
13639:
13632:
13631:
13630:
13626:
13622:
13600:
13592:
13588:
13582:
13578:
13574:
13570:
13564:
13559:
13556:
13553:
13549:
13544:
13540:
13536:
13531:
13528:
13525:
13521:
13517:
13513:
13510:
13505:
13501:
13494:
13491:
13488:
13484:
13477:
13471:
13468:
13461:
13460:
13459:
13456:
13431:
13428:
13422:
13419:
13416:
13413:
13410:
13407:
13404:
13400:
13396:
13393:
13387:
13381:
13374:
13373:
13372:
13357:
13349:
13339:
13335:
13309:
13305:
13301:
13298:
13295:
13290:
13286:
13279:
13271:
13267:
13260:
13257:
13249:
13245:
13238:
13230:
13226:
13219:
13214:
13211:
13208:
13204:
13200:
13187:
13181:
13171:
13168:
13161:
13160:
13159:
13155:
13151:
13147:
13102:
13097:
13093:
13085:
13079:
13073:
13067:
13064:
13061:
13058:
13053:
13049:
13041:
13035:
13029:
13023:
13020:
13013:
13012:
13011:
13004:
13002:
12991:
12987:
12976:
12972:
12946:
12943:
12940:
12936:
12932:
12927:
12924:
12921:
12917:
12913:
12910:
12907:
12902:
12899:
12895:
12891:
12888:
12885:
12878:
12877:
12876:
12869:
12847:
12842:
12837:
12831:
12827:
12821:
12817:
12812:
12808:
12805:
12801:
12796:
12790:
12786:
12782:
12778:
12774:
12771:
12768:
12759:
12755:
12746:
12742:
12738:
12730:
12726:
12717:
12713:
12707:
12704:
12700:
12696:
12689:
12688:
12687:
12669:
12662:
12659:
12653:
12650:
12646:
12642:
12637:
12634:
12625:
12621:
12612:
12608:
12604:
12596:
12592:
12583:
12579:
12573:
12570:
12566:
12562:
12556:
12552:
12543:
12539:
12530:
12526:
12522:
12514:
12510:
12501:
12497:
12492:
12484:
12483:
12482:
12473:
12471:
12467:
12462:
12460:
12459:external legs
12453:
12449:
12443:
12436:
12432:
12431:normalization
12413:
12405:
12399:
12395:
12392:
12389:
12382:
12378:
12371:
12367:
12362:
12359:
12356:
12349:
12344:
12340:
12331:
12327:
12321:
12317:
12304:
12301:
12296:
12292:
12287:
12280:
12276:
12272:
12266:
12260:
12253:
12252:
12251:
12234:
12230:
12226:
12223:
12218:
12213:
12209:
12203:
12200:
12195:
12190:
12186:
12182:
12179:
12175:
12170:
12166:
12163:
12157:
12154:
12148:
12143:
12139:
12133:
12127:
12124:
12118:
12114:
12108:
12102:
12099:
12095:
12089:
12081:
12080:
12079:
12076:
12063:
12039:
12034:
12030:
12021:
12017:
12011:
12007:
11999:
11995:
11987:
11986:
11985:
11974:
11967:
11910:
11906:
11899:
11895:
11891:
11885:
11879:
11872:
11871:
11870:
11853:
11847:
11841:
11838:
11835:
11832:
11826:
11820:
11817:
11810:
11809:
11808:
11805:
11801:
11797:
11793:
11786:
11780:
11778:
11775:
11767:
11757:
11754:
11745:
11741:
11736:
11731:
11727:
11718:
11713:
11709:
11702:
11698:
11691:
11687:
11680:
11676:
11669:
11665:
11658:
11654:
11637:
11636:
11635:
11625:
11620:
11616:
11587:s where each
11581:
11543:
11542:vacuum bubble
11530:
11503:
11501:
11496:
11487:
11472:
11464:
11459:
11450:
11439:
11434:
11431:
11408:
11399:
11396:
11381:
11361:
11342:
11308:
11303:
11299:
11289:
11285:
11281:
11276:
11272:
11265:
11255:
11254:
11253:
11251:
11241:
11237:
11232:
11219:
11215:
11209:
11204:
11203:
11202:
11195:
11168:
11162:
11134:
11126:
11121:
11117:
11113:
11104:
11099:
11095:
11091:
11082:
11077:
11073:
11069:
11060:
11055:
11051:
11047:
11042:
11035:
11034:
11033:
11025:
11021:
11011:
11008:
11002:
10990:
10983:
10953:
10949:
10946:
10938:
10934:
10927:
10919:
10915:
10908:
10900:
10896:
10889:
10881:
10877:
10870:
10865:
10862:
10858:
10854:
10848:
10844:
10835:
10831:
10824:
10816:
10812:
10805:
10797:
10793:
10786:
10778:
10774:
10767:
10763:
10755:
10754:
10753:
10746:
10742:
10738:
10729:
10709:
10705:
10702:
10699:
10696:
10693:
10687:
10684:
10680:
10675:
10672:
10669:
10663:
10660:
10656:
10651:
10648:
10645:
10642:
10638:
10630:
10626:
10622:
10618:
10614:
10609:
10606:
10602:
10594:
10593:
10592:
10560:
10557:
10554:
10549:
10545:
10541:
10533:
10529:
10525:
10520:
10516:
10512:
10507:
10503:
10499:
10494:
10490:
10483:
10475:
10471:
10464:
10456:
10452:
10445:
10437:
10433:
10426:
10418:
10414:
10407:
10400:
10396:
10390:
10386:
10380:
10376:
10370:
10366:
10361:
10354:
10351:
10347:
10342:
10337:
10332:
10325:
10319:
10315:
10308:
10304:
10298:
10294:
10290:
10287:
10280:
10279:
10278:
10261:
10256:
10252:
10245:
10242:
10238:
10233:
10230:
10225:
10217:
10212:
10204:
10201:
10198:
10191:
10190:
10189:
10181:
10162:
10157:
10152:
10148:
10144:
10139:
10136:
10133:
10130:
10127:
10124:
10121:
10115:
10112:
10109:
10106:
10100:
10094:
10091:
10088:
10085:
10079:
10075:
10070:
10067:
10063:
10059:
10051:
10050:
10049:
10046:
10039:
10027:
10015:
10007:
9977:
9971:
9968:
9964:
9960:
9955:
9951:
9945:
9941:
9935:
9931:
9926:
9918:
9917:
9916:
9909:
9886:
9883:
9878:
9874:
9869:
9865:
9861:
9853:
9835:
9831:
9827:
9819:
9816:
9813:
9810:
9804:
9801:
9798:
9795:
9792:
9789:
9786:
9783:
9775:
9771:
9765:
9761:
9757:
9754:
9750:
9746:
9739:
9735:
9729:
9725:
9721:
9718:
9714:
9708:
9705:
9701:
9697:
9691:
9687:
9682:
9679:
9675:
9671:
9663:
9662:
9661:
9636:
9632:
9629:
9626:
9623:
9617:
9613:
9606:
9603:
9589:
9586:
9578:
9575:
9572:
9569:
9566:
9563:
9555:
9552:
9549:
9546:
9540:
9537:
9534:
9531:
9528:
9525:
9522:
9516:
9513:
9510:
9505:
9501:
9495:
9491:
9487:
9484:
9480:
9472:
9468:
9462:
9459:
9455:
9449:
9446:
9443:
9435:
9431:
9421:
9407:
9390:
9386:
9383:
9376:
9373:
9370:
9365:
9361:
9355:
9351:
9347:
9344:
9340:
9336:
9333:
9330:
9323:
9322:
9321:
9307:
9298:
9284:
9263:
9249:
9227:
9222:
9218:
9208:
9204:
9200:
9195:
9191:
9184:
9174:
9169:
9165:
9155:
9151:
9147:
9142:
9138:
9131:
9125:
9118:
9113:
9109:
9099:
9095:
9091:
9086:
9082:
9075:
9065:
9060:
9056:
9046:
9042:
9038:
9033:
9029:
9022:
9016:
9009:
9004:
9000:
8990:
8986:
8982:
8977:
8973:
8966:
8956:
8951:
8947:
8937:
8933:
8929:
8924:
8920:
8913:
8907:
8903:
8894:
8890:
8883:
8875:
8871:
8864:
8856:
8852:
8845:
8837:
8833:
8826:
8822:
8814:
8813:
8812:
8791:
8786:
8782:
8776:
8770:
8766:
8762:
8757:
8753:
8748:
8744:
8736:
8733:
8730:
8726:
8722:
8719:
8715:
8706:
8703:
8699:
8692:
8689:
8681:
8677:
8670:
8666:
8658:
8657:
8656:
8630:
8621:
8617:
8610:
8607:
8599:
8595:
8588:
8580:
8576:
8569:
8565:
8557:
8556:
8555:
8553:
8547:
8537:
8511:
8507:
8503:
8498:
8492:
8479:
8478:
8477:
8456:
8450:
8447:
8444:
8438:
8427:
8415:
8414:
8413:
8408:
8377:
8374:
8370:
8363:
8357:
8351:
8345:
8341:
8335:
8325:
8313:
8312:
8311:
8293:
8290:
8287:
8282:
8272:
8260:
8259:
8258:
8250:
8238:
8234:
8205:
8201:
8192:
8188:
8184:
8176:
8172:
8163:
8159:
8150:
8146:
8137:
8133:
8127:
8119:
8116:
8110:
8104:
8098:
8091:
8090:
8089:
8074:
8070:
8035:
8031:
8024:
8021:
8018:
8015:
8007:
8003:
7996:
7993:
7990:
7982:
7978:
7971:
7968:
7965:
7962:
7952:
7948:
7939:
7936:
7932:
7929:
7923:
7920:
7916:
7908:
7905:
7898:
7892:
7886:
7882:
7874:
7873:
7872:
7851:
7847:
7843:
7834:
7831:
7827:
7824:
7818:
7815:
7811:
7803:
7800:
7793:
7787:
7781:
7777:
7769:
7768:
7767:
7738:
7734:
7730:
7725:
7721:
7715:
7711:
7705:
7701:
7696:
7688:
7687:
7686:
7682:
7678:
7657:
7654:
7650:
7643:
7640:
7635:
7629:
7625:
7620:
7612:
7611:
7610:
7602:
7600:
7595:
7589:
7559:
7544:
7540:
7531:
7527:
7523:
7515:
7511:
7502:
7498:
7492:
7488:
7468:
7455:
7451:
7442:
7438:
7431:
7428:
7420:
7416:
7409:
7405:
7397:
7396:
7395:
7392:
7388:
7384:
7380:
7376:
7369:
7365:
7355:
7351:
7338:
7313:
7306:
7298:
7284:
7281:
7230:
7227:
7221:
7218:
7214:
7210:
7205:
7202:
7193:
7189:
7182:
7179:
7171:
7167:
7160:
7155:
7152:
7148:
7144:
7138:
7134:
7125:
7121:
7114:
7111:
7103:
7099:
7092:
7088:
7080:
7079:
7078:
7061:
7055:
7050:
7046:
7039:
7034:
7029:
7025:
7021:
7014:
7010:
7006:
7002:
6996:
6992:
6988:
6981:
6977:
6971:
6966:
6962:
6956:
6952:
6945:
6942:
6936:
6931:
6927:
6922:
6914:
6913:
6912:
6903:
6898:
6877:
6871:
6866:
6861:
6858:
6853:
6848:
6845:
6840:
6836:
6828:
6827:
6826:
6798:
6793:
6789:
6782:
6777:
6772:
6768:
6764:
6757:
6753:
6747:
6744:
6738:
6730:
6726:
6721:
6715:
6711:
6707:
6704:
6701:
6696:
6689:
6683:
6677:
6669:
6665:
6659:
6655:
6649:
6645:
6639:
6636:
6630:
6626:
6623:
6620:
6613:
6612:
6611:
6589:
6586:
6583:
6573:
6572:
6571:
6547:
6544:
6541:
6538:
6534:
6530:
6525:
6522:
6517:
6514:
6510:
6502:
6501:
6500:
6480:
6477:
6474:
6470:
6466:
6461:
6458:
6454:
6446:
6445:
6444:
6440:
6436:
6429:
6425:
6419:
6415:
6394:
6391:
6388:
6377:
6369:
6368:
6367:
6346:
6340:
6336:
6332:
6328:
6323:
6319:
6315:
6309:
6305:
6301:
6298:
6295:
6290:
6286:
6282:
6278:
6273:
6269:
6265:
6259:
6255:
6251:
6248:
6242:
6229:
6228:
6227:
6210:
6204:
6200:
6194:
6191:
6187:
6183:
6178:
6174:
6166:
6165:
6164:
6159:
6155:
6149:
6126:
6121:
6114:
6108:
6104:
6097:
6093:
6086:
6083:
6075:
6071:
6067:
6064:
6057:
6056:
6055:
6038:
6033:
6029:
6024:
6019:
6009:
6005:
5999:
5988:
5985:
5979:
5976:
5973:
5966:
5965:
5964:
5961:
5957:
5953:
5929:
5926:
5920:
5914:
5908:
5903:
5899:
5892:
5889:
5881:
5877:
5873:
5870:
5863:
5862:
5861:
5855:
5854:even function
5849:
5845:
5825:
5818:
5815:
5809:
5806:
5801:
5793:
5787:
5780:
5779:
5778:
5771:
5751:
5712:
5706:
5701:
5694:
5688:
5684:
5677:
5673:
5666:
5663:
5655:
5651:
5647:
5644:
5637:
5636:
5635:
5614:
5608:
5604:
5598:
5593:
5589:
5566:
5562:
5555:
5552:
5549:
5546:
5536:
5533:
5530:
5515:
5511:
5504:
5501:
5498:
5495:
5485:
5470:
5466:
5459:
5456:
5453:
5450:
5431:
5427:
5423:
5420:
5413:
5412:
5411:
5408:
5405:
5401:
5383:
5379:
5357:
5351:
5336:
5330:
5327:
5321:
5315:
5302:
5299:
5288:
5285:
5282:
5275:
5271:
5268:
5261:
5260:
5259:
5256:
5241:
5233:
5220:
5199:
5165:
5159:
5156:
5153:
5149:
5142:
5136:
5131:
5127:
5123:
5118:
5115:
5112:
5109:
5105:
5098:
5092:
5084:
5076:
5073:
5065:
5062:
5056:
5053:
5047:
5041:
5034:
5033:
5032:
5030:
5029:Fourier modes
5020:
5012:
5003:
4998:
4996:
4977:
4972:
4968:
4964:
4955:
4951:
4944:
4941:
4933:
4929:
4922:
4918:
4914:
4910:
4906:
4900:
4897:
4891:
4888:
4884:
4880:
4875:
4872:
4863:
4859:
4852:
4849:
4841:
4837:
4830:
4825:
4822:
4818:
4814:
4804:
4803:
4802:
4784:
4779:
4775:
4771:
4762:
4758:
4751:
4748:
4740:
4736:
4729:
4725:
4721:
4717:
4713:
4710:
4707:
4698:
4694:
4687:
4684:
4676:
4672:
4665:
4660:
4657:
4653:
4647:
4642:
4638:
4630:
4629:
4628:
4625:
4620:
4616:
4609:
4605:
4577:
4573:
4570:
4564:
4561:
4557:
4551:
4546:
4542:
4534:
4533:
4532:
4515:
4511:
4506:
4502:
4497:
4492:
4484:
4479:
4468:
4465:
4459:
4456:
4453:
4446:
4445:
4444:
4432:
4430:
4429:Wick rotation
4426:
4422:
4421:path integral
4412:
4389:
4386:
4382:
4379:
4373:
4370:
4367:
4363:
4356:
4353:
4350:
4347:
4344:
4341:
4338:
4334:
4324:
4316:
4313:
4305:
4300:
4296:
4289:
4286:
4281:
4273:
4270:
4257:
4248:
4242:
4232:
4231:
4230:
4213:
4205:
4202:
4193:
4189:
4181:
4173:
4169:
4160:
4157:
4150:
4144:
4140:
4133:
4125:
4122:
4109:
4099:
4093:
4084:
4080:
4072:
4060:
4054:
4047:
4046:
4045:
4034:
4031:
4028:
4027:
4026:
4003:
4000:
3996:
3993:
3987:
3984:
3981:
3977:
3970:
3967:
3964:
3959:
3955:
3947:
3944:
3940:
3936:
3933:
3922:
3914:
3911:
3903:
3898:
3894:
3887:
3884:
3879:
3871:
3868:
3859:
3855:
3848:
3840:
3836:
3825:
3824:
3823:
3806:
3800:
3792:
3789:
3780:
3776:
3769:
3761:
3757:
3746:
3743:
3736:
3731:
3727:
3719:
3716:
3703:
3694:
3688:
3683:
3679:
3672:
3660:
3654:
3647:
3646:
3645:
3643:
3616:
3613:
3604:
3600:
3593:
3585:
3581:
3573:
3570:
3563:
3558:
3554:
3550:
3547:
3540:
3537:
3524:
3515:
3509:
3504:
3500:
3496:
3493:
3487:
3475:
3469:
3460:
3456:
3455:
3432:
3425:
3422:
3413:
3409:
3400:
3397:
3390:
3384:
3380:
3371:
3368:
3355:
3345:
3337:
3333:
3325:
3319:
3313:
3309:
3301:
3289:
3283:
3278:
3275:
3269:
3265:
3260:
3255:
3251:
3247:
3241:
3238:
3231:
3223:
3220:
3211:
3203:
3196:
3188:
3187:
3186:
3162:
3154:
3148:
3136:
3129:
3124:
3097:
3093:
3084:
3080:
3071:
3068:
3065:
3061:
3051:
3050:
3049:
3019:
3015:
3011:
3006:
3002:
2992:
2986:
2983:
2980:
2975:
2971:
2963:
2962:
2961:
2959:
2952:Feynman rules
2949:
2946:
2941:
2937:
2932:
2912:
2907:
2902:
2898:
2894:
2888:
2874:
2869:
2866:
2863:
2859:
2845:
2833:
2829:
2825:
2820:
2816:
2812:
2806:
2792:
2787:
2784:
2781:
2777:
2762:
2761:
2760:
2754:
2749:
2740:
2721:
2712:
2705:
2694:
2691:
2688:
2684:
2680:
2676:
2671:
2666:
2662:
2658:
2652:
2638:
2633:
2630:
2627:
2623:
2618:
2606:
2600:
2596:
2590:
2586:
2582:
2577:
2572:
2569:
2566:
2562:
2557:
2550:
2547:
2541:
2537:
2524:
2521:
2518:
2514:
2510:
2507:
2500:
2499:
2498:
2474:
2465:
2458:
2447:
2444:
2441:
2437:
2433:
2429:
2424:
2419:
2415:
2411:
2405:
2391:
2386:
2383:
2380:
2376:
2371:
2359:
2353:
2349:
2343:
2339:
2335:
2330:
2325:
2322:
2319:
2315:
2310:
2303:
2300:
2293:
2285:
2282:
2266:
2263:
2260:
2256:
2252:
2249:
2242:
2241:
2240:
2238:
2234:
2230:
2200:
2192:
2184:
2180:
2176:
2173:
2165:
2157:
2153:
2149:
2146:
2142:
2131:
2128:
2121:
2120:
2119:
2117:
2098:
2089:
2085:
2081:
2076:
2072:
2065:
2057:
2049:
2045:
2031:
2023:
2019:
2010:
2007:
2000:
1999:
1998:
1993:
1989:
1963:
1946:
1930:
1916:
1908:
1907:
1906:
1892:
1882:
1879:
1873:
1872:
1871:
1870:interaction:
1869:
1860:
1851:
1848:
1846:
1841:
1827:
1824:
1821:
1818:
1815:
1811:
1810:
1809:
1807:
1803:
1799:
1794:
1790:
1787:
1784:
1781:
1770:
1761:
1756:
1750:
1742:
1733:
1731:
1726:
1722:
1718:
1713:
1709:
1707:
1703:
1699:
1695:
1691:
1687:
1682:
1680:
1675:
1665:
1659:
1651:
1647:
1644:
1640:
1636:
1634:
1630:
1624:
1622:
1621:Feynman rules
1618:
1614:
1610:
1604:
1602:
1598:
1594:
1590:
1586:
1582:
1572:
1570:
1569:Freeman Dyson
1566:
1562:
1557:
1555:
1551:
1547:
1538:
1536:
1532:
1527:
1525:
1521:
1517:
1513:
1509:
1505:
1501:
1497:
1492:
1490:
1486:
1482:
1476:
1474:
1470:
1466:
1462:
1458:
1453:
1451:
1447:
1443:
1439:
1435:
1430:
1428:
1424:
1420:
1416:
1411:
1407:
1403:
1399:
1391:
1387:
1383:
1379:
1375:
1371:
1367:
1364:, made of an
1363:
1358:
1342:
1337:
1335:
1330:
1328:
1323:
1322:
1320:
1319:
1313:
1308:
1305:
1303:
1300:
1298:
1295:
1293:
1290:
1288:
1285:
1283:
1282:Zamolodchikov
1280:
1278:
1277:Zamolodchikov
1275:
1273:
1270:
1268:
1265:
1263:
1260:
1258:
1255:
1253:
1250:
1248:
1245:
1243:
1240:
1238:
1235:
1233:
1230:
1228:
1225:
1223:
1220:
1218:
1215:
1213:
1210:
1208:
1205:
1203:
1200:
1198:
1195:
1193:
1190:
1188:
1185:
1183:
1180:
1178:
1175:
1173:
1170:
1168:
1165:
1163:
1160:
1158:
1155:
1153:
1150:
1148:
1145:
1143:
1140:
1138:
1135:
1133:
1130:
1128:
1125:
1123:
1120:
1118:
1115:
1113:
1110:
1108:
1105:
1103:
1100:
1098:
1095:
1093:
1090:
1088:
1085:
1083:
1080:
1078:
1075:
1073:
1070:
1068:
1065:
1063:
1060:
1058:
1055:
1053:
1050:
1048:
1045:
1043:
1040:
1038:
1035:
1033:
1030:
1028:
1025:
1023:
1020:
1018:
1015:
1013:
1010:
1008:
1005:
1003:
1000:
998:
995:
993:
990:
988:
985:
983:
980:
978:
975:
973:
970:
968:
965:
963:
960:
958:
955:
953:
950:
948:
945:
943:
940:
938:
935:
933:
930:
928:
925:
923:
920:
918:
915:
913:
910:
908:
905:
903:
900:
898:
895:
893:
890:
888:
885:
883:
880:
878:
875:
873:
870:
868:
865:
863:
860:
858:
855:
853:
850:
848:
845:
843:
840:
838:
835:
833:
830:
828:
825:
823:
820:
818:
815:
813:
810:
808:
805:
803:
800:
798:
795:
793:
790:
788:
785:
783:
780:
778:
775:
773:
770:
768:
765:
763:
760:
758:
755:
753:
750:
748:
745:
743:
740:
738:
735:
733:
730:
728:
725:
723:
720:
718:
715:
713:
710:
708:
705:
703:
700:
698:
695:
693:
690:
688:
685:
683:
680:
678:
675:
673:
670:
668:
665:
663:
660:
658:
655:
653:
650:
648:
645:
643:
640:
638:
635:
633:
630:
628:
625:
623:
620:
618:
615:
613:
610:
609:
602:
601:
594:
591:
589:
586:
584:
581:
579:
578:Supersymmetry
576:
574:
573:String theory
571:
570:
564:
563:
556:
553:
551:
548:
546:
543:
541:
538:
537:
534:
529:
528:
521:
518:
516:
513:
511:
508:
506:
503:
501:
498:
496:
493:
491:
488:
487:
481:
480:
473:
470:
468:
465:
463:
460:
458:
455:
453:
450:
448:
445:
443:
440:
438:
435:
433:
430:
428:
425:
423:
420:
418:
415:
413:
410:
408:
405:
403:
400:
398:
395:
393:
390:
388:
385:
383:
380:
378:
375:
374:
368:
367:
360:
357:
355:
352:
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156:
155:antiparticles
152:
148:
144:
141:Feynman used
139:
137:
133:
129:
128:Frank Wilczek
125:
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109:
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93:
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24367:
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24346:
24333:
24326:
24288:Joan Feynman
24265:
24254:
24243:
24232:
24221:
24210:
24199:
24188:
24177:
24122:Parton model
24081:
24016:
23824:
23759:
23698:
23685:. Retrieved
23681:the original
23675:
23652:
23631:
23595:
23586:
23572:
23552:
23544:www.fnal.gov
23543:
23534:
23513:January 2017
23504:
23490:
23470:
23450:
23442:www.fnal.gov
23441:
23432:
23420:. Retrieved
23417:Freakonomics
23416:
23403:
23392:. Retrieved
23388:
23378:
23362:
23342:
23335:
23316:
23310:
23295:
23290:
23272:
23268:
23263:
23254:
23248:
23239:
23233:
23223:February 26,
23221:. Retrieved
23218:The Atlantic
23217:
23207:
23164:
23160:
23154:
23145:
23139:. Retrieved
23135:the original
23106:
23102:
23092:
23081:. Retrieved
23076:
23067:
23043:
23037:
23024:
23004:
22939:
22918:
22912:
22906:
22898:
22878:
22866:bound states
22862:
22841:
22834:
22822:
22812:
22804:
22794:
22787:
22697:
22514:
22448:
22444:
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21016:
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19763:
19756:
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19284:
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19268:
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19088:
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18104:
17900:
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17281:
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16768:
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16293:
16284:
16270:
16258:
16238:
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16222:
16214:
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15857:
15656:
15650:
15643:
15639:
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15534:
15457:
15450:
15447:
15135:
15122:
14952:
14595:
14589:
14585:
14558:
14434:
14429:
14422:
14297:
14282:
14233:
14076:
14065:
14058:
13915:
13833:
13760:
13624:
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13447:
13358:
13344:
13337:
13333:
13330:
13156:
13149:
13145:
13118:
13005:
13000:
12997:
12985:
12974:
12970:
12963:
12870:
12862:
12684:
12479:
12463:
12458:
12451:
12447:
12441:
12434:
12430:
12428:
12249:
12074:
12064:
12057:
11969:
11965:
11928:
11868:
11803:
11791:
11789:
11784:
11779:
11776:
11765:
11758:
11755:
11751:
11743:
11739:
11729:
11725:
11711:
11707:
11700:
11696:
11694:can go from
11689:
11685:
11678:
11674:
11667:
11663:
11656:
11652:
11624:automorphism
11621:
11617:
11613:4! × 4! × 2!
11582:
11541:
11531:
11504:
11499:
11497:
11493:
11473:
11462:
11451:
11437:
11435:
11432:
11409:
11405:
11397:
11379:
11362:
11340:
11327:
11249:
11247:
11239:
11235:
11217:
11213:
11207:
11196:
11190:s is called
11174:
11160:
11149:
11023:
11019:
11009:
11006:
11003:
10985:
10981:
10970:
10747:
10740:
10736:
10728:power series
10725:
10575:
10276:
10187:
10179:
10044:
10037:
10025:
10013:
10005:
9993:
9907:
9903:
9656:Dividing by
9655:
9299:
9255:
9248:propagator.
9246:
8810:
8646:
8551:
8549:
8540:Wick theorem
8530:
8475:
8406:
8392:
8309:
8256:
8236:
8229:
8226:
8075:
8068:
8065:
7870:
7757:
7680:
7676:
7673:
7608:
7596:
7587:
7582:
7393:
7386:
7382:
7378:
7374:
7367:
7360:
7353:
7346:
7336:
7319:
7304:
7296:
7282:
7279:
7248:
7076:
6901:
6899:
6892:
6816:
6605:
6565:
6498:
6438:
6434:
6427:
6423:
6420:
6413:
6409:
6361:
6225:
6157:
6153:
6150:
6143:
6053:
5962:
5955:
5951:
5947:
5847:
5843:
5840:
5772:
5749:
5727:
5629:
5409:
5403:
5399:
5381:
5377:
5372:
5257:
5242:
5231:
5197:
5180:
5026:
5023:On a lattice
5010:
4999:
4994:
4992:
4799:
4626:
4618:
4614:
4607:
4603:
4592:
4530:
4438:
4425:ground state
4418:
4410:
4228:
4043:
4024:
3821:
3639:
3180:
3157:
3152:
3131:
3127:
3036:
2955:
2944:
2930:
2927:
2753:Dyson series
2747:
2736:
2489:
2215:
2113:
1978:
1888:
1880:
1877:
1865:
1849:
1842:
1839:
1806:gauge bosons
1795:
1791:
1788:
1785:
1782:
1779:
1768:
1759:
1754:
1748:
1714:
1710:
1697:
1693:
1689:
1685:
1683:
1678:
1671:
1643:regularizing
1637:
1626:
1606:
1601:Sidney Drell
1578:
1565:Dyson graphs
1564:
1560:
1558:
1549:
1544:
1528:
1493:
1477:
1454:
1431:
1419:Dyson series
1395:
1311:
1157:Stueckelberg
887:Jona-Lasinio
462:Vacuum state
447:Quantization
416:
284:Gauge theory
264:Strong force
249:Field theory
226:
177:perturbative
174:
159:
140:
116:David Kaiser
103:
97:
76:
68:
67:, antiquark
64:
63:pair (quark
52:
40:
32:
24371:(2013 film)
24361:(2001 play)
24350:(1996 film)
23922:Self-energy
23912:Landau pole
23876:Positronium
23851:Dual photon
23764:Brady Haran
21690:as well as
21060:∈ {1,2,...,
14289:determinant
11559:2 × 4! × 4!
11028:factors in
10184:Interaction
10048:should be:
8088:integrals.
7316:Monte Carlo
6817:The factor
3185:-matrix is
1736:Description
1704:instead of
1518:, Feynman,
1423:eigenstates
1410:Hamiltonian
1380:into three
1297:Zinn-Justin
1147:Sommerfield
1072:Pomeranchuk
1037:Osterwalder
1032:Oppenheimer
962:Łopuszański
782:Fredenhagen
583:Technicolor
24387:Categories
23988:Lamb shift
23917:QED vacuum
23687:2011-01-28
23615:Diagrammar
23422:9 February
23394:2022-06-23
23367:Jorge Cham
23269:Diagrammar
23141:2021-11-12
23083:2020-06-16
23046:(2): 156.
23016:References
22979:Propagator
22908:PhD Comics
22019:Scattering
16208:. Since a
15868:in pairs:
13359:The field
11719:If a line
11661:goes from
11646:to vertex
11638:If a line
11402:Loop order
11194:th order.
10030:different
9994:where the
9911:different
5773:The field
3064:propagator
3062:A bosonic
2940:propagator
1874:e + e → 2γ
1845:propagator
1633:Lagrangian
1512:Hans Bethe
1469:propagator
1406:free field
1292:Zimmermann
1187:Vainshtein
1062:Polchinski
922:Kontsevich
867:Iliopoulos
842:Heisenberg
667:Bogoliubov
605:Scientists
442:Propagator
324:T-symmetry
319:P-symmetry
314:C-symmetry
302:Symmetries
259:Weak force
243:Background
45:annihilate
24317:Namesakes
24017:See also:
23956:Processes
23844:Particles
23813:Formalism
23592:"Quantum"
23147:electron.
23131:120117564
22895:antiquark
22743:−
22671:⟨
22668:⟩
22637:∫
22603:⟨
22600:⟩
22570:−
22557:−
22541:δ
22538:∫
22473:−
22411:⟨
22408:⟩
22390:∫
22338:λ
22276:λ
22158:ϕ
22139:ϕ
22120:ϕ
22101:ϕ
21980:⋯
21907:⋯
21824:∫
21798:⋯
21721:∑
21717:−
21642:∑
21566:⋯
21519:⋯
21442:−
21398:∫
21392:∞
21383:∫
21339:∧
21336:⋯
21320:∧
21288:∧
21285:⋯
21269:∧
21253:∧
21133:⋯
20957:⋯
20918:−
20915:⋯
20892:−
20882:∞
20873:∫
20869:⋯
20864:∞
20855:∫
20835:⋯
20725:⋯
20632:∫
20545:−
20491:∫
20477:∫
20424:⋅
20375:∫
20361:∫
20272:−
20230:∫
20144:−
20138:⋅
20089:∫
20057:⋅
20011:−
19988:∫
19944:∧
19926:∧
19707:−
19701:⋅
19684:−
19632:−
19607:∫
19491:−
19456:−
19431:∫
19304:∫
19246:τ
19207:˙
19188:τ
19179:∫
19175:−
19160:∫
19157:τ
19149:∞
19140:∫
19124:Δ
19060:τ
19042:−
19010:τ
19007:π
18989:τ
18976:−
18968:τ
18960:∞
18951:∫
18935:Δ
18909:τ
18867:τ
18864:−
18854:∞
18845:∫
18754:η
18738:¯
18735:η
18726:η
18721:μ
18717:∂
18710:¯
18707:η
18699:μ
18695:∂
18691:
18596:ν
18581:μ
18571:μ
18556:ν
18552:∂
18525:ν
18515:μ
18511:∂
18505:ν
18495:μ
18491:∂
18487:
18481:∫
18436:η
18426:¯
18423:η
18411:η
18406:μ
18395:μ
18391:∂
18384:¯
18381:η
18370:∫
18336:μ
18325:μ
18321:∂
18278:α
18273:μ
18262:μ
18258:∂
18219:μ
18209:μ
18205:∂
18137:∂
18129:∂
18116:∫
18066:∂
18058:∂
18019:δ
18016:∫
18007:∫
17971:∂
17963:∂
17924:δ
17921:∫
17912:∫
17864:∫
17861:π
17826:∫
17762:θ
17729:∫
17719:θ
17712:∫
17680:∫
17629:θ
17599:δ
17584:∫
17580:θ
17574:∫
17542:∫
17491:θ
17461:δ
17457:θ
17451:∫
17436:∫
17404:∫
17337:−
17321:−
17316:μ
17306:μ
17302:∂
17293:δ
17228:ν
17218:μ
17207:λ
17204:−
17199:ν
17196:μ
17161:δ
17134:ν
17115:μ
17061:μ
17051:μ
17047:∂
17037:λ
17034:−
17029:ν
17019:μ
17015:∂
17009:ν
16999:μ
16995:∂
16974:∫
16945:λ
16877:ν
16874:μ
16840:δ
16813:ν
16794:μ
16752:ν
16742:μ
16738:∂
16732:ν
16722:μ
16718:∂
16714:∫
16658:μ
16648:μ
16644:∂
16632:−
16567:−
16551:−
16546:μ
16536:μ
16532:∂
16523:δ
16520:∫
16456:ν
16446:ν
16442:∂
16436:μ
16426:μ
16422:∂
16418:−
16413:ν
16403:μ
16399:∂
16393:ν
16383:μ
16379:∂
16358:−
16355:∫
16347:ν
16344:μ
16334:ν
16331:μ
16311:∫
16167:−
16154:⋅
16151:γ
16127:−
16109:δ
16076:∏
16059:−
16024:∑
15996:ψ
15993:⋯
15971:ψ
15949:¯
15946:ψ
15940:⋯
15918:¯
15915:ψ
15890:¯
15887:ψ
15830:−
15806:⋅
15803:γ
15777:δ
15765:−
15759:⋅
15756:γ
15727:δ
15707:ψ
15687:¯
15684:ψ
15614:ψ
15603:−
15598:μ
15588:μ
15584:γ
15569:¯
15566:ψ
15554:∫
15520:ψ
15509:−
15504:μ
15500:∂
15494:μ
15490:γ
15478:¯
15475:ψ
15469:∫
15424:−
15403:∗
15374:∗
15366:∂
15362:∂
15351:∂
15347:∂
15324:ϕ
15319:∗
15315:ϕ
15254:−
15244:∗
15226:ϕ
15217:∗
15213:ϕ
15198:∗
15194:ϕ
15187:ϕ
15182:∗
15171:ϕ
15163:∗
15159:ϕ
15150:∫
15103:−
15081:¯
15078:η
15069:η
15050:¯
15047:η
15041:∂
15037:∂
15029:η
15026:∂
15022:∂
14999:ψ
14993:¯
14990:ψ
14932:η
14924:−
14913:¯
14910:η
14904:−
14873:ψ
14863:¯
14860:ψ
14848:η
14840:−
14829:¯
14826:η
14820:−
14812:η
14804:−
14793:ψ
14772:−
14761:¯
14758:η
14746:¯
14743:ψ
14727:∫
14721:ψ
14711:¯
14708:ψ
14696:η
14690:¯
14687:ψ
14678:ψ
14672:¯
14669:η
14660:ψ
14651:¯
14648:ψ
14637:∫
14511:ψ
14496:ψ
14486:ψ
14454:∫
14381:ψ
14371:¯
14368:ψ
14350:ψ
14337:¯
14334:ψ
14309:∫
14189:
14168:∂
14164:∂
14129:∂
14125:∂
14093:ϕ
14001:η
13998:−
13986:ϕ
13956:∫
13947:η
13944:−
13941:ϕ
13935:δ
13886:−
13866:∫
13854:−
13845:δ
13806:ϕ
13778:∏
13740:ϕ
13719:ϕ
13701:ϕ
13680:∫
13655:∫
13550:∏
13472:ρ
13469:∫
13432:ϕ
13423:ϕ
13417:∫
13397:∫
13299:⋯
13258:⋯
13205:∑
13172:
13080:ϕ
13065:∫
13036:ϕ
13021:∫
12944:ρ
12941:−
12922:−
12911:ϕ
12900:−
12892:∫
12818:∑
12809:
12783:∑
12772:ϕ
12743:ϕ
12739:⋯
12714:ϕ
12705:−
12697:∫
12663:ϕ
12651:−
12643:∫
12638:ϕ
12609:ϕ
12605:⋯
12580:ϕ
12571:−
12563:∫
12527:ϕ
12523:⋯
12498:ϕ
12472:in 1957.
12396:
12390:∝
12368:∑
12363:
12310:∞
12288:∑
12277:∏
12273:∝
12235:⋯
12227:⋯
12167:⋯
11996:∏
11896:∑
11892:∝
11839:
11833:≡
11792:connected
11360:remains.
11266:δ
11222:from the
11043:λ
10950:ϕ
10928:ϕ
10909:ϕ
10890:ϕ
10871:ϕ
10863:−
10855:∫
10825:ϕ
10806:ϕ
10787:ϕ
10768:ϕ
10706:⋯
10623:−
10607:−
10484:δ
10465:ϕ
10446:ϕ
10427:ϕ
10408:ϕ
10362:∫
10348:λ
10320:ϕ
10295:∫
10253:ϕ
10239:λ
10231:ϕ
10226:μ
10222:∂
10218:ϕ
10213:μ
10209:∂
10205:∫
10137:⋅
10131:⋅
10125:⋅
10122:…
10113:−
10101:⋅
10092:−
9961:⋯
9817:−
9805:⋅
9802:…
9796:⋅
9790:⋅
9755:−
9747:∫
9719:−
9698:∫
9618:−
9607:π
9587:⋅
9579:…
9573:⋅
9567:⋅
9553:−
9541:⋅
9538:…
9532:⋅
9526:⋅
9485:−
9450:∫
9428:∂
9418:∂
9387:π
9345:−
9337:∫
9308:ϕ
9285:ϕ
9264:ϕ
9201:−
9185:δ
9148:−
9132:δ
9092:−
9076:δ
9039:−
9023:δ
8983:−
8967:δ
8930:−
8914:δ
8884:ϕ
8865:ϕ
8846:ϕ
8827:ϕ
8763:−
8745:δ
8727:∏
8723:∑
8693:ϕ
8690:⋯
8671:ϕ
8611:ϕ
8608:⋯
8589:ϕ
8570:ϕ
8487:Δ
8476:So that:
8451:δ
8433:Δ
8424:∂
8358:ϕ
8346:ϕ
8336:μ
8332:∂
8326:μ
8322:∂
8288:ϕ
8283:μ
8279:∂
8273:μ
8269:∂
8189:δ
8185:⋯
8160:δ
8134:δ
8120:π
8099:δ
8066:But near
8025:
8016:⋯
7997:
7972:
7966:−
7933:−
7924:δ
7899:ϕ
7887:ϕ
7828:−
7819:δ
7794:ϕ
7782:ϕ
7712:ϕ
7702:ϕ
7636:ϕ
7626:ϕ
7528:ϕ
7524:⋯
7499:ϕ
7489:∑
7480:∞
7477:→
7432:ϕ
7429:⋯
7410:ϕ
7231:ϕ
7219:−
7211:∫
7206:ϕ
7183:ϕ
7180:⋯
7161:ϕ
7153:−
7145:∫
7115:ϕ
7112:⋯
7093:ϕ
7026:ϕ
7007:−
6993:∏
6978:ϕ
6972:∗
6963:ϕ
6937:−
6928:∫
6769:ϕ
6727:ϕ
6722:∫
6712:∏
6705:ϕ
6684:ϕ
6670:∗
6666:ϕ
6646:∑
6627:
6621:∫
6539:−
6523:−
6329:⋯
6299:∫
6279:⋯
6249:∫
6109:ϕ
6072:∫
6025:ϕ
6020:μ
6016:∂
6010:∗
6006:ϕ
6000:μ
5996:∂
5980:∫
5927:−
5921:ϕ
5909:ϕ
5878:∫
5816:−
5810:ϕ
5802:∗
5788:ϕ
5689:ϕ
5652:∫
5605:ϕ
5599:∗
5590:ϕ
5556:
5550:−
5534:⋯
5505:
5499:−
5460:
5454:−
5428:∫
5331:ϕ
5328:−
5316:ϕ
5292:⟩
5280:⟨
5276:∑
5240:instead.
5137:ϕ
5128:∫
5116:⋅
5093:ϕ
5077:π
5057:∫
5042:ϕ
4945:ϕ
4942:⋯
4923:ϕ
4901:ϕ
4881:∫
4876:ϕ
4853:ϕ
4850:⋯
4831:ϕ
4815:∫
4752:ϕ
4749:⋯
4730:ϕ
4711:ϕ
4688:ϕ
4685:⋯
4666:ϕ
4639:∫
4574:ϕ
4543:∫
4498:ϕ
4493:μ
4489:∂
4485:ϕ
4480:μ
4476:∂
4460:∫
4383:−
4368:−
4345:−
4339:γ
4317:π
4290:∫
4282:_
4261:¯
4258:ψ
4243:ψ
4194:ν
4174:μ
4151:ψ
4145:ν
4141:γ
4134:_
4113:¯
4110:ψ
4094:ψ
4085:μ
4081:γ
4064:¯
4061:ψ
3997:−
3982:−
3948:ν
3945:μ
3934:−
3915:π
3888:∫
3880:_
3860:ν
3841:μ
3801:_
3781:ν
3762:μ
3737:ψ
3732:ν
3728:γ
3707:¯
3704:ψ
3689:ψ
3684:μ
3680:γ
3664:¯
3661:ψ
3605:ν
3586:μ
3564:ψ
3559:ν
3555:γ
3528:¯
3525:ψ
3510:ψ
3505:μ
3501:γ
3479:¯
3476:ψ
3414:ν
3391:ψ
3385:ν
3381:γ
3359:¯
3356:ψ
3338:μ
3320:ψ
3314:μ
3310:γ
3293:¯
3290:ψ
3248:∫
3085:μ
3020:μ
3012:ψ
3007:μ
3003:γ
2996:¯
2993:ψ
2984:−
2860:∏
2846:±
2834:∑
2778:∏
2759:-matrix,
2700:∞
2685:∑
2681:≡
2624:∏
2583:∫
2563:∏
2530:∞
2515:∑
2453:∞
2438:∑
2434:≡
2377:∏
2336:∫
2316:∏
2283:−
2272:∞
2257:∑
2193:τ
2177:τ
2169:∞
2161:∞
2158:−
2154:∫
2147:−
2061:∞
2058:−
2055:→
2035:∞
2029:→
1960:⟩
1934:⟨
1772:and time
1646:integrals
1520:Schwinger
1450:off-shell
1442:conserved
1227:Wetterich
1212:Weisskopf
1162:Sudarshan
1112:Schwinger
1022:Nishijima
987:Maldacena
952:Leutwyler
912:Kinoshita
812:Goldstone
802:Gell-Mann
717:Doplicher
484:Equations
185:canonical
170:variables
166:integrals
61:antiquark
24408:Diagrams
24347:Infinity
24301:(nephew)
24292:(sister)
23890:Concepts
23871:Positron
23856:Electron
23766:for the
23748:JaxoDraw
23280:Archived
23056:Archived
22947:See also
22853:coherent
22178:⟩
22097:⟨
20585:′
20510:′
19954:′
19905:" gives
19881:′
19851:′
19813:Jacobian
19796:, while
19714:′
19691:′
19648:′
19621:′
19564:′
19498:′
19445:′
19249:′
17811:is just
17179:′
17154:⟩
17146:′
17106:⟨
16858:′
16833:⟩
16825:′
16785:⟨
16016:⟩
15986:′
15880:⟨
15793:′
15743:′
15720:⟩
15700:′
15677:⟨
15637:linking
15328:⟩
15310:⟨
15003:⟩
14983:⟨
14293:Pfaffian
14106:⟩
14089:⟨
13818:⟩
13773:⟨
13745:⟩
13726:⟨
13601:⟩
13545:⟨
13537:⟩
13518:⟨
12868:factor.
12553:⟩
12493:⟨
10845:⟩
10764:⟨
10158:⟩
10145:⟨
10076:⟩
10060:⟨
10034:leaving
9978:⟩
9927:⟨
9875:⟩
9862:⟨
9688:⟩
9672:⟨
8904:⟩
8823:⟨
8716:⟩
8667:⟨
8631:⟩
8566:⟨
8371:⟩
8342:⟨
7940:′
7917:⟩
7909:′
7883:⟨
7835:′
7812:⟩
7804:′
7778:⟨
7722:⟩
7697:⟨
7651:⟩
7644:′
7621:⟨
7452:⟩
7406:⟨
7135:⟩
7089:⟨
6410:so that
5384:⟩
5376:⟨
4973:⟩
4911:⟨
4780:⟩
4718:⟨
4390:′
4274:′
4206:′
4161:′
4126:′
4004:′
3872:′
3793:′
3747:′
3720:′
3617:′
3574:′
3541:′
3426:′
3401:′
3372:′
3279:′
2497:, it is
1802:fermions
1766:• space
1755:(orange)
1674:particle
1524:Tomonaga
1496:infinite
1438:momentum
1378:strongly
1242:Wightman
1207:Weinberg
1197:Virasoro
1177:Tomonaga
1172:Thirring
1167:Symanzik
1127:Semenoff
1102:Schrader
1067:Polyakov
982:Majorana
917:Klebanov
872:Ivanenko
862:'t Hooft
832:Guralnik
777:Fröhlich
772:Fritzsch
767:Frampton
682:Buchholz
627:Bargmann
617:Anderson
397:Crossing
151:electron
147:positron
37:positron
35:) and a
29:electron
24310:Related
24171:"
23607:Sources
23199:2895311
23179:Bibcode
23111:Bibcode
22802:√
22311:
22295:
21052:
21033:
19903:wedging
19785:
19768:
19086:√
18775:loops.
16935:
16923:
16253:
16241:
14581:
14569:
14277:
14265:
14256:
14244:
13001:sources
12994:Sources
12452:sources
11650:, then
11578:
11566:
11564:=
11562:
11550:
11384:
11370:
11164:1,2,3,4
7341:
7327:
7309:
7293:
7287:
7270:
5754:
5740:
5407:means.
5202:
5188:
2755:of the
2751:of the
2114:In the
1988:-matrix
1983:is the
1760:(black)
1386:W boson
1252:Wilczek
1217:Wentzel
1192:Veltman
1137:Shirkov
1132:Shifman
1122:Seiberg
1107:Schwarz
1087:Rubakov
1007:Naimark
957:Lipatov
947:Lehmann
927:Kreimer
907:Kendall
797:Gelfand
792:Glashow
752:Feynman
732:Faddeev
727:Englert
697:Coleman
687:Cachazo
672:Brodsky
657:Bjorken
647:Berezin
637:Belavin
377:Anomaly
235:History
196:-matrix
94:in 1984
24359:
24299:
24290:
24281:Family
24272:(2005)
24270:
24261:(1999)
24259:
24250:(1999)
24248:
24239:(1997)
24237:
24228:(1988)
24226:
24217:(1985)
24215:
24206:(1985)
24204:
24195:(1965)
24193:
24184:(1964)
24182:
24173:(1959)
24075:Career
23866:Photon
23707:
23661:
23638:
23350:
23323:
23302:
23277:online
23197:
23129:
22940:Qantum
22868:, the
22837:series
22362:|
22356:|
22033:Wigner
21066:. The
21015:... +
18297:where
14559:where
14443:, and
14423:where
13331:where
12964:where
12440:= exp
12058:where
10576:Where
8227:where
7591:|
7585:|
7583:where
6893:where
5852:is an
5765:
5373:where
4593:where
4229:where
3822:where
2942:) and
2928:where
2216:where
1979:where
1725:graphs
1679:vertex
1522:, and
1434:energy
1388:and a
1374:weakly
1307:Zumino
1272:Yukawa
1262:Witten
1257:Wilson
1247:Wigner
1182:Tyutin
1142:Skyrme
1092:Ruelle
1057:Plefka
1052:Peskin
1042:Parisi
1002:Møller
992:Migdal
977:Maiani
972:Lüders
937:Landau
932:Kuraev
902:Källén
892:Jordan
877:Jackiw
817:Gribov
707:DeWitt
702:Dashen
692:Callan
662:Bleuer
632:Becchi
622:Anselm
49:photon
24161:Works
23195:S2CID
23169:arXiv
23127:S2CID
23059:(PDF)
23034:(PDF)
22996:Notes
22891:quark
19901:and "
14263:Spin
14242:Spin
13343:,...,
11929:with
6226:then
5217:from
4419:In a
1749:(red)
1516:Dyson
1390:gluon
1382:pions
1302:Zuber
1152:Stora
1117:Segal
1097:Salam
1082:Proca
1077:Popov
1047:Pauli
1027:Oehme
1017:Neveu
1012:Nambu
997:Mills
882:Jaffe
857:Hagen
852:Higgs
827:Gupta
822:Gross
807:Glimm
787:Furry
757:Fierz
747:Fermi
742:Fayet
737:Fadin
722:Dyson
712:Dirac
677:Brout
652:Bethe
612:Adler
371:Tools
73:gluon
57:quark
23705:ISBN
23659:ISBN
23636:ISBN
23424:2024
23348:ISBN
23321:ISBN
23300:ISBN
23225:2013
22523:and
21672:<
21626:>
21055:for
20340:′ =
19811:The
19761:and
16194:and
15862:and
15648:and
15127:and
14612:and
14602:and
13920:and
13629:is:
11716:too.
11438:loop
11250:line
11197:The
8401:and
7674:for
6416:= ±1
6412:det
5630:For
5391:and
4597:and
3640:The
1889:The
1866:The
1832:and
1814:spin
1719:and
1599:and
1587:and
1510:and
1473:spin
1440:are
1436:and
1376:and
1368:and
1362:kaon
1267:Yang
1237:Wick
1232:Weyl
1222:Wess
1202:Ward
897:Jost
847:Hepp
837:Haag
762:Fock
642:Bell
102:, a
24357:QED
23619:doi
23187:doi
23119:doi
23048:doi
21635:and
20796:+ 1
20335:to
18312:det
18119:det
18048:det
17953:det
17359:det
14593:).
14588:= −
14291:or
14186:log
13169:log
12988:(0)
12806:exp
12444:= 1
12393:exp
12360:exp
12072:to
11705:to
11672:to
11622:An
11430:s.
10040:− 4
10028:− 3
10016:− 2
10008:− 1
8071:= 0
8022:cos
7994:cos
7969:cos
7460:lim
7381:),
6911:is
6624:exp
6581:det
6432:to
6362:If
6237:det
5856:of
5553:cos
5502:cos
5457:cos
5234:→ 0
5013:→ 0
2958:QED
2041:lim
2015:lim
1798:QED
1796:In
1627:In
1563:or
1467:'s
1400:in
1287:Zee
967:Low
942:Lee
138:."
98:In
24389::
23762:.
23758:.
23594:.
23542:.
23440:.
23415:.
23387:.
23369:,
23216:.
23193:.
23185:.
23177:.
23165:27
23163:.
23144:.
23125:.
23117:.
23107:76
23105:.
23101:.
23075:.
23054:.
23044:93
23042:.
23036:.
22938:.
22318::
22078:λφ
22067:+
21031:=
21008:+
21001:=
20346:vp
20344:+
20324:;
19766:=
19755:+
19751:=
19595:,
19589:+
19414:+
19271:.
19110:.
18688:Tr
18484:Tr
17669:,
17393::
16914:.
16509::
16225:,
16213:,
14971:.
14590:ψψ
14586:ψψ
14432:,
13924:,
13367::
13154:.
11836:ln
11701:v′
11679:v′
11648:v′
11580:.
11575:48
11553:4!
11395:.
9915::
9660:,
8554::
7679:≠
7385:(−
7312:.
7283:dk
6819:dk
6610:.
6148:.
5954:,−
5770:.
5743:2π
5730:dk
5191:2π
5031::
1836:).
1834:•~
1830:~•
1762:,
1603::
1537:.
1491:.
1475:.
1366:up
126:.
69:q̄
43:)
24167:"
24060:e
24053:t
24046:v
23798:e
23791:t
23784:v
23770:.
23713:.
23690:.
23667:.
23644:.
23625:.
23621::
23598:.
23546:.
23444:.
23426:.
23397:.
23356:.
23329:.
23227:.
23201:.
23189::
23181::
23171::
23121::
23113::
23086:.
23050::
22893:–
22818:φ
22813:φ
22805:m
22797:k
22790:M
22773:.
22766:4
22763:1
22757:)
22751:2
22747:m
22738:2
22734:k
22729:(
22724:=
22719:E
22704:k
22700:k
22683:.
22678:|
22674:k
22665:k
22661:|
22654:E
22651:2
22646:k
22643:d
22634:=
22631:k
22628:d
22624:E
22621:d
22616:|
22612:k
22609:,
22606:E
22597:k
22594:,
22591:E
22587:|
22583:)
22578:2
22574:m
22565:2
22561:k
22552:2
22548:E
22544:(
22525:E
22521:k
22517:k
22500:.
22494:2
22490:m
22486:=
22481:2
22477:k
22468:2
22464:E
22449:k
22447:,
22445:E
22440:k
22423:.
22418:|
22414:k
22405:k
22401:|
22396:k
22393:d
22377:k
22373:k
22366:M
22359:M
22335:i
22332:=
22329:M
22316:M
22307:k
22303:/
22299:i
22273:i
22266:2
22261:4
22257:k
22253:i
22244:2
22239:3
22235:k
22231:i
22222:2
22217:2
22213:k
22209:i
22200:2
22195:1
22191:k
22187:i
22182:=
22174:)
22169:4
22165:k
22161:(
22155:)
22150:3
22146:k
22142:(
22136:)
22131:2
22127:k
22123:(
22117:)
22112:1
22108:k
22104:(
22083:λ
22069:m
22065:n
22060:m
22056:n
22048:)
22046:x
22044:(
22042:φ
22012:i
22010:v
21991:n
21987:v
21983:d
21975:2
21971:v
21967:d
21961:1
21957:v
21953:d
21945:1
21942:+
21939:n
21934:)
21928:n
21924:D
21918:n
21914:v
21910:+
21902:1
21898:D
21892:1
21888:v
21884:+
21879:0
21875:D
21869:0
21865:v
21860:(
21855:1
21847:x
21844:e
21841:l
21838:p
21835:m
21832:i
21829:s
21820:!
21817:n
21814:=
21806:n
21802:D
21793:0
21789:D
21784:1
21769:u
21752:.
21746:i
21742:v
21736:n
21731:1
21728:=
21725:i
21714:1
21711:=
21706:0
21702:v
21675:1
21667:i
21663:v
21657:n
21652:1
21649:=
21646:i
21629:0
21621:i
21617:v
21590:,
21586:u
21583:d
21577:n
21573:v
21569:d
21561:1
21557:v
21553:d
21546:)
21540:n
21536:D
21530:n
21526:v
21522:+
21514:2
21510:D
21504:2
21500:v
21496:+
21491:1
21487:D
21481:1
21477:v
21473:+
21468:0
21464:D
21458:0
21454:v
21449:(
21445:u
21438:e
21432:n
21428:u
21421:x
21418:e
21415:l
21412:p
21409:m
21406:i
21403:s
21387:0
21356:.
21350:n
21346:u
21342:d
21331:1
21327:u
21323:d
21315:0
21311:u
21307:d
21304:=
21299:n
21295:v
21291:d
21280:2
21276:v
21272:d
21264:1
21260:v
21256:d
21250:u
21247:d
21241:n
21237:u
21210:.
21204:i
21200:u
21196:d
21193:=
21190:)
21185:i
21181:v
21177:u
21174:(
21171:d
21147:n
21143:u
21139:d
21136:+
21128:1
21124:u
21120:d
21117:+
21112:0
21108:u
21104:d
21101:=
21098:u
21095:d
21079:n
21075:v
21070:i
21068:v
21064:}
21062:n
21058:i
21048:u
21044:/
21039:i
21037:u
21028:i
21026:v
21019:n
21017:u
21013:1
21010:u
21006:0
21003:u
20999:u
20993:i
20991:u
20974:.
20968:n
20964:u
20960:d
20952:0
20948:u
20944:d
20936:n
20932:D
20926:n
20922:u
20910:0
20906:D
20900:0
20896:u
20888:e
20877:0
20859:0
20851:=
20843:n
20839:D
20830:1
20826:D
20820:0
20816:D
20811:1
20794:n
20771:2
20767:m
20763:+
20758:2
20754:)
20748:n
20744:p
20740:+
20737:k
20734:(
20730:1
20717:2
20713:m
20709:+
20704:2
20700:)
20694:1
20690:p
20686:+
20683:k
20680:(
20676:1
20666:2
20662:m
20658:+
20653:2
20649:k
20644:1
20638:k
20635:d
20611:p
20593:v
20590:d
20582:k
20578:d
20570:2
20565:)
20559:2
20555:p
20551:)
20548:v
20542:1
20539:(
20536:v
20533:+
20528:2
20524:m
20520:+
20514:2
20506:k
20501:(
20496:1
20486:1
20481:0
20473:=
20470:v
20467:d
20463:k
20460:d
20452:2
20447:)
20441:2
20437:p
20433:v
20430:+
20427:k
20421:p
20418:v
20415:2
20412:+
20407:2
20403:m
20399:+
20394:2
20390:k
20385:(
20380:1
20370:1
20365:0
20342:k
20338:k
20333:k
20326:v
20322:v
20305:v
20302:d
20293:2
20287:)
20281:B
20278:)
20275:v
20269:1
20266:(
20263:+
20260:A
20257:v
20252:(
20246:1
20239:1
20234:0
20226:=
20220:B
20217:A
20213:1
20194:v
20177:v
20174:d
20166:2
20161:)
20155:2
20151:p
20147:v
20141:k
20135:p
20132:2
20129:v
20126:+
20121:2
20117:m
20113:+
20108:2
20104:k
20099:(
20094:1
20086:=
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20018:(
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19998:v
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19968:.
19951:t
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19935:=
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19923:u
19920:d
19916:u
19886:,
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19862:=
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19841:=
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19829:(
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19136:=
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18478:=
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18316:(
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17855:=
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17709:=
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17571:=
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17470:)
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17448:)
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17433:=
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17416:)
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17143:k
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17126:)
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17057:A
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17005:A
16990:(
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16971:=
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16929:/
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16912:k
16894:.
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16870:g
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16855:k
16851:+
16848:k
16844:(
16837:=
16829:)
16822:k
16818:(
16809:A
16805:)
16802:k
16799:(
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16771:A
16748:A
16728:A
16711:=
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16639:(
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16614:f
16597:.
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16590:D
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16467:.
16462:)
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16432:A
16409:A
16389:A
16374:(
16367:2
16364:1
16352:=
16340:F
16327:F
16320:4
16317:1
16308:=
16305:S
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15968:)
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15937:)
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15790:k
15786:+
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15780:(
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15736:+
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15730:(
15724:=
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15713:k
15710:(
15704:)
15697:k
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15663:k
15659:)
15657:k
15655:(
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15646:)
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15640:ψ
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15610:)
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15550:=
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15485:(
15433:.
15427:1
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15416:=
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15340:Z
15337:1
15332:=
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15279:(
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15269:D
15262:h
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15240:h
15235:e
15229:=
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15203:h
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15174:+
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15154:e
15136:M
15130:η
15125:η
15106:1
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15095:=
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15087:=
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15059:Z
15015:Z
15012:1
15007:=
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14963:η
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14955:η
14938:,
14927:1
14920:M
14900:e
14896:)
14893:M
14890:(
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14880:D
14876:=
14870:D
14854:D
14843:1
14836:M
14816:)
14807:1
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14789:(
14785:M
14781:)
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14752:+
14736:(
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14634:=
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14618:M
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14599:ψ
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14575:/
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14565:ψ
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14540:)
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14534:(
14530:f
14527:f
14524:a
14521:f
14518:P
14514:=
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14473:A
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14398:(
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14384:=
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14318:M
14313:e
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14219:.
14213:)
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14194:(
14180:)
14177:x
14174:(
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14159:=
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14150:[
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14141:)
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14115:1
14110:=
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14031:x
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13992:x
13989:(
13981:(
13976:)
13973:x
13970:(
13967:h
13964:i
13960:e
13953:=
13950:)
13938:(
13922:η
13918:φ
13901:k
13898:d
13892:)
13889:y
13883:x
13880:(
13877:k
13874:i
13870:e
13863:=
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13857:y
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13848:(
13810:x
13800:x
13796:h
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13788:e
13782:x
13737:h
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13722:=
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13710:)
13707:x
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13695:x
13692:(
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13652:=
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13646:h
13643:[
13640:Z
13627:)
13625:x
13623:(
13621:h
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13589:y
13583:i
13579:h
13575:i
13571:e
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13557:=
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13522:e
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13489:i
13485:e
13481:)
13478:y
13475:(
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13450:h
13429:D
13420:h
13414:i
13411:+
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13405:i
13401:e
13394:=
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13388:h
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13382:Z
13369:h
13365:h
13361:h
13354:n
13350:)
13347:n
13345:k
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13334:C
13315:)
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13280:C
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13239:h
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13223:(
13220:h
13215:C
13212:,
13209:n
13201:=
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13191:]
13188:h
13185:[
13182:Z
13177:(
13152:)
13150:k
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13137:H
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13103:k
13098:d
13094:d
13089:)
13086:k
13083:(
13077:)
13074:k
13071:(
13068:h
13062:=
13059:x
13054:d
13050:d
13045:)
13042:x
13039:(
13033:)
13030:x
13027:(
13024:h
13008:h
12986:δ
12981:k
12977:)
12975:k
12973:(
12971:δ
12966:ρ
12947:V
12937:e
12933:=
12928:T
12925:H
12918:e
12914:=
12908:D
12903:S
12896:e
12889:=
12886:Z
12873:Z
12865:E
12848:.
12843:)
12838:)
12832:i
12828:C
12822:i
12813:(
12802:(
12797:)
12791:i
12787:E
12779:(
12775:=
12769:D
12765:)
12760:n
12756:x
12752:(
12747:n
12736:)
12731:1
12727:x
12723:(
12718:1
12708:S
12701:e
12670:.
12660:D
12654:S
12647:e
12635:D
12631:)
12626:n
12622:x
12618:(
12613:n
12602:)
12597:1
12593:x
12589:(
12584:1
12574:S
12567:e
12557:=
12549:)
12544:n
12540:x
12536:(
12531:n
12520:)
12515:1
12511:x
12507:(
12502:1
12455:J
12442:W
12438:0
12435:Z
12414:.
12409:]
12406:J
12403:[
12400:W
12383:i
12379:C
12372:i
12357:=
12350:!
12345:i
12341:n
12332:i
12328:n
12322:i
12318:C
12305:0
12302:=
12297:i
12293:n
12281:i
12270:]
12267:J
12264:[
12261:Z
12231:)
12224:+
12219:2
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12196:+
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12075:Z
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12035:i
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11981:k
11979:D
11975:!
11972:i
11970:n
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11959:D
11954:i
11952:C
11947:i
11945:n
11940:i
11938:C
11933:k
11931:D
11911:k
11907:D
11900:k
11889:]
11886:J
11883:[
11880:Z
11854:.
11851:]
11848:J
11845:[
11842:Z
11830:]
11827:J
11824:[
11821:W
11818:i
11804:Z
11772:X
11768:!
11766:n
11761:X
11746:)
11744:l
11742:(
11740:M
11732:)
11730:l
11728:(
11726:M
11721:l
11714:)
11712:v
11710:(
11708:N
11703:)
11699:(
11697:N
11692:)
11690:l
11688:(
11686:M
11681:)
11677:(
11675:N
11670:)
11668:v
11666:(
11664:N
11659:)
11657:l
11655:(
11653:M
11644:v
11640:l
11632:N
11628:M
11605:X
11601:X
11597:X
11593:X
11589:X
11585:X
11572:/
11569:1
11556:/
11546:X
11538:X
11534:X
11527:X
11519:X
11515:X
11511:X
11507:X
11484:k
11480:k
11476:k
11469:k
11463:R
11454:k
11447:X
11442:X
11428:X
11424:X
11420:X
11416:X
11412:X
11393:k
11389:k
11380:k
11376:/
11373:1
11365:X
11358:k
11354:k
11350:k
11346:k
11341:k
11339:−
11335:k
11331:k
11309:2
11304:1
11300:k
11295:)
11290:2
11286:k
11282:+
11277:1
11273:k
11269:(
11242:)
11240:k
11238:(
11236:φ
11230:.
11228:k
11224:X
11220:)
11218:k
11216:(
11214:φ
11208:n
11206:4
11199:n
11192:n
11188:X
11185:n
11181:X
11177:X
11161:k
11156:X
11152:X
11135:.
11127:2
11122:4
11118:k
11114:1
11105:2
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11096:k
11092:1
11083:2
11078:2
11074:k
11070:1
11061:2
11056:1
11052:k
11048:1
11030:X
11026:)
11024:k
11022:(
11020:φ
11015:X
11010:X
11007:e
10999:k
10995:X
10991:)
10988:i
10986:k
10984:(
10982:φ
10977:k
10973:k
10954:Z
10947:D
10944:)
10939:4
10935:k
10931:(
10925:)
10920:3
10916:k
10912:(
10906:)
10901:2
10897:k
10893:(
10887:)
10882:1
10878:k
10874:(
10866:S
10859:e
10849:=
10841:)
10836:4
10832:k
10828:(
10822:)
10817:3
10813:k
10809:(
10803:)
10798:2
10794:k
10790:(
10784:)
10779:1
10775:k
10771:(
10750:X
10743:)
10741:k
10739:(
10737:φ
10732:X
10710:)
10703:+
10700:X
10697:X
10694:X
10688:!
10685:3
10681:1
10676:+
10673:X
10670:X
10664:!
10661:2
10657:1
10652:+
10649:X
10646:+
10643:1
10639:(
10631:F
10627:S
10619:e
10615:=
10610:S
10603:e
10589:λ
10585:S
10580:F
10578:S
10561:.
10558:X
10555:+
10550:F
10546:S
10542:=
10539:)
10534:4
10530:k
10526:+
10521:3
10517:k
10513:+
10508:2
10504:k
10500:+
10495:1
10491:k
10487:(
10481:)
10476:4
10472:k
10468:(
10462:)
10457:3
10453:k
10449:(
10443:)
10438:2
10434:k
10430:(
10424:)
10419:1
10415:k
10411:(
10401:4
10397:k
10391:3
10387:k
10381:2
10377:k
10371:1
10367:k
10355:!
10352:4
10343:+
10338:2
10333:|
10329:)
10326:k
10323:(
10316:|
10309:2
10305:k
10299:k
10291:=
10288:S
10262:.
10257:4
10246:!
10243:4
10234:+
10202:=
10199:S
10163:n
10153:2
10149:x
10140:1
10134:3
10128:5
10119:)
10116:3
10110:n
10107:2
10104:(
10098:)
10095:1
10089:n
10086:2
10083:(
10080:=
10071:n
10068:2
10064:x
10045:x
10038:n
10036:2
10032:x
10026:n
10024:2
10020:x
10014:n
10012:2
10006:n
10004:2
10000:x
9996:x
9972:n
9969:2
9965:x
9956:3
9952:x
9946:2
9942:x
9936:1
9932:x
9913:x
9908:n
9906:2
9887:a
9884:1
9879:=
9870:2
9866:x
9836:n
9832:a
9828:1
9823:)
9820:1
9814:n
9811:2
9808:(
9799:5
9793:3
9787:1
9784:=
9776:2
9772:/
9766:2
9762:x
9758:a
9751:e
9740:2
9736:/
9730:2
9726:x
9722:a
9715:e
9709:n
9706:2
9702:x
9692:=
9683:n
9680:2
9676:x
9658:I
9637:2
9633:1
9630:+
9627:n
9624:2
9614:a
9604:2
9590:2
9576:2
9570:2
9564:2
9559:)
9556:1
9550:n
9547:2
9544:(
9535:5
9529:3
9523:1
9517:=
9514:x
9511:d
9506:2
9502:/
9496:2
9492:x
9488:a
9481:e
9473:n
9469:2
9463:n
9460:2
9456:x
9447:=
9444:I
9436:n
9432:a
9422:n
9391:a
9384:2
9377:=
9374:x
9371:d
9366:2
9362:/
9356:2
9352:x
9348:a
9341:e
9334:=
9331:I
9228:2
9223:2
9219:k
9214:)
9209:3
9205:k
9196:2
9192:k
9188:(
9175:2
9170:1
9166:k
9161:)
9156:4
9152:k
9143:1
9139:k
9135:(
9126:+
9119:2
9114:2
9110:k
9105:)
9100:4
9096:k
9087:2
9083:k
9079:(
9066:2
9061:3
9057:k
9052:)
9047:3
9043:k
9034:1
9030:k
9026:(
9017:+
9010:2
9005:3
9001:k
8996:)
8991:4
8987:k
8978:3
8974:k
8970:(
8957:2
8952:1
8948:k
8943:)
8938:2
8934:k
8925:1
8921:k
8917:(
8908:=
8900:)
8895:4
8891:k
8887:(
8881:)
8876:3
8872:k
8868:(
8862:)
8857:2
8853:k
8849:(
8843:)
8838:1
8834:k
8830:(
8792:2
8787:i
8783:k
8777:)
8771:j
8767:k
8758:i
8754:k
8749:(
8737:j
8734:,
8731:i
8720:=
8712:)
8707:n
8704:2
8700:k
8696:(
8687:)
8682:1
8678:k
8674:(
8653:φ
8649:φ
8627:)
8622:n
8618:k
8614:(
8605:)
8600:2
8596:k
8592:(
8586:)
8581:1
8577:k
8573:(
8534:i
8512:2
8508:k
8504:i
8499:=
8496:)
8493:k
8490:(
8460:)
8457:x
8454:(
8448:i
8445:=
8442:)
8439:x
8436:(
8428:2
8410:Δ
8403:y
8399:x
8395:x
8378:0
8375:=
8367:)
8364:y
8361:(
8355:)
8352:x
8349:(
8294:0
8291:=
8247:k
8243:π
8239:)
8237:k
8235:(
8232:D
8230:δ
8211:)
8206:d
8202:k
8198:(
8193:D
8182:)
8177:2
8173:k
8169:(
8164:D
8156:)
8151:1
8147:k
8143:(
8138:D
8128:d
8124:)
8117:2
8114:(
8111:=
8108:)
8105:k
8102:(
8086:k
8082:π
8078:π
8069:k
8046:)
8041:)
8036:d
8032:k
8028:(
8019:+
8013:)
8008:2
8004:k
8000:(
7991:+
7988:)
7983:1
7979:k
7975:(
7963:d
7958:(
7953:2
7949:1
7944:)
7937:k
7930:k
7927:(
7921:=
7913:)
7906:k
7902:(
7896:)
7893:k
7890:(
7852:2
7848:k
7844:1
7839:)
7832:k
7825:k
7822:(
7816:=
7808:)
7801:k
7797:(
7791:)
7788:k
7785:(
7764:k
7760:V
7739:2
7735:k
7731:V
7726:=
7716:k
7706:k
7683:′
7681:k
7677:k
7658:0
7655:=
7641:k
7630:k
7588:C
7564:|
7560:C
7556:|
7550:)
7545:n
7541:x
7537:(
7532:C
7521:)
7516:1
7512:x
7508:(
7503:C
7493:C
7473:|
7469:C
7465:|
7456:=
7448:)
7443:n
7439:x
7435:(
7426:)
7421:1
7417:x
7413:(
7389:)
7387:k
7383:φ
7379:k
7377:(
7375:φ
7370:)
7368:x
7366:(
7363:C
7361:φ
7356:)
7354:k
7352:(
7349:C
7347:φ
7337:k
7333:/
7330:1
7322:k
7305:k
7301:/
7297:V
7280:k
7276:/
7273:1
7265:k
7260:k
7258:φ
7253:k
7251:φ
7228:D
7222:S
7215:e
7203:D
7199:)
7194:n
7190:x
7186:(
7177:)
7172:1
7168:x
7164:(
7156:S
7149:e
7139:=
7131:)
7126:n
7122:x
7118:(
7109:)
7104:1
7100:x
7096:(
7062:.
7056:k
7051:d
7047:d
7040:2
7035:|
7030:k
7022:|
7015:2
7011:k
7003:e
6997:k
6989:=
6982:k
6967:k
6957:2
6953:k
6946:2
6943:1
6932:k
6923:e
6908:k
6906:φ
6895:L
6878:,
6872:d
6867:)
6862:L
6859:1
6854:(
6849:=
6846:k
6841:d
6837:d
6823:k
6799:k
6794:d
6790:d
6783:2
6778:|
6773:k
6765:|
6758:2
6754:k
6748:2
6745:i
6739:e
6731:k
6716:k
6708:=
6702:D
6697:)
6693:)
6690:k
6687:(
6681:)
6678:k
6675:(
6660:2
6656:k
6650:k
6640:2
6637:i
6631:(
6608:k
6590:1
6587:=
6584:A
6568:π
6548:x
6545:k
6542:i
6535:e
6531:=
6526:1
6518:x
6515:k
6511:A
6481:x
6478:k
6475:i
6471:e
6467:=
6462:x
6459:k
6455:A
6441:)
6439:k
6437:(
6435:φ
6430:)
6428:x
6426:(
6424:φ
6414:A
6395:I
6392:=
6389:A
6383:T
6378:A
6364:A
6347:.
6341:n
6337:y
6333:d
6324:2
6320:y
6316:d
6310:1
6306:y
6302:d
6296:=
6291:n
6287:x
6283:d
6274:2
6270:x
6266:d
6260:1
6256:x
6252:d
6246:)
6243:A
6240:(
6211:,
6205:j
6201:x
6195:j
6192:i
6188:A
6184:=
6179:i
6175:y
6160:)
6158:x
6156:(
6154:φ
6146:k
6127:2
6122:|
6118:)
6115:k
6112:(
6105:|
6098:2
6094:k
6087:2
6084:1
6076:k
6068:=
6065:S
6039:x
6034:d
6030:d
5989:2
5986:1
5977:=
5974:S
5958:)
5956:k
5952:k
5950:(
5933:)
5930:k
5924:(
5918:)
5915:k
5912:(
5904:2
5900:k
5893:2
5890:1
5882:k
5874:=
5871:S
5858:k
5850:)
5848:k
5846:(
5844:φ
5826:.
5822:)
5819:k
5813:(
5807:=
5798:)
5794:k
5791:(
5775:φ
5758:)
5750:V
5746:/
5736:(
5713:.
5707:2
5702:|
5698:)
5695:k
5692:(
5685:|
5678:2
5674:k
5667:2
5664:1
5656:k
5648:=
5645:S
5632:k
5615:.
5609:k
5594:k
5584:)
5577:)
5572:)
5567:d
5563:k
5559:(
5547:1
5542:(
5537:+
5531:+
5526:)
5521:)
5516:2
5512:k
5508:(
5496:1
5491:(
5486:+
5481:)
5476:)
5471:1
5467:k
5463:(
5451:1
5446:(
5439:(
5432:k
5424:=
5421:S
5404:φ
5400:μ
5397:∂
5393:y
5389:x
5382:y
5380:,
5378:x
5358:,
5352:2
5346:)
5340:)
5337:y
5334:(
5325:)
5322:x
5319:(
5311:(
5303:2
5300:1
5289:y
5286:,
5283:x
5272:=
5269:S
5253:k
5249:k
5245:k
5238:k
5232:a
5227:k
5223:k
5215:π
5211:k
5207:k
5198:a
5194:/
5183:k
5166:.
5160:x
5157:k
5154:i
5150:e
5146:)
5143:k
5140:(
5132:k
5124:=
5119:x
5113:k
5110:i
5106:e
5102:)
5099:k
5096:(
5085:d
5081:)
5074:2
5071:(
5066:k
5063:d
5054:=
5051:)
5048:x
5045:(
5017:a
5011:a
5006:a
4978:.
4969:0
4965:|
4961:)
4956:n
4952:x
4948:(
4939:)
4934:1
4930:x
4926:(
4919:|
4915:0
4907:=
4898:D
4892:S
4889:i
4885:e
4873:D
4869:)
4864:n
4860:x
4856:(
4847:)
4842:1
4838:x
4834:(
4826:S
4823:i
4819:e
4785:,
4776:B
4772:|
4768:)
4763:n
4759:x
4755:(
4746:)
4741:1
4737:x
4733:(
4726:|
4722:A
4714:=
4708:D
4704:)
4699:n
4695:x
4691:(
4682:)
4677:1
4673:x
4669:(
4661:S
4658:i
4654:e
4648:B
4643:A
4621:)
4619:B
4617:(
4615:φ
4610:)
4608:A
4606:(
4604:φ
4599:B
4595:A
4578:,
4571:D
4565:S
4562:i
4558:e
4552:B
4547:A
4516:.
4512:x
4507:d
4503:d
4469:2
4466:1
4457:=
4454:S
4441:d
4394:)
4387:x
4380:x
4377:(
4374:p
4371:i
4364:e
4357:0
4354:i
4351:+
4348:m
4342:p
4335:i
4325:4
4321:)
4314:2
4311:(
4306:p
4301:4
4297:d
4287:=
4278:)
4271:x
4267:(
4252:)
4249:x
4246:(
4214:,
4210:)
4203:x
4199:(
4190:A
4185:)
4182:x
4179:(
4170:A
4165:)
4158:x
4154:(
4130:)
4123:x
4119:(
4103:)
4100:x
4097:(
4076:)
4073:x
4070:(
4055:N
4008:)
4001:x
3994:x
3991:(
3988:k
3985:i
3978:e
3971:0
3968:i
3965:+
3960:2
3956:k
3941:g
3937:i
3923:4
3919:)
3912:2
3909:(
3904:k
3899:4
3895:d
3885:=
3876:)
3869:x
3865:(
3856:A
3852:)
3849:x
3846:(
3837:A
3807:,
3797:)
3790:x
3786:(
3777:A
3773:)
3770:x
3767:(
3758:A
3751:)
3744:x
3740:(
3724:)
3717:x
3713:(
3698:)
3695:x
3692:(
3676:)
3673:x
3670:(
3655:N
3621:)
3614:x
3610:(
3601:A
3597:)
3594:x
3591:(
3582:A
3578:)
3571:x
3567:(
3551:e
3548:i
3545:)
3538:x
3534:(
3519:)
3516:x
3513:(
3497:e
3494:i
3491:)
3488:x
3485:(
3470:N
3433:.
3430:)
3423:x
3419:(
3410:A
3405:)
3398:x
3394:(
3376:)
3369:x
3365:(
3349:)
3346:x
3343:(
3334:A
3329:)
3326:x
3323:(
3305:)
3302:x
3299:(
3284:T
3276:x
3270:4
3266:d
3261:x
3256:4
3252:d
3242:!
3239:2
3232:2
3228:)
3224:e
3221:i
3218:(
3212:=
3207:)
3204:2
3201:(
3197:S
3183:S
3169:i
3167:x
3163:)
3160:i
3158:x
3156:(
3153:ψ
3143:i
3141:x
3137:)
3134:i
3132:x
3130:(
3128:ψ
3122:;
3119:i
3117:x
3103:)
3098:i
3094:x
3090:(
3081:A
3056:j
3054:x
3045:μ
3043:A
3039:ψ
3016:A
2987:g
2981:=
2976:v
2972:L
2945:A
2931:N
2913:,
2908:)
2903:j
2899:x
2895:(
2889:V
2883:L
2875:n
2870:1
2867:=
2864:j
2854:N
2849:)
2843:(
2838:A
2830:=
2826:)
2821:j
2817:x
2813:(
2807:V
2801:L
2793:n
2788:1
2785:=
2782:j
2772:T
2757:S
2748:S
2743:n
2722:.
2716:)
2713:n
2710:(
2706:S
2695:0
2692:=
2689:n
2677:}
2672:)
2667:j
2663:x
2659:(
2653:V
2647:L
2639:n
2634:1
2631:=
2628:j
2619:{
2613:T
2607:)
2601:j
2597:x
2591:4
2587:d
2578:n
2573:1
2570:=
2567:j
2558:(
2551:!
2548:n
2542:n
2538:i
2525:0
2522:=
2519:n
2511:=
2508:S
2494:V
2492:L
2475:.
2469:)
2466:n
2463:(
2459:S
2448:0
2445:=
2442:n
2430:}
2425:)
2420:j
2416:x
2412:(
2406:V
2400:H
2392:n
2387:1
2384:=
2381:j
2372:{
2366:T
2360:)
2354:j
2350:x
2344:4
2340:d
2331:n
2326:1
2323:=
2320:j
2311:(
2304:!
2301:n
2294:n
2290:)
2286:i
2280:(
2267:0
2264:=
2261:n
2253:=
2250:S
2225:T
2220:V
2218:H
2201:.
2196:)
2190:(
2185:V
2181:H
2174:d
2166:+
2150:i
2143:e
2137:T
2132:=
2129:S
2099:.
2095:)
2090:1
2086:t
2082:,
2077:2
2073:t
2069:(
2066:U
2050:1
2046:t
2032:+
2024:2
2020:t
2011:=
2008:S
1995:U
1986:S
1981:S
1964:,
1956:i
1951:|
1947:S
1943:|
1938:f
1931:=
1925:i
1922:f
1917:S
1774:t
1769:x
1662:d
1654:d
1340:e
1333:t
1326:v
205:S
194:S
77:g
75:(
65:q
59:–
53:γ
51:(
41:e
39:(
33:e
31:(
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