7016:
14753:
7975:
6532:
8487:
6312:
12758:
7547:
26036:
7011:{\displaystyle {\begin{aligned}K(x_{f},t_{f};x_{i},t_{i})&=Qe^{\frac {iS_{\text{c}}}{\hbar }}\prod _{j=1}^{\infty }{\frac {j\pi }{\sqrt {2}}}\int da_{j}\exp {\left({\frac {i}{2\hbar }}a_{j}^{2}{\frac {m}{2}}\left({\frac {(j\pi )^{2}}{t_{f}-t_{i}}}-\omega ^{2}(t_{f}-t_{i})\right)\right)}\\&=e^{\frac {iS_{\text{c}}}{\hbar }}Q\prod _{j=1}^{\infty }\left(1-\left({\frac {\omega (t_{f}-t_{i})}{j\pi }}\right)^{2}\right)^{-{\frac {1}{2}}}\end{aligned}}}
8051:
5922:
23552:, the "sum over histories" interpretation, the path integral is taken to be fundamental, and reality is viewed as a single indistinguishable "class" of paths that all share the same events. For this interpretation, it is crucial to understand what exactly an event is. The sum-over-histories method gives identical results to canonical quantum mechanics, and Sinha and Sorkin claim the interpretation explains the
12272:
7970:{\displaystyle {\begin{aligned}K(x_{f},t_{f};x_{i},t_{i})&=\left({\frac {m\omega }{2\pi i\hbar \sin \omega T}}\right)^{\frac {1}{2}}\exp {\left({\frac {i}{\hbar }}{\tfrac {1}{2}}m\omega {\frac {(x_{i}^{2}+x_{f}^{2})\cos \omega T-2x_{i}x_{f}}{\sin \omega T}}\right)}\\&=\sum _{n=0}^{\infty }\exp {\left(-{\frac {iE_{n}T}{\hbar }}\right)}\psi _{n}(x_{f})\psi _{n}(x_{i})^{*}~.\end{aligned}}}
19302:
7516:
3800:
8482:{\displaystyle K(x_{f},t_{f};x_{i},t_{i})=\left({\frac {m\omega }{\pi \hbar }}\right)^{\frac {1}{2}}e^{\frac {-i\omega T}{2}}\left(1-e^{-2i\omega T}\right)^{-{\frac {1}{2}}}\exp {\left(-{\frac {m\omega }{2\hbar }}\left(\left(x_{i}^{2}+x_{f}^{2}\right){\frac {1+e^{-2i\omega T}}{1-e^{-2i\omega T}}}-{\frac {4x_{i}x_{f}e^{-i\omega T}}{1-e^{-2i\omega T}}}\right)\right)}.}
20460:
15996:, this would not have been such a big problem – it would just have been a bad choice of coordinates. But the lack of symmetry means that the infinite quantities must be cut off, and the bad coordinates make it nearly impossible to cut off the theory without spoiling the symmetry. This makes it difficult to extract the physical predictions, which require a
9309:
21009:
2870:
6307:{\displaystyle {\begin{aligned}S_{\text{c}}&=\int _{t_{i}}^{t_{f}}{\mathcal {L}}\,dt=\int _{t_{i}}^{t_{f}}\left({\tfrac {1}{2}}m{\dot {x}}^{2}-{\tfrac {1}{2}}m\omega ^{2}x^{2}\right)\,dt\\&={\frac {1}{2}}m\omega \left({\frac {(x_{i}^{2}+x_{f}^{2})\cos \omega (t_{f}-t_{i})-2x_{i}x_{f}}{\sin \omega (t_{f}-t_{i})}}\right)~.\end{aligned}}}
10419:
13289:
12753:{\displaystyle \langle x_{f}|e^{-{\frac {i}{\hbar }}{\hat {H}}(t-t_{1})}F_{1}({\hat {x}})e^{-{\frac {i}{\hbar }}{\hat {H}}(t_{1}-t_{2})}F_{2}({\hat {x}})e^{-{\frac {i}{\hbar }}{\hat {H}}(t_{2})}|x_{i}\rangle =\int _{x(0)=x_{i}}^{x(t)=x_{f}}{\mathcal {D}}F_{1}(x(t_{1}))F_{2}(x(t_{2}))e^{{\frac {i}{\hbar }}\int dtL(x(t),{\dot {x}}(t))}}
10212:
4307:
21499:, but in the sense that the transformed value of the field at any given point under an infinitesimal transformation would only depend on the field configuration over an arbitrarily small neighborhood of the point in question). Let's also assume that the action is local in the sense that it is the integral over spacetime of a
19043:
6521:
7234:
5911:
14191:
Much of the study of quantum field theories from the path-integral perspective, in both the mathematics and physics literatures, is done in the
Euclidean setting, that is, after a Wick rotation. In particular, there are various results showing that if a Euclidean field theory with suitable properties
20734:
16003:
The problem of lost symmetry also appears in classical mechanics, where the
Hamiltonian formulation also superficially singles out time. The Lagrangian formulation makes the relativistic invariance apparent. In the same way, the path integral is manifestly relativistic. It reproduces the Schrödinger
5022:
in the numerator, the time slicing produces convolved
Gaussians, just as before. Now, however, the convolution product is marginally singular, since it requires careful limits to evaluate the oscillating integrals. To make the factors well defined, the easiest way is to add a small imaginary part to
4861:
The proportionality constant is not really determined by the time-slicing approach, only the ratio of values for different endpoint choices is determined. The proportionality constant should be chosen to ensure that between each two time slices the time evolution is quantum-mechanically unitary, but
2295:
possible paths of the system in between the initial and final states, including those that are absurd by classical standards. In calculating the probability amplitude for a single particle to go from one space-time coordinate to another, it is correct to include paths in which the particle describes
20114:
19674:
17199:
The integral above is not trivial to interpret because of the square root. Fortunately, there is a heuristic trick. The sum is over the relativistic arc length of the path of an oscillating quantity, and like the nonrelativistic path integral should be interpreted as slightly rotated into imaginary
16884:
It is also possible to reexpress the nonrelativistic time evolution in terms of propagators going toward the past, since the Schrödinger equation is time-reversible. The past propagator is the same as the future propagator except for the obvious difference that it vanishes in the future, and in the
1150:
The
Hamiltonian is a function of the position and momentum at one time, and it determines the position and momentum a little later. The Lagrangian is a function of the position now and the position a little later (or, equivalently for infinitesimal time separations, it is a function of the position
18668:
Unlike the nonrelativistic case, it is impossible to produce a relativistic theory of local particle propagation without including antiparticles. All local differential operators have inverses that are nonzero outside the light cone, meaning that it is impossible to keep a particle from travelling
16011:
Further, different choices of canonical variables lead to very different-seeming formulations of the same theory. The transformations between the variables can be very complicated, but the path integral makes them into reasonably straightforward changes of integration variables. For these reasons,
2238:
Feynman showed that Dirac's quantum action was, for most cases of interest, simply equal to the classical action, appropriately discretized. This means that the classical action is the phase acquired by quantum evolution between two fixed endpoints. He proposed to recover all of quantum mechanics
17033:
For a nonrelativistic theory, the time as measured along the path of a moving particle and the time as measured by an outside observer are the same. In relativity, this is no longer true. For a relativistic theory the propagator should be defined as the sum over all paths that travel between two
13685:
3293:
20272:
18347:
The second term has a nonrelativistic limit also, but this limit is concentrated on frequencies that are negative. The second pole is dominated by contributions from paths where the proper time and the coordinate time are ticking in an opposite sense, which means that the second term is to be
4865:
The result has a probability interpretation. The sum over all paths of the exponential factor can be seen as the sum over each path of the probability of selecting that path. The probability is the product over each segment of the probability of selecting that segment, so that each segment is
1088:
16015:
The price of a path integral representation is that the unitarity of a theory is no longer self-evident, but it can be proven by changing variables to some canonical representation. The path integral itself also deals with larger mathematical spaces than is usual, which requires more careful
18547:
12068:
14380:
9056:
22535:
18946:
17194:
12910:
4097:
20753:
8672:
8850:
2672:
9700:
and this is the Schrödinger equation. The normalization of the path integral needs to be fixed in exactly the same way as in the free particle case. An arbitrary continuous potential does not affect the normalization, although singular potentials require careful treatment.
11201:
9588:
3550:
18186:
3803:
10227:
17893:
12260:
14022:
3807:
3806:
3802:
3801:
5407:
14192:
can be constructed, one can then undo the Wick rotation to recover the physical, Lorentzian theory. On the other hand, it is much more difficult to give a meaning to path integrals (even
Euclidean path integrals) in quantum field theory than in quantum mechanics.
13093:
16016:
mathematics, not all of which has been fully worked out. The path integral historically was not immediately accepted, partly because it took many years to incorporate fermions properly. This required physicists to invent an entirely new mathematical object – the
3808:
19334:
Since this formulation of quantum mechanics is analogous to classical action principle, one might expect that identities concerning the action in classical mechanics would have quantum counterparts derivable from a functional integral. This is often the case.
22265:
20261:
17736:
10045:
18660:
So in the relativistic case, the
Feynman path-integral representation of the propagator includes paths going backwards in time, which describe antiparticles. The paths that contribute to the relativistic propagator go forward and backwards in time, and the
4108:
9695:
19297:{\displaystyle \operatorname {P} (B\mid A)={\frac {\sum _{F\subset A\cap B}\left|\int {\mathcal {D}}\varphi O_{\text{in}}e^{i{\mathcal {S}}}F\right|^{2}}{\sum _{F\subset A}\left|\int {\mathcal {D}}\varphi O_{\text{in}}e^{i{\mathcal {S}}}F\right|^{2}}},}
11398:
just set it equal to 1. In physics, such a quantity is "equal to 1 as an operator identity". In mathematics, it "weakly converges to 1". In either case, it is 1 in any expectation value, or when averaged over any interval, or for all practical purpose.
10030:
10528:
9044:
2368:
16027:
The integration variables in the path integral are subtly non-commuting. The value of the product of two field operators at what looks like the same point depends on how the two points are ordered in space and time. This makes some naive identities
23725:, the path integral formation can be used to determine an effective action and pre-exponential model to see the effect of dissipation on tunnelling. From this model, tunneling rates of macroscopic systems (at finite temperatures) can be predicted.
21115:
19926:
7223:
16424:
14496:, i.e., make time imaginary, and integrate over all possible beginning-ending configurations. The Wick-rotated path integral—described in the previous subsection, with the ordinary action replaced by its "Euclidean" counterpart—now resembles the
21887:
17347:
19768:
23079:
and the path integral formulation, an ordering prescription is required to resolve the ambiguity in the correspondence between non-commutative operators and the commutative functions that appear in path integrands. For example, the operator
17480:
11074:
16044:
The nonrelativistic formulation is traditionally given in terms of particle paths, not fields. There, the path integral in the usual variables, with fixed boundary conditions, gives the probability amplitude for a particle to go from point
23690:
17013:
5628:
3805:
12925:
from real to imaginary times. In the setting of quantum field theory, the Wick rotation changes the geometry of space-time from
Lorentzian to Euclidean; as a result, Wick-rotated path integrals are often called Euclidean path integrals.
11865:
16853:
5505:
17599:
7511:{\displaystyle K(x_{f},t_{f};x_{i},t_{i})=Qe^{\frac {iS_{\text{c}}}{\hbar }}{\sqrt {\frac {\omega (t_{f}-t_{i})}{\sin \omega (t_{f}-t_{i})}}}=e^{\frac {iS_{c}}{\hbar }}{\sqrt {\frac {m\omega }{2\pi i\hbar \sin \omega (t_{f}-t_{i})}}}.}
13529:
6327:
2015:
representation it is reinterpreted as a quantity associated to the path. In the limit that one takes a large power of this operator, one reconstructs the full quantum evolution between two states, the early one with a fixed value of
16040:
In relativistic theories, there is both a particle and field representation for every theory. The field representation is a sum over all field configurations, and the particle representation is a sum over different particle paths.
22861:
18335:
16209:
11302:
5697:
15882:
Both the Schrödinger and
Heisenberg approaches to quantum mechanics single out time and are not in the spirit of relativity. For example, the Heisenberg approach requires that scalar field operators obey the commutation relation
1265:
20567:
14153:
10958:
19937:
19561:
4451:
2536:
4665:
21250:
9415:
4572:
20455:{\displaystyle \langle F\rangle _{J}={\frac {\int {\mathcal {D}}\varphi Fe^{i\left({\mathcal {S}}+\langle J,\varphi \rangle \right)}}{\int {\mathcal {D}}\varphi e^{i\left({\mathcal {S}}+\langle J,\varphi \rangle \right)}}}.}
18961:
here is a concise way to represent the infinite-dimensional integral over all possible field configurations on all of space-time. As stated above, the unadorned path integral in the denominator ensures proper normalization.
13548:
14642:
10759:
3146:
124:
18380:
17356:
of the exponential of minus the length. This can be given a probability interpretation. The sum over all paths is a probability average over a path constructed step by step. The total number of steps is proportional to
14558:
Clearly, such a deep analogy between quantum mechanics and statistical mechanics cannot be dependent on the formulation. In the canonical formulation, one sees that the unitary evolution operator of a state is given by
23538:
23162:
9830:
9326:. In this case, to lowest order the potential energy is constant, and only the kinetic energy contribution is nontrivial. (This separation of the kinetic and potential energy terms in the exponent is essentially the
9304:{\displaystyle \psi (y;t+\varepsilon )=\int _{-\infty }^{\infty }\psi (x;t)\int _{x(t)=x}^{x(t+\varepsilon )=y}e^{i\int _{t}^{t+\varepsilon }{\bigl (}{\frac {1}{2}}{\dot {x}}^{2}-V(x){\bigr )}dt}Dx(t)\,dx\qquad (1)}
3135:
1104:
is the generator of time translations. This means that the state at a slightly later time differs from the state at the current time by the result of acting with the
Hamiltonian operator (multiplied by the negative
14206:
3794:
21004:{\displaystyle F\leftG=(-i)^{n}{\frac {\partial ^{k_{1}}}{\partial x_{1}^{k_{1}}}}{\frac {\delta }{\delta J(x_{1})}}\cdots {\frac {\partial ^{k_{n}}}{\partial x_{n}^{k_{n}}}}{\frac {\delta }{\delta J(x_{n})}}G.}
10836:
1432:
22376:
18821:
17040:
12770:
8926:
3930:
19399:
16531:, the propagator can be evaluated either explicitly from the path integral or by noting that the Schrödinger equation is a diffusion equation in imaginary time, and the solution must be a normalized Gaussian:
10544:, at the same moment in time, the manipulations above are only heuristic, because the multiplication rules for these quantities is just as noncommuting in the path integral as it is in the operator formalism.
7103:
3912:
18009:
5208:
2654:
17611:
is a weight factor, the relative importance of paths of different proper time. By the translation symmetry in proper time, this weight can only be an exponential factor and can be absorbed into the constant
9049:
The path integral reproduces the Schrödinger equation for the initial and final state even when a potential is present. This is easiest to see by taking a path-integral over infinitesimally separated times.
8515:
5013:
2865:{\displaystyle \int \limits _{-\infty }^{+\infty }\cdots \int \limits _{-\infty }^{+\infty }\exp \left({\frac {i}{\hbar }}\int _{t_{a}}^{t_{b}}L{\big (}x(t),v(t){\big )}\,dt\right)\,dx_{0}\,\cdots \,dx_{n},}
11901:
8706:
4856:
21491:
for the classical case? Does it have a quantum analog as well? Yes, but with a caveat. The functional measure would have to be invariant under the one parameter group of symmetry transformation as well.
21441:
18638:
16626:
10660:
1493:
1326:
7552:
6537:
5927:
5123:
4753:
10414:{\displaystyle \int \psi _{0}(x)\int _{u(0)=x}-\left(\int \left({\frac {d}{dt}}{\frac {\partial S}{\partial {\dot {u}}}}-{\frac {\partial S}{\partial u}}\right)\varepsilon (t)\,dt\right)e^{iS}\,Du\,}
11089:
9445:
3395:
21784:
18024:
11506:
25054:
Discusses the definition of Path
Integrals for systems whose kinematical variables are the generators of a real separable, connected Lie group with irreducible, square integrable representations.
23575:
Whereas in quantum mechanics the path integral formulation is fully equivalent to other formulations, it may be that it can be extended to quantum gravity, which would make it different from the
22365:
17775:
13879:
13284:{\displaystyle \psi (x,t)={\frac {1}{Z}}\int _{\mathbf {x} (0)=x}e^{-S_{\mathrm {Euclidean} }(\mathbf {x} ,{\dot {\mathbf {x} }})/\hbar }\psi _{0}(\mathbf {x} (t))\,{\mathcal {D}}\mathbf {x} \,}
22138:
5223:
18665:
of this is that the amplitude for a free particle to travel between two points includes amplitudes for the particle to fluctuate into an antiparticle, travel back in time, then forward again.
16858:
which is the reciprocal of the operator that annihilates the wavefunction in the Schrödinger equation, which wouldn't have come out right if the proportionality factor weren't constant in the
11734:
11318:
is a rapidly fluctuating statistical quantity, whose average value is 1, i.e. a normalized "Gaussian process". The fluctuations of such a quantity can be described by a statistical Lagrangian
21687:
19316:
is the superposition of all incoming states that could lead to the states we are interested in. In particular, this could be a state corresponding to the state of the Universe just after the
11649:
in Schrödinger quantum mechanics. One may, however, solve this problem by transforming the time-sliced flat-space path integral to curved space using a multivalued coordinate transformation (
22922:
18643:
which when factored, produces opposite-sign infinitesimal terms in each factor. This is the mathematically precise form of the relativistic particle propagator, free of any ambiguities. The
16720:
12076:
9717:
variables, but it is instructive to see this directly. The direct approach shows that the expectation values calculated from the path integral reproduce the usual ones of quantum mechanics.
9023:
the singularity is removed and a time-sliced approximation exists, which is exactly integrable, since it can be made harmonic by a simple coordinate transformation, as discovered in 1979 by
21330:
16004:
equation, the Heisenberg equations of motion, and the canonical commutation relations and shows that they are compatible with relativity. It extends the Heisenberg-type operator algebra to
11381:
21578:
10207:{\displaystyle \int \psi _{0}(x)\int _{u(0)=x}\left(\int {\frac {\partial S}{\partial u}}\varepsilon +{\frac {\partial S}{\partial {\dot {u}}}}{\dot {\varepsilon }}\,dt\right)e^{iS}\,Du\,}
2132:
will vary periodically with a very high frequency about the value zero, as a result of which its integral will be practically zero. The only important part in the domain of integration of
23044:
22149:
20125:
17622:
22061:
16522:
14716:
14433:
4302:{\displaystyle K(x-y;T)=\int _{x(0)=x}^{x(T)=y}\prod _{t}\exp \left(-{\tfrac {1}{2}}\left({\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}\right)^{2}\varepsilon \right)\,{\mathcal {D}}x,}
3582:
9018:
1825:
14474:
13346:
13031:
1889:
23244:
23203:
13081:
9603:
23454:
23413:
23349:
23308:
19485:
18732:
Much of the formal study of QFT is devoted to the properties of the resulting functional integral, and much effort (not yet entirely successful) has been made toward making these
4931:
21952:
9905:
1988:
15975:
10441:
13864:
3804:
3641:
3381:
Thus, in contrast to classical mechanics, not only does the stationary path contribute, but actually all virtual paths between the initial and the final point also contribute.
21023:
19829:
13719:
10584:
do not commute. In the path integral, these are just integration variables and they have no obvious ordering. Feynman discovered that the non-commutativity is still present.
7118:
18706:
is itself a function of space and time, and the square brackets are a reminder that the action depends on all the field's values everywhere, not just some particular value.
16242:
21795:
1656:
17225:
14721:
which is precisely the partition function of statistical mechanics for the same system at the temperature quoted earlier. One aspect of this equivalence was also known to
13827:
3604:
2161:. We can apply this argument to each of the variables of integration ... and obtain the result that the only important part in the domain of integration is that for which
1714:
19693:
9031:. The combination of a path-dependent time transformation and a coordinate transformation is an important tool to solve many path integrals and is called generically the
17375:
14186:
14061:
11646:
1358:
18681:
application to quantum field theory, in which the "paths" or histories being considered are not the motions of a single particle, but the possible time evolutions of a
10964:
13805:
13750:
23836:
18767:) makes the functional integral even more like a statistical partition function and also tames some of the mathematical difficulties of working with these integrals.
11591:
1909:
factor contains all the dynamical information – it pushes the state forward in time. The first part and the last part are just Fourier transforms to change to a pure
975:
is obscure in the formulation. The path-integral approach has proven to be equivalent to the other formalisms of quantum mechanics and quantum field theory. Thus, by
23616:
19530:
19320:, although for actual calculation this can be simplified using heuristic methods. Since this expression is a quotient of path integrals, it is naturally normalised.
16902:
5531:
11748:
16750:
11645:
For a particle in curved space the kinetic term depends on the position, and the above time slicing cannot be applied, this being a manifestation of the notorious
6516:{\displaystyle S=S_{\text{c}}+\sum _{n=1}^{\infty }{\tfrac {1}{2}}a_{n}^{2}{\frac {m}{2}}\left({\frac {(n\pi )^{2}}{t_{f}-t_{i}}}-\omega ^{2}(t_{f}-t_{i})\right).}
5429:
17499:
12978:
10560:, which is now reinterpreted as the condition for constructive interference. This can be shown using the method of stationary phase applied to the propagator. As
23922:
23372:
23267:
13354:
11635:
13778:
12952:
10587:
To see this, consider the simplest path integral, the brownian walk. This is not yet quantum mechanics, so in the path-integral the action is not multiplied by
5906:{\displaystyle x_{\text{c}}(t)=x_{i}{\frac {\sin \omega (t_{f}-t)}{\sin \omega (t_{f}-t_{i})}}+x_{f}{\frac {\sin \omega (t-t_{i})}{\sin \omega (t_{f}-t_{i})}}.}
3681:
3661:
2231:
Dirac's work did not provide a precise prescription to calculate the sum over paths, and he did not show that one could recover the Schrödinger equation or the
18798:
11650:
20729:{\displaystyle F={\frac {\partial ^{k_{1}}}{\partial x_{1}^{k_{1}}}}\varphi (x_{1})\cdots {\frac {\partial ^{k_{n}}}{\partial x_{n}^{k_{n}}}}\varphi (x_{n}),}
8936:
Feynman's time-sliced approximation does not, however, exist for the most important quantum-mechanical path integrals of atoms, due to the singularity of the
26280:
22546:
20109:{\displaystyle {\frac {\delta ^{n}Z}{\delta J(x_{1})\cdots \delta J(x_{n})}}=i^{n}\,Z\,\left\langle \varphi (x_{1})\cdots \varphi (x_{n})\right\rangle _{J},}
19669:{\displaystyle \left\langle {\frac {\delta F}{\delta \varphi }}\right\rangle =-i\left\langle F{\frac {\delta {\mathcal {S}}}{\delta \varphi }}\right\rangle }
19408:; the equation means that the action is stationary under small changes in the field configuration). The quantum analogues of these equations are called the
18227:
17030:
in Green's function can either be the energy if the paths are going toward the future, or the negative of the energy if the paths are going toward the past.
16063:
11212:
3924:
in the exponential, so that large deviations are suppressed by small numbers, not by cancelling oscillatory contributions. The amplitude (or Kernel) reads:
1176:
19416:
14069:
10854:
9597:
is diffusive from the free particle propagation, with an extra infinitesimal rotation in phase that slowly varies from point to point from the potential:
5128:
which, with the same normalization as before (not the sum-squares normalization – this function has a divergent norm), obeys a free Schrödinger equation:
4367:
2418:
14476:, the path of minimum action dominates the integral, because the phase of any path away from this fluctuates rapidly and different contributions cancel.
13680:{\displaystyle {\frac {1}{Z}}\int _{\mathbf {x} (0)=x}f(\mathbf {x} )e^{-{\frac {m}{2}}\int |{\dot {\mathbf {x} }}|^{2}dt}\,{\mathcal {D}}\mathbf {x} \,}
13539:
is from the context of quantum field theory, where the change from real to imaginary time changes the space-time geometry from Lorentzian to Euclidean.)
4587:
21126:
13721:
includes all the remaining dependence of the integrand on the path. This integral has a rigorous mathematical interpretation as integration against the
9336:
4484:
3288:{\displaystyle \exp \left({\frac {i}{\hbar }}\varepsilon \sum _{j=1}^{n+1}L\left({\tilde {x}}_{j},{\frac {x_{j}-x_{j-1}}{\varepsilon }},j\right)\right)}
25265:
19805:
24502:
This course, designed for mathematicians, is a rigorous introduction to perturbative quantum field theory, using the language of functional integrals.
1166:
In quantum mechanics, the Legendre transform is hard to interpret, because the motion is not over a definite trajectory. In classical mechanics, with
26341:
21470:
region, for example inside a double light-cone. This gives a more mathematically precise and physically rigorous definition of quantum field theory.
18542:{\displaystyle K(p)={\frac {i}{p_{0}-{\sqrt {{\vec {p}}^{2}+m^{2}}}+i\varepsilon }}+{\frac {i}{p_{0}-{\sqrt {{\vec {p}}^{2}+m^{2}}}-i\varepsilon }}.}
15002:
14565:
10682:
1147:. So the Hamiltonian is different in different frames, and this type of symmetry is not apparent in the original formulation of quantum mechanics.
54:
23567:
have attempted to make more rigorous the notion of extracting a classical-like "coarse-grained" history from the space of all possible histories.
25833:
14375:{\displaystyle Z=\int e^{\frac {i{\mathcal {S}}}{\hbar }}\,{\mathcal {D}}\mathbf {x} \quad {\text{where }}{\mathcal {S}}=\int _{0}^{t_{f}}L\,dt}
979:
either approach from the other, problems associated with one or the other approach (as exemplified by Lorentz covariance or unitarity) go away.
23459:
23083:
21361:
15992:
covariant, but the symmetry is not apparent in intermediate stages. If naive field-theory calculations did not produce infinite answers in the
15869:
9726:
16896:. In this case, the interpretation is that these are the quantities to convolve the final wavefunction so as to get the initial wavefunction:
3024:
1940:
to be expressed as a simple function along each path. This function is the quantum analog of the classical action. This observation is due to
25466:
23076:
18669:
faster than light. Such a particle cannot have a Green's function that is only nonzero in the future in a relativistically invariant theory.
390:
22530:{\displaystyle \langle q(x)\rangle +i\langle Fq(x)\rangle =\langle q(x)\rangle +i\left\langle F\partial _{\mu }j^{\mu }(x)\right\rangle =0.}
18941:{\displaystyle \langle F\rangle ={\frac {\int {\mathcal {D}}\varphi Fe^{i{\mathcal {S}}}}{\int {\mathcal {D}}\varphi e^{i{\mathcal {S}}}}}.}
17189:{\displaystyle K(x-y,\mathrm {T} )=\int _{x(0)=x}^{x(\mathrm {T} )=y}e^{i\int _{0}^{\mathrm {T} }{\sqrt {{\dot {x}}^{2}}}-\alpha \,d\tau }.}
12905:{\displaystyle \langle F\rangle ={\frac {\int {\mathcal {D}}F(\phi )e^{{\frac {i}{\hbar }}S}}{\int {\mathcal {D}}e^{{\frac {i}{\hbar }}S}}}}
4092:{\displaystyle K(x-y;T)=\int _{x(0)=x}^{x(T)=y}\exp \left(-\int _{0}^{T}{\frac {{\dot {x}}^{2}}{2}}\,\mathrm {d} t\right)\,{\mathcal {D}}x.}
3689:
14159:
Although the Wick-rotated Schrödinger equation does not have a direct physical meaning, interesting properties of the Schrödinger operator
5516:
10781:
10564:
decreases, the exponential in the integral oscillates rapidly in the complex domain for any change in the action. Thus, in the limit that
1381:
26214:
23735:
14922:
8861:
8667:{\displaystyle K(x_{f},t_{f};x_{i},t_{i})=\left({\frac {m\omega }{\pi \hbar }}\right)^{\frac {1}{2}}e^{\frac {-i\omega T}{2}}\cdot R(T).}
891:
22274:
and its first partial derivatives! More general Lagrangians would require a modification to this definition!). We're not insisting that
21598:
If we don't assume any special boundary conditions, this would not be a "true" symmetry in the true sense of the term in general unless
19348:
10556:
by many orders of magnitude, we typically have destructive interference other than in the vicinity of those trajectories satisfying the
9709:
Since the states obey the Schrödinger equation, the path integral must reproduce the Heisenberg equations of motion for the averages of
7027:
3853:
26330:
26236:
18740:
17920:
14497:
12063:{\displaystyle \langle x_{f}|e^{-{\frac {i}{\hbar }}{\hat {H}}(t-t')}F({\hat {x}})e^{-{\frac {i}{\hbar }}{\hat {H}}(t')}|x_{i}\rangle }
8845:{\displaystyle e^{\frac {-i\omega T}{2}}e^{-in\omega T}=e^{-i\omega T\left({\frac {1}{2}}+n\right)}\quad {\text{for }}n=0,1,2,\ldots .}
5134:
2584:
2327:
when the Hamiltonian is at most quadratic in the momentum. An amplitude computed according to Feynman's principles will also obey the
26004:
24496:
16729:-space the proportionality factor here is constant in time, as will be verified in a moment. The Fourier transform in time, extending
4942:
2979:(more precisely, since one must work with a continuous spectrum, the respective densities) to find the quantum mechanical particle at
2194:, which is just the action function, which classical mechanics requires to be stationary for small variations in all the intermediate
26072:
23602:
can be modeled by using the path integral formation to determine the action of the trajectory through a potential barrier. Using the
18752:
14962:
18348:
interpreted as the antiparticle. The nonrelativistic analysis shows that with this form the antiparticle still has positive energy.
17034:
points in a fixed proper time, as measured along the path (these paths describe the trajectory of a particle in space and in time):
25315:
18208:
space just means multiplying by the propagator, the second term is suppressed and the first term is enhanced. For frequencies near
4768:
2373:
One common approach to deriving the path integral formula is to divide the time interval into small pieces. Once this is done, the
142:
30:
This article is about a formulation of quantum mechanics. For integrals along a path, also known as line or contour integrals, see
23892:
21387:
18567:
18191:
For states where one nonrelativistic particle is present, the initial wavefunction has a frequency distribution concentrated near
16537:
10841:
This shows that the random walk is not differentiable, since the ratio that defines the derivative diverges with probability one.
10597:
1443:
1276:
971:(this is related to conservation of probability; the probabilities of all physically possible outcomes must add up to one) of the
26182:
26016:
23553:
18662:
5037:
4676:
4478:. The result is easy to evaluate by taking the Fourier transform of both sides, so that the convolutions become multiplications:
1091:
These are five of the infinitely many paths available for a particle to move from point A at time t to point B at time t’(>t).
939:(time and space components of quantities enter equations in the same way) is easier to achieve than in the operator formalism of
598:
11196:{\displaystyle -={\frac {{\big (}x(t+\varepsilon )-x(t){\big )}^{2}}{\varepsilon }}\approx {\frac {\varepsilon }{\varepsilon }}}
9583:{\displaystyle \psi (y;t+\varepsilon )\approx \int \psi (x;t)e^{-i\varepsilon V(x)}e^{\frac {i(x-y)^{2}}{2\varepsilon }}\,dx\,.}
3545:{\displaystyle \psi (x,t)={\frac {1}{Z}}\int _{\mathbf {x} (0)=x}{\mathcal {D}}\mathbf {x} \,e^{iS}\psi _{0}(\mathbf {x} (t))\,}
25700:
25258:
18181:{\displaystyle 2p_{0}K(p)={\frac {i}{p_{0}-{\sqrt {{\vec {p}}^{2}+m^{2}}}}}+{\frac {i}{p_{0}+{\sqrt {{\vec {p}}^{2}+m^{2}}}}}.}
10435:. The conclusion is that this first order variation is zero for an arbitrary initial state and at any arbitrary point in time:
9846:
at each separate time is a separate integration variable. So it is legitimate to change variables in the integral by shifting:
371:
19555:(after a Wick rotation, followed by a Wick rotation back) to get the following Schwinger–Dyson equations for the expectation:
14658:. If one makes a Wick rotation here, and finds the amplitude to go from any state, back to the same state in (imaginary) time
26225:
25634:
25182:
25159:
25134:
25047:
24981:
24959:
24939:
24916:
24888:
24854:
24830:
24808:
24789:
24770:
24743:
24693:
24674:
24629:
24599:
24537:
24362:
18552:
Without these terms, the pole contribution could not be unambiguously evaluated when taking the inverse Fourier transform of
17888:{\displaystyle K(p)=\int _{0}^{\infty }e^{-\mathrm {T} p^{2}-\mathrm {T} \alpha }\,d\mathrm {T} ={\frac {1}{p^{2}+\alpha }},}
16865:
The infinitesimal term in the denominator is a small positive number, which guarantees that the inverse Fourier transform in
9431:
locally by an amount proportional to the potential energy. The second term is the free particle propagator, corresponding to
951:
descriptions of the same quantum system. Another advantage is that it is in practice easier to guess the correct form of the
554:
17:
23843:
21711:
14017:{\displaystyle \psi (x,t)=\int e^{-\int V(\mathbf {x} (t))\,dt/\hbar }\,\psi _{0}(\mathbf {x} (t))\,d\mu _{x}(\mathbf {x} )}
11412:
4862:
a more illuminating way to fix the normalization is to consider the path integral as a description of a stochastic process.
25557:
5402:{\displaystyle \psi _{t}(y)=\int \psi _{0}(x)K(x-y;t)\,dx=\int \psi _{0}(x)\int _{x(0)=x}^{x(t)=y}e^{iS}\,{\mathcal {D}}x,}
477:
22303:
15727:
26105:
23755:
23549:
18351:
The proper way to express this mathematically is that, adding a small suppression factor in proper time, the limit where
1073:
22072:
12255:{\displaystyle \int _{x(0)=x_{i}}^{x(t)=x_{f}}{\mathcal {D}}F(x(t'))e^{{\frac {i}{\hbar }}\int dtL(x(t),{\dot {x}}(t))}}
8855:
Comparison to the above eigenstate expansion yields the standard energy spectrum for the simple harmonic oscillator,
26269:
25204:
24651:
15862:
11671:
1041:
also declaring it to be the "fundamental mathematical construct of modern quantum mechanics and quantum field theory".
21634:
11665:
Sometimes (e.g. a particle moving in curved space) we also have measure-theoretic factors in the functional integral:
11596:
and in quantum mechanics, the extra imaginary unit in the action converts this to the canonical commutation relation,
25288:
25251:
24253:
23765:
22882:
16869:
will be nonzero only for future times. For past times, the inverse Fourier transform contour closes toward values of
16649:
10424:
But this was just a shift of integration variables, which doesn't change the value of the integral for any choice of
6317:
Next, expand the deviation from the classical path as a Fourier series, and calculate the contribution to the action
137:
22260:{\displaystyle j^{\mu }(x)=f^{\mu }(x)-{\frac {\partial }{\partial (\partial _{\mu }\varphi )}}{\mathcal {L}}(x)Q\,}
21462:
The path integrals are usually thought of as being the sum of all paths through an infinite space–time. However, in
21261:
20256:{\displaystyle Z^{,i_{1}\cdots i_{n}}=i^{n}Z\left\langle \varphi ^{i_{1}}\cdots \varphi ^{i_{n}}\right\rangle _{J},}
17731:{\displaystyle K(x-y)=\int _{0}^{\infty }e^{-{\frac {(x-y)^{2}}{\mathrm {T} }}-\alpha \mathrm {T} }\,d\mathrm {T} .}
11324:
3812:
The diagram shows the contribution to the path integral of a free particle for a set of paths, eventually drawing a
25398:
21509:
15492:
9032:
2011:
representation the quantity that is being summed over the intermediate states is an obscure matrix element, in the
226:
24607:
The historical reference written by the inventor of the path integral formulation himself and one of his students.
26426:
26354:
26192:
25971:
25680:
25675:
25383:
22933:
18374:
slightly, so that the inverse Fourier transform will pick up a small decay factor in one of the time directions:
14765:
4873:
The probability interpretation gives a natural normalization choice. The path integral should be defined so that
4866:
probabilistically independently chosen. The fact that the answer is a Gaussian spreading linearly in time is the
2332:
2317:
1101:
1081:
964:
884:
333:
313:
181:
21963:
16466:
14668:
14409:
13535:
Note the sign change between this and the normal action, where the potential energy term is negative. (The term
3558:
26411:
25983:
25655:
24299:
23584:
9690:{\displaystyle {\frac {\partial \psi }{\partial t}}=i\cdot \left({\tfrac {1}{2}}\nabla ^{2}-V(x)\right)\psi \,}
8971:
2384:
paths, which in one dimension is a product of ordinary integrals. For the motion of the particle from position
2232:
1776:
573:
303:
21335:
If the functional measure is not translationally invariant, it might be possible to express it as the product
14438:
13298:
12983:
1843:
26431:
26421:
26203:
26065:
25961:
25738:
25660:
25532:
25428:
25298:
24992:
23208:
23167:
21610:
15855:
15040:
14879:
14839:
13036:
9314:
Since the time separation is infinitesimal and the cancelling oscillations become severe for large values of
995:
613:
351:
251:
10025:{\displaystyle \int \psi _{0}(x)\int _{u(0)=x}e^{iS(u+\varepsilon ,{\dot {u}}+{\dot {\varepsilon }})}\,Du\,}
26406:
26115:
25695:
25629:
25624:
25595:
25293:
19409:
19329:
17765:
contribution is a Gaussian, gives whose Fourier transform is another Gaussian with reciprocal width. So in
15597:
15522:
14869:
10523:{\displaystyle \left\langle \psi _{0}\left|{\frac {\delta S}{\delta x}}(t)\right|\psi _{0}\right\rangle =0}
2042:
Crucially, Dirac identified the effect of the classical limit on the quantum form of the action principle:
549:
544:
515:
366:
147:
23418:
23377:
23313:
23272:
21495:
Let's just assume for simplicity here that the symmetry in question is local (not local in the sense of a
19444:
17742:
4879:
3389:
In terms of the wave function in the position representation, the path integral formula reads as follows:
25763:
25670:
21905:
21479:
21110:{\displaystyle \left\langle {\frac {\delta {\mathcal {S}}}{\delta \varphi (x)}}+J(x)\right\rangle _{J}=0}
19921:{\displaystyle Z=\int {\mathcal {D}}\varphi e^{i\left({\mathcal {S}}+\langle J,\varphi \rangle \right)}.}
19339:
15035:
14972:
10557:
1957:
1156:
1064:, whose 1933 paper gave birth to path integral formulation. The complete method was developed in 1948 by
920:. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or
913:
583:
328:
318:
21360:
is a translationally invariant measure. This is true, for example, for nonlinear sigma models where the
18729:. In principle, one integrates Feynman's amplitude over the class of all possible field configurations.
15889:
7218:{\displaystyle \prod _{j=1}^{\infty }\left(1-{\frac {x^{2}}{j^{2}}}\right)={\frac {\sin \pi x}{\pi x}},}
1719:
and if this is also interpreted as a matrix multiplication, the sum over all states integrates over all
26416:
26150:
26039:
25801:
25609:
25580:
25126:
21463:
18014:
This expression can be interpreted in the nonrelativistic limit, where it is convenient to split it by
16419:{\displaystyle \psi _{T}(y)=\int _{x}\psi _{0}(x)K(x,y;T)\,dx=\int ^{x(T)=y}\psi _{0}(x(0))e^{iS}\,Dx.}
16005:
14874:
14200:
The path integral is just the generalization of the integral above to all quantum mechanical problems—
13832:
3609:
877:
529:
500:
22297:
is. And we also assume the even stronger assumption that the functional measure is locally invariant:
21882:{\displaystyle \langle Q\rangle +i\left\langle F\int _{\partial V}f^{\mu }\,ds_{\mu }\right\rangle =0}
20477:
is viewed as a functional distribution (this shouldn't be taken too literally as an interpretation of
15707:
1537:. So consider two states separated in time and act with the operator corresponding to the Lagrangian:
955:
of a theory, which naturally enters the path integrals (for interactions of a certain type, these are
26247:
26135:
26110:
25823:
25690:
25614:
25575:
25512:
25448:
25341:
21500:
17342:{\displaystyle K(x-y,\mathrm {T} )=e^{-\alpha \mathrm {T} }\int _{x(0)=x}^{x(\mathrm {T} )=y}e^{-L}.}
15025:
14967:
13757:
13693:
1128:
1077:
1072:. The original motivation stemmed from the desire to obtain a quantum-mechanical formulation for the
1057:
952:
534:
495:
448:
346:
206:
19763:{\displaystyle \left\langle F_{,i}\right\rangle =-i\left\langle F{\mathcal {S}}_{,i}\right\rangle .}
2219:– the classical limit – the path integral is dominated by solutions that are in the neighborhood of
26155:
26058:
25890:
25870:
25860:
25850:
25806:
25366:
25058:
23060:
21374:
is some topologically nontrivial space, the concept of a translation does not even make any sense.
19809:
18794:
18784:
18686:
17475:{\displaystyle K(x-y,\mathrm {T} )=e^{-\alpha \mathrm {T} }e^{-{\frac {(x-y)^{2}}{\mathrm {T} }}}.}
15527:
15152:
15045:
14982:
14942:
14552:
2346:) as a weighted sum of all possible histories of the system from the initial to the final state. A
2300:, curves in which the particle shoots off into outer space and flies back again, and so forth. The
1543:
991:
618:
24372:
17485:
The usual definition of the relativistic propagator only asks for the amplitude is to travel from
15517:
14729:
after Wick rotation. Note, however, that the Euclidean path integral is actually in the form of a
13810:
11654:
11069:{\displaystyle =x{\frac {dx}{dt}}=x(t+\varepsilon ){\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}}
3587:
1671:
26380:
26125:
26100:
25585:
25496:
25192:
23760:
21696:. This property is called the invariance of the measure, and this does not hold in general. (See
18983:
16021:
15807:
15267:
15075:
15070:
14912:
13870:
10568:
goes to zero, only points where the classical action does not vary contribute to the propagator.
2945:, which, apart from a nonessential factor, is directly the product of the probability amplitudes
2377:
tells us that the noncommutativity of the kinetic and potential energy operators can be ignored.
1152:
1135:. The Hamiltonian indicates how to march forward in time, but the time is different in different
1121:
787:
505:
413:
25999:
25547:
24169:
18365:
limit of the second term must vanish. In the Fourier transform, this means shifting the pole in
16881:. The infinitesimal term can be interpreted as an infinitesimal rotation toward imaginary time.
14162:
14031:
4936:
This condition normalizes the Gaussian and produces a kernel that obeys the diffusion equation:
2328:
1334:
999:
463:
361:
127:
26120:
25910:
25685:
25665:
25590:
25443:
24451:
24443:
23685:{\displaystyle \Gamma =A_{\mathrm {o} }\exp \left(-{\frac {S_{\mathrm {eff} }}{\hbar }}\right)}
20541:
19540:
19428:
17008:{\displaystyle G_{\text{B}}(p,E)={\frac {-i}{-E-{\frac {i{\vec {p}}^{2}}{2m}}+i\varepsilon }}.}
15677:
15657:
15147:
15080:
14977:
14937:
10576:
The formulation of the path integral does not make it clear at first sight that the quantities
9024:
5623:{\displaystyle {\mathcal {L}}={\tfrac {1}{2}}m{\dot {x}}^{2}-{\tfrac {1}{2}}m\omega ^{2}x^{2}.}
2324:
2316:. Contributions from paths wildly different from the classical trajectory may be suppressed by
1069:
1034:
940:
792:
510:
338:
308:
271:
25486:
13783:
13728:
11860:{\displaystyle S=\int \left({\frac {m}{2}}g_{ij}{\dot {x}}^{i}{\dot {x}}^{j}-V(x)\right)\,dt,}
9045:
Relation between Schrödinger's equation and the path integral formulation of quantum mechanics
4870:, which can be interpreted as the first historical evaluation of a statistical path integral.
2369:
Relation between Schrödinger's equation and the path integral formulation of quantum mechanics
2287:
In order to find the overall probability amplitude for a given process, then, one adds up, or
418:
26258:
26145:
25935:
25433:
25413:
24267:
23064:
22876:, we can forget about all the boundary conditions and locality assumptions. We'd simply have
20482:
19813:
19432:
19405:
18760:
18744:
17362:
16848:{\displaystyle G_{\text{F}}(p,E)={\frac {-i}{E-{\frac {{\vec {p}}^{2}}{2m}}+i\varepsilon }},}
15562:
15417:
15113:
15050:
14957:
14819:
14480:
11528:
5500:{\displaystyle i{\frac {\partial }{\partial t}}\psi _{t}=-{\frac {\nabla ^{2}}{2}}\psi _{t}.}
4867:
2255:
1510:
1007:
948:
925:
261:
246:
24350:
19496:
19431:(we'll assume this for the rest of this article, although this does not hold for, let's say
17594:{\displaystyle K(x-y)=\int _{0}^{\infty }K(x-y,\mathrm {T} )W(\mathrm {T} )\,d\mathrm {T} ,}
26359:
26160:
26140:
25966:
25895:
25840:
25570:
25378:
25336:
25074:
25007:
24723:
24566:
24551:
24517:
24466:
24404:
24384:
24321:
24279:
24140:
23740:
23557:
23067:. In fact, renormalization is the major obstruction to making path integrals well-defined.
21488:
20518:
20478:
19552:
18776:
15812:
15587:
15118:
14952:
14744:
13524:{\displaystyle S_{\mathrm {Euclidean} }(\mathbf {x} ,{\dot {\mathbf {x} }})=\int \left\,dt}
3015:
2343:
2339:
2280:
1068:. Some preliminaries were worked out earlier in his doctoral work under the supervision of
1030:
987:
986:
processes, and this provided the basis for the grand synthesis of the 1970s, which unified
578:
490:
216:
173:
21892:
where the integral is over the boundary. This is the quantum analog of Noether's theorem.
16744:
to be zero for negative times, gives Green's function, or the frequency-space propagator:
14864:
14722:
14483:
follows. Considering only paths that begin and end in the same configuration, perform the
822:
8:
25951:
25920:
25865:
25845:
25753:
25710:
25565:
25476:
25403:
25393:
25305:
23075:
Regardless of whether one works in configuration space or phase space, when equating the
21015:
18733:
18341:
15822:
15142:
12957:
11516:
9327:
2942:
2374:
1117:
932:
921:
917:
677:
485:
403:
231:
211:
163:
25226:
25078:
25035:
25011:
24727:
24570:
24521:
24470:
24388:
24325:
24283:
24144:
23786:
23354:
23249:
15532:
15257:
11602:
1120:
between frequency and energy, and the general relation is consistent with that plus the
26436:
26335:
26011:
25880:
25778:
25471:
25418:
25310:
25097:
25062:
25023:
24928:
24865:
24846:
Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets
24749:
24704:
24663:
24588:
24408:
24337:
24311:
23980:
23871:
23714:
23599:
22856:{\displaystyle q(x)\leftZ+J(x)Q\leftZ=\partial _{\mu }j^{\mu }(x)\leftZ+J(x)Q\leftZ=0.}
18330:{\displaystyle 2mK_{\text{NR}}(p)={\frac {i}{(p_{0}-m)-{\frac {{\vec {p}}^{2}}{2m}}}}.}
16204:{\displaystyle K(x,y;T)=\langle y;T\mid x;0\rangle =\int _{x(0)=x}^{x(T)=y}e^{iS}\,Dx.}
16017:
15832:
15757:
15687:
15567:
15552:
15482:
15442:
15432:
15392:
15307:
15197:
15157:
14927:
14889:
14844:
14832:
14809:
14804:
14726:
14501:
13763:
13087:.) If we repeat the derivation of the path-integral formula in this setting, we obtain
12937:
11297:{\displaystyle f(t)={\frac {{\big (}x(t+\varepsilon )-x(t){\big )}^{2}}{\varepsilon }}}
3666:
3646:
1136:
1132:
1003:
936:
398:
323:
256:
168:
1260:{\displaystyle \varepsilon H=p(t){\big (}q(t+\varepsilon )-q(t){\big )}-\varepsilon L}
26310:
26165:
26021:
25930:
25900:
25828:
25791:
25786:
25768:
25733:
25723:
25527:
25423:
25388:
25371:
25274:
25200:
25178:
25155:
25130:
25119:
25102:
25063:"The correspondence principle in the statistical interpretation of quantum mechanics"
25043:
25027:
24977:
24955:
24935:
24912:
24884:
24850:
24826:
24804:
24785:
24766:
24753:
24739:
24689:
24670:
24647:
24625:
24595:
24533:
24478:
24412:
24358:
24341:
24291:
24249:
23923:"Finally We May Have a Path to the Fundamental Theory of Physics… and It's Beautiful"
23718:
23603:
23579:
model. Feynman had some success in this direction, and his work has been extended by
21698:
21618:
20526:
18015:
16029:
15988:, and this is not a relativistically invariant concept. The results of a calculation
15787:
15717:
15602:
15592:
15437:
15372:
15342:
15312:
15172:
15167:
14997:
14917:
14907:
14799:
14148:{\displaystyle \hbar {\frac {\partial }{\partial t}}\psi (x,t)=-{\hat {H}}\psi (x,t)}
10953:{\displaystyle =x{\frac {dx}{dt}}=x(t){\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}}
10676:
is fluctuating, and the derivative is defined as the limit of a discrete difference.
8937:
5517:
Propagator § Basic examples: propagator of free particle and harmonic oscillator
1731:
909:
832:
807:
747:
742:
642:
608:
588:
186:
45:
25221:
24424:
24024:
16020:– which also allowed changes of variables to be done naturally, as well as allowing
15502:
4446:{\displaystyle K(x-y;T)=G_{\varepsilon }*G_{\varepsilon }*\cdots *G_{\varepsilon },}
2531:{\displaystyle t_{a}=t_{0}<t_{1}<\cdots <t_{n-1}<t_{n}<t_{n+1}=t_{b}}
2114:, as varying continuously while the other ones are fixed. Owing to the smallness of
1920:
Another way of saying this is that since the Hamiltonian is naturally a function of
25748:
25743:
25600:
25481:
25170:
25092:
25082:
25015:
24731:
24639:
24574:
24525:
24474:
24392:
24329:
24287:
24148:
23745:
18780:
18657:, which in the Minkowski version is a small exponential suppression of long paths.
16877:
propagates the particle into the future and is the reason for the subscript "F" on
15827:
15742:
15722:
15692:
15642:
15462:
15452:
15447:
15332:
15317:
15247:
14932:
14859:
14849:
14784:
14386:
4660:{\displaystyle {\tilde {G}}_{\varepsilon }(p)=e^{-{\frac {\varepsilon p^{2}}{2}}},}
2272:
2220:
2198:
s. This shows the way in which equation (11) goes over into classical results when
1160:
837:
827:
817:
717:
697:
682:
652:
520:
408:
21245:{\displaystyle {\frac {\delta {\mathcal {S}}}{\delta \varphi (x)}}\leftZ+J(x)Z=0,}
11079:
In elementary calculus, the two are only different by an amount that goes to 0 as
9410:{\displaystyle e^{-i\varepsilon V(x)}e^{i{\frac {{\dot {x}}^{2}}{2}}\varepsilon }}
4567:{\displaystyle {\tilde {K}}(p;T)={\tilde {G}}_{\varepsilon }(p)^{T/\varepsilon }.}
4316:
is interpreted as a finite collection of integrations at each integer multiple of
2258:
is given by adding together the contributions of all paths in configuration space.
26095:
26081:
25978:
25905:
25885:
25855:
25818:
25813:
25718:
25542:
25148:
25114:
24818:
24617:
24545:
The 1942 thesis. Also includes Dirac's 1933 paper and Feynman's 1948 publication.
24507:
24400:
24170:"The sum over all possibilities: The path integral formulation of quantum theory"
23918:
23897:
23722:
23580:
22290:
21371:
19684:
18682:
17365:, the result of many independent steps is a Gaussian of variance proportional to
15997:
15993:
15772:
15767:
15737:
15732:
15702:
15662:
15632:
15612:
15512:
15402:
15397:
15362:
15347:
15322:
15302:
15282:
15212:
15202:
15123:
15085:
14987:
14947:
14779:
14757:
14752:
2347:
2248:
2213:
1065:
1053:
1049:
1038:
1018:
862:
732:
712:
458:
298:
24946:
Highly readable textbook; introduction to relativistic QFT for particle physics.
24784:. Proceedings Of The 9Th International Conference. World Scientific Publishing.
24302:; DeWitt-Morette, Cécile (1995). "A new perspective on Functional Integration".
19777:
EL equations. The time ordering is taken before the time derivatives inside the
18755:, and the two are essentially mathematically identical except for the factor of
13542:
Now, the contribution of the kinetic energy to the path integral is as follows:
25956:
25925:
25915:
25537:
25517:
25346:
25067:
Proceedings of the National Academy of Sciences of the United States of America
24840:
24642:(1993). "Most of the Good Stuff". In Brown, Laurie M.; Rigden, John S. (eds.).
24447:
21614:
21496:
15797:
15747:
15667:
15652:
15637:
15617:
15582:
15577:
15572:
15557:
15542:
15507:
15497:
15422:
15357:
15327:
15262:
15227:
15217:
15192:
15063:
15020:
14884:
14637:{\displaystyle |\alpha ;t\rangle =e^{-{\frac {iHt}{\hbar }}}|\alpha ;0\rangle }
13753:
13722:
10754:{\displaystyle {\frac {dx}{dt}}={\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}}
9028:
3140:
and the abovementioned "zigzagging" corresponds to the appearance of the terms
2323:
Feynman showed that this formulation of quantum mechanics is equivalent to the
2313:
1167:
1140:
1106:
1045:
797:
757:
737:
707:
687:
637:
603:
453:
443:
236:
25237:
25231:
24844:
24735:
24578:
24355:
Path Integrals in Physics Volume 1: Stochastic Process & Quantum Mechanics
24153:
24128:
18982:
The answer to this is a number between 0 and 1, which can be interpreted as a
15547:
10035:
The change in the integral from the shift is, to first infinitesimal order in
3833:
as an integral over all paths. For a free-particle action (for simplicity let
3825:
The path integral representation gives the quantum amplitude to go from point
924:, over an infinity of quantum-mechanically possible trajectories to compute a
26400:
26348:
25875:
25728:
25619:
25522:
25438:
25408:
25361:
24703:
Grosche, Christian (1992). "An Introduction into the Feynman Path Integral".
24492:
24263:
23576:
21622:
21447:
20486:
19436:
18790:
of field configurations (which only depends locally on the fields), then the
18764:
15802:
15792:
15782:
15777:
15697:
15682:
15477:
15472:
15297:
15292:
15252:
15222:
15187:
15177:
15108:
15103:
14484:
14389:
of the classical problem in which one investigates the path starting at time
13084:
12922:
7109:
5520:
5031:. Then the same convolution argument as before gives the propagation kernel:
5028:
2309:
2276:
1947:
Dirac further noted that one could square the time-evolution operator in the
1022:
857:
852:
782:
752:
722:
593:
539:
266:
241:
119:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }
31:
23717:, in which the systems and surroundings must be modeled together. Using the
2034:. The result is a sum over paths with a phase, which is the quantum action.
26301:
25758:
25356:
25351:
25106:
24613:
24232:
19797:
18791:
17907:
to be imaginary gives the usual relativistic propagator, up to a factor of
15837:
15762:
15647:
15622:
15607:
15537:
15457:
15412:
15407:
15287:
15277:
15272:
15237:
15232:
15207:
15162:
15030:
14992:
14814:
14794:
13869:
We then have a rigorous version of the Feynman path integral, known as the
3813:
2380:
For a particle in a smooth potential, the path integral is approximated by
2351:
2223:
of the action. The classical path arises naturally in the classical limit.
1155:, and the condition that determines the classical equations of motion (the
847:
842:
777:
762:
727:
221:
25087:
24375:(1972). "Feynman's path integral: Definition without limiting procedure".
11512:
2666:
An approximation for the path integral can be computed as proportional to
25796:
25143:
24969:
24900:
Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral
24870:
24316:
23564:
23533:{\displaystyle {\frac {1}{2}}({\hat {q}}{\hat {p}}+{\hat {p}}{\hat {q}})}
23157:{\displaystyle {\frac {1}{2}}({\hat {q}}{\hat {p}}+{\hat {p}}{\hat {q}})}
16457:
15817:
15752:
15712:
15672:
15627:
15487:
15387:
15382:
15367:
15352:
15337:
9825:{\displaystyle \int \psi _{0}(x)\int _{x(0)=x}e^{iS(x,{\dot {x}})}\,Dx\,}
4351:
3299:
2244:
1144:
1011:
944:
943:. Unlike previous methods, the path integral allows one to easily change
812:
767:
702:
657:
16012:
the Feynman path integral has made earlier formalisms largely obsolete.
14435:
denotes the integration measure over all paths. In the classical limit,
3130:{\displaystyle L(x,{\dot {x}})=T-V={\frac {1}{2}}m|{\dot {x}}|^{2}-V(x)}
26369:
25019:
24709:
24420:
24396:
23750:
19801:
19544:
16215:
15467:
15377:
15242:
15182:
14854:
14789:
2890:
is the Lagrangian of the one-dimensional system with position variable
1941:
1061:
983:
802:
772:
692:
667:
662:
647:
25166:
A great introduction to Path Integrals (Chapter 1) and QFT in general.
18970:
Strictly speaking, the only question that can be asked in physics is:
18677:
However, the path integral formulation is also extremely important in
16460:, the final state is the initial state convolved with the propagator:
11519:, and the (euclideanized) canonical commutation relations in physics.
11083:
goes to 0. But in this case, the difference between the two is not 0:
3789:{\displaystyle S=\int dt\,L(\mathbf {x} (t),{\dot {\mathbf {x} }}(t))}
1044:
The basic idea of the path integral formulation can be traced back to
25650:
25331:
24864:
MacKenzie, Richard (2000). "Path Integral Methods and Applications".
24333:
23785:
For a simplified, step-by-step derivation of the above relation, see
23543:
18722:
9720:
Start by considering the path integral with some fixed initial state
2338:
The path integral formulation of quantum field theory represents the
2297:
1661:
If the multiplications implicit in this formula are reinterpreted as
1026:
968:
931:
This formulation has proven crucial to the subsequent development of
293:
25243:
24511:
23059:
Path integrals as they are defined here require the introduction of
10831:{\displaystyle x(t+\varepsilon )-x(t)\approx {\sqrt {\varepsilon }}}
1427:{\displaystyle \varepsilon L=\varepsilon p{\dot {q}}-\varepsilon H,}
23876:
23588:
23563:
Some advocates of interpretations of quantum mechanics emphasizing
21485:
19774:
19317:
15427:
11881:. This measure cannot be expressed as a functional multiplying the
11870:
then it means that each spatial slice is multiplied by the measure
8921:{\displaystyle E_{n}=\left(n+{\tfrac {1}{2}}\right)\hbar \omega ~.}
7541:. One may write this propagator in terms of energy eigenstates as
3302:
approximating the time integral, which are finally integrated over
2288:
1017:
The path integral has impacted a wide array of sciences, including
972:
672:
26050:
23985:
21613:
that generates the one parameter group in question. We could have
19394:{\displaystyle {\frac {\delta {\mathcal {S}}}{\delta \varphi }}=0}
17898:
which is the Euclidean propagator for a scalar particle. Rotating
17493:, after summing over all the possible proper times it could take:
14551:. Strictly speaking, though, this is the partition function for a
7098:{\displaystyle Q={\sqrt {\frac {m}{2\pi i\hbar (t_{f}-t_{i})}}}~.}
3907:{\displaystyle S=\int {\frac {{\dot {x}}^{2}}{2}}\,\mathrm {d} t,}
1905:
This is not very different from just ordinary time evolution: the
1037:. It has been called the "most powerful formula in physics", with
1006:
with an imaginary diffusion constant, and the path integral is an
25232:
A mathematically rigorous approach to perturbative path integrals
24529:
18004:{\displaystyle K(p)={\frac {i}{p_{0}^{2}-{\vec {p}}^{2}-m^{2}}}.}
16008:, which are new relations difficult to see in the old formalism.
5203:{\displaystyle {\frac {d}{dt}}K(x;T)=i{\frac {\nabla ^{2}}{2}}K.}
2649:{\displaystyle \varepsilon =\Delta t={\frac {t_{b}-t_{a}}{n+1}}.}
1116:). For states with a definite energy, this is a statement of the
17219:
can be evaluated when the sum is over paths in Euclidean space:
14063:
satisfies the Wick-rotated version of the Schrödinger equation,
11522:
For a general statistical action, a similar argument shows that
5008:{\displaystyle {\frac {d}{dt}}K(x;T)={\frac {\nabla ^{2}}{2}}K.}
1087:
24761:
Inomata, Akira; Kuratsuji, Hiroshi; Gerry, Christopher (1992).
14725:
who remarked that the equation named after him looked like the
2927:
th time step, if the time integral is approximated by a sum of
2381:
1533:
is only definite on states that are indefinite with respect to
23870:
Hari Dass, N. D. (2020-03-28). "Dirac and the Path Integral".
23798:
For a brief account of the origins of these difficulties, see
2212:
That is, in the limit of action that is large compared to the
24243:
4851:{\displaystyle K(x-y;T)\propto e^{-{\frac {(x-y)^{2}}{2T}}}.}
26248:
Feynman's Lost Lecture: The Motion of Planets Around the Sun
25197:
Quantum Mechanics Formalism, Methodologies, and Applications
21436:{\displaystyle {\hat {\mathcal {S}}}={\mathcal {S}}-i\ln M.}
18685:
over all space. The action is referred to technically as a
18633:{\displaystyle K(p)={\frac {i}{p^{2}-m^{2}+i\varepsilon }},}
16621:{\displaystyle K(x,y;T)\propto e^{\frac {im(x-y)^{2}}{2T}}.}
12929:
10655:{\displaystyle S=\int \left({\frac {dx}{dt}}\right)^{2}\,dt}
4320:. Each factor in the product is a Gaussian as a function of
1488:{\displaystyle {\dot {q}}={\frac {\partial H}{\partial p}},}
1321:{\displaystyle p={\frac {\partial L}{\partial {\dot {q}}}},}
24803:. Oxford Mathematical Monographs. Oxford University Press.
24552:"Space-Time Approach to Non-Relativistic Quantum Mechanics"
22866:
The above two equations are the Ward–Takahashi identities.
14195:
11890:
measure because they belong to entirely different classes.
5217:
s will also obey the same equation, by linearity. Defining
5118:{\displaystyle K(x-y;T)\propto e^{\frac {i(x-y)^{2}}{2T}},}
4748:{\displaystyle {\tilde {K}}(p;T)=e^{-{\frac {Tp^{2}}{2}}}.}
1529:
can be interpreted as noncommuting operators. The operator
24881:
Mathematical Feynman path integrals and their applications
24722:. Graduate Texts in Mathematics. Vol. 267. Springer.
24688:. Springer Tracts in Modern Physics 145. Springer-Verlag.
24244:
Albeverio, S.; Hoegh-Krohn, R. & Mazzucchi, S (2008).
23054:
2139:
is thus that for which a comparatively large variation in
2103:
tends to zero. Let us now picture one of the intermediate
23540:
for the same respective choice of ordering prescription.
21454:= 0, we get the entire set of Schwinger–Dyson equations.
19338:
In the language of functional analysis, we can write the
14504:
with inverse temperature proportional to imaginary time,
11893:
10764:
The distance that a random walk moves is proportional to
3920:
To do this, it is convenient to start without the factor
2046:...we see that the integrand in (11) must be of the form
1131:, which is a more fundamental quantity in the context of
1127:
The Hamiltonian in classical mechanics is derived from a
25227:
Path Integrals in Quantum Theories: A Pedagogic 1st Step
25125:, The Quantum Theory of Fields, vol. 1, Cambridge:
23893:"How Our Reality May Be a Sum of All Possible Realities"
23787:
Path Integrals in Quantum Theories: A Pedagogic 1st Step
17361:, and each step is less likely the longer it is. By the
1993:
and this gives the time-evolution operator between time
24801:
The Feynman Integral and Feynman's Operational Calculus
24763:
Path Integrals and Coherent States of SU(2) and SU(1,1)
18739:
Such a functional integral is extremely similar to the
16218:. Superposing different values of the initial position
4762:, and it is a Gaussian again with reciprocal variance:
3341:
is an arbitrary value of the interval corresponding to
2291:, the amplitude of the 3rd postulate over the space of
2251:
of a complex number called the "probability amplitude".
2165:
is stationary for small variations in all intermediate
1928:, exponentiating this quantity and changing basis from
24993:"A Sum-over-histories Account of an EPR(B) Experiment"
24270:(1983). "Quantum tunnelling in a dissipative system".
21779:{\displaystyle \int {\mathcal {D}}\varphi \,Q\left=0,}
17018:
Given the nearly identical only change is the sign of
11501:{\displaystyle =x{\frac {dx}{dt}}-{\frac {dx}{dt}}x=1}
9642:
8890:
7701:
6372:
6065:
6028:
5583:
5546:
4209:
2150:. This part is the neighbourhood of a point for which
26281:
Perfectly Reasonable Deviations from the Beaten Track
24760:
24298:
24248:. Lecture Notes in Mathematics 523. Springer-Verlag.
23619:
23462:
23421:
23380:
23357:
23316:
23275:
23252:
23211:
23170:
23086:
22936:
22885:
22549:
22379:
22306:
22152:
22075:
21966:
21908:
21798:
21714:
21637:
21512:
21390:
21264:
21129:
21026:
20756:
20570:
20275:
20128:
19940:
19832:
19696:
19564:
19499:
19447:
19351:
19046:
18824:
18763:
the integral to an imaginary time variable (called a
18570:
18383:
18230:
18027:
17923:
17778:
17625:
17502:
17378:
17228:
17043:
16905:
16753:
16652:
16540:
16469:
16245:
16066:
15892:
14671:
14568:
14441:
14412:
14209:
14165:
14072:
14034:
13882:
13835:
13813:
13786:
13766:
13731:
13696:
13551:
13357:
13301:
13096:
13039:
12986:
12960:
12940:
12773:
12275:
12079:
11904:
11751:
11674:
11605:
11531:
11415:
11327:
11215:
11092:
10967:
10857:
10784:
10685:
10600:
10444:
10230:
10048:
9908:
9729:
9606:
9448:
9339:
9059:
8974:
8864:
8709:
8694:: All terms in this expansion get multiplied by the
8518:
8054:
7550:
7237:
7121:
7030:
6535:
6330:
5925:
5700:
5534:
5525:
The Lagrangian for the simple harmonic oscillator is
5432:
5226:
5137:
5040:
4945:
4882:
4771:
4679:
4590:
4487:
4370:
4111:
3933:
3856:
3692:
3669:
3649:
3612:
3590:
3561:
3398:
3149:
3027:
2675:
2587:
2421:
2357:
1960:
1846:
1779:
1674:
1546:
1446:
1384:
1337:
1279:
1179:
1100:
In quantum mechanics, as in classical mechanics, the
57:
24665:
Quantum Physics: A Functional Integral Point of View
23583:
and others. Approaches that use this method include
22360:{\displaystyle \int {\mathcal {D}}\varphi \,q(x)=0.}
21466:
we would restrict everything to lie within a finite
16873:
where there is no singularity. This guarantees that
12266:
This generalizes to multiple operators, for example
5644:
as the classical trajectory plus some perturbation,
1151:
and velocity). The relation between the two is by a
24624:. Mineola, NY: Dover Publications. pp. 29–31.
24513:
Feynman's Thesis — A New Approach to Quantum Theory
23993:
23063:. Changing the scale of the regulator leads to the
19804:of the field configurations (which has at least an
1837:Finally, the last factor in this interpretation is
1143:, while the Hamiltonian is the time component of a
25147:
25118:
24927:
24662:
24587:
23684:
23544:Path integral in quantum-mechanical interpretation
23532:
23448:
23407:
23366:
23343:
23302:
23261:
23238:
23197:
23156:
23038:
22916:
22855:
22529:
22359:
22259:
22133:{\displaystyle q(x)=\partial _{\mu }j^{\mu }(x)\,}
22132:
22055:
21946:
21881:
21778:
21681:
21572:
21435:
21324:
21244:
21109:
21003:
20728:
20454:
20255:
20108:
19920:
19762:
19668:
19524:
19479:
19393:
19296:
18940:
18632:
18541:
18329:
18180:
18003:
17887:
17769:-space, the propagator can be reexpressed simply:
17730:
17593:
17474:
17341:
17188:
17007:
16847:
16714:
16620:
16516:
16418:
16203:
15969:
14710:
14636:
14468:
14427:
14374:
14180:
14147:
14055:
14016:
13858:
13821:
13799:
13772:
13744:
13713:
13679:
13523:
13340:
13283:
13075:
13025:
12972:
12946:
12904:
12752:
12254:
12062:
11859:
11728:
11629:
11585:
11500:
11375:
11296:
11195:
11068:
10952:
10830:
10753:
10654:
10571:
10522:
10413:
10206:
10024:
9824:
9689:
9582:
9409:
9303:
9012:
8920:
8844:
8700:factor in the front, yielding terms of the form
8666:
8481:
7969:
7510:
7217:
7097:
7010:
6515:
6306:
5905:
5622:
5499:
5401:
5202:
5117:
5007:
4925:
4850:
4747:
4659:
4566:
4445:
4301:
4091:
3906:
3788:
3675:
3655:
3635:
3598:
3576:
3544:
3287:
3129:
2864:
2648:
2530:
2154:is stationary with respect to small variations in
1982:
1883:
1819:
1708:
1650:
1498:and the partial derivative now is with respect to
1487:
1426:
1352:
1320:
1259:
967:. Possible downsides of the approach include that
118:
24952:Techniques & Applications of Path Integration
24611:
24348:
24114:
23979:Bernstein, Jeremy (2010-04-20). "Another Dirac".
23945:
22270:(this is assuming the Lagrangian only depends on
17914:and an ambiguity, which will be clarified below:
12921:It is very common in path integrals to perform a
11729:{\displaystyle \int \mu e^{iS}\,{\mathcal {D}}x.}
10547:
9439:they are additive; in any case one has with (1):
7108:Using the infinite-product representation of the
26398:
24684:Grosche, Christian & Steiner, Frank (1998).
21682:{\displaystyle \int {\mathcal {D}}\varphi \,Q=0}
18204:. When convolving with the propagator, which in
13758:Einstein's mathematical model of Brownian motion
8965:by another path-dependent pseudo-time parameter
25154:(Second ed.). Princeton University Press.
24799:Johnson, Gerald W.; Lapidus, Michel L. (2002).
24262:
24215:
22917:{\displaystyle \left\langle Q\right\rangle =0.}
20553:is defined to be the operator that substitutes
18340:This is the expression for the nonrelativistic
17745:. Taking a Fourier transform over the variable
16715:{\displaystyle K(p;T)=e^{\frac {iTp^{2}}{2m}},}
26342:Quantum Man: Richard Feynman's Life in Science
25240:| PBS Space Time. July 7, 2017. (Video, 15:48)
24909:Path Integrals Methods in Quantum Field Theory
24897:
24798:
24683:
24452:"Solution of the path integral for the H-atom"
24371:
21325:{\displaystyle {\mathcal {S}}_{,i}Z+J_{i}Z=0.}
21120:we get the "master" Schwinger–Dyson equation:
18647:term introduces a small imaginary part to the
17352:This describes a sum over all paths of length
11376:{\displaystyle {\mathcal {L}}=(f(t)-1)^{2}\,,}
9435:times a diffusion process. To lowest order in
2261:The contribution of a path is proportional to
26066:
25259:
24849:(4th ed.). Singapore: World Scientific.
24246:Mathematical Theory of Feynman Path Integrals
24108:
23351:, or Weyl ordering prescription; conversely,
21573:{\displaystyle Q=\partial _{\mu }f^{\mu }(x)}
18972:What fraction of states satisfying condition
15863:
11277:
11236:
11163:
11122:
9256:
9205:
8961:at the origin. Only after replacing the time
5691:. The classical trajectory can be written as
5510:
5018:For oscillatory path integrals, ones with an
2810:
2776:
1638:
1607:
1567:
1557:
1331:where the partial derivative with respect to
1243:
1203:
885:
24902:(2nd ed.). Singapore: World Scientific.
24779:
24585:
24442:
24044:
22470:
22446:
22440:
22413:
22404:
22380:
22285:is the generator of a symmetry (i.e. we are
21814:
21799:
21473:
20436:
20424:
20367:
20355:
20283:
20276:
19905:
19893:
19323:
18831:
18825:
16118:
16094:
14631:
14583:
12780:
12774:
12528:
12276:
12057:
11905:
11739:This factor is needed to restore unitarity.
11660:
5916:This trajectory yields the classical action
5419:obeys the free Schrödinger equation just as
4581:is another Gaussian of reciprocal variance:
1832:evolve an infinitesimal time into the future
1756:. That is the action on the Hilbert space –
113:
87:
26215:QED: The Strange Theory of Light and Matter
25169:
24990:
24823:A Modern Approach to Functional Integration
24782:Path Integrals--New Trends And Perspectives
24193:
24126:
23736:Static forces and virtual-particle exchange
23039:{\displaystyle \int d^{d}x\,J(x)Q\leftZ=0.}
21377:In that case, we would have to replace the
12916:
11640:
10536:If the action contains terms that multiply
10533:this is the Heisenberg equation of motion.
9038:
2226:
2126:varying extremely rapidly. This means that
1095:
1056:. This idea was extended to the use of the
982:The path integral also relates quantum and
26073:
26059:
25266:
25252:
24991:Sinha, Sukanya; Sorkin, Rafael D. (1991).
24974:Functional Integration and Quantum Physics
24660:
24102:
22056:{\displaystyle q(x)=\delta ^{(d)}(X-y)Q\,}
18759:in the exponent in Feynman's postulate 3.
16517:{\displaystyle \psi _{T}=\psi _{0}*K(;T).}
16429:For a spatially homogeneous system, where
15870:
15856:
14751:
14711:{\displaystyle Z=\operatorname {Tr} \left}
14428:{\displaystyle {\mathcal {D}}\mathbf {x} }
10848:is ambiguous, with two possible meanings:
9899:, since the endpoints are not integrated:
3917:the integral can be evaluated explicitly.
3577:{\displaystyle {\mathcal {D}}\mathbf {x} }
3018:of the one-dimensional system considered,
2362:
2354:contribution to the transition amplitude.
2235:from this rule. This was done by Feynman.
1936:at each step allows the matrix element of
892:
878:
26237:What Do You Care What Other People Think?
25096:
25086:
25057:
24878:
24869:
24863:
24708:
24638:
24432:Physikalische Zeitschrift der Sowjetunion
24315:
24204:
24152:
23984:
23978:
23967:
23875:
23869:
22953:
22320:
22256:
22129:
22052:
21931:
21854:
21728:
21651:
20044:
20031:
18358:of the first term must vanish, while the
17845:
17716:
17579:
17174:
16406:
16321:
16191:
14365:
14254:
13989:
13958:
13941:
13676:
13663:
13514:
13280:
13267:
12930:Wick rotation and the Feynman–Kac formula
11847:
11712:
11369:
10645:
10410:
10403:
10378:
10203:
10196:
10171:
10021:
10014:
9821:
9814:
9686:
9576:
9569:
9284:
9013:{\displaystyle s=\int {\frac {dt}{r(t)}}}
8492:One may absorb all terms after the first
6104:
5983:
5385:
5295:
4910:
4361:with copies of itself at adjacent times:
4285:
4102:Splitting the integral into time slices:
4075:
4061:
3892:
3736:
3541:
3469:
2845:
2841:
2827:
2815:
1820:{\displaystyle e^{-i\varepsilon H(p,q)},}
1375:fixed. The inverse Legendre transform is
25191:
25113:
24949:
24839:
24237:Mathematical Integrals in Quantum Nature
23822:
23713:. This form is specifically useful in a
23070:
21692:for any polynomially-bounded functional
19679:for any polynomially-bounded functional
18672:
14469:{\displaystyle {\mathcal {S}}\gg \hbar }
14196:Path integral and the partition function
13341:{\displaystyle S_{\mathrm {Euclidean} }}
13026:{\displaystyle e^{-it{\hat {H}}/\hbar }}
9318:, the path integral has most weight for
4350:. The multiple integrals are a repeated
3798:
2146:produces only a very small variation in
2023:and the later one with a fixed value of
1884:{\displaystyle e^{ipq(t+\varepsilon )},}
1170:in time, the Legendre transform becomes
1086:
1010:of a method for summing up all possible
24817:
24780:Janke, W.; Pelster, Axel, eds. (2008).
24702:
24549:
24506:
24491:
24349:Chaichian, M.; Demichev, A. P. (2001).
24056:
23999:
23917:
23834:
23239:{\displaystyle qp+{\frac {i\hbar }{2}}}
23198:{\displaystyle qp-{\frac {i\hbar }{2}}}
23055:Need for regulators and renormalization
21595:(and possibly the spacetime position).
21381:in this equation by another functional
18221:, the dominant first term has the form
15980:for two simultaneous spatial positions
14736:
13076:{\displaystyle e^{-t{\hat {H}}/\hbar }}
10552:If the variation in the action exceeds
2325:canonical approach to quantum mechanics
14:
26399:
24906:
24377:Communications in Mathematical Physics
24167:
23610:) can be determined to be of the form
19812:for the functional measure), then the
19773:These equations are the analog of the
18998:. In terms of path integration, since
17761:separately, and because each separate
14500:of statistical mechanics defined in a
11894:Expectation values and matrix elements
9884:is a different shift at each time but
9704:
4577:The Fourier transform of the Gaussian
1052:for solving problems in diffusion and
26054:
25273:
25247:
24968:
24925:
24898:Müller-Kirsten, Harald J. W. (2012).
24586:Feynman, R. P.; Hibbs, A. R. (1965).
24425:"The Lagrangian in Quantum Mechanics"
24419:
24357:. Taylor & Francis. p. 1ff.
24231:
24091:
24011:
23956:
23865:
23863:
23269:depending on whether one chooses the
18770:
13752:. The Wiener measure, constructed by
12764:and to the general expectation value
5213:This means that any superposition of
1665:multiplications, the first factor is
1509:In quantum mechanics, the state is a
1029:. In physics, it is a foundation for
26183:There's Plenty of Room at the Bottom
25034:
24717:
24622:Quantum Mechanics and Path Integrals
24590:Quantum Mechanics and Path Integrals
24079:
24067:
23890:
23799:
23594:
23449:{\displaystyle {\hat {p}}{\hat {q}}}
23408:{\displaystyle {\hat {q}}{\hat {p}}}
23344:{\displaystyle {\hat {p}}{\hat {q}}}
23303:{\displaystyle {\hat {q}}{\hat {p}}}
21895:Now, let's assume even further that
19480:{\displaystyle e^{i{\mathcal {S}}},}
11390:derived from extremizing the action
9420:The first term rotates the phase of
9330:.) The exponential of the action is
8931:
4926:{\displaystyle \int K(x-y;T)\,dy=1.}
4456:where the number of convolutions is
26080:
25175:Path Integrals in Quantum Mechanics
25142:
24954:. New York: John Wiley & Sons.
24497:"Geometry and Quantum Field Theory"
23550:interpretation of quantum mechanics
21947:{\displaystyle Q=\int d^{d}x\,q(x)}
18965:
3584:denotes integration over all paths
2923:corresponds to the position at the
2350:is a graphical representation of a
2335:corresponding to the given action.
1983:{\displaystyle e^{i\varepsilon S},}
24:
26270:The Pleasure of Finding Things Out
26226:Surely You're Joking, Mr. Feynman!
25150:Quantum Field Theory in a Nutshell
24686:Handbook of Feynman Path Integrals
24661:Glimm, J. & Jaffe, A. (1981).
23860:
23666:
23663:
23660:
23632:
23620:
23570:
22697:
22488:
22312:
22230:
22210:
22203:
22199:
22101:
21836:
21720:
21643:
21542:
21521:
21410:
21395:
21291:
21268:
21138:
21041:
20927:
20910:
20845:
20828:
20676:
20659:
20606:
20589:
20407:
20384:
20338:
20303:
19876:
19853:
19738:
19636:
19458:
19360:
19249:
19212:
19144:
19107:
19047:
18916:
18898:
18875:
18845:
17850:
17836:
17818:
17804:
17721:
17710:
17697:
17657:
17584:
17572:
17558:
17534:
17461:
17420:
17398:
17308:
17270:
17248:
17139:
17105:
17063:
15970:{\displaystyle =i\delta ^{3}(x-y)}
15912:
14444:
14415:
14275:
14257:
14231:
14082:
14078:
13666:
13388:
13385:
13382:
13379:
13376:
13373:
13370:
13367:
13364:
13348:is the Euclidean action, given by
13332:
13329:
13326:
13323:
13320:
13317:
13314:
13311:
13308:
13270:
13193:
13190:
13187:
13184:
13181:
13178:
13175:
13172:
13169:
12856:
12794:
12595:
12141:
11715:
11554:
11546:
11330:
10352:
10344:
10320:
10312:
10141:
10133:
10115:
10107:
9654:
9618:
9610:
9100:
9095:
7847:
7138:
6905:
6651:
6526:This means that the propagator is
6366:
5978:
5537:
5470:
5442:
5438:
5388:
5180:
4985:
4288:
4078:
4063:
3894:
3564:
3459:
2716:
2708:
2692:
2684:
2594:
2358:Path integral in quantum mechanics
2037:
1473:
1465:
1297:
1289:
424:Sum-over-histories (path integral)
110:
84:
40:Part of a series of articles about
25:
26448:
25215:
25040:Path Integrals on Group Manifolds
24720:Quantum Theory for Mathematicians
24646:. American Institute of Physics.
24022:
23972:
23766:Path integrals in polymer science
23672:
23227:
23186:
23164:can be translated back as either
21014:Then, from the properties of the
19808:because of the assumption of the
19435:), and if we assume that after a
14610:
14463:
14248:
14188:can be extracted by studying it.
13953:
13859:{\displaystyle \mathbf {x} (0)=x}
13232:
13068:
13018:
12880:
12830:
12689:
12478:
12389:
12307:
12191:
12009:
11936:
8906:
8599:
8266:
8135:
7887:
7695:
7645:
7460:
7433:
7323:
7228:the propagator can be written as
7053:
6880:
6710:
6629:
3636:{\displaystyle \mathbf {x} (0)=x}
3166:
2737:
1511:superposition of different states
61:
26035:
26034:
25238:Feynman's Infinite Quantum Paths
24239:. The Nucleus. pp. 189–209.
23837:"Dynamic Vortex Mott Transition"
21446:If we expand this equation as a
19539:, it goes to zero faster than a
18344:of a free Schrödinger particle.
16631:Taking the Fourier transform in
16222:with an arbitrary initial state
14453:
14421:
14343:
14323:
14284:
14263:
14240:
14007:
13973:
13925:
13837:
13815:
13704:
13672:
13633:
13596:
13568:
13493:
13452:
13409:
13398:
13276:
13251:
13214:
13203:
13134:
11386:and the equations of motion for
3820:
3764:
3744:
3711:
3700:
3663:is a normalization factor. Here
3614:
3592:
3570:
3525:
3496:
3485:
3465:
3436:
3384:
2342:(corresponding to the classical
2304:assigns to all these amplitudes
2118:, we shall then in general have
27:Formulation of quantum mechanics
26355:Feynman Prize in Nanotechnology
26193:The Feynman Lectures on Physics
26106:Wheeler–Feynman absorber theory
25042:. Singapore: World Scientific.
24765:. Singapore: World Scientific.
24304:Journal of Mathematical Physics
24224:
24209:
24198:
24187:
24161:
24120:
24115:Feynman, Hibbs & Styer 2010
24096:
24085:
24073:
24061:
24050:
24038:
24016:
24005:
23792:
23779:
23756:Wheeler–Feynman absorber theory
23585:causal dynamical triangulations
23554:Einstein–Podolsky–Rosen paradox
21457:
18561:. The terms can be recombined:
14267:
13756:gives a rigorous foundation to
13714:{\displaystyle f(\mathbf {x} )}
10572:Canonical commutation relations
10217:which, integrating by parts in
9291:
8806:
2239:from the following postulates:
2233:canonical commutation relations
2173:has for its classical analogue
1074:Wheeler–Feynman absorber theory
25984:Relativistic quantum mechanics
25199:. Cambridge University Press.
25000:Foundations of Physics Letters
24934:. Cambridge University Press.
24911:. Cambridge University Press.
23961:
23950:
23939:
23911:
23884:
23828:
23816:
23527:
23521:
23509:
23494:
23482:
23473:
23440:
23428:
23399:
23387:
23335:
23323:
23294:
23282:
23151:
23145:
23133:
23118:
23106:
23097:
23027:
23021:
22984:
22981:
22975:
22969:
22963:
22957:
22900:
22894:
22844:
22838:
22801:
22798:
22792:
22786:
22780:
22774:
22765:
22759:
22722:
22716:
22690:
22684:
22647:
22644:
22638:
22632:
22626:
22620:
22611:
22605:
22568:
22562:
22559:
22553:
22513:
22507:
22467:
22461:
22458:
22452:
22437:
22431:
22428:
22422:
22401:
22395:
22392:
22386:
22348:
22342:
22339:
22333:
22330:
22324:
22253:
22247:
22241:
22235:
22222:
22206:
22191:
22185:
22169:
22163:
22126:
22120:
22094:
22088:
22085:
22079:
22049:
22046:
22040:
22034:
22028:
22016:
22011:
22005:
21994:
21991:
21985:
21979:
21976:
21970:
21941:
21935:
21811:
21805:
21764:
21758:
21670:
21664:
21661:
21655:
21567:
21561:
21535:
21532:
21526:
21516:
21399:
21294:
21282:
21230:
21224:
21218:
21212:
21203:
21197:
21157:
21151:
21087:
21081:
21072:
21066:
21060:
21054:
20995:
20989:
20980:
20967:
20898:
20885:
20816:
20806:
20800:
20794:
20720:
20707:
20650:
20637:
20580:
20574:
20418:
20412:
20349:
20343:
20320:
20314:
20193:
20187:
20168:
20162:
20089:
20076:
20067:
20054:
20041:
20035:
20015:
20009:
20003:
19990:
19978:
19965:
19887:
19881:
19842:
19836:
19647:
19641:
19625:
19619:
19584:
19578:
19517:
19511:
19469:
19463:
19371:
19365:
19274:
19268:
19260:
19254:
19236:
19230:
19169:
19163:
19155:
19149:
19131:
19125:
19065:
19053:
18927:
18921:
18886:
18880:
18862:
18856:
18580:
18574:
18497:
18427:
18393:
18387:
18299:
18284:
18265:
18253:
18247:
18145:
18084:
18050:
18044:
17970:
17933:
17927:
17788:
17782:
17757:can be done for each value of
17686:
17673:
17641:
17629:
17576:
17568:
17562:
17542:
17518:
17506:
17450:
17437:
17402:
17382:
17312:
17304:
17290:
17284:
17252:
17232:
17109:
17101:
17087:
17081:
17067:
17047:
16967:
16928:
16916:
16808:
16776:
16764:
16668:
16656:
16594:
16581:
16562:
16544:
16508:
16499:
16401:
16395:
16381:
16378:
16372:
16366:
16345:
16339:
16318:
16300:
16294:
16288:
16262:
16256:
16186:
16180:
16158:
16152:
16138:
16132:
16088:
16070:
15964:
15952:
15933:
15930:
15924:
15905:
15899:
15893:
15051:Renormalization group equation
14618:
14570:
14457:
14449:
14362:
14359:
14353:
14333:
14327:
14319:
14288:
14280:
14244:
14236:
14172:
14142:
14130:
14121:
14106:
14094:
14050:
14038:
14011:
14003:
13986:
13983:
13977:
13969:
13938:
13935:
13929:
13921:
13898:
13886:
13847:
13841:
13708:
13700:
13645:
13625:
13600:
13592:
13578:
13572:
13506:
13503:
13497:
13489:
13473:
13468:
13462:
13444:
13419:
13394:
13264:
13261:
13255:
13247:
13224:
13199:
13144:
13138:
13112:
13100:
13057:
13007:
12980:, the time-evolution operator
12894:
12888:
12867:
12861:
12844:
12838:
12817:
12811:
12805:
12799:
12745:
12742:
12736:
12718:
12712:
12706:
12676:
12673:
12660:
12654:
12641:
12638:
12625:
12619:
12606:
12600:
12575:
12569:
12548:
12542:
12514:
12508:
12495:
12489:
12462:
12456:
12447:
12432:
12406:
12400:
12373:
12367:
12358:
12343:
12324:
12318:
12290:
12247:
12244:
12238:
12220:
12214:
12208:
12178:
12175:
12164:
12158:
12152:
12146:
12121:
12115:
12094:
12088:
12043:
12037:
12026:
12020:
11993:
11987:
11978:
11970:
11953:
11947:
11919:
11839:
11833:
11707:
11701:
11687:
11681:
11618:
11606:
11437:
11416:
11360:
11350:
11344:
11338:
11271:
11265:
11256:
11244:
11225:
11219:
11157:
11151:
11142:
11130:
11111:
11105:
11099:
11093:
11057:
11051:
11042:
11030:
11021:
11009:
10974:
10968:
10941:
10935:
10926:
10914:
10905:
10899:
10864:
10858:
10815:
10809:
10800:
10788:
10742:
10736:
10727:
10715:
10548:Stationary-phase approximation
10491:
10485:
10375:
10369:
10267:
10261:
10250:
10244:
10085:
10079:
10068:
10062:
10009:
9967:
9945:
9939:
9928:
9922:
9809:
9788:
9766:
9760:
9749:
9743:
9675:
9669:
9548:
9535:
9520:
9514:
9494:
9482:
9470:
9452:
9363:
9357:
9298:
9292:
9281:
9275:
9251:
9245:
9163:
9151:
9137:
9131:
9120:
9108:
9081:
9063:
9004:
8998:
8658:
8652:
8574:
8522:
8110:
8058:
7948:
7934:
7921:
7908:
7757:
7721:
7610:
7558:
7498:
7472:
7402:
7376:
7362:
7336:
7293:
7241:
7082:
7056:
6959:
6933:
6835:
6809:
6759:
6749:
6595:
6543:
6502:
6476:
6426:
6416:
6284:
6258:
6218:
6192:
6180:
6144:
5894:
5868:
5854:
5835:
5804:
5778:
5764:
5745:
5717:
5711:
5361:
5355:
5341:
5335:
5324:
5318:
5292:
5274:
5268:
5262:
5243:
5237:
5168:
5156:
5091:
5078:
5062:
5044:
4976:
4964:
4907:
4889:
4823:
4810:
4793:
4775:
4704:
4692:
4686:
4616:
4610:
4598:
4544:
4537:
4525:
4512:
4500:
4494:
4392:
4374:
4258:
4252:
4243:
4231:
4173:
4167:
4153:
4147:
4133:
4115:
3995:
3989:
3975:
3969:
3955:
3937:
3783:
3780:
3774:
3754:
3748:
3740:
3721:
3696:
3624:
3618:
3538:
3535:
3529:
3521:
3506:
3481:
3446:
3440:
3414:
3402:
3216:
3124:
3118:
3102:
3084:
3052:
3031:
2805:
2799:
2790:
2784:
1873:
1861:
1809:
1797:
1698:
1692:
1633:
1621:
1602:
1596:
1587:
1575:
1238:
1232:
1223:
1211:
1198:
1192:
994:of a fluctuating field near a
574:Relativistic quantum mechanics
106:
99:
80:
13:
1:
26204:The Character of Physical Law
25962:Quantum statistical mechanics
25739:Quantum differential calculus
25661:Delayed-choice quantum eraser
25429:Symmetry in quantum mechanics
25222:Path integral on Scholarpedia
24669:. New York: Springer-Verlag.
24510:(2005) . Brown, L. M. (ed.).
23946:Chaichian & Demichev 2001
23835:Vinokur, V. M. (2015-02-27).
23809:
16035:
14880:Spontaneous symmetry breaking
14840:Symmetry in quantum mechanics
14733:statistical mechanics model.
13083:. (This change is known as a
5027:. This is closely related to
3318:with the integration measure
2247:for an event is given by the
1651:{\displaystyle e^{i{\big }}.}
996:second-order phase transition
614:Quantum statistical mechanics
24976:. New York: Academic Press.
24479:10.1016/0370-2693(79)90280-6
24292:10.1016/0003-4916(83)90202-6
23891:Wood, Charlie (2023-02-06).
23374:can be translated to either
16527:For a free particle of mass
16236:constructs the final state:
13822:{\displaystyle \mathbf {x} }
11898:Matrix elements of the kind
9593:As mentioned, the spread in
9033:Duru–Kleinert transformation
4758:The Fourier transform gives
3599:{\displaystyle \mathbf {x} }
2916:considered (see below), and
1709:{\displaystyle e^{-ipq(t)},}
7:
25764:Quantum stochastic calculus
25754:Quantum measurement problem
25676:Mach–Zehnder interferometer
25177:. Oxford University Press.
24644:Memories Of Richard Feynman
24216:Caldeira & Leggett 1983
23927:writings.stephenwolfram.com
23728:
23704:and pre-exponential factor
20489:complications here!), then
18747:. Indeed, it is sometimes
16643:produces another Gaussian:
13780:indicates that the measure
11402:Defining the time order to
1913:basis from an intermediate
914:stationary action principle
584:Quantum information science
10:
26453:
26151:Quantum cellular automaton
25127:Cambridge University Press
24127:Zinn-Justin, Jean (2009).
23772:
23695:with the effective action
23049:
21477:
21464:local quantum field theory
20485:analogue, because we have
20481:, unlike its Wick-rotated
19327:
15998:careful limiting procedure
14875:Explicit symmetry breaking
14181:{\displaystyle {\hat {H}}}
14056:{\displaystyle \psi (x,t)}
9042:
5514:
5511:Simple harmonic oscillator
2366:
2099:, which remains finite as
1353:{\displaystyle {\dot {q}}}
29:
26323:
26294:
26174:
26131:Path integral formulation
26088:
26030:
25992:
25944:
25824:Quantum complexity theory
25802:Quantum cellular automata
25777:
25709:
25643:
25556:
25505:
25492:Path integral formulation
25459:
25324:
25281:
24736:10.1007/978-1-4614-7116-5
24594:. New York: McGraw-Hill.
24579:10.1103/RevModPhys.20.367
24559:Reviews of Modern Physics
24154:10.4249/scholarpedia.8674
21474:Ward–Takahashi identities
19429:translationally invariant
19410:Schwinger–Dyson equations
19404:(the left-hand side is a
19324:Schwinger–Dyson equations
15041:Bargmann–Wigner equations
14968:Path Integral Formulation
11661:Measure-theoretic factors
11647:operator ordering problem
2202:becomes extremely small.
1517:, or different values of
1513:with different values of
906:path integral formulation
26156:Rogers Commission Report
26116:Hellmann–Feynman theorem
25891:Quantum machine learning
25871:Quantum key distribution
25861:Quantum image processing
25851:Quantum error correction
25701:Wheeler's delayed choice
25234:via animation on YouTube
24926:Ryder, Lewis H. (1985).
24825:. New York: Birkhäuser.
24045:Duru & Kleinert 1979
21591:only depends locally on
19819:of the source fields is
19810:translational invariance
19340:Euler–Lagrange equations
19330:Schwinger–Dyson equation
18795:vacuum expectation value
18736:mathematically precise.
17743:Schwinger representation
16022:constrained quantization
15046:Schwinger-Dyson equation
14553:statistical field theory
13800:{\displaystyle \mu _{x}}
13745:{\displaystyle \mu _{x}}
12917:Euclidean path integrals
11641:Particle in curved space
9039:The Schrödinger equation
8677:One may finally expand
3683:is the action, given by
2227:Feynman's interpretation
1157:Euler–Lagrange equations
1096:Quantum action principle
1060:in quantum mechanics by
1021:, quantum field theory,
992:statistical field theory
619:Quantum machine learning
372:Wheeler's delayed-choice
26382:The Challenger Disaster
26126:Feynman parametrization
25807:Quantum finite automata
24950:Schulman, L S. (1981).
24718:Hall, Brian C. (2013).
24550:Feynman, R. P. (1948).
24194:Sinha & Sorkin 1991
24168:Pössel, Markus (2006).
22869:Now for the case where
21480:Ward–Takahashi identity
19800:) is an element of the
18984:conditional probability
18976:also satisfy condition
18761:Analytically continuing
16862:-space representation.
15076:Electroweak interaction
15071:Quantum electrodynamics
15036:Wheeler–DeWitt equation
14913:Background field method
11586:{\displaystyle \left=1}
10558:Euler–Lagrange equation
9328:Trotter product formula
7021:for some normalization
2659:This process is called
2541:can be divided up into
2375:Trotter product formula
2363:Time-slicing derivation
1153:Legendre transformation
1122:superposition principle
1084:) as a starting point.
947:between very different
329:Leggett–Garg inequality
26427:Differential equations
26121:Feynman slash notation
25911:Quantum neural network
24907:Rivers, R. J. (1987).
24879:Mazzucchi, S. (2009).
24103:Glimm & Jaffe 1981
23686:
23606:, the tunneling rate (
23534:
23450:
23409:
23368:
23345:
23304:
23263:
23240:
23199:
23158:
23040:
22918:
22857:
22531:
22361:
22261:
22134:
22057:
21948:
21883:
21780:
21683:
21574:
21437:
21326:
21246:
21111:
21005:
20730:
20456:
20257:
20110:
19922:
19764:
19670:
19526:
19525:{\displaystyle e^{-H}}
19481:
19433:nonlinear sigma models
19395:
19298:
18942:
18634:
18543:
18331:
18182:
18005:
17889:
17732:
17595:
17476:
17343:
17190:
17009:
16849:
16716:
16622:
16518:
16444:is only a function of
16420:
16205:
16006:operator product rules
15971:
15081:Quantum chromodynamics
14938:Effective field theory
14712:
14638:
14470:
14429:
14376:
14182:
14149:
14057:
14018:
13860:
13823:
13807:is supported on paths
13801:
13774:
13746:
13715:
13681:
13525:
13342:
13285:
13077:
13027:
12974:
12948:
12906:
12754:
12256:
12064:
11861:
11730:
11631:
11587:
11502:
11377:
11298:
11197:
11070:
10954:
10832:
10755:
10656:
10524:
10415:
10208:
10026:
9826:
9691:
9584:
9411:
9305:
9014:
8922:
8846:
8668:
8483:
7971:
7851:
7512:
7219:
7142:
7099:
7012:
6909:
6655:
6517:
6370:
6308:
5907:
5624:
5501:
5403:
5204:
5119:
5009:
4927:
4852:
4749:
4661:
4568:
4447:
4303:
4093:
3908:
3817:
3790:
3677:
3657:
3637:
3600:
3578:
3546:
3289:
3200:
3131:
2866:
2720:
2696:
2650:
2532:
2210:
1984:
1885:
1821:
1730:, and so it takes the
1710:
1652:
1489:
1428:
1354:
1322:
1261:
1139:. The Lagrangian is a
1092:
1070:John Archibald Wheeler
1035:quantum chromodynamics
961:Feynman path integrals
941:canonical quantization
120:
26412:Statistical mechanics
26259:The Meaning of It All
26146:One-electron universe
26111:Bethe–Feynman formula
25936:Quantum teleportation
25449:Wave–particle duality
25088:10.1073/pnas.14.2.178
24499:. MIT OpenCourseWare.
23687:
23556:without resorting to
23535:
23451:
23410:
23369:
23346:
23305:
23264:
23241:
23200:
23159:
23071:Ordering prescription
23065:renormalization group
23041:
22919:
22858:
22532:
22362:
22262:
22135:
22058:
21949:
21884:
21781:
21684:
21575:
21438:
21327:
21247:
21112:
21006:
20731:
20483:statistical mechanics
20457:
20258:
20111:
19923:
19814:generating functional
19765:
19671:
19527:
19482:
19406:functional derivative
19396:
19307:where the functional
19299:
18943:
18745:statistical mechanics
18673:Functionals of fields
18635:
18544:
18332:
18183:
18006:
17890:
17733:
17596:
17477:
17363:central limit theorem
17344:
17191:
17010:
16850:
16717:
16623:
16519:
16421:
16206:
15972:
15026:Klein–Gordon equation
14958:LSZ reduction formula
14713:
14651:is evolved from time
14639:
14481:statistical mechanics
14471:
14430:
14377:
14183:
14150:
14058:
14019:
13861:
13824:
13802:
13775:
13747:
13716:
13682:
13526:
13343:
13286:
13078:
13028:
12975:
12949:
12907:
12755:
12257:
12065:
11862:
11731:
11632:
11588:
11503:
11378:
11299:
11198:
11071:
10955:
10833:
10756:
10657:
10525:
10416:
10209:
10027:
9827:
9692:
9585:
9412:
9306:
9015:
8923:
8847:
8669:
8509:, thereby obtaining
8484:
7980:Using the identities
7972:
7831:
7513:
7220:
7122:
7100:
7013:
6889:
6635:
6518:
6350:
6309:
5908:
5633:Write its trajectory
5625:
5502:
5404:
5205:
5120:
5010:
4928:
4868:central limit theorem
4853:
4750:
4662:
4569:
4448:
4304:
4094:
3909:
3811:
3791:
3678:
3658:
3638:
3601:
3579:
3547:
3290:
3174:
3132:
2986:in the initial state
2867:
2700:
2676:
2651:
2533:
2312:, or argument of the
2256:probability amplitude
2044:
1985:
1896:change basis back to
1886:
1822:
1711:
1653:
1521:, and the quantities
1490:
1429:
1355:
1323:
1262:
1090:
1048:, who introduced the
1008:analytic continuation
912:that generalizes the
314:Elitzur–Vaidman
304:Davisson–Germer
121:
18:Feynman path integral
26432:Mathematical physics
26422:Quantum field theory
26161:Feynman checkerboard
26141:Sticky bead argument
25967:Quantum field theory
25896:Quantum metamaterial
25841:Quantum cryptography
25571:Consistent histories
24930:Quantum Field Theory
24883:. World Scientific.
24516:. World Scientific.
23741:Feynman checkerboard
23617:
23460:
23419:
23378:
23355:
23314:
23273:
23250:
23209:
23168:
23084:
22934:
22883:
22547:
22377:
22370:Then, we would have
22304:
22150:
22073:
21964:
21906:
21899:is a local integral
21796:
21712:
21635:
21605:or something. Here,
21510:
21388:
21364:is diffeomorphic to
21351:is a functional and
21262:
21127:
21024:
21016:functional integrals
20754:
20568:
20273:
20126:
19938:
19830:
19694:
19562:
19547:for large values of
19497:
19445:
19349:
19044:
18822:
18799:polynomially bounded
18777:quantum field theory
18734:functional integrals
18710:such given function
18568:
18381:
18228:
18025:
17921:
17776:
17623:
17500:
17376:
17226:
17041:
16903:
16751:
16650:
16538:
16467:
16456:, the integral is a
16243:
16064:
15890:
15119:Theory of everything
14953:Lattice field theory
14923:Correlation function
14745:Quantum field theory
14737:Quantum field theory
14669:
14566:
14479:The connection with
14439:
14410:
14207:
14163:
14070:
14032:
13880:
13833:
13811:
13784:
13764:
13729:
13694:
13549:
13355:
13299:
13094:
13037:
12984:
12958:
12938:
12771:
12273:
12077:
11902:
11749:
11672:
11651:nonholonomic mapping
11603:
11529:
11413:
11406:the operator order:
11325:
11213:
11090:
10965:
10855:
10782:
10683:
10598:
10442:
10228:
10046:
9906:
9727:
9604:
9446:
9337:
9057:
8972:
8862:
8707:
8516:
8052:
7548:
7235:
7119:
7028:
6533:
6328:
5923:
5698:
5532:
5430:
5224:
5135:
5038:
4943:
4880:
4769:
4677:
4588:
4485:
4368:
4109:
3931:
3854:
3690:
3667:
3647:
3610:
3588:
3559:
3396:
3147:
3025:
2673:
2585:
2578:, of fixed duration
2419:
2412:, the time sequence
2344:correlation function
2340:transition amplitude
2329:Schrödinger equation
1958:
1844:
1777:
1672:
1544:
1444:
1382:
1335:
1277:
1177:
1031:lattice gauge theory
1000:Schrödinger equation
988:quantum field theory
908:is a description in
579:Quantum field theory
491:Consistent histories
128:Schrödinger equation
55:
26407:Concepts in physics
26101:Feynman–Kac formula
25952:Quantum fluctuation
25921:Quantum programming
25881:Quantum logic gates
25866:Quantum information
25846:Quantum electronics
25306:Classical mechanics
25079:1928PNAS...14..178V
25012:1991FoPhL...4..303S
24728:2013qtm..book.....H
24571:1948RvMP...20..367F
24522:2005ftna.book.....B
24471:1979PhLB...84..185D
24389:1972CMaPh..28...47D
24326:1995JMP....36.2237C
24284:1983AnPhy.149..374C
24145:2009SchpJ...4.8674Z
23761:Feynman–Kac formula
22289:insisting upon the
21702:for more details.)
20951:
20869:
20743:is a functional of
20700:
20630:
20536:is a functional of
18727:field configuration
17959:
17808:
17661:
17538:
17322:
17200:time. The function
17144:
17119:
16214:This is called the
16168:
15098:Incomplete theories
14396:and ending at time
14315:
13871:Feynman–Kac formula
12973:{\displaystyle -it}
12592:
12138:
11517:stochastic calculus
11511:This is called the
9705:Equations of motion
9202:
9173:
9104:
8314:
8296:
7756:
7738:
6730:
6397:
6179:
6161:
6021:
5975:
5371:
5023:the time increment
4183:
4034:
4005:
3345:, e.g. its center,
3000:in the final state
2943:functional integral
2770:
2207:Dirac (1933), p. 69
2169:s. ... We see that
1745:to change basis to
1118:de Broglie relation
935:, because manifest
933:theoretical physics
922:functional integral
918:classical mechanics
367:Stern–Gerlach
164:Classical mechanics
26336:Cargo cult science
26005:in popular culture
25787:Quantum algorithms
25635:Von Neumann–Wigner
25615:Objective collapse
25311:Old quantum theory
25020:10.1007/BF00665892
24397:10.1007/BF02099371
24373:DeWitt-Morette, C.
24032:221A Lecture Notes
23715:dissipative system
23682:
23600:Quantum tunnelling
23530:
23446:
23405:
23367:{\displaystyle qp}
23364:
23341:
23300:
23262:{\displaystyle qp}
23259:
23236:
23195:
23154:
23077:operator formalism
23036:
22914:
22853:
22527:
22357:
22257:
22130:
22053:
21944:
21879:
21776:
21679:
21628:Let's also assume
21583:for some function
21570:
21484:Now how about the
21433:
21370:. However, if the
21322:
21242:
21107:
21001:
20930:
20848:
20726:
20679:
20609:
20561:. For example, if
20452:
20253:
20106:
19918:
19760:
19666:
19553:integrate by parts
19522:
19490:which now becomes
19477:
19417:functional measure
19391:
19294:
19200:
19095:
18938:
18771:Expectation values
18753:partition function
18741:partition function
18695:, where the field
18630:
18539:
18327:
18178:
18001:
17945:
17885:
17794:
17728:
17647:
17591:
17524:
17472:
17339:
17276:
17186:
17128:
17073:
17005:
16845:
16712:
16618:
16514:
16416:
16201:
16124:
16018:Grassmann variable
15967:
14963:Partition function
14890:Topological charge
14810:General relativity
14805:Special relativity
14727:diffusion equation
14708:
14634:
14502:canonical ensemble
14498:partition function
14466:
14425:
14372:
14294:
14178:
14145:
14053:
14014:
13856:
13819:
13797:
13770:
13742:
13711:
13677:
13521:
13338:
13281:
13073:
13023:
12970:
12944:
12902:
12750:
12534:
12252:
12080:
12060:
11857:
11726:
11630:{\displaystyle =i}
11627:
11583:
11498:
11373:
11294:
11193:
11066:
10950:
10828:
10751:
10652:
10520:
10411:
10204:
10022:
9822:
9687:
9651:
9580:
9407:
9301:
9182:
9123:
9087:
9010:
8918:
8899:
8842:
8664:
8479:
8300:
8282:
8045:, this amounts to
7967:
7965:
7742:
7724:
7710:
7508:
7215:
7095:
7008:
7006:
6716:
6513:
6383:
6381:
6304:
6302:
6165:
6147:
6074:
6037:
5993:
5947:
5903:
5674:and the action as
5620:
5592:
5555:
5497:
5399:
5327:
5200:
5115:
5005:
4923:
4848:
4745:
4670:and the result is
4657:
4564:
4443:
4299:
4218:
4193:
4139:
4089:
4020:
3961:
3904:
3818:
3786:
3673:
3653:
3633:
3596:
3574:
3542:
3285:
3127:
2862:
2742:
2646:
2528:
1980:
1881:
1817:
1706:
1648:
1485:
1424:
1350:
1318:
1257:
1133:special relativity
1093:
1004:diffusion equation
937:Lorentz covariance
555:Von Neumann–Wigner
535:Objective-collapse
334:Mach–Zehnder
324:Leggett inequality
319:Franck–Hertz
169:Old quantum theory
116:
26417:Quantum mechanics
26394:
26393:
26386:
26365:
26311:Charles Hirshberg
26166:Feynman sprinkler
26048:
26047:
26022:Quantum mysticism
26000:Schrödinger's cat
25931:Quantum simulator
25901:Quantum metrology
25829:Quantum computing
25792:Quantum amplifier
25769:Quantum spacetime
25734:Quantum cosmology
25724:Quantum chemistry
25424:Scattering theory
25372:Zero-point energy
25367:Degenerate levels
25275:Quantum mechanics
25184:978-0-19-856674-8
25161:978-0-691-14034-6
25136:978-0-521-55001-7
25049:978-981-02-3355-6
24983:978-0-8218-6941-3
24961:978-0-486-44528-1
24941:978-0-521-33859-2
24918:978-0-521-25979-8
24890:978-981-283-690-8
24856:978-981-238-107-1
24832:978-0-8176-4790-2
24810:978-0-19-851572-2
24791:978-981-283-726-4
24772:978-981-02-0656-7
24745:978-1-4614-7115-8
24695:978-3-540-57135-3
24676:978-0-387-90562-4
24640:Gell-Mann, Murray
24631:978-0-486-47722-0
24608:
24601:978-0-07-020650-2
24546:
24539:978-981-256-366-8
24503:
24421:Dirac, Paul A. M.
24364:978-0-7503-0801-4
24272:Annals of Physics
23719:Langevin equation
23675:
23604:WKB approximation
23595:Quantum tunneling
23524:
23512:
23497:
23485:
23471:
23443:
23431:
23402:
23390:
23338:
23326:
23297:
23285:
23234:
23193:
23148:
23136:
23121:
23109:
23095:
23011:
22828:
22749:
22674:
22595:
22293:), but just that
22226:
21699:anomaly (physics)
21617:as well, such as
21489:Noether's theorem
21468:causally complete
21402:
21187:
21161:
21064:
20984:
20953:
20902:
20871:
20784:
20702:
20632:
20527:Fourier transform
20447:
20007:
19659:
19596:
19383:
19289:
19227:
19185:
19122:
19074:
18933:
18625:
18534:
18522:
18500:
18464:
18452:
18430:
18322:
18319:
18302:
18244:
18173:
18170:
18148:
18112:
18109:
18087:
18016:partial fractions
17996:
17973:
17880:
17701:
17465:
17166:
17157:
17000:
16988:
16970:
16913:
16840:
16828:
16811:
16761:
16706:
16612:
15880:
15879:
14943:Expectation value
14918:BRST quantization
14865:Poincaré symmetry
14820:Yang–Mills theory
14800:Quantum mechanics
14723:Erwin Schrödinger
14613:
14350:
14271:
14251:
14175:
14124:
14089:
13773:{\displaystyle x}
13640:
13619:
13560:
13459:
13441:
13416:
13221:
13126:
13060:
13010:
12947:{\displaystyle t}
12900:
12883:
12833:
12733:
12692:
12492:
12481:
12459:
12403:
12392:
12370:
12321:
12310:
12235:
12194:
12023:
12012:
11990:
11950:
11939:
11818:
11799:
11774:
11742:For instance, if
11570:
11566:
11487:
11464:
11434:
11394:corresponding to
11292:
11191:
11178:
11064:
11001:
10948:
10891:
10826:
10749:
10704:
10633:
10483:
10359:
10336:
10332:
10307:
10168:
10157:
10153:
10122:
10006:
9991:
9806:
9650:
9625:
9566:
9400:
9388:
9230:
9218:
9025:İsmail Hakkı Duru
9008:
8938:Coulomb potential
8932:Coulomb potential
8914:
8898:
8810:
8791:
8733:
8643:
8617:
8603:
8463:
8382:
8270:
8233:
8179:
8153:
8139:
7959:
7890:
7813:
7709:
7698:
7675:
7661:
7503:
7502:
7436:
7407:
7406:
7326:
7319:
7210:
7176:
7091:
7087:
7086:
7000:
6971:
6883:
6876:
6794:
6739:
6714:
6671:
6670:
6632:
6625:
6461:
6406:
6380:
6344:
6296:
6288:
6129:
6073:
6052:
6036:
5937:
5898:
5808:
5708:
5591:
5570:
5554:
5482:
5449:
5192:
5151:
5109:
4997:
4959:
4841:
4738:
4689:
4650:
4601:
4528:
4497:
4354:of this Gaussian
4265:
4217:
4184:
4059:
4047:
3890:
3878:
3809:
3771:
3718:
3676:{\displaystyle S}
3656:{\displaystyle Z}
3503:
3428:
3267:
3219:
3169:
3097:
3078:
3049:
3014:is the classical
2941:, this becomes a
2740:
2641:
2548:smaller segments
2221:stationary points
2056:is a function of
1732:Fourier transform
1480:
1456:
1409:
1347:
1313:
1309:
1163:has an extremum.
926:quantum amplitude
910:quantum mechanics
902:
901:
609:Scattering theory
589:Quantum computing
362:Schrödinger's cat
294:Bell's inequality
102:
77:
46:Quantum mechanics
16:(Redirected from
26444:
26384:
26376:
26363:
26316:
26307:
26287:
26276:
26265:
26254:
26243:
26232:
26221:
26210:
26199:
26188:
26075:
26068:
26061:
26052:
26051:
26038:
26037:
25749:Quantum geometry
25744:Quantum dynamics
25601:Superdeterminism
25533:Rarita–Schwinger
25482:Matrix mechanics
25337:Bra–ket notation
25268:
25261:
25254:
25245:
25244:
25210:
25188:
25165:
25153:
25139:
25124:
25110:
25100:
25090:
25059:Van Vleck, J. H.
25053:
25031:
24997:
24987:
24965:
24945:
24933:
24922:
24903:
24894:
24875:
24873:
24871:quant-ph/0004090
24860:
24836:
24819:Klauder, John R.
24814:
24795:
24776:
24757:
24714:
24712:
24699:
24680:
24668:
24657:
24635:
24612:Feynman, R. P.;
24606:
24605:
24593:
24582:
24556:
24544:
24543:
24501:
24500:
24488:
24486:
24485:
24456:
24439:
24429:
24416:
24368:
24345:
24334:10.1063/1.531039
24319:
24317:funct-an/9602005
24310:(5): 2137–2340.
24295:
24259:
24240:
24218:
24213:
24207:
24202:
24196:
24191:
24185:
24184:
24182:
24181:
24165:
24159:
24158:
24156:
24124:
24118:
24117:, pp. 29–31
24112:
24106:
24100:
24094:
24089:
24083:
24077:
24071:
24065:
24059:
24054:
24048:
24042:
24036:
24035:
24029:
24020:
24014:
24009:
24003:
23997:
23991:
23990:
23988:
23976:
23970:
23965:
23959:
23954:
23948:
23943:
23937:
23936:
23934:
23933:
23919:Wolfram, Stephen
23915:
23909:
23908:
23906:
23905:
23888:
23882:
23881:
23879:
23867:
23858:
23857:
23855:
23854:
23848:
23842:. Archived from
23841:
23832:
23826:
23820:
23803:
23796:
23790:
23783:
23746:Berezin integral
23712:
23703:
23691:
23689:
23688:
23683:
23681:
23677:
23676:
23671:
23670:
23669:
23653:
23637:
23636:
23635:
23609:
23539:
23537:
23536:
23531:
23526:
23525:
23517:
23514:
23513:
23505:
23499:
23498:
23490:
23487:
23486:
23478:
23472:
23464:
23455:
23453:
23452:
23447:
23445:
23444:
23436:
23433:
23432:
23424:
23414:
23412:
23411:
23406:
23404:
23403:
23395:
23392:
23391:
23383:
23373:
23371:
23370:
23365:
23350:
23348:
23347:
23342:
23340:
23339:
23331:
23328:
23327:
23319:
23309:
23307:
23306:
23301:
23299:
23298:
23290:
23287:
23286:
23278:
23268:
23266:
23265:
23260:
23245:
23243:
23242:
23237:
23235:
23230:
23222:
23204:
23202:
23201:
23196:
23194:
23189:
23181:
23163:
23161:
23160:
23155:
23150:
23149:
23141:
23138:
23137:
23129:
23123:
23122:
23114:
23111:
23110:
23102:
23096:
23088:
23045:
23043:
23042:
23037:
23017:
23013:
23012:
23010:
22999:
22949:
22948:
22923:
22921:
22920:
22915:
22907:
22903:
22875:
22862:
22860:
22859:
22854:
22834:
22830:
22829:
22827:
22816:
22755:
22751:
22750:
22748:
22737:
22715:
22714:
22705:
22704:
22680:
22676:
22675:
22673:
22662:
22601:
22597:
22596:
22594:
22583:
22536:
22534:
22533:
22528:
22520:
22516:
22506:
22505:
22496:
22495:
22366:
22364:
22363:
22358:
22316:
22315:
22296:
22284:
22273:
22266:
22264:
22263:
22258:
22234:
22233:
22227:
22225:
22218:
22217:
22198:
22184:
22183:
22162:
22161:
22139:
22137:
22136:
22131:
22119:
22118:
22109:
22108:
22062:
22060:
22059:
22054:
22015:
22014:
21953:
21951:
21950:
21945:
21927:
21926:
21898:
21888:
21886:
21885:
21880:
21872:
21868:
21867:
21866:
21853:
21852:
21843:
21842:
21785:
21783:
21782:
21777:
21757:
21753:
21752:
21751:
21724:
21723:
21695:
21688:
21686:
21685:
21680:
21647:
21646:
21608:
21604:
21594:
21590:
21586:
21579:
21577:
21576:
21571:
21560:
21559:
21550:
21549:
21525:
21524:
21442:
21440:
21439:
21434:
21414:
21413:
21404:
21403:
21398:
21393:
21380:
21369:
21359:
21355:
21350:
21346:
21342:
21331:
21329:
21328:
21323:
21312:
21311:
21281:
21280:
21272:
21271:
21251:
21249:
21248:
21243:
21193:
21189:
21188:
21186:
21175:
21162:
21160:
21143:
21142:
21141:
21131:
21116:
21114:
21113:
21108:
21100:
21099:
21094:
21090:
21065:
21063:
21046:
21045:
21044:
21034:
21010:
21008:
21007:
21002:
20985:
20983:
20979:
20978:
20956:
20954:
20952:
20950:
20949:
20948:
20938:
20925:
20924:
20923:
20922:
20908:
20903:
20901:
20897:
20896:
20874:
20872:
20870:
20868:
20867:
20866:
20856:
20843:
20842:
20841:
20840:
20826:
20824:
20823:
20790:
20786:
20785:
20783:
20772:
20746:
20742:
20735:
20733:
20732:
20727:
20719:
20718:
20703:
20701:
20699:
20698:
20697:
20687:
20674:
20673:
20672:
20671:
20657:
20649:
20648:
20633:
20631:
20629:
20628:
20627:
20617:
20604:
20603:
20602:
20601:
20587:
20560:
20556:
20552:
20546:
20539:
20535:
20524:
20516:
20515:
20476:
20469:
20461:
20459:
20458:
20453:
20448:
20446:
20445:
20444:
20443:
20439:
20411:
20410:
20388:
20387:
20377:
20376:
20375:
20374:
20370:
20342:
20341:
20307:
20306:
20296:
20291:
20290:
20262:
20260:
20259:
20254:
20249:
20248:
20243:
20239:
20238:
20237:
20236:
20235:
20218:
20217:
20216:
20215:
20183:
20182:
20161:
20160:
20159:
20158:
20146:
20145:
20115:
20113:
20112:
20107:
20102:
20101:
20096:
20092:
20088:
20087:
20066:
20065:
20030:
20029:
20008:
20006:
20002:
20001:
19977:
19976:
19957:
19953:
19952:
19942:
19927:
19925:
19924:
19919:
19914:
19913:
19912:
19908:
19880:
19879:
19857:
19856:
19818:
19806:affine structure
19795:
19788:
19781:
19769:
19767:
19766:
19761:
19756:
19752:
19751:
19750:
19742:
19741:
19717:
19713:
19712:
19687:this looks like
19682:
19675:
19673:
19672:
19667:
19665:
19661:
19660:
19658:
19650:
19640:
19639:
19629:
19601:
19597:
19595:
19587:
19570:
19550:
19538:
19531:
19529:
19528:
19523:
19521:
19520:
19486:
19484:
19483:
19478:
19473:
19472:
19462:
19461:
19427:turns out to be
19426:
19422:
19400:
19398:
19397:
19392:
19384:
19382:
19374:
19364:
19363:
19353:
19315:
19303:
19301:
19300:
19295:
19290:
19288:
19287:
19286:
19281:
19277:
19264:
19263:
19253:
19252:
19229:
19228:
19225:
19216:
19215:
19199:
19183:
19182:
19181:
19176:
19172:
19159:
19158:
19148:
19147:
19124:
19123:
19120:
19111:
19110:
19094:
19072:
19036:
19035:
19033:
19032:
19025:
19022:
18997:
18979:
18975:
18966:As a probability
18960:
18956:
18947:
18945:
18944:
18939:
18934:
18932:
18931:
18930:
18920:
18919:
18902:
18901:
18891:
18890:
18889:
18879:
18878:
18849:
18848:
18838:
18814:
18813:
18804:
18789:
18783:is given by the
18758:
18720:
18705:
18694:
18656:
18646:
18639:
18637:
18636:
18631:
18626:
18624:
18614:
18613:
18601:
18600:
18587:
18560:
18548:
18546:
18545:
18540:
18535:
18533:
18523:
18521:
18520:
18508:
18507:
18502:
18501:
18493:
18489:
18484:
18483:
18470:
18465:
18463:
18453:
18451:
18450:
18438:
18437:
18432:
18431:
18423:
18419:
18414:
18413:
18400:
18373:
18364:
18357:
18342:Green's function
18336:
18334:
18333:
18328:
18323:
18321:
18320:
18318:
18310:
18309:
18304:
18303:
18295:
18291:
18277:
18276:
18260:
18246:
18245:
18242:
18220:
18207:
18203:
18187:
18185:
18184:
18179:
18174:
18172:
18171:
18169:
18168:
18156:
18155:
18150:
18149:
18141:
18137:
18132:
18131:
18118:
18113:
18111:
18110:
18108:
18107:
18095:
18094:
18089:
18088:
18080:
18076:
18071:
18070:
18057:
18040:
18039:
18010:
18008:
18007:
18002:
17997:
17995:
17994:
17993:
17981:
17980:
17975:
17974:
17966:
17958:
17953:
17940:
17913:
17906:
17894:
17892:
17891:
17886:
17881:
17879:
17872:
17871:
17858:
17853:
17844:
17843:
17839:
17831:
17830:
17821:
17807:
17802:
17768:
17764:
17760:
17756:
17737:
17735:
17734:
17729:
17724:
17715:
17714:
17713:
17702:
17700:
17695:
17694:
17693:
17671:
17660:
17655:
17615:
17610:
17600:
17598:
17597:
17592:
17587:
17575:
17561:
17537:
17532:
17492:
17488:
17481:
17479:
17478:
17473:
17468:
17467:
17466:
17464:
17459:
17458:
17457:
17435:
17425:
17424:
17423:
17401:
17368:
17360:
17355:
17348:
17346:
17345:
17340:
17335:
17334:
17321:
17311:
17299:
17275:
17274:
17273:
17251:
17218:
17195:
17193:
17192:
17187:
17182:
17181:
17167:
17165:
17164:
17159:
17158:
17150:
17146:
17143:
17142:
17136:
17118:
17108:
17096:
17066:
17029:
17026:, the parameter
17025:
17021:
17014:
17012:
17011:
17006:
17001:
16999:
16989:
16987:
16979:
16978:
16977:
16972:
16971:
16963:
16955:
16943:
16935:
16915:
16914:
16911:
16895:
16888:
16880:
16876:
16872:
16868:
16861:
16854:
16852:
16851:
16846:
16841:
16839:
16829:
16827:
16819:
16818:
16813:
16812:
16804:
16800:
16791:
16783:
16763:
16762:
16759:
16743:
16728:
16721:
16719:
16718:
16713:
16708:
16707:
16705:
16697:
16696:
16695:
16679:
16642:
16627:
16625:
16624:
16619:
16614:
16613:
16611:
16603:
16602:
16601:
16573:
16530:
16523:
16521:
16520:
16515:
16492:
16491:
16479:
16478:
16455:
16443:
16425:
16423:
16422:
16417:
16405:
16404:
16365:
16364:
16355:
16354:
16287:
16286:
16277:
16276:
16255:
16254:
16235:
16221:
16210:
16208:
16207:
16202:
16190:
16189:
16167:
16147:
16056:
16052:
16048:
15987:
15983:
15976:
15974:
15973:
15968:
15951:
15950:
15920:
15919:
15872:
15865:
15858:
14933:Effective action
14860:Lorentz symmetry
14785:Electromagnetism
14755:
14741:
14740:
14717:
14715:
14714:
14709:
14707:
14703:
14702:
14661:
14657:
14650:
14647:where the state
14643:
14641:
14640:
14635:
14621:
14616:
14615:
14614:
14609:
14598:
14573:
14550:
14549:
14547:
14546:
14541:
14538:
14522:
14520:
14519:
14514:
14511:
14495:
14475:
14473:
14472:
14467:
14456:
14448:
14447:
14434:
14432:
14431:
14426:
14424:
14419:
14418:
14405:
14395:
14381:
14379:
14378:
14373:
14352:
14351:
14346:
14341:
14326:
14314:
14313:
14312:
14302:
14287:
14279:
14278:
14272:
14269:
14266:
14261:
14260:
14253:
14252:
14247:
14243:
14235:
14234:
14224:
14187:
14185:
14184:
14179:
14177:
14176:
14168:
14154:
14152:
14151:
14146:
14126:
14125:
14117:
14090:
14088:
14077:
14062:
14060:
14059:
14054:
14023:
14021:
14020:
14015:
14010:
14002:
14001:
13976:
13968:
13967:
13957:
13956:
13952:
13928:
13865:
13863:
13862:
13857:
13840:
13828:
13826:
13825:
13820:
13818:
13806:
13804:
13803:
13798:
13796:
13795:
13779:
13777:
13776:
13771:
13760:. The subscript
13751:
13749:
13748:
13743:
13741:
13740:
13720:
13718:
13717:
13712:
13707:
13686:
13684:
13683:
13678:
13675:
13670:
13669:
13662:
13661:
13654:
13653:
13648:
13642:
13641:
13636:
13631:
13628:
13620:
13612:
13599:
13588:
13587:
13571:
13561:
13553:
13530:
13528:
13527:
13522:
13513:
13509:
13496:
13482:
13481:
13476:
13461:
13460:
13455:
13450:
13447:
13442:
13434:
13418:
13417:
13412:
13407:
13401:
13393:
13392:
13391:
13347:
13345:
13344:
13339:
13337:
13336:
13335:
13290:
13288:
13287:
13282:
13279:
13274:
13273:
13254:
13246:
13245:
13236:
13235:
13231:
13223:
13222:
13217:
13212:
13206:
13198:
13197:
13196:
13154:
13153:
13137:
13127:
13119:
13082:
13080:
13079:
13074:
13072:
13071:
13067:
13062:
13061:
13053:
13032:
13030:
13029:
13024:
13022:
13021:
13017:
13012:
13011:
13003:
12979:
12977:
12976:
12971:
12953:
12951:
12950:
12945:
12911:
12909:
12908:
12903:
12901:
12899:
12898:
12897:
12884:
12876:
12860:
12859:
12849:
12848:
12847:
12834:
12826:
12798:
12797:
12787:
12759:
12757:
12756:
12751:
12749:
12748:
12735:
12734:
12726:
12693:
12685:
12672:
12671:
12653:
12652:
12637:
12636:
12618:
12617:
12599:
12598:
12591:
12590:
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9898:
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9259:
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8321:
8319:
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7798:
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7634:
7626:
7609:
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7596:
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7540:
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7509:
7504:
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7432:
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7408:
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7307:
7292:
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7279:
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7266:
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7141:
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7101:
7096:
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7039:
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7017:
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7009:
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7003:
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6993:
6987:
6983:
6982:
6981:
6976:
6972:
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6962:
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6945:
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6885:
6884:
6879:
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6852:
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6833:
6821:
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6808:
6807:
6795:
6793:
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6779:
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6715:
6713:
6702:
6688:
6687:
6672:
6666:
6665:
6657:
6654:
6649:
6634:
6633:
6628:
6627:
6626:
6623:
6613:
6594:
6593:
6581:
6580:
6568:
6567:
6555:
6554:
6522:
6520:
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6514:
6509:
6505:
6501:
6500:
6488:
6487:
6475:
6474:
6462:
6460:
6459:
6458:
6446:
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6435:
6434:
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6414:
6407:
6399:
6396:
6391:
6382:
6373:
6369:
6364:
6346:
6345:
6342:
6320:
6313:
6311:
6310:
6305:
6303:
6294:
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6283:
6282:
6270:
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6236:
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6217:
6216:
6204:
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6178:
6173:
6160:
6155:
6142:
6130:
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6114:
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6038:
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6020:
6019:
6018:
6008:
6007:
6006:
5982:
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5974:
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5897:
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5892:
5880:
5879:
5857:
5853:
5852:
5824:
5822:
5821:
5809:
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5790:
5789:
5767:
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5734:
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5709:
5706:
5690:
5673:
5643:
5629:
5627:
5626:
5621:
5616:
5615:
5606:
5605:
5593:
5584:
5578:
5577:
5572:
5571:
5563:
5556:
5547:
5541:
5540:
5506:
5504:
5503:
5498:
5493:
5492:
5483:
5478:
5477:
5468:
5460:
5459:
5450:
5448:
5437:
5422:
5418:
5408:
5406:
5405:
5400:
5392:
5391:
5384:
5383:
5370:
5350:
5317:
5316:
5261:
5260:
5236:
5235:
5216:
5209:
5207:
5206:
5201:
5193:
5188:
5187:
5178:
5152:
5150:
5139:
5124:
5122:
5121:
5116:
5111:
5110:
5108:
5100:
5099:
5098:
5073:
5026:
5021:
5014:
5012:
5011:
5006:
4998:
4993:
4992:
4983:
4960:
4958:
4947:
4932:
4930:
4929:
4924:
4857:
4855:
4854:
4849:
4844:
4843:
4842:
4840:
4832:
4831:
4830:
4808:
4761:
4754:
4752:
4751:
4746:
4741:
4740:
4739:
4734:
4733:
4732:
4719:
4691:
4690:
4682:
4666:
4664:
4663:
4658:
4653:
4652:
4651:
4646:
4645:
4644:
4631:
4609:
4608:
4603:
4602:
4594:
4580:
4573:
4571:
4570:
4565:
4560:
4559:
4555:
4536:
4535:
4530:
4529:
4521:
4499:
4498:
4490:
4477:
4476:
4474:
4473:
4468:
4465:
4452:
4450:
4449:
4444:
4439:
4438:
4420:
4419:
4407:
4406:
4360:
4349:
4345:
4334:
4319:
4315:
4308:
4306:
4305:
4300:
4292:
4291:
4284:
4280:
4276:
4275:
4270:
4266:
4261:
4226:
4219:
4210:
4192:
4182:
4162:
4098:
4096:
4095:
4090:
4082:
4081:
4074:
4070:
4066:
4060:
4055:
4054:
4049:
4048:
4040:
4036:
4033:
4028:
4004:
3984:
3923:
3913:
3911:
3910:
3905:
3897:
3891:
3886:
3885:
3880:
3879:
3871:
3867:
3846:
3839:
3832:
3828:
3810:
3795:
3793:
3792:
3787:
3773:
3772:
3767:
3762:
3747:
3720:
3719:
3714:
3709:
3703:
3682:
3680:
3679:
3674:
3662:
3660:
3659:
3654:
3642:
3640:
3639:
3634:
3617:
3605:
3603:
3602:
3597:
3595:
3583:
3581:
3580:
3575:
3573:
3568:
3567:
3551:
3549:
3548:
3543:
3528:
3520:
3519:
3510:
3509:
3505:
3504:
3499:
3494:
3488:
3468:
3463:
3462:
3456:
3455:
3439:
3429:
3421:
3377:
3376:
3374:
3373:
3370:
3367:
3344:
3340:
3333:
3317:
3310:
3294:
3292:
3291:
3286:
3284:
3280:
3279:
3275:
3268:
3263:
3262:
3261:
3243:
3242:
3232:
3227:
3226:
3221:
3220:
3212:
3199:
3188:
3170:
3162:
3136:
3134:
3133:
3128:
3111:
3110:
3105:
3099:
3098:
3090:
3087:
3079:
3071:
3051:
3050:
3042:
3013:
3006:
2999:
2992:
2985:
2978:
2977:
2940:
2930:
2926:
2922:
2915:
2900:
2889:
2871:
2869:
2868:
2863:
2858:
2857:
2840:
2839:
2826:
2822:
2814:
2813:
2780:
2779:
2769:
2768:
2767:
2757:
2756:
2755:
2741:
2733:
2719:
2711:
2695:
2687:
2655:
2653:
2652:
2647:
2642:
2640:
2629:
2628:
2627:
2615:
2614:
2604:
2577:
2566:
2547:
2537:
2535:
2534:
2529:
2527:
2526:
2514:
2513:
2495:
2494:
2482:
2481:
2457:
2456:
2444:
2443:
2431:
2430:
2411:
2404:
2397:
2390:
2270:
2266:
2218:
2208:
2201:
2197:
2193:
2189:
2188:
2187:
2178:
2172:
2168:
2164:
2160:
2153:
2149:
2145:
2138:
2131:
2117:
2113:
2106:
2102:
2098:
2055:
2051:
2033:
2022:
2014:
2010:
2006:
1996:
1989:
1987:
1986:
1981:
1976:
1975:
1951:representation:
1950:
1939:
1935:
1931:
1927:
1923:
1916:
1912:
1908:
1899:
1890:
1888:
1887:
1882:
1877:
1876:
1826:
1824:
1823:
1818:
1813:
1812:
1765:
1761:
1758:change basis to
1755:
1744:
1729:
1715:
1713:
1712:
1707:
1702:
1701:
1657:
1655:
1654:
1649:
1644:
1643:
1642:
1641:
1611:
1610:
1571:
1570:
1561:
1560:
1536:
1532:
1528:
1524:
1520:
1516:
1505:
1501:
1494:
1492:
1491:
1486:
1481:
1479:
1471:
1463:
1458:
1457:
1449:
1433:
1431:
1430:
1425:
1411:
1410:
1402:
1374:
1359:
1357:
1356:
1351:
1349:
1348:
1340:
1327:
1325:
1324:
1319:
1314:
1312:
1311:
1310:
1302:
1295:
1287:
1266:
1264:
1263:
1258:
1247:
1246:
1207:
1206:
1137:reference frames
1115:
957:coordinate space
894:
887:
880:
521:Superdeterminism
174:Bra–ket notation
125:
123:
122:
117:
109:
104:
103:
95:
83:
78:
76:
65:
37:
36:
21:
26452:
26451:
26447:
26446:
26445:
26443:
26442:
26441:
26397:
26396:
26395:
26390:
26374:
26319:
26314:
26305:
26290:
26285:
26274:
26263:
26252:
26241:
26230:
26219:
26208:
26197:
26186:
26170:
26096:Feynman diagram
26084:
26082:Richard Feynman
26079:
26049:
26044:
26026:
26012:Wigner's friend
25988:
25979:Quantum gravity
25940:
25926:Quantum sensing
25906:Quantum network
25886:Quantum machine
25856:Quantum imaging
25819:Quantum circuit
25814:Quantum channel
25773:
25719:Quantum biology
25705:
25681:Elitzur–Vaidman
25656:Davisson–Germer
25639:
25591:Hidden-variable
25581:de Broglie–Bohm
25558:Interpretations
25552:
25501:
25455:
25342:Complementarity
25320:
25277:
25272:
25218:
25213:
25207:
25193:Deshmukh, P. C.
25185:
25171:Zinn Justin, J.
25162:
25137:
25050:
24995:
24984:
24962:
24942:
24919:
24891:
24857:
24841:Kleinert, Hagen
24833:
24811:
24792:
24773:
24746:
24696:
24677:
24654:
24632:
24602:
24554:
24540:
24483:
24481:
24459:Physics Letters
24454:
24448:Kleinert, Hagen
24427:
24365:
24264:Caldeira, A. O.
24256:
24227:
24222:
24221:
24214:
24210:
24203:
24199:
24192:
24188:
24179:
24177:
24174:Einstein Online
24166:
24162:
24129:"Path integral"
24125:
24121:
24113:
24109:
24101:
24097:
24090:
24086:
24082:, Theorem 20.3.
24078:
24074:
24070:, Section 20.3.
24066:
24062:
24055:
24051:
24043:
24039:
24027:
24025:"Path Integral"
24021:
24017:
24010:
24006:
23998:
23994:
23977:
23973:
23966:
23962:
23955:
23951:
23944:
23940:
23931:
23929:
23916:
23912:
23903:
23901:
23898:Quanta Magazine
23889:
23885:
23868:
23861:
23852:
23850:
23846:
23839:
23833:
23829:
23821:
23817:
23812:
23807:
23806:
23802:, Section 20.6.
23797:
23793:
23784:
23780:
23775:
23770:
23731:
23723:Brownian motion
23711:
23705:
23702:
23696:
23659:
23658:
23654:
23652:
23648:
23644:
23631:
23630:
23626:
23618:
23615:
23614:
23607:
23597:
23573:
23571:Quantum gravity
23546:
23516:
23515:
23504:
23503:
23489:
23488:
23477:
23476:
23463:
23461:
23458:
23457:
23435:
23434:
23423:
23422:
23420:
23417:
23416:
23394:
23393:
23382:
23381:
23379:
23376:
23375:
23356:
23353:
23352:
23330:
23329:
23318:
23317:
23315:
23312:
23311:
23289:
23288:
23277:
23276:
23274:
23271:
23270:
23251:
23248:
23247:
23223:
23221:
23210:
23207:
23206:
23182:
23180:
23169:
23166:
23165:
23140:
23139:
23128:
23127:
23113:
23112:
23101:
23100:
23087:
23085:
23082:
23081:
23073:
23057:
23052:
23003:
22998:
22991:
22987:
22944:
22940:
22935:
22932:
22931:
22927:Alternatively,
22890:
22886:
22884:
22881:
22880:
22870:
22820:
22815:
22808:
22804:
22741:
22736:
22729:
22725:
22710:
22706:
22700:
22696:
22666:
22661:
22654:
22650:
22587:
22582:
22575:
22571:
22548:
22545:
22544:
22540:Alternatively,
22501:
22497:
22491:
22487:
22483:
22479:
22378:
22375:
22374:
22311:
22310:
22305:
22302:
22301:
22294:
22291:gauge principle
22275:
22271:
22229:
22228:
22213:
22209:
22202:
22197:
22179:
22175:
22157:
22153:
22151:
22148:
22147:
22114:
22110:
22104:
22100:
22074:
22071:
22070:
22004:
22000:
21965:
21962:
21961:
21922:
21918:
21907:
21904:
21903:
21896:
21862:
21858:
21848:
21844:
21835:
21831:
21827:
21823:
21797:
21794:
21793:
21744:
21740:
21736:
21732:
21719:
21718:
21713:
21710:
21709:
21693:
21642:
21641:
21636:
21633:
21632:
21615:antiderivations
21606:
21599:
21592:
21588:
21584:
21555:
21551:
21545:
21541:
21520:
21519:
21511:
21508:
21507:
21482:
21476:
21460:
21409:
21408:
21394:
21392:
21391:
21389:
21386:
21385:
21378:
21372:target manifold
21365:
21353:
21352:
21348:
21340:
21336:
21307:
21303:
21273:
21267:
21266:
21265:
21263:
21260:
21259:
21179:
21174:
21167:
21163:
21144:
21137:
21136:
21132:
21130:
21128:
21125:
21124:
21095:
21047:
21040:
21039:
21035:
21033:
21032:
21028:
21027:
21025:
21022:
21021:
20974:
20970:
20960:
20955:
20944:
20940:
20939:
20934:
20926:
20918:
20914:
20913:
20909:
20907:
20892:
20888:
20878:
20873:
20862:
20858:
20857:
20852:
20844:
20836:
20832:
20831:
20827:
20825:
20819:
20815:
20776:
20771:
20764:
20760:
20755:
20752:
20751:
20744:
20740:
20714:
20710:
20693:
20689:
20688:
20683:
20675:
20667:
20663:
20662:
20658:
20656:
20644:
20640:
20623:
20619:
20618:
20613:
20605:
20597:
20593:
20592:
20588:
20586:
20569:
20566:
20565:
20558:
20554:
20548:
20544:
20537:
20533:
20522:
20512:
20502:
20491:
20490:
20467:
20466:
20406:
20405:
20404:
20400:
20396:
20392:
20383:
20382:
20378:
20337:
20336:
20335:
20331:
20327:
20323:
20302:
20301:
20297:
20295:
20286:
20282:
20274:
20271:
20270:
20244:
20231:
20227:
20226:
20222:
20211:
20207:
20206:
20202:
20201:
20197:
20196:
20178:
20174:
20154:
20150:
20141:
20137:
20133:
20129:
20127:
20124:
20123:
20097:
20083:
20079:
20061:
20057:
20050:
20046:
20045:
20025:
20021:
19997:
19993:
19972:
19968:
19958:
19948:
19944:
19943:
19941:
19939:
19936:
19935:
19875:
19874:
19873:
19869:
19865:
19861:
19852:
19851:
19831:
19828:
19827:
19816:
19793:
19787:
19779:
19778:
19743:
19737:
19736:
19735:
19731:
19727:
19705:
19701:
19697:
19695:
19692:
19691:
19685:deWitt notation
19680:
19651:
19635:
19634:
19630:
19628:
19615:
19611:
19588:
19571:
19569:
19565:
19563:
19560:
19559:
19548:
19536:
19504:
19500:
19498:
19495:
19494:
19457:
19456:
19452:
19448:
19446:
19443:
19442:
19420:
19419:
19375:
19359:
19358:
19354:
19352:
19350:
19347:
19346:
19332:
19326:
19314:
19308:
19282:
19248:
19247:
19243:
19239:
19224:
19220:
19211:
19210:
19206:
19202:
19201:
19189:
19184:
19177:
19143:
19142:
19138:
19134:
19119:
19115:
19106:
19105:
19101:
19097:
19096:
19078:
19073:
19071:
19045:
19042:
19041:
19026:
19023:
19012:
19011:
19009:
18999:
18987:
18977:
18973:
18968:
18954:
18952:
18915:
18914:
18910:
18906:
18897:
18896:
18892:
18874:
18873:
18869:
18865:
18844:
18843:
18839:
18837:
18823:
18820:
18819:
18807:
18806:
18802:
18787:
18773:
18756:
18711:
18696:
18690:
18675:
18648:
18644:
18609:
18605:
18596:
18592:
18591:
18586:
18569:
18566:
18565:
18559:
18553:
18516:
18512:
18503:
18492:
18491:
18490:
18488:
18479:
18475:
18474:
18469:
18446:
18442:
18433:
18422:
18421:
18420:
18418:
18409:
18405:
18404:
18399:
18382:
18379:
18378:
18372:
18366:
18359:
18352:
18311:
18305:
18294:
18293:
18292:
18290:
18272:
18268:
18264:
18259:
18241:
18237:
18229:
18226:
18225:
18215:
18209:
18205:
18198:
18192:
18164:
18160:
18151:
18140:
18139:
18138:
18136:
18127:
18123:
18122:
18117:
18103:
18099:
18090:
18079:
18078:
18077:
18075:
18066:
18062:
18061:
18056:
18035:
18031:
18026:
18023:
18022:
17989:
17985:
17976:
17965:
17964:
17963:
17954:
17949:
17944:
17939:
17922:
17919:
17918:
17908:
17905:
17899:
17867:
17863:
17862:
17857:
17849:
17835:
17826:
17822:
17817:
17813:
17809:
17803:
17798:
17777:
17774:
17773:
17766:
17762:
17758:
17746:
17720:
17709:
17696:
17689:
17685:
17672:
17670:
17666:
17662:
17656:
17651:
17624:
17621:
17620:
17613:
17605:
17583:
17571:
17557:
17533:
17528:
17501:
17498:
17497:
17490:
17486:
17460:
17453:
17449:
17436:
17434:
17430:
17426:
17419:
17412:
17408:
17397:
17377:
17374:
17373:
17366:
17358:
17353:
17327:
17323:
17307:
17300:
17280:
17269:
17262:
17258:
17247:
17227:
17224:
17223:
17201:
17160:
17149:
17148:
17147:
17145:
17138:
17137:
17132:
17124:
17120:
17104:
17097:
17077:
17062:
17042:
17039:
17038:
17027:
17023:
17019:
16980:
16973:
16962:
16961:
16960:
16956:
16954:
16944:
16936:
16934:
16910:
16906:
16904:
16901:
16900:
16890:
16889:is replaced by
16886:
16878:
16874:
16870:
16866:
16859:
16820:
16814:
16803:
16802:
16801:
16799:
16792:
16784:
16782:
16758:
16754:
16752:
16749:
16748:
16730:
16726:
16698:
16691:
16687:
16680:
16678:
16674:
16651:
16648:
16647:
16632:
16604:
16597:
16593:
16574:
16572:
16568:
16539:
16536:
16535:
16528:
16487:
16483:
16474:
16470:
16468:
16465:
16464:
16445:
16430:
16388:
16384:
16360:
16356:
16335:
16331:
16282:
16278:
16272:
16268:
16250:
16246:
16244:
16241:
16240:
16229:
16223:
16219:
16173:
16169:
16148:
16128:
16065:
16062:
16061:
16054:
16050:
16046:
16038:
15994:continuum limit
15985:
15981:
15946:
15942:
15915:
15911:
15891:
15888:
15887:
15876:
15847:
15846:
15845:
15843:
15137:
15129:
15128:
15124:Quantum gravity
15099:
15091:
15090:
15086:Higgs mechanism
15066:
15056:
15055:
15031:Proca equations
15016:
15008:
15007:
15003:Wightman Axioms
14988:Renormalization
14948:Feynman diagram
14903:
14895:
14894:
14835:
14825:
14824:
14775:
14760:
14758:Feynman diagram
14739:
14692:
14688:
14684:
14670:
14667:
14666:
14659:
14652:
14648:
14617:
14599:
14597:
14593:
14589:
14569:
14567:
14564:
14563:
14542:
14539:
14534:
14527:
14526:
14524:
14515:
14512:
14509:
14508:
14506:
14505:
14487:
14452:
14443:
14442:
14440:
14437:
14436:
14420:
14414:
14413:
14411:
14408:
14407:
14404:
14397:
14390:
14342:
14340:
14339:
14322:
14308:
14304:
14303:
14298:
14283:
14274:
14273:
14268:
14262:
14256:
14255:
14239:
14230:
14229:
14225:
14223:
14219:
14208:
14205:
14204:
14198:
14167:
14166:
14164:
14161:
14160:
14116:
14115:
14081:
14076:
14071:
14068:
14067:
14033:
14030:
14029:
14006:
13997:
13993:
13972:
13963:
13959:
13948:
13924:
13911:
13907:
13881:
13878:
13877:
13836:
13834:
13831:
13830:
13814:
13812:
13809:
13808:
13791:
13787:
13785:
13782:
13781:
13765:
13762:
13761:
13736:
13732:
13730:
13727:
13726:
13703:
13695:
13692:
13691:
13671:
13665:
13664:
13649:
13644:
13643:
13632:
13630:
13629:
13624:
13611:
13607:
13603:
13595:
13567:
13566:
13562:
13552:
13550:
13547:
13546:
13492:
13477:
13472:
13471:
13451:
13449:
13448:
13443:
13433:
13432:
13428:
13408:
13406:
13405:
13397:
13363:
13362:
13358:
13356:
13353:
13352:
13307:
13306:
13302:
13300:
13297:
13296:
13275:
13269:
13268:
13250:
13241:
13237:
13227:
13213:
13211:
13210:
13202:
13168:
13167:
13163:
13159:
13155:
13133:
13132:
13128:
13118:
13095:
13092:
13091:
13063:
13052:
13051:
13044:
13040:
13038:
13035:
13034:
13033:is replaced by
13013:
13002:
13001:
12991:
12987:
12985:
12982:
12981:
12959:
12956:
12955:
12939:
12936:
12935:
12932:
12919:
12875:
12874:
12870:
12855:
12854:
12850:
12825:
12824:
12820:
12793:
12792:
12788:
12786:
12772:
12769:
12768:
12725:
12724:
12684:
12683:
12679:
12667:
12663:
12648:
12644:
12632:
12628:
12613:
12609:
12594:
12593:
12585:
12581:
12565:
12558:
12554:
12538:
12522:
12518:
12513:
12502:
12498:
12484:
12483:
12473:
12469:
12465:
12451:
12450:
12441:
12437:
12426:
12422:
12413:
12409:
12395:
12394:
12384:
12380:
12376:
12362:
12361:
12352:
12348:
12337:
12333:
12313:
12312:
12302:
12298:
12294:
12289:
12283:
12279:
12274:
12271:
12270:
12227:
12226:
12186:
12185:
12181:
12167:
12140:
12139:
12131:
12127:
12111:
12104:
12100:
12084:
12078:
12075:
12074:
12051:
12047:
12042:
12029:
12015:
12014:
12004:
12000:
11996:
11982:
11981:
11962:
11942:
11941:
11931:
11927:
11923:
11918:
11912:
11908:
11903:
11900:
11899:
11896:
11883:
11882:
11874:
11872:
11871:
11821:
11810:
11809:
11808:
11802:
11791:
11790:
11789:
11780:
11776:
11766:
11765:
11761:
11750:
11747:
11746:
11714:
11713:
11694:
11690:
11673:
11670:
11669:
11663:
11643:
11604:
11601:
11600:
11558:
11557:
11553:
11545:
11543:
11536:
11532:
11530:
11527:
11526:
11479:
11471:
11469:
11456:
11448:
11446:
11426:
11425:
11414:
11411:
11410:
11395:
11391:
11387:
11363:
11359:
11329:
11328:
11326:
11323:
11322:
11308:
11282:
11276:
11275:
11274:
11235:
11234:
11233:
11231:
11214:
11211:
11210:
11183:
11168:
11162:
11161:
11160:
11121:
11120:
11119:
11117:
11091:
11088:
11087:
11080:
11026:
11024:
10993:
10985:
10983:
10966:
10963:
10962:
10910:
10908:
10883:
10875:
10873:
10856:
10853:
10852:
10845:
10821:
10783:
10780:
10779:
10768:
10766:
10765:
10711:
10709:
10696:
10688:
10686:
10684:
10681:
10680:
10666:
10639:
10625:
10617:
10615:
10611:
10610:
10599:
10596:
10595:
10588:
10581:
10577:
10574:
10565:
10561:
10553:
10550:
10541:
10537:
10503:
10499:
10475:
10467:
10465:
10464:
10460:
10454:
10450:
10449:
10445:
10443:
10440:
10439:
10425:
10394:
10390:
10351:
10343:
10341:
10324:
10323:
10319:
10311:
10309:
10299:
10294:
10293:
10289:
10285:
10281:
10257:
10253:
10238:
10234:
10229:
10226:
10225:
10218:
10187:
10183:
10160:
10159:
10145:
10144:
10140:
10132:
10130:
10114:
10106:
10104:
10100:
10096:
10075:
10071:
10056:
10052:
10047:
10044:
10043:
10036:
9998:
9997:
9983:
9982:
9960:
9956:
9935:
9931:
9916:
9912:
9907:
9904:
9903:
9885:
9874:
9847:
9836:
9798:
9797:
9781:
9777:
9756:
9752:
9737:
9733:
9728:
9725:
9724:
9714:
9710:
9707:
9657:
9653:
9641:
9640:
9636:
9617:
9609:
9607:
9605:
9602:
9601:
9594:
9558:
9551:
9547:
9531:
9529:
9525:
9501:
9497:
9447:
9444:
9443:
9436:
9432:
9421:
9391:
9380:
9379:
9378:
9376:
9372:
9368:
9344:
9340:
9338:
9335:
9334:
9323:
9319:
9315:
9255:
9254:
9233:
9222:
9221:
9220:
9210:
9204:
9203:
9191:
9186:
9178:
9174:
9147:
9127:
9099:
9091:
9058:
9055:
9054:
9047:
9041:
8994:
8986:
8984:
8973:
8970:
8969:
8962:
8952:
8949:
8944:
8943:
8941:
8940:
8934:
8889:
8882:
8878:
8869:
8865:
8863:
8860:
8859:
8807:
8783:
8782:
8778:
8765:
8761:
8740:
8736:
8716:
8714:
8710:
8708:
8705:
8704:
8695:
8689:
8678:
8626:
8624:
8620:
8609:
8595:
8587:
8585:
8581:
8580:
8568:
8564:
8555:
8551:
8542:
8538:
8529:
8525:
8517:
8514:
8513:
8499:
8493:
8444:
8440:
8433:
8417:
8413:
8407:
8403:
8397:
8393:
8389:
8387:
8363:
8359:
8352:
8333:
8329:
8322:
8320:
8309:
8304:
8291:
8286:
8281:
8277:
8276:
8272:
8262:
8254:
8252:
8248:
8244:
8243:
8225:
8221:
8198:
8194:
8187:
8183:
8182:
8162:
8160:
8156:
8145:
8131:
8123:
8121:
8117:
8116:
8104:
8100:
8091:
8087:
8078:
8074:
8065:
8061:
8053:
8050:
8049:
8030:
8027:
8024:
8023:
8021:
8015:
7999:
7996:
7993:
7992:
7990:
7981:
7964:
7963:
7951:
7947:
7941:
7937:
7928:
7924:
7915:
7911:
7902:
7898:
7877:
7873:
7869:
7867:
7863:
7859:
7858:
7846:
7835:
7822:
7821:
7799:
7792:
7788:
7782:
7778:
7751:
7746:
7733:
7728:
7720:
7718:
7700:
7690:
7689:
7685:
7684:
7667:
7635:
7627:
7625:
7621:
7620:
7613:
7604:
7600:
7591:
7587:
7578:
7574:
7565:
7561:
7551:
7549:
7546:
7545:
7538:
7531:
7522:
7492:
7488:
7479:
7475:
7450:
7442:
7439:
7426:
7422:
7418:
7416:
7412:
7396:
7392:
7383:
7379:
7366:
7356:
7352:
7343:
7339:
7332:
7329:
7316:
7312:
7308:
7306:
7302:
7287:
7283:
7274:
7270:
7261:
7257:
7248:
7244:
7236:
7233:
7232:
7202:
7188:
7186:
7170:
7166:
7160:
7156:
7154:
7147:
7143:
7137:
7126:
7120:
7117:
7116:
7076:
7072:
7063:
7059:
7043:
7037:
7029:
7026:
7025:
7005:
7004:
6992:
6988:
6977:
6963:
6953:
6949:
6940:
6936:
6929:
6927:
6923:
6922:
6915:
6911:
6910:
6904:
6893:
6873:
6869:
6865:
6863:
6859:
6850:
6849:
6829:
6825:
6816:
6812:
6803:
6799:
6787:
6783:
6774:
6770:
6769:
6762:
6758:
6748:
6746:
6745:
6741:
6731:
6725:
6720:
6706:
6701:
6700:
6696:
6695:
6683:
6679:
6658:
6656:
6650:
6639:
6622:
6618:
6614:
6612:
6608:
6598:
6589:
6585:
6576:
6572:
6563:
6559:
6550:
6546:
6536:
6534:
6531:
6530:
6496:
6492:
6483:
6479:
6470:
6466:
6454:
6450:
6441:
6437:
6436:
6429:
6425:
6415:
6413:
6412:
6408:
6398:
6392:
6387:
6371:
6365:
6354:
6341:
6337:
6329:
6326:
6325:
6318:
6301:
6300:
6278:
6274:
6265:
6261:
6248:
6241:
6237:
6231:
6227:
6212:
6208:
6199:
6195:
6174:
6169:
6156:
6151:
6143:
6141:
6137:
6121:
6112:
6111:
6093:
6089:
6083:
6079:
6064:
6055:
6044:
6043:
6042:
6027:
6026:
6022:
6014:
6010:
6009:
6002:
5998:
5997:
5977:
5976:
5968:
5964:
5963:
5956:
5952:
5951:
5940:
5934:
5930:
5926:
5924:
5921:
5920:
5888:
5884:
5875:
5871:
5858:
5848:
5844:
5825:
5823:
5817:
5813:
5798:
5794:
5785:
5781:
5768:
5752:
5748:
5735:
5733:
5727:
5723:
5705:
5701:
5699:
5696:
5695:
5685:
5675:
5659:
5645:
5634:
5611:
5607:
5601:
5597:
5582:
5573:
5562:
5561:
5560:
5545:
5536:
5535:
5533:
5530:
5529:
5523:
5513:
5488:
5484:
5473:
5469:
5467:
5455:
5451:
5441:
5436:
5431:
5428:
5427:
5420:
5417:
5413:
5387:
5386:
5376:
5372:
5351:
5331:
5312:
5308:
5256:
5252:
5231:
5227:
5225:
5222:
5221:
5214:
5183:
5179:
5177:
5143:
5138:
5136:
5133:
5132:
5101:
5094:
5090:
5074:
5072:
5068:
5039:
5036:
5035:
5024:
5019:
4988:
4984:
4982:
4951:
4946:
4944:
4941:
4940:
4881:
4878:
4877:
4833:
4826:
4822:
4809:
4807:
4803:
4799:
4770:
4767:
4766:
4759:
4728:
4724:
4720:
4718:
4714:
4710:
4681:
4680:
4678:
4675:
4674:
4640:
4636:
4632:
4630:
4626:
4622:
4604:
4593:
4592:
4591:
4589:
4586:
4585:
4578:
4551:
4547:
4543:
4531:
4520:
4519:
4518:
4489:
4488:
4486:
4483:
4482:
4469:
4466:
4461:
4460:
4458:
4457:
4434:
4430:
4415:
4411:
4402:
4398:
4369:
4366:
4365:
4359:
4355:
4347:
4336:
4321:
4317:
4313:
4287:
4286:
4271:
4227:
4225:
4221:
4220:
4208:
4204:
4200:
4188:
4163:
4143:
4110:
4107:
4106:
4077:
4076:
4062:
4050:
4039:
4038:
4037:
4035:
4029:
4024:
4016:
4012:
3985:
3965:
3932:
3929:
3928:
3921:
3893:
3881:
3870:
3869:
3868:
3866:
3855:
3852:
3851:
3841:
3834:
3830:
3826:
3823:
3799:
3763:
3761:
3760:
3743:
3710:
3708:
3707:
3699:
3691:
3688:
3687:
3668:
3665:
3664:
3648:
3645:
3644:
3613:
3611:
3608:
3607:
3591:
3589:
3586:
3585:
3569:
3563:
3562:
3560:
3557:
3556:
3524:
3515:
3511:
3495:
3493:
3492:
3484:
3474:
3470:
3464:
3458:
3457:
3435:
3434:
3430:
3420:
3397:
3394:
3393:
3387:
3371:
3368:
3366:
3355:
3350:
3349:
3347:
3346:
3342:
3339:
3335:
3331:
3325:
3319:
3316:
3312:
3309:
3303:
3251:
3247:
3238:
3234:
3233:
3231:
3222:
3211:
3210:
3209:
3208:
3204:
3189:
3178:
3161:
3160:
3156:
3148:
3145:
3144:
3106:
3101:
3100:
3089:
3088:
3083:
3070:
3041:
3040:
3026:
3023:
3022:
3011:
3005:
3001:
2998:
2994:
2991:
2987:
2984:
2980:
2974:
2967:
2960:
2953:
2947:
2946:
2935:
2928:
2924:
2921:
2917:
2902:
2891:
2876:
2853:
2849:
2835:
2831:
2809:
2808:
2775:
2774:
2763:
2759:
2758:
2751:
2747:
2746:
2732:
2731:
2727:
2712:
2704:
2688:
2680:
2674:
2671:
2670:
2630:
2623:
2619:
2610:
2606:
2605:
2603:
2586:
2583:
2582:
2568:
2565:
2554:
2549:
2542:
2522:
2518:
2503:
2499:
2490:
2486:
2471:
2467:
2452:
2448:
2439:
2435:
2426:
2422:
2420:
2417:
2416:
2410:
2406:
2403:
2399:
2396:
2392:
2389:
2385:
2371:
2365:
2360:
2348:Feynman diagram
2283:along the path.
2268:
2262:
2249:squared modulus
2229:
2216:
2214:Planck constant
2209:
2206:
2199:
2195:
2186:
2181:
2180:
2179:
2176:
2175:
2174:
2170:
2166:
2162:
2159:
2155:
2151:
2147:
2144:
2140:
2137:
2133:
2127:
2115:
2112:
2108:
2104:
2100:
2097:
2088:
2079:
2072:
2065:
2057:
2053:
2047:
2040:
2038:Classical limit
2024:
2017:
2012:
2008:
2007:. While in the
1998:
1994:
1965:
1961:
1959:
1956:
1955:
1948:
1937:
1933:
1929:
1925:
1921:
1914:
1910:
1906:
1900:at a later time
1897:
1851:
1847:
1845:
1842:
1841:
1784:
1780:
1778:
1775:
1774:
1763:
1759:
1746:
1735:
1720:
1679:
1675:
1673:
1670:
1669:
1637:
1636:
1606:
1605:
1566:
1565:
1556:
1555:
1551:
1547:
1545:
1542:
1541:
1534:
1530:
1526:
1522:
1518:
1514:
1503:
1499:
1472:
1464:
1462:
1448:
1447:
1445:
1442:
1441:
1401:
1400:
1383:
1380:
1379:
1361:
1339:
1338:
1336:
1333:
1332:
1301:
1300:
1296:
1288:
1286:
1278:
1275:
1274:
1242:
1241:
1202:
1201:
1178:
1175:
1174:
1110:
1098:
1080:(rather than a
1066:Richard Feynman
1054:Brownian motion
1050:Wiener integral
1039:Stephen Wolfram
1019:polymer physics
898:
869:
868:
867:
632:
624:
623:
569:
568:Advanced topics
561:
560:
559:
511:Hidden-variable
501:de Broglie–Bohm
480:
478:Interpretations
470:
469:
468:
438:
430:
429:
428:
386:
378:
377:
376:
343:
299:CHSH inequality
288:
280:
279:
278:
207:Complementarity
201:
193:
192:
191:
159:
130:
105:
94:
93:
79:
69:
64:
56:
53:
52:
35:
28:
23:
22:
15:
12:
11:
5:
26450:
26440:
26439:
26434:
26429:
26424:
26419:
26414:
26409:
26392:
26391:
26389:
26388:
26378:
26367:
26357:
26352:
26345:
26338:
26333:
26327:
26325:
26321:
26320:
26318:
26317:
26308:
26298:
26296:
26292:
26291:
26289:
26288:
26277:
26266:
26255:
26244:
26233:
26222:
26211:
26200:
26189:
26178:
26176:
26172:
26171:
26169:
26168:
26163:
26158:
26153:
26148:
26143:
26138:
26133:
26128:
26123:
26118:
26113:
26108:
26103:
26098:
26092:
26090:
26086:
26085:
26078:
26077:
26070:
26063:
26055:
26046:
26045:
26043:
26042:
26031:
26028:
26027:
26025:
26024:
26019:
26014:
26009:
26008:
26007:
25996:
25994:
25990:
25989:
25987:
25986:
25981:
25976:
25975:
25974:
25964:
25959:
25957:Casimir effect
25954:
25948:
25946:
25942:
25941:
25939:
25938:
25933:
25928:
25923:
25918:
25916:Quantum optics
25913:
25908:
25903:
25898:
25893:
25888:
25883:
25878:
25873:
25868:
25863:
25858:
25853:
25848:
25843:
25838:
25837:
25836:
25826:
25821:
25816:
25811:
25810:
25809:
25799:
25794:
25789:
25783:
25781:
25775:
25774:
25772:
25771:
25766:
25761:
25756:
25751:
25746:
25741:
25736:
25731:
25726:
25721:
25715:
25713:
25707:
25706:
25704:
25703:
25698:
25693:
25691:Quantum eraser
25688:
25683:
25678:
25673:
25668:
25663:
25658:
25653:
25647:
25645:
25641:
25640:
25638:
25637:
25632:
25627:
25622:
25617:
25612:
25607:
25606:
25605:
25604:
25603:
25588:
25583:
25578:
25573:
25568:
25562:
25560:
25554:
25553:
25551:
25550:
25545:
25540:
25535:
25530:
25525:
25520:
25515:
25509:
25507:
25503:
25502:
25500:
25499:
25494:
25489:
25484:
25479:
25474:
25469:
25463:
25461:
25457:
25456:
25454:
25453:
25452:
25451:
25446:
25436:
25431:
25426:
25421:
25416:
25411:
25406:
25401:
25396:
25391:
25386:
25381:
25376:
25375:
25374:
25369:
25364:
25359:
25349:
25347:Density matrix
25344:
25339:
25334:
25328:
25326:
25322:
25321:
25319:
25318:
25313:
25308:
25303:
25302:
25301:
25291:
25285:
25283:
25279:
25278:
25271:
25270:
25263:
25256:
25248:
25242:
25241:
25235:
25229:
25224:
25217:
25216:External links
25214:
25212:
25211:
25206:978-1316512258
25205:
25189:
25183:
25167:
25160:
25146:(2010-02-21).
25140:
25135:
25111:
25073:(2): 178–188.
25055:
25048:
25032:
25006:(4): 303–335.
24988:
24982:
24966:
24960:
24947:
24940:
24923:
24917:
24904:
24895:
24889:
24876:
24861:
24855:
24837:
24831:
24815:
24809:
24796:
24790:
24777:
24771:
24758:
24744:
24715:
24710:hep-th/9302097
24700:
24694:
24681:
24675:
24658:
24653:978-0883188705
24652:
24636:
24630:
24609:
24600:
24583:
24565:(2): 367–387.
24547:
24538:
24508:Feynman, R. P.
24504:
24489:
24465:(2): 185–188.
24440:
24417:
24369:
24363:
24351:"Introduction"
24346:
24300:Cartier, P. C.
24296:
24278:(2): 374–456.
24268:Leggett, A. J.
24260:
24254:
24241:
24228:
24226:
24223:
24220:
24219:
24208:
24205:Gell-Mann 1993
24197:
24186:
24160:
24119:
24107:
24095:
24084:
24072:
24060:
24049:
24037:
24015:
24004:
23992:
23971:
23968:Van Vleck 1928
23960:
23949:
23938:
23921:(2020-04-14).
23910:
23883:
23859:
23827:
23814:
23813:
23811:
23808:
23805:
23804:
23791:
23777:
23776:
23774:
23771:
23769:
23768:
23763:
23758:
23753:
23748:
23743:
23738:
23732:
23730:
23727:
23709:
23700:
23693:
23692:
23680:
23674:
23668:
23665:
23662:
23657:
23651:
23647:
23643:
23640:
23634:
23629:
23625:
23622:
23596:
23593:
23572:
23569:
23545:
23542:
23529:
23523:
23520:
23511:
23508:
23502:
23496:
23493:
23484:
23481:
23475:
23470:
23467:
23442:
23439:
23430:
23427:
23401:
23398:
23389:
23386:
23363:
23360:
23337:
23334:
23325:
23322:
23296:
23293:
23284:
23281:
23258:
23255:
23233:
23229:
23226:
23220:
23217:
23214:
23192:
23188:
23185:
23179:
23176:
23173:
23153:
23147:
23144:
23135:
23132:
23126:
23120:
23117:
23108:
23105:
23099:
23094:
23091:
23072:
23069:
23056:
23053:
23051:
23048:
23047:
23046:
23035:
23032:
23029:
23026:
23023:
23020:
23016:
23009:
23006:
23002:
22997:
22994:
22990:
22986:
22983:
22980:
22977:
22974:
22971:
22968:
22965:
22962:
22959:
22956:
22952:
22947:
22943:
22939:
22925:
22924:
22913:
22910:
22906:
22902:
22899:
22896:
22893:
22889:
22864:
22863:
22852:
22849:
22846:
22843:
22840:
22837:
22833:
22826:
22823:
22819:
22814:
22811:
22807:
22803:
22800:
22797:
22794:
22791:
22788:
22785:
22782:
22779:
22776:
22773:
22770:
22767:
22764:
22761:
22758:
22754:
22747:
22744:
22740:
22735:
22732:
22728:
22724:
22721:
22718:
22713:
22709:
22703:
22699:
22695:
22692:
22689:
22686:
22683:
22679:
22672:
22669:
22665:
22660:
22657:
22653:
22649:
22646:
22643:
22640:
22637:
22634:
22631:
22628:
22625:
22622:
22619:
22616:
22613:
22610:
22607:
22604:
22600:
22593:
22590:
22586:
22581:
22578:
22574:
22570:
22567:
22564:
22561:
22558:
22555:
22552:
22538:
22537:
22526:
22523:
22519:
22515:
22512:
22509:
22504:
22500:
22494:
22490:
22486:
22482:
22478:
22475:
22472:
22469:
22466:
22463:
22460:
22457:
22454:
22451:
22448:
22445:
22442:
22439:
22436:
22433:
22430:
22427:
22424:
22421:
22418:
22415:
22412:
22409:
22406:
22403:
22400:
22397:
22394:
22391:
22388:
22385:
22382:
22368:
22367:
22356:
22353:
22350:
22347:
22344:
22341:
22338:
22335:
22332:
22329:
22326:
22323:
22319:
22314:
22309:
22268:
22267:
22255:
22252:
22249:
22246:
22243:
22240:
22237:
22232:
22224:
22221:
22216:
22212:
22208:
22205:
22201:
22196:
22193:
22190:
22187:
22182:
22178:
22174:
22171:
22168:
22165:
22160:
22156:
22141:
22140:
22128:
22125:
22122:
22117:
22113:
22107:
22103:
22099:
22096:
22093:
22090:
22087:
22084:
22081:
22078:
22064:
22063:
22051:
22048:
22045:
22042:
22039:
22036:
22033:
22030:
22027:
22024:
22021:
22018:
22013:
22010:
22007:
22003:
21999:
21996:
21993:
21990:
21987:
21984:
21981:
21978:
21975:
21972:
21969:
21955:
21954:
21943:
21940:
21937:
21934:
21930:
21925:
21921:
21917:
21914:
21911:
21890:
21889:
21878:
21875:
21871:
21865:
21861:
21857:
21851:
21847:
21841:
21838:
21834:
21830:
21826:
21822:
21819:
21816:
21813:
21810:
21807:
21804:
21801:
21789:which implies
21787:
21786:
21775:
21772:
21769:
21766:
21763:
21760:
21756:
21750:
21747:
21743:
21739:
21735:
21731:
21727:
21722:
21717:
21690:
21689:
21678:
21675:
21672:
21669:
21666:
21663:
21660:
21657:
21654:
21650:
21645:
21640:
21581:
21580:
21569:
21566:
21563:
21558:
21554:
21548:
21544:
21540:
21537:
21534:
21531:
21528:
21523:
21518:
21515:
21497:gauge symmetry
21478:Main article:
21475:
21472:
21459:
21456:
21444:
21443:
21432:
21429:
21426:
21423:
21420:
21417:
21412:
21407:
21401:
21397:
21333:
21332:
21321:
21318:
21315:
21310:
21306:
21302:
21299:
21296:
21293:
21290:
21287:
21284:
21279:
21276:
21270:
21253:
21252:
21241:
21238:
21235:
21232:
21229:
21226:
21223:
21220:
21217:
21214:
21211:
21208:
21205:
21202:
21199:
21196:
21192:
21185:
21182:
21178:
21173:
21170:
21166:
21159:
21156:
21153:
21150:
21147:
21140:
21135:
21118:
21117:
21106:
21103:
21098:
21093:
21089:
21086:
21083:
21080:
21077:
21074:
21071:
21068:
21062:
21059:
21056:
21053:
21050:
21043:
21038:
21031:
21012:
21011:
21000:
20997:
20994:
20991:
20988:
20982:
20977:
20973:
20969:
20966:
20963:
20959:
20947:
20943:
20937:
20933:
20929:
20921:
20917:
20912:
20906:
20900:
20895:
20891:
20887:
20884:
20881:
20877:
20865:
20861:
20855:
20851:
20847:
20839:
20835:
20830:
20822:
20818:
20814:
20811:
20808:
20805:
20802:
20799:
20796:
20793:
20789:
20782:
20779:
20775:
20770:
20767:
20763:
20759:
20737:
20736:
20725:
20722:
20717:
20713:
20709:
20706:
20696:
20692:
20686:
20682:
20678:
20670:
20666:
20661:
20655:
20652:
20647:
20643:
20639:
20636:
20626:
20622:
20616:
20612:
20608:
20600:
20596:
20591:
20585:
20582:
20579:
20576:
20573:
20540:, then for an
20510:
20500:
20465:Basically, if
20463:
20462:
20451:
20442:
20438:
20435:
20432:
20429:
20426:
20423:
20420:
20417:
20414:
20409:
20403:
20399:
20395:
20391:
20386:
20381:
20373:
20369:
20366:
20363:
20360:
20357:
20354:
20351:
20348:
20345:
20340:
20334:
20330:
20326:
20322:
20319:
20316:
20313:
20310:
20305:
20300:
20294:
20289:
20285:
20281:
20278:
20264:
20263:
20252:
20247:
20242:
20234:
20230:
20225:
20221:
20214:
20210:
20205:
20200:
20195:
20192:
20189:
20186:
20181:
20177:
20173:
20170:
20167:
20164:
20157:
20153:
20149:
20144:
20140:
20136:
20132:
20117:
20116:
20105:
20100:
20095:
20091:
20086:
20082:
20078:
20075:
20072:
20069:
20064:
20060:
20056:
20053:
20049:
20043:
20040:
20037:
20034:
20028:
20024:
20020:
20017:
20014:
20011:
20005:
20000:
19996:
19992:
19989:
19986:
19983:
19980:
19975:
19971:
19967:
19964:
19961:
19956:
19951:
19947:
19929:
19928:
19917:
19911:
19907:
19904:
19901:
19898:
19895:
19892:
19889:
19886:
19883:
19878:
19872:
19868:
19864:
19860:
19855:
19850:
19847:
19844:
19841:
19838:
19835:
19782:
19771:
19770:
19759:
19755:
19749:
19746:
19740:
19734:
19730:
19726:
19723:
19720:
19716:
19711:
19708:
19704:
19700:
19677:
19676:
19664:
19657:
19654:
19649:
19646:
19643:
19638:
19633:
19627:
19624:
19621:
19618:
19614:
19610:
19607:
19604:
19600:
19594:
19591:
19586:
19583:
19580:
19577:
19574:
19568:
19551:, then we can
19533:
19532:
19519:
19516:
19513:
19510:
19507:
19503:
19488:
19487:
19476:
19471:
19468:
19465:
19460:
19455:
19451:
19402:
19401:
19390:
19387:
19381:
19378:
19373:
19370:
19367:
19362:
19357:
19328:Main article:
19325:
19322:
19312:
19305:
19304:
19293:
19285:
19280:
19276:
19273:
19270:
19267:
19262:
19259:
19256:
19251:
19246:
19242:
19238:
19235:
19232:
19223:
19219:
19214:
19209:
19205:
19198:
19195:
19192:
19188:
19180:
19175:
19171:
19168:
19165:
19162:
19157:
19154:
19151:
19146:
19141:
19137:
19133:
19130:
19127:
19118:
19114:
19109:
19104:
19100:
19093:
19090:
19087:
19084:
19081:
19077:
19070:
19067:
19064:
19061:
19058:
19055:
19052:
19049:
18967:
18964:
18949:
18948:
18937:
18929:
18926:
18923:
18918:
18913:
18909:
18905:
18900:
18895:
18888:
18885:
18882:
18877:
18872:
18868:
18864:
18861:
18858:
18855:
18852:
18847:
18842:
18836:
18833:
18830:
18827:
18815:, is given by
18772:
18769:
18689:of the field:
18674:
18671:
18663:interpretation
18641:
18640:
18629:
18623:
18620:
18617:
18612:
18608:
18604:
18599:
18595:
18590:
18585:
18582:
18579:
18576:
18573:
18557:
18550:
18549:
18538:
18532:
18529:
18526:
18519:
18515:
18511:
18506:
18499:
18496:
18487:
18482:
18478:
18473:
18468:
18462:
18459:
18456:
18449:
18445:
18441:
18436:
18429:
18426:
18417:
18412:
18408:
18403:
18398:
18395:
18392:
18389:
18386:
18370:
18338:
18337:
18326:
18317:
18314:
18308:
18301:
18298:
18289:
18286:
18283:
18280:
18275:
18271:
18267:
18263:
18258:
18255:
18252:
18249:
18240:
18236:
18233:
18213:
18196:
18189:
18188:
18177:
18167:
18163:
18159:
18154:
18147:
18144:
18135:
18130:
18126:
18121:
18116:
18106:
18102:
18098:
18093:
18086:
18083:
18074:
18069:
18065:
18060:
18055:
18052:
18049:
18046:
18043:
18038:
18034:
18030:
18012:
18011:
18000:
17992:
17988:
17984:
17979:
17972:
17969:
17962:
17957:
17952:
17948:
17943:
17938:
17935:
17932:
17929:
17926:
17903:
17896:
17895:
17884:
17878:
17875:
17870:
17866:
17861:
17856:
17852:
17848:
17842:
17838:
17834:
17829:
17825:
17820:
17816:
17812:
17806:
17801:
17797:
17793:
17790:
17787:
17784:
17781:
17739:
17738:
17727:
17723:
17719:
17712:
17708:
17705:
17699:
17692:
17688:
17684:
17681:
17678:
17675:
17669:
17665:
17659:
17654:
17650:
17646:
17643:
17640:
17637:
17634:
17631:
17628:
17602:
17601:
17590:
17586:
17582:
17578:
17574:
17570:
17567:
17564:
17560:
17556:
17553:
17550:
17547:
17544:
17541:
17536:
17531:
17527:
17523:
17520:
17517:
17514:
17511:
17508:
17505:
17483:
17482:
17471:
17463:
17456:
17452:
17448:
17445:
17442:
17439:
17433:
17429:
17422:
17418:
17415:
17411:
17407:
17404:
17400:
17396:
17393:
17390:
17387:
17384:
17381:
17350:
17349:
17338:
17333:
17330:
17326:
17320:
17317:
17314:
17310:
17306:
17303:
17298:
17295:
17292:
17289:
17286:
17283:
17279:
17272:
17268:
17265:
17261:
17257:
17254:
17250:
17246:
17243:
17240:
17237:
17234:
17231:
17197:
17196:
17185:
17180:
17177:
17173:
17170:
17163:
17156:
17153:
17141:
17135:
17131:
17127:
17123:
17117:
17114:
17111:
17107:
17103:
17100:
17095:
17092:
17089:
17086:
17083:
17080:
17076:
17072:
17069:
17065:
17061:
17058:
17055:
17052:
17049:
17046:
17016:
17015:
17004:
16998:
16995:
16992:
16986:
16983:
16976:
16969:
16966:
16959:
16953:
16950:
16947:
16942:
16939:
16933:
16930:
16927:
16924:
16921:
16918:
16909:
16856:
16855:
16844:
16838:
16835:
16832:
16826:
16823:
16817:
16810:
16807:
16798:
16795:
16790:
16787:
16781:
16778:
16775:
16772:
16769:
16766:
16757:
16723:
16722:
16711:
16704:
16701:
16694:
16690:
16686:
16683:
16677:
16673:
16670:
16667:
16664:
16661:
16658:
16655:
16629:
16628:
16617:
16610:
16607:
16600:
16596:
16592:
16589:
16586:
16583:
16580:
16577:
16571:
16567:
16564:
16561:
16558:
16555:
16552:
16549:
16546:
16543:
16525:
16524:
16513:
16510:
16507:
16504:
16501:
16498:
16495:
16490:
16486:
16482:
16477:
16473:
16427:
16426:
16415:
16412:
16409:
16403:
16400:
16397:
16394:
16391:
16387:
16383:
16380:
16377:
16374:
16371:
16368:
16363:
16359:
16353:
16350:
16347:
16344:
16341:
16338:
16334:
16330:
16327:
16324:
16320:
16317:
16314:
16311:
16308:
16305:
16302:
16299:
16296:
16293:
16290:
16285:
16281:
16275:
16271:
16267:
16264:
16261:
16258:
16253:
16249:
16227:
16212:
16211:
16200:
16197:
16194:
16188:
16185:
16182:
16179:
16176:
16172:
16166:
16163:
16160:
16157:
16154:
16151:
16146:
16143:
16140:
16137:
16134:
16131:
16127:
16123:
16120:
16117:
16114:
16111:
16108:
16105:
16102:
16099:
16096:
16093:
16090:
16087:
16084:
16081:
16078:
16075:
16072:
16069:
16037:
16034:
15978:
15977:
15966:
15963:
15960:
15957:
15954:
15949:
15945:
15941:
15938:
15935:
15932:
15929:
15926:
15923:
15918:
15914:
15910:
15907:
15904:
15901:
15898:
15895:
15878:
15877:
15875:
15874:
15867:
15860:
15852:
15849:
15848:
15841:
15840:
15835:
15830:
15825:
15820:
15815:
15810:
15805:
15800:
15795:
15790:
15785:
15780:
15775:
15770:
15765:
15760:
15755:
15750:
15745:
15740:
15735:
15730:
15725:
15720:
15715:
15710:
15705:
15700:
15695:
15690:
15685:
15680:
15675:
15670:
15665:
15660:
15655:
15650:
15645:
15640:
15635:
15630:
15625:
15620:
15615:
15610:
15605:
15600:
15595:
15590:
15585:
15580:
15575:
15570:
15565:
15560:
15555:
15550:
15545:
15540:
15535:
15530:
15525:
15520:
15515:
15510:
15505:
15500:
15495:
15490:
15485:
15480:
15475:
15470:
15465:
15460:
15455:
15450:
15445:
15440:
15435:
15430:
15425:
15420:
15415:
15410:
15405:
15400:
15395:
15390:
15385:
15380:
15375:
15370:
15365:
15360:
15355:
15350:
15345:
15340:
15335:
15330:
15325:
15320:
15315:
15310:
15305:
15300:
15295:
15290:
15285:
15280:
15275:
15270:
15265:
15260:
15255:
15250:
15245:
15240:
15235:
15230:
15225:
15220:
15215:
15210:
15205:
15200:
15195:
15190:
15185:
15180:
15175:
15170:
15165:
15160:
15155:
15150:
15145:
15139:
15138:
15135:
15134:
15131:
15130:
15127:
15126:
15121:
15116:
15111:
15106:
15100:
15097:
15096:
15093:
15092:
15089:
15088:
15083:
15078:
15073:
15067:
15064:Standard Model
15062:
15061:
15058:
15057:
15054:
15053:
15048:
15043:
15038:
15033:
15028:
15023:
15021:Dirac equation
15017:
15014:
15013:
15010:
15009:
15006:
15005:
15000:
14998:Wick's theorem
14995:
14990:
14985:
14983:Regularization
14980:
14975:
14970:
14965:
14960:
14955:
14950:
14945:
14940:
14935:
14930:
14925:
14920:
14915:
14910:
14904:
14901:
14900:
14897:
14896:
14893:
14892:
14887:
14885:Noether charge
14882:
14877:
14872:
14870:Gauge symmetry
14867:
14862:
14857:
14852:
14847:
14842:
14836:
14831:
14830:
14827:
14826:
14823:
14822:
14817:
14812:
14807:
14802:
14797:
14792:
14787:
14782:
14776:
14773:
14772:
14769:
14768:
14762:
14761:
14756:
14748:
14747:
14738:
14735:
14719:
14718:
14706:
14701:
14698:
14695:
14691:
14687:
14683:
14680:
14677:
14674:
14645:
14644:
14633:
14630:
14627:
14624:
14620:
14612:
14608:
14605:
14602:
14596:
14592:
14588:
14585:
14582:
14579:
14576:
14572:
14532:
14465:
14462:
14459:
14455:
14451:
14446:
14423:
14417:
14402:
14383:
14382:
14371:
14368:
14364:
14361:
14358:
14355:
14349:
14345:
14338:
14335:
14332:
14329:
14325:
14321:
14318:
14311:
14307:
14301:
14297:
14293:
14290:
14286:
14282:
14277:
14265:
14259:
14250:
14246:
14242:
14238:
14233:
14228:
14222:
14218:
14215:
14212:
14197:
14194:
14174:
14171:
14157:
14156:
14144:
14141:
14138:
14135:
14132:
14129:
14123:
14120:
14114:
14111:
14108:
14105:
14102:
14099:
14096:
14093:
14087:
14084:
14080:
14075:
14052:
14049:
14046:
14043:
14040:
14037:
14026:
14025:
14013:
14009:
14005:
14000:
13996:
13992:
13988:
13985:
13982:
13979:
13975:
13971:
13966:
13962:
13955:
13951:
13947:
13944:
13940:
13937:
13934:
13931:
13927:
13923:
13920:
13917:
13914:
13910:
13906:
13903:
13900:
13897:
13894:
13891:
13888:
13885:
13855:
13852:
13849:
13846:
13843:
13839:
13817:
13794:
13790:
13769:
13754:Norbert Wiener
13739:
13735:
13723:Wiener measure
13710:
13706:
13702:
13699:
13688:
13687:
13674:
13668:
13660:
13657:
13652:
13647:
13639:
13635:
13627:
13623:
13618:
13615:
13610:
13606:
13602:
13598:
13594:
13591:
13586:
13583:
13580:
13577:
13574:
13570:
13565:
13559:
13556:
13533:
13532:
13520:
13517:
13512:
13508:
13505:
13502:
13499:
13495:
13491:
13488:
13485:
13480:
13475:
13470:
13467:
13464:
13458:
13454:
13446:
13440:
13437:
13431:
13427:
13424:
13421:
13415:
13411:
13404:
13400:
13396:
13390:
13387:
13384:
13381:
13378:
13375:
13372:
13369:
13366:
13361:
13334:
13331:
13328:
13325:
13322:
13319:
13316:
13313:
13310:
13305:
13293:
13292:
13278:
13272:
13266:
13263:
13260:
13257:
13253:
13249:
13244:
13240:
13234:
13230:
13226:
13220:
13216:
13209:
13205:
13201:
13195:
13192:
13189:
13186:
13183:
13180:
13177:
13174:
13171:
13166:
13162:
13158:
13152:
13149:
13146:
13143:
13140:
13136:
13131:
13125:
13122:
13117:
13114:
13111:
13108:
13105:
13102:
13099:
13070:
13066:
13059:
13056:
13050:
13047:
13043:
13020:
13016:
13009:
13006:
13000:
12997:
12994:
12990:
12969:
12966:
12963:
12943:
12934:If we replace
12931:
12928:
12918:
12915:
12914:
12913:
12896:
12893:
12890:
12887:
12882:
12879:
12873:
12869:
12866:
12863:
12858:
12853:
12846:
12843:
12840:
12837:
12832:
12829:
12823:
12819:
12816:
12813:
12810:
12807:
12804:
12801:
12796:
12791:
12785:
12782:
12779:
12776:
12762:
12761:
12747:
12744:
12741:
12738:
12732:
12729:
12723:
12720:
12717:
12714:
12711:
12708:
12705:
12702:
12699:
12696:
12691:
12688:
12682:
12678:
12675:
12670:
12666:
12662:
12659:
12656:
12651:
12647:
12643:
12640:
12635:
12631:
12627:
12624:
12621:
12616:
12612:
12608:
12605:
12602:
12597:
12588:
12584:
12580:
12577:
12574:
12571:
12568:
12561:
12557:
12553:
12550:
12547:
12544:
12541:
12537:
12533:
12530:
12525:
12521:
12516:
12510:
12505:
12501:
12497:
12491:
12488:
12480:
12477:
12472:
12468:
12464:
12458:
12455:
12449:
12444:
12440:
12434:
12429:
12425:
12421:
12416:
12412:
12408:
12402:
12399:
12391:
12388:
12383:
12379:
12375:
12369:
12366:
12360:
12355:
12351:
12345:
12340:
12336:
12332:
12329:
12326:
12320:
12317:
12309:
12306:
12301:
12297:
12292:
12286:
12282:
12278:
12264:
12263:
12249:
12246:
12243:
12240:
12234:
12231:
12225:
12222:
12219:
12216:
12213:
12210:
12207:
12204:
12201:
12198:
12193:
12190:
12184:
12180:
12177:
12173:
12170:
12166:
12163:
12160:
12157:
12154:
12151:
12148:
12143:
12134:
12130:
12126:
12123:
12120:
12117:
12114:
12107:
12103:
12099:
12096:
12093:
12090:
12087:
12083:
12070:take the form
12059:
12054:
12050:
12045:
12039:
12035:
12032:
12028:
12022:
12019:
12011:
12008:
12003:
11999:
11995:
11989:
11986:
11980:
11977:
11972:
11968:
11965:
11961:
11958:
11955:
11949:
11946:
11938:
11935:
11930:
11926:
11921:
11915:
11911:
11907:
11895:
11892:
11868:
11867:
11856:
11853:
11850:
11845:
11841:
11838:
11835:
11832:
11829:
11824:
11817:
11814:
11805:
11798:
11795:
11786:
11783:
11779:
11773:
11770:
11764:
11760:
11757:
11754:
11737:
11736:
11725:
11722:
11717:
11709:
11706:
11703:
11700:
11697:
11693:
11689:
11686:
11683:
11680:
11677:
11662:
11659:
11642:
11639:
11638:
11637:
11626:
11623:
11620:
11617:
11614:
11611:
11608:
11594:
11593:
11582:
11579:
11575:
11565:
11562:
11556:
11551:
11548:
11542:
11539:
11535:
11509:
11508:
11497:
11494:
11491:
11485:
11482:
11477:
11474:
11468:
11462:
11459:
11454:
11451:
11445:
11442:
11439:
11433:
11430:
11424:
11421:
11418:
11384:
11383:
11372:
11366:
11362:
11358:
11355:
11352:
11349:
11346:
11343:
11340:
11337:
11332:
11305:
11304:
11291:
11285:
11279:
11273:
11270:
11267:
11264:
11261:
11258:
11255:
11252:
11249:
11246:
11243:
11238:
11230:
11227:
11224:
11221:
11218:
11204:
11203:
11190:
11187:
11182:
11177:
11171:
11165:
11159:
11156:
11153:
11150:
11147:
11144:
11141:
11138:
11135:
11132:
11129:
11124:
11116:
11113:
11110:
11107:
11104:
11101:
11098:
11095:
11077:
11076:
11063:
11059:
11056:
11053:
11050:
11047:
11044:
11041:
11038:
11035:
11032:
11029:
11023:
11020:
11017:
11014:
11011:
11008:
11005:
10999:
10996:
10991:
10988:
10982:
10979:
10976:
10973:
10970:
10960:
10947:
10943:
10940:
10937:
10934:
10931:
10928:
10925:
10922:
10919:
10916:
10913:
10907:
10904:
10901:
10898:
10895:
10889:
10886:
10881:
10878:
10872:
10869:
10866:
10863:
10860:
10839:
10838:
10825:
10820:
10817:
10814:
10811:
10808:
10805:
10802:
10799:
10796:
10793:
10790:
10787:
10762:
10761:
10748:
10744:
10741:
10738:
10735:
10732:
10729:
10726:
10723:
10720:
10717:
10714:
10708:
10702:
10699:
10694:
10691:
10663:
10662:
10651:
10648:
10642:
10637:
10631:
10628:
10623:
10620:
10614:
10609:
10606:
10603:
10573:
10570:
10549:
10546:
10531:
10530:
10519:
10516:
10512:
10506:
10502:
10497:
10493:
10490:
10487:
10481:
10478:
10473:
10470:
10463:
10457:
10453:
10448:
10422:
10421:
10409:
10406:
10400:
10397:
10393:
10388:
10384:
10381:
10377:
10374:
10371:
10368:
10364:
10357:
10354:
10349:
10346:
10340:
10331:
10328:
10322:
10317:
10314:
10305:
10302:
10298:
10292:
10288:
10284:
10280:
10275:
10272:
10269:
10266:
10263:
10260:
10256:
10252:
10249:
10246:
10241:
10237:
10233:
10215:
10214:
10202:
10199:
10193:
10190:
10186:
10181:
10177:
10174:
10167:
10164:
10152:
10149:
10143:
10138:
10135:
10129:
10126:
10120:
10117:
10112:
10109:
10103:
10099:
10093:
10090:
10087:
10084:
10081:
10078:
10074:
10070:
10067:
10064:
10059:
10055:
10051:
10033:
10032:
10020:
10017:
10011:
10005:
10002:
9996:
9990:
9987:
9981:
9978:
9975:
9972:
9969:
9966:
9963:
9959:
9953:
9950:
9947:
9944:
9941:
9938:
9934:
9930:
9927:
9924:
9919:
9915:
9911:
9833:
9832:
9820:
9817:
9811:
9805:
9802:
9796:
9793:
9790:
9787:
9784:
9780:
9774:
9771:
9768:
9765:
9762:
9759:
9755:
9751:
9748:
9745:
9740:
9736:
9732:
9706:
9703:
9698:
9697:
9685:
9681:
9677:
9674:
9671:
9668:
9665:
9660:
9656:
9649:
9646:
9639:
9635:
9632:
9629:
9623:
9620:
9615:
9612:
9591:
9590:
9579:
9575:
9572:
9564:
9561:
9554:
9550:
9546:
9543:
9540:
9537:
9534:
9528:
9522:
9519:
9516:
9513:
9510:
9507:
9504:
9500:
9496:
9493:
9490:
9487:
9484:
9481:
9478:
9475:
9472:
9469:
9466:
9463:
9460:
9457:
9454:
9451:
9418:
9417:
9404:
9399:
9394:
9387:
9384:
9375:
9371:
9365:
9362:
9359:
9356:
9353:
9350:
9347:
9343:
9312:
9311:
9300:
9297:
9294:
9290:
9287:
9283:
9280:
9277:
9274:
9271:
9266:
9263:
9258:
9253:
9250:
9247:
9244:
9241:
9236:
9229:
9226:
9217:
9214:
9207:
9200:
9197:
9194:
9189:
9185:
9181:
9177:
9171:
9168:
9165:
9162:
9159:
9156:
9153:
9150:
9145:
9142:
9139:
9136:
9133:
9130:
9126:
9122:
9119:
9116:
9113:
9110:
9107:
9102:
9097:
9094:
9090:
9086:
9083:
9080:
9077:
9074:
9071:
9068:
9065:
9062:
9043:Main article:
9040:
9037:
9029:Hagen Kleinert
9021:
9020:
9006:
9003:
9000:
8997:
8992:
8989:
8983:
8980:
8977:
8933:
8930:
8929:
8928:
8917:
8911:
8908:
8904:
8897:
8894:
8888:
8885:
8881:
8877:
8872:
8868:
8853:
8852:
8841:
8838:
8835:
8832:
8829:
8826:
8823:
8820:
8817:
8814:
8802:
8798:
8795:
8790:
8787:
8781:
8777:
8774:
8771:
8768:
8764:
8760:
8755:
8752:
8749:
8746:
8743:
8739:
8732:
8728:
8725:
8722:
8719:
8713:
8688:in powers of
8675:
8674:
8663:
8660:
8657:
8654:
8651:
8648:
8642:
8638:
8635:
8632:
8629:
8623:
8616:
8613:
8607:
8601:
8598:
8593:
8590:
8584:
8579:
8576:
8571:
8567:
8563:
8558:
8554:
8550:
8545:
8541:
8537:
8532:
8528:
8524:
8521:
8490:
8489:
8478:
8473:
8468:
8459:
8456:
8453:
8450:
8447:
8443:
8439:
8436:
8429:
8426:
8423:
8420:
8416:
8410:
8406:
8400:
8396:
8392:
8386:
8378:
8375:
8372:
8369:
8366:
8362:
8358:
8355:
8348:
8345:
8342:
8339:
8336:
8332:
8328:
8325:
8318:
8312:
8307:
8303:
8299:
8294:
8289:
8285:
8280:
8275:
8268:
8265:
8260:
8257:
8251:
8247:
8242:
8239:
8232:
8229:
8224:
8219:
8213:
8210:
8207:
8204:
8201:
8197:
8193:
8190:
8186:
8178:
8174:
8171:
8168:
8165:
8159:
8152:
8149:
8143:
8137:
8134:
8129:
8126:
8120:
8115:
8112:
8107:
8103:
8099:
8094:
8090:
8086:
8081:
8077:
8073:
8068:
8064:
8060:
8057:
7978:
7977:
7962:
7954:
7950:
7944:
7940:
7936:
7931:
7927:
7923:
7918:
7914:
7910:
7905:
7901:
7895:
7889:
7885:
7880:
7876:
7872:
7866:
7862:
7857:
7854:
7849:
7844:
7841:
7838:
7834:
7830:
7827:
7825:
7823:
7818:
7811:
7808:
7805:
7802:
7795:
7791:
7785:
7781:
7777:
7774:
7771:
7768:
7765:
7762:
7759:
7754:
7749:
7745:
7741:
7736:
7731:
7727:
7723:
7717:
7714:
7708:
7705:
7697:
7694:
7688:
7683:
7680:
7674:
7671:
7665:
7659:
7656:
7653:
7650:
7647:
7644:
7641:
7638:
7633:
7630:
7624:
7619:
7616:
7614:
7612:
7607:
7603:
7599:
7594:
7590:
7586:
7581:
7577:
7573:
7568:
7564:
7560:
7557:
7554:
7553:
7536:
7529:
7519:
7518:
7507:
7500:
7495:
7491:
7487:
7482:
7478:
7474:
7471:
7468:
7465:
7462:
7459:
7456:
7453:
7448:
7445:
7435:
7429:
7425:
7421:
7415:
7411:
7404:
7399:
7395:
7391:
7386:
7382:
7378:
7375:
7372:
7369:
7364:
7359:
7355:
7351:
7346:
7342:
7338:
7335:
7325:
7315:
7311:
7305:
7301:
7298:
7295:
7290:
7286:
7282:
7277:
7273:
7269:
7264:
7260:
7256:
7251:
7247:
7243:
7240:
7226:
7225:
7214:
7208:
7205:
7200:
7197:
7194:
7191:
7185:
7181:
7173:
7169:
7163:
7159:
7153:
7150:
7146:
7140:
7135:
7132:
7129:
7125:
7106:
7105:
7094:
7084:
7079:
7075:
7071:
7066:
7062:
7058:
7055:
7052:
7049:
7046:
7042:
7036:
7033:
7019:
7018:
6999:
6996:
6991:
6986:
6980:
6975:
6969:
6966:
6961:
6956:
6952:
6948:
6943:
6939:
6935:
6932:
6926:
6921:
6918:
6914:
6907:
6902:
6899:
6896:
6892:
6888:
6882:
6872:
6868:
6862:
6858:
6855:
6853:
6851:
6846:
6841:
6837:
6832:
6828:
6824:
6819:
6815:
6811:
6806:
6802:
6798:
6790:
6786:
6782:
6777:
6773:
6765:
6761:
6757:
6754:
6751:
6744:
6738:
6735:
6728:
6723:
6719:
6712:
6709:
6705:
6699:
6694:
6691:
6686:
6682:
6678:
6675:
6669:
6664:
6661:
6653:
6648:
6645:
6642:
6638:
6631:
6621:
6617:
6611:
6607:
6604:
6601:
6599:
6597:
6592:
6588:
6584:
6579:
6575:
6571:
6566:
6562:
6558:
6553:
6549:
6545:
6542:
6539:
6538:
6524:
6523:
6512:
6508:
6504:
6499:
6495:
6491:
6486:
6482:
6478:
6473:
6469:
6465:
6457:
6453:
6449:
6444:
6440:
6432:
6428:
6424:
6421:
6418:
6411:
6405:
6402:
6395:
6390:
6386:
6379:
6376:
6368:
6363:
6360:
6357:
6353:
6349:
6340:
6336:
6333:
6321:, which gives
6315:
6314:
6299:
6292:
6286:
6281:
6277:
6273:
6268:
6264:
6260:
6257:
6254:
6251:
6244:
6240:
6234:
6230:
6226:
6223:
6220:
6215:
6211:
6207:
6202:
6198:
6194:
6191:
6188:
6185:
6182:
6177:
6172:
6168:
6164:
6159:
6154:
6150:
6146:
6140:
6136:
6133:
6128:
6125:
6120:
6117:
6115:
6113:
6110:
6107:
6102:
6096:
6092:
6086:
6082:
6078:
6072:
6069:
6063:
6058:
6051:
6048:
6041:
6035:
6032:
6025:
6017:
6013:
6005:
6001:
5996:
5992:
5989:
5986:
5980:
5971:
5967:
5959:
5955:
5950:
5946:
5943:
5941:
5933:
5929:
5928:
5914:
5913:
5902:
5896:
5891:
5887:
5883:
5878:
5874:
5870:
5867:
5864:
5861:
5856:
5851:
5847:
5843:
5840:
5837:
5834:
5831:
5828:
5820:
5816:
5812:
5806:
5801:
5797:
5793:
5788:
5784:
5780:
5777:
5774:
5771:
5766:
5763:
5760:
5755:
5751:
5747:
5744:
5741:
5738:
5730:
5726:
5722:
5719:
5716:
5713:
5704:
5683:
5657:
5631:
5630:
5619:
5614:
5610:
5604:
5600:
5596:
5590:
5587:
5581:
5576:
5569:
5566:
5559:
5553:
5550:
5544:
5539:
5512:
5509:
5508:
5507:
5496:
5491:
5487:
5481:
5476:
5472:
5466:
5463:
5458:
5454:
5447:
5444:
5440:
5435:
5415:
5410:
5409:
5398:
5395:
5390:
5382:
5379:
5375:
5369:
5366:
5363:
5360:
5357:
5354:
5349:
5346:
5343:
5340:
5337:
5334:
5330:
5326:
5323:
5320:
5315:
5311:
5307:
5304:
5301:
5298:
5294:
5291:
5288:
5285:
5282:
5279:
5276:
5273:
5270:
5267:
5264:
5259:
5255:
5251:
5248:
5245:
5242:
5239:
5234:
5230:
5211:
5210:
5199:
5196:
5191:
5186:
5182:
5176:
5173:
5170:
5167:
5164:
5161:
5158:
5155:
5149:
5146:
5142:
5126:
5125:
5114:
5107:
5104:
5097:
5093:
5089:
5086:
5083:
5080:
5077:
5071:
5067:
5064:
5061:
5058:
5055:
5052:
5049:
5046:
5043:
5016:
5015:
5004:
5001:
4996:
4991:
4987:
4981:
4978:
4975:
4972:
4969:
4966:
4963:
4957:
4954:
4950:
4934:
4933:
4922:
4919:
4916:
4913:
4909:
4906:
4903:
4900:
4897:
4894:
4891:
4888:
4885:
4859:
4858:
4847:
4839:
4836:
4829:
4825:
4821:
4818:
4815:
4812:
4806:
4802:
4798:
4795:
4792:
4789:
4786:
4783:
4780:
4777:
4774:
4756:
4755:
4744:
4737:
4731:
4727:
4723:
4717:
4713:
4709:
4706:
4703:
4700:
4697:
4694:
4688:
4685:
4668:
4667:
4656:
4649:
4643:
4639:
4635:
4629:
4625:
4621:
4618:
4615:
4612:
4607:
4600:
4597:
4575:
4574:
4563:
4558:
4554:
4550:
4546:
4542:
4539:
4534:
4527:
4524:
4517:
4514:
4511:
4508:
4505:
4502:
4496:
4493:
4454:
4453:
4442:
4437:
4433:
4429:
4426:
4423:
4418:
4414:
4410:
4405:
4401:
4397:
4394:
4391:
4388:
4385:
4382:
4379:
4376:
4373:
4357:
4346:with variance
4310:
4309:
4298:
4295:
4290:
4283:
4279:
4274:
4269:
4264:
4260:
4257:
4254:
4251:
4248:
4245:
4242:
4239:
4236:
4233:
4230:
4224:
4216:
4213:
4207:
4203:
4199:
4196:
4191:
4187:
4181:
4178:
4175:
4172:
4169:
4166:
4161:
4158:
4155:
4152:
4149:
4146:
4142:
4138:
4135:
4132:
4129:
4126:
4123:
4120:
4117:
4114:
4100:
4099:
4088:
4085:
4080:
4073:
4069:
4065:
4058:
4053:
4046:
4043:
4032:
4027:
4023:
4019:
4015:
4011:
4008:
4003:
4000:
3997:
3994:
3991:
3988:
3983:
3980:
3977:
3974:
3971:
3968:
3964:
3960:
3957:
3954:
3951:
3948:
3945:
3942:
3939:
3936:
3915:
3914:
3903:
3900:
3896:
3889:
3884:
3877:
3874:
3865:
3862:
3859:
3822:
3819:
3797:
3796:
3785:
3782:
3779:
3776:
3770:
3766:
3759:
3756:
3753:
3750:
3746:
3742:
3739:
3735:
3732:
3729:
3726:
3723:
3717:
3713:
3706:
3702:
3698:
3695:
3672:
3652:
3632:
3629:
3626:
3623:
3620:
3616:
3594:
3572:
3566:
3553:
3552:
3540:
3537:
3534:
3531:
3527:
3523:
3518:
3514:
3508:
3502:
3498:
3491:
3487:
3483:
3480:
3477:
3473:
3467:
3461:
3454:
3451:
3448:
3445:
3442:
3438:
3433:
3427:
3424:
3419:
3416:
3413:
3410:
3407:
3404:
3401:
3386:
3383:
3361:
3353:
3337:
3329:
3323:
3314:
3307:
3296:
3295:
3283:
3278:
3274:
3271:
3266:
3260:
3257:
3254:
3250:
3246:
3241:
3237:
3230:
3225:
3218:
3215:
3207:
3203:
3198:
3195:
3192:
3187:
3184:
3181:
3177:
3173:
3168:
3165:
3159:
3155:
3152:
3138:
3137:
3126:
3123:
3120:
3117:
3114:
3109:
3104:
3096:
3093:
3086:
3082:
3077:
3074:
3069:
3066:
3063:
3060:
3057:
3054:
3048:
3045:
3039:
3036:
3033:
3030:
3003:
2996:
2989:
2982:
2972:
2965:
2958:
2951:
2919:
2873:
2872:
2861:
2856:
2852:
2848:
2844:
2838:
2834:
2830:
2825:
2821:
2818:
2812:
2807:
2804:
2801:
2798:
2795:
2792:
2789:
2786:
2783:
2778:
2773:
2766:
2762:
2754:
2750:
2745:
2739:
2736:
2730:
2726:
2723:
2718:
2715:
2710:
2707:
2703:
2699:
2694:
2691:
2686:
2683:
2679:
2657:
2656:
2645:
2639:
2636:
2633:
2626:
2622:
2618:
2613:
2609:
2602:
2599:
2596:
2593:
2590:
2560:
2552:
2539:
2538:
2525:
2521:
2517:
2512:
2509:
2506:
2502:
2498:
2493:
2489:
2485:
2480:
2477:
2474:
2470:
2466:
2463:
2460:
2455:
2451:
2447:
2442:
2438:
2434:
2429:
2425:
2408:
2401:
2394:
2387:
2367:Main article:
2364:
2361:
2359:
2356:
2314:complex number
2285:
2284:
2259:
2252:
2228:
2225:
2204:
2182:
2157:
2142:
2135:
2110:
2093:
2084:
2077:
2070:
2061:
2039:
2036:
1991:
1990:
1979:
1974:
1971:
1968:
1964:
1901:
1892:
1891:
1880:
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1850:
1833:
1828:
1827:
1816:
1811:
1808:
1805:
1802:
1799:
1796:
1793:
1790:
1787:
1783:
1766:
1717:
1716:
1705:
1700:
1697:
1694:
1691:
1688:
1685:
1682:
1678:
1659:
1658:
1647:
1640:
1635:
1632:
1629:
1626:
1623:
1620:
1617:
1614:
1609:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1569:
1564:
1559:
1554:
1550:
1496:
1495:
1484:
1478:
1475:
1470:
1467:
1461:
1455:
1452:
1435:
1434:
1423:
1420:
1417:
1414:
1408:
1405:
1399:
1396:
1393:
1390:
1387:
1346:
1343:
1329:
1328:
1317:
1308:
1305:
1299:
1294:
1291:
1285:
1282:
1268:
1267:
1256:
1253:
1250:
1245:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1205:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1168:discretization
1159:) is that the
1141:Lorentz scalar
1107:imaginary unit
1097:
1094:
1046:Norbert Wiener
900:
899:
897:
896:
889:
882:
874:
871:
870:
866:
865:
860:
855:
850:
845:
840:
835:
830:
825:
820:
815:
810:
805:
800:
795:
790:
785:
780:
775:
770:
765:
760:
755:
750:
745:
740:
735:
730:
725:
720:
715:
710:
705:
700:
695:
690:
685:
680:
675:
670:
665:
660:
655:
650:
645:
640:
634:
633:
630:
629:
626:
625:
622:
621:
616:
611:
606:
604:Density matrix
601:
596:
591:
586:
581:
576:
570:
567:
566:
563:
562:
558:
557:
552:
547:
542:
537:
532:
527:
526:
525:
524:
523:
508:
503:
498:
493:
488:
482:
481:
476:
475:
472:
471:
467:
466:
461:
456:
451:
446:
440:
439:
436:
435:
432:
431:
427:
426:
421:
416:
411:
406:
401:
395:
394:
393:
387:
384:
383:
380:
379:
375:
374:
369:
364:
358:
357:
356:
355:
354:
352:Delayed-choice
347:Quantum eraser
342:
341:
336:
331:
326:
321:
316:
311:
306:
301:
296:
290:
289:
286:
285:
282:
281:
277:
276:
275:
274:
264:
259:
254:
249:
244:
239:
237:Quantum number
234:
229:
224:
219:
214:
209:
203:
202:
199:
198:
195:
194:
190:
189:
184:
178:
177:
176:
171:
166:
160:
157:
156:
153:
152:
151:
150:
145:
140:
132:
131:
126:
115:
112:
108:
101:
98:
92:
89:
86:
82:
75:
72:
68:
63:
60:
49:
48:
42:
41:
26:
9:
6:
4:
3:
2:
26449:
26438:
26435:
26433:
26430:
26428:
26425:
26423:
26420:
26418:
26415:
26413:
26410:
26408:
26405:
26404:
26402:
26387:
26383:
26379:
26377:
26372:
26368:
26366:
26362:
26358:
26356:
26353:
26351:
26350:
26349:Tuva or Bust!
26346:
26344:
26343:
26339:
26337:
26334:
26332:
26329:
26328:
26326:
26322:
26312:
26309:
26303:
26300:
26299:
26297:
26293:
26283:
26282:
26278:
26272:
26271:
26267:
26261:
26260:
26256:
26250:
26249:
26245:
26239:
26238:
26234:
26228:
26227:
26223:
26217:
26216:
26212:
26206:
26205:
26201:
26195:
26194:
26190:
26184:
26180:
26179:
26177:
26173:
26167:
26164:
26162:
26159:
26157:
26154:
26152:
26149:
26147:
26144:
26142:
26139:
26137:
26134:
26132:
26129:
26127:
26124:
26122:
26119:
26117:
26114:
26112:
26109:
26107:
26104:
26102:
26099:
26097:
26094:
26093:
26091:
26087:
26083:
26076:
26071:
26069:
26064:
26062:
26057:
26056:
26053:
26041:
26033:
26032:
26029:
26023:
26020:
26018:
26015:
26013:
26010:
26006:
26003:
26002:
26001:
25998:
25997:
25995:
25991:
25985:
25982:
25980:
25977:
25973:
25970:
25969:
25968:
25965:
25963:
25960:
25958:
25955:
25953:
25950:
25949:
25947:
25943:
25937:
25934:
25932:
25929:
25927:
25924:
25922:
25919:
25917:
25914:
25912:
25909:
25907:
25904:
25902:
25899:
25897:
25894:
25892:
25889:
25887:
25884:
25882:
25879:
25877:
25876:Quantum logic
25874:
25872:
25869:
25867:
25864:
25862:
25859:
25857:
25854:
25852:
25849:
25847:
25844:
25842:
25839:
25835:
25832:
25831:
25830:
25827:
25825:
25822:
25820:
25817:
25815:
25812:
25808:
25805:
25804:
25803:
25800:
25798:
25795:
25793:
25790:
25788:
25785:
25784:
25782:
25780:
25776:
25770:
25767:
25765:
25762:
25760:
25757:
25755:
25752:
25750:
25747:
25745:
25742:
25740:
25737:
25735:
25732:
25730:
25729:Quantum chaos
25727:
25725:
25722:
25720:
25717:
25716:
25714:
25712:
25708:
25702:
25699:
25697:
25696:Stern–Gerlach
25694:
25692:
25689:
25687:
25684:
25682:
25679:
25677:
25674:
25672:
25669:
25667:
25664:
25662:
25659:
25657:
25654:
25652:
25649:
25648:
25646:
25642:
25636:
25633:
25631:
25630:Transactional
25628:
25626:
25623:
25621:
25620:Quantum logic
25618:
25616:
25613:
25611:
25608:
25602:
25599:
25598:
25597:
25594:
25593:
25592:
25589:
25587:
25584:
25582:
25579:
25577:
25574:
25572:
25569:
25567:
25564:
25563:
25561:
25559:
25555:
25549:
25546:
25544:
25541:
25539:
25536:
25534:
25531:
25529:
25526:
25524:
25521:
25519:
25516:
25514:
25511:
25510:
25508:
25504:
25498:
25495:
25493:
25490:
25488:
25485:
25483:
25480:
25478:
25475:
25473:
25470:
25468:
25465:
25464:
25462:
25458:
25450:
25447:
25445:
25442:
25441:
25440:
25439:Wave function
25437:
25435:
25432:
25430:
25427:
25425:
25422:
25420:
25417:
25415:
25414:Superposition
25412:
25410:
25409:Quantum state
25407:
25405:
25402:
25400:
25397:
25395:
25392:
25390:
25387:
25385:
25382:
25380:
25377:
25373:
25370:
25368:
25365:
25363:
25362:Excited state
25360:
25358:
25355:
25354:
25353:
25350:
25348:
25345:
25343:
25340:
25338:
25335:
25333:
25330:
25329:
25327:
25323:
25317:
25314:
25312:
25309:
25307:
25304:
25300:
25297:
25296:
25295:
25292:
25290:
25287:
25286:
25284:
25280:
25276:
25269:
25264:
25262:
25257:
25255:
25250:
25249:
25246:
25239:
25236:
25233:
25230:
25228:
25225:
25223:
25220:
25219:
25208:
25202:
25198:
25194:
25190:
25186:
25180:
25176:
25172:
25168:
25163:
25157:
25152:
25151:
25145:
25141:
25138:
25132:
25128:
25123:
25122:
25116:
25112:
25108:
25104:
25099:
25094:
25089:
25084:
25080:
25076:
25072:
25068:
25064:
25060:
25056:
25051:
25045:
25041:
25037:
25033:
25029:
25025:
25021:
25017:
25013:
25009:
25005:
25001:
24994:
24989:
24985:
24979:
24975:
24971:
24967:
24963:
24957:
24953:
24948:
24943:
24937:
24932:
24931:
24924:
24920:
24914:
24910:
24905:
24901:
24896:
24892:
24886:
24882:
24877:
24872:
24867:
24862:
24858:
24852:
24848:
24847:
24842:
24838:
24834:
24828:
24824:
24820:
24816:
24812:
24806:
24802:
24797:
24793:
24787:
24783:
24778:
24774:
24768:
24764:
24759:
24755:
24751:
24747:
24741:
24737:
24733:
24729:
24725:
24721:
24716:
24711:
24706:
24701:
24697:
24691:
24687:
24682:
24678:
24672:
24667:
24666:
24659:
24655:
24649:
24645:
24641:
24637:
24633:
24627:
24623:
24619:
24615:
24610:
24603:
24597:
24592:
24591:
24584:
24580:
24576:
24572:
24568:
24564:
24560:
24553:
24548:
24541:
24535:
24531:
24527:
24523:
24519:
24515:
24514:
24509:
24505:
24498:
24494:
24490:
24480:
24476:
24472:
24468:
24464:
24460:
24453:
24449:
24445:
24441:
24437:
24433:
24426:
24422:
24418:
24414:
24410:
24406:
24402:
24398:
24394:
24390:
24386:
24382:
24378:
24374:
24370:
24366:
24360:
24356:
24352:
24347:
24343:
24339:
24335:
24331:
24327:
24323:
24318:
24313:
24309:
24305:
24301:
24297:
24293:
24289:
24285:
24281:
24277:
24273:
24269:
24265:
24261:
24257:
24255:9783540769569
24251:
24247:
24242:
24238:
24234:
24233:Ahmad, Ishfaq
24230:
24229:
24217:
24212:
24206:
24201:
24195:
24190:
24175:
24171:
24164:
24155:
24150:
24146:
24142:
24138:
24134:
24130:
24123:
24116:
24111:
24105:, Chapter 19.
24104:
24099:
24093:
24088:
24081:
24076:
24069:
24064:
24058:
24053:
24047:, Chapter 13.
24046:
24041:
24033:
24026:
24019:
24013:
24008:
24001:
23996:
23987:
23982:
23975:
23969:
23964:
23958:
23953:
23947:
23942:
23928:
23924:
23920:
23914:
23900:
23899:
23894:
23887:
23878:
23873:
23866:
23864:
23849:on 2017-08-12
23845:
23838:
23831:
23824:
23823:Weinberg 2002
23819:
23815:
23801:
23795:
23788:
23782:
23778:
23767:
23764:
23762:
23759:
23757:
23754:
23752:
23749:
23747:
23744:
23742:
23739:
23737:
23734:
23733:
23726:
23724:
23720:
23716:
23708:
23699:
23678:
23655:
23649:
23645:
23641:
23638:
23627:
23623:
23613:
23612:
23611:
23605:
23601:
23592:
23590:
23586:
23582:
23578:
23577:Hilbert space
23568:
23566:
23561:
23559:
23555:
23551:
23541:
23518:
23506:
23500:
23491:
23479:
23468:
23465:
23437:
23425:
23396:
23384:
23361:
23358:
23332:
23320:
23291:
23279:
23256:
23253:
23231:
23224:
23218:
23215:
23212:
23190:
23183:
23177:
23174:
23171:
23142:
23130:
23124:
23115:
23103:
23092:
23089:
23078:
23068:
23066:
23062:
23033:
23030:
23024:
23018:
23014:
23007:
23004:
23000:
22995:
22992:
22988:
22978:
22972:
22966:
22960:
22954:
22950:
22945:
22941:
22937:
22930:
22929:
22928:
22911:
22908:
22904:
22897:
22891:
22887:
22879:
22878:
22877:
22873:
22867:
22850:
22847:
22841:
22835:
22831:
22824:
22821:
22817:
22812:
22809:
22805:
22795:
22789:
22783:
22777:
22771:
22768:
22762:
22756:
22752:
22745:
22742:
22738:
22733:
22730:
22726:
22719:
22711:
22707:
22701:
22693:
22687:
22681:
22677:
22670:
22667:
22663:
22658:
22655:
22651:
22641:
22635:
22629:
22623:
22617:
22614:
22608:
22602:
22598:
22591:
22588:
22584:
22579:
22576:
22572:
22565:
22556:
22550:
22543:
22542:
22541:
22524:
22521:
22517:
22510:
22502:
22498:
22492:
22484:
22480:
22476:
22473:
22464:
22455:
22449:
22443:
22434:
22425:
22419:
22416:
22410:
22407:
22398:
22389:
22383:
22373:
22372:
22371:
22354:
22351:
22345:
22336:
22327:
22321:
22317:
22307:
22300:
22299:
22298:
22292:
22288:
22282:
22278:
22250:
22244:
22238:
22219:
22214:
22194:
22188:
22180:
22176:
22172:
22166:
22158:
22154:
22146:
22145:
22144:
22123:
22115:
22111:
22105:
22097:
22091:
22082:
22076:
22069:
22068:
22067:
22043:
22037:
22031:
22025:
22022:
22019:
22008:
22001:
21997:
21988:
21982:
21973:
21967:
21960:
21959:
21958:
21938:
21932:
21928:
21923:
21919:
21915:
21912:
21909:
21902:
21901:
21900:
21893:
21876:
21873:
21869:
21863:
21859:
21855:
21849:
21845:
21839:
21832:
21828:
21824:
21820:
21817:
21808:
21802:
21792:
21791:
21790:
21773:
21770:
21767:
21761:
21754:
21748:
21745:
21741:
21737:
21733:
21729:
21725:
21715:
21708:
21707:
21706:
21703:
21701:
21700:
21676:
21673:
21667:
21658:
21652:
21648:
21638:
21631:
21630:
21629:
21626:
21624:
21623:supersymmetry
21620:
21616:
21612:
21602:
21596:
21564:
21556:
21552:
21546:
21538:
21529:
21513:
21506:
21505:
21504:
21502:
21498:
21493:
21490:
21487:
21481:
21471:
21469:
21465:
21455:
21453:
21449:
21448:Taylor series
21430:
21427:
21424:
21421:
21418:
21415:
21405:
21384:
21383:
21382:
21375:
21373:
21368:
21363:
21358:
21345:
21339:
21319:
21316:
21313:
21308:
21304:
21300:
21297:
21288:
21285:
21277:
21274:
21258:
21257:
21256:
21239:
21236:
21233:
21227:
21221:
21215:
21209:
21206:
21200:
21194:
21190:
21183:
21180:
21176:
21171:
21168:
21164:
21154:
21148:
21145:
21133:
21123:
21122:
21121:
21104:
21101:
21096:
21091:
21084:
21078:
21075:
21069:
21057:
21051:
21048:
21036:
21029:
21020:
21019:
21018:
21017:
20998:
20992:
20986:
20975:
20971:
20964:
20961:
20957:
20945:
20941:
20935:
20931:
20919:
20915:
20904:
20893:
20889:
20882:
20879:
20875:
20863:
20859:
20853:
20849:
20837:
20833:
20820:
20812:
20809:
20803:
20797:
20791:
20787:
20780:
20777:
20773:
20768:
20765:
20761:
20757:
20750:
20749:
20748:
20723:
20715:
20711:
20704:
20694:
20690:
20684:
20680:
20668:
20664:
20653:
20645:
20641:
20634:
20624:
20620:
20614:
20610:
20598:
20594:
20583:
20577:
20571:
20564:
20563:
20562:
20551:
20543:
20530:
20528:
20520:
20513:
20506:
20499:
20495:
20488:
20487:time ordering
20484:
20480:
20475:
20472:
20449:
20440:
20433:
20430:
20427:
20421:
20415:
20401:
20397:
20393:
20389:
20379:
20371:
20364:
20361:
20358:
20352:
20346:
20332:
20328:
20324:
20317:
20311:
20308:
20298:
20292:
20287:
20279:
20269:
20268:
20267:
20250:
20245:
20240:
20232:
20228:
20223:
20219:
20212:
20208:
20203:
20198:
20190:
20184:
20179:
20175:
20171:
20165:
20155:
20151:
20147:
20142:
20138:
20134:
20130:
20122:
20121:
20120:
20103:
20098:
20093:
20084:
20080:
20073:
20070:
20062:
20058:
20051:
20047:
20038:
20032:
20026:
20022:
20018:
20012:
19998:
19994:
19987:
19984:
19981:
19973:
19969:
19962:
19959:
19954:
19949:
19945:
19934:
19933:
19932:
19915:
19909:
19902:
19899:
19896:
19890:
19884:
19870:
19866:
19862:
19858:
19848:
19845:
19839:
19833:
19826:
19825:
19824:
19822:
19815:
19811:
19807:
19803:
19799:
19790:
19786:
19776:
19757:
19753:
19747:
19744:
19732:
19728:
19724:
19721:
19718:
19714:
19709:
19706:
19702:
19698:
19690:
19689:
19688:
19686:
19662:
19655:
19652:
19644:
19631:
19622:
19616:
19612:
19608:
19605:
19602:
19598:
19592:
19589:
19581:
19575:
19572:
19566:
19558:
19557:
19556:
19554:
19546:
19542:
19514:
19508:
19505:
19501:
19493:
19492:
19491:
19474:
19466:
19453:
19449:
19441:
19440:
19439:
19438:
19437:Wick rotation
19434:
19430:
19425:
19418:
19413:
19411:
19407:
19388:
19385:
19379:
19376:
19368:
19355:
19345:
19344:
19343:
19341:
19336:
19331:
19321:
19319:
19311:
19291:
19283:
19278:
19271:
19265:
19257:
19244:
19240:
19233:
19221:
19217:
19207:
19203:
19196:
19193:
19190:
19186:
19178:
19173:
19166:
19160:
19152:
19139:
19135:
19128:
19116:
19112:
19102:
19098:
19091:
19088:
19085:
19082:
19079:
19075:
19068:
19062:
19059:
19056:
19050:
19040:
19039:
19038:
19037:, this means
19030:
19020:
19016:
19007:
19003:
18995:
18991:
18986:, written as
18985:
18981:
18963:
18959:
18935:
18924:
18911:
18907:
18903:
18893:
18883:
18870:
18866:
18859:
18853:
18850:
18840:
18834:
18828:
18818:
18817:
18816:
18811:
18800:
18796:
18793:
18786:
18782:
18778:
18768:
18766:
18765:Wick rotation
18762:
18754:
18750:
18746:
18742:
18737:
18735:
18730:
18728:
18724:
18718:
18714:
18709:
18703:
18699:
18693:
18688:
18684:
18680:
18670:
18666:
18664:
18658:
18655:
18651:
18627:
18621:
18618:
18615:
18610:
18606:
18602:
18597:
18593:
18588:
18583:
18577:
18571:
18564:
18563:
18562:
18556:
18536:
18530:
18527:
18524:
18517:
18513:
18509:
18504:
18494:
18485:
18480:
18476:
18471:
18466:
18460:
18457:
18454:
18447:
18443:
18439:
18434:
18424:
18415:
18410:
18406:
18401:
18396:
18390:
18384:
18377:
18376:
18375:
18369:
18362:
18355:
18349:
18345:
18343:
18324:
18315:
18312:
18306:
18296:
18287:
18281:
18278:
18273:
18269:
18261:
18256:
18250:
18238:
18234:
18231:
18224:
18223:
18222:
18219:
18212:
18202:
18195:
18175:
18165:
18161:
18157:
18152:
18142:
18133:
18128:
18124:
18119:
18114:
18104:
18100:
18096:
18091:
18081:
18072:
18067:
18063:
18058:
18053:
18047:
18041:
18036:
18032:
18028:
18021:
18020:
18019:
18017:
17998:
17990:
17986:
17982:
17977:
17967:
17960:
17955:
17950:
17946:
17941:
17936:
17930:
17924:
17917:
17916:
17915:
17912:
17902:
17882:
17876:
17873:
17868:
17864:
17859:
17854:
17846:
17840:
17832:
17827:
17823:
17814:
17810:
17799:
17795:
17791:
17785:
17779:
17772:
17771:
17770:
17754:
17750:
17744:
17725:
17717:
17706:
17703:
17690:
17682:
17679:
17676:
17667:
17663:
17652:
17648:
17644:
17638:
17635:
17632:
17626:
17619:
17618:
17617:
17608:
17588:
17580:
17565:
17554:
17551:
17548:
17545:
17539:
17529:
17525:
17521:
17515:
17512:
17509:
17503:
17496:
17495:
17494:
17469:
17454:
17446:
17443:
17440:
17431:
17427:
17416:
17413:
17409:
17405:
17394:
17391:
17388:
17385:
17379:
17372:
17371:
17370:
17364:
17336:
17331:
17328:
17324:
17318:
17315:
17301:
17296:
17293:
17287:
17281:
17277:
17266:
17263:
17259:
17255:
17244:
17241:
17238:
17235:
17229:
17222:
17221:
17220:
17216:
17212:
17208:
17204:
17183:
17178:
17175:
17171:
17168:
17161:
17154:
17151:
17133:
17129:
17125:
17121:
17115:
17112:
17098:
17093:
17090:
17084:
17078:
17074:
17070:
17059:
17056:
17053:
17050:
17044:
17037:
17036:
17035:
17031:
17002:
16996:
16993:
16990:
16984:
16981:
16974:
16964:
16957:
16951:
16948:
16945:
16940:
16937:
16931:
16925:
16922:
16919:
16907:
16899:
16898:
16897:
16894:
16882:
16863:
16842:
16836:
16833:
16830:
16824:
16821:
16815:
16805:
16796:
16793:
16788:
16785:
16779:
16773:
16770:
16767:
16755:
16747:
16746:
16745:
16741:
16737:
16733:
16709:
16702:
16699:
16692:
16688:
16684:
16681:
16675:
16671:
16665:
16662:
16659:
16653:
16646:
16645:
16644:
16640:
16636:
16615:
16608:
16605:
16598:
16590:
16587:
16584:
16578:
16575:
16569:
16565:
16559:
16556:
16553:
16550:
16547:
16541:
16534:
16533:
16532:
16511:
16505:
16502:
16496:
16493:
16488:
16484:
16480:
16475:
16471:
16463:
16462:
16461:
16459:
16453:
16449:
16441:
16437:
16433:
16413:
16410:
16407:
16398:
16392:
16389:
16385:
16375:
16369:
16361:
16357:
16351:
16348:
16342:
16336:
16332:
16328:
16325:
16322:
16315:
16312:
16309:
16306:
16303:
16297:
16291:
16283:
16279:
16273:
16269:
16265:
16259:
16251:
16247:
16239:
16238:
16237:
16233:
16226:
16217:
16198:
16195:
16192:
16183:
16177:
16174:
16170:
16164:
16161:
16155:
16149:
16144:
16141:
16135:
16129:
16125:
16121:
16115:
16112:
16109:
16106:
16103:
16100:
16097:
16091:
16085:
16082:
16079:
16076:
16073:
16067:
16060:
16059:
16058:
16042:
16033:
16031:
16025:
16023:
16019:
16013:
16009:
16007:
16001:
15999:
15995:
15991:
15961:
15958:
15955:
15947:
15943:
15939:
15936:
15927:
15921:
15916:
15908:
15902:
15896:
15886:
15885:
15884:
15873:
15868:
15866:
15861:
15859:
15854:
15853:
15851:
15850:
15844:
15839:
15836:
15834:
15831:
15829:
15826:
15824:
15821:
15819:
15816:
15814:
15813:Zamolodchikov
15811:
15809:
15808:Zamolodchikov
15806:
15804:
15801:
15799:
15796:
15794:
15791:
15789:
15786:
15784:
15781:
15779:
15776:
15774:
15771:
15769:
15766:
15764:
15761:
15759:
15756:
15754:
15751:
15749:
15746:
15744:
15741:
15739:
15736:
15734:
15731:
15729:
15726:
15724:
15721:
15719:
15716:
15714:
15711:
15709:
15706:
15704:
15701:
15699:
15696:
15694:
15691:
15689:
15686:
15684:
15681:
15679:
15676:
15674:
15671:
15669:
15666:
15664:
15661:
15659:
15656:
15654:
15651:
15649:
15646:
15644:
15641:
15639:
15636:
15634:
15631:
15629:
15626:
15624:
15621:
15619:
15616:
15614:
15611:
15609:
15606:
15604:
15601:
15599:
15596:
15594:
15591:
15589:
15586:
15584:
15581:
15579:
15576:
15574:
15571:
15569:
15566:
15564:
15561:
15559:
15556:
15554:
15551:
15549:
15546:
15544:
15541:
15539:
15536:
15534:
15531:
15529:
15526:
15524:
15521:
15519:
15516:
15514:
15511:
15509:
15506:
15504:
15501:
15499:
15496:
15494:
15491:
15489:
15486:
15484:
15481:
15479:
15476:
15474:
15471:
15469:
15466:
15464:
15461:
15459:
15456:
15454:
15451:
15449:
15446:
15444:
15441:
15439:
15436:
15434:
15431:
15429:
15426:
15424:
15421:
15419:
15416:
15414:
15411:
15409:
15406:
15404:
15401:
15399:
15396:
15394:
15391:
15389:
15386:
15384:
15381:
15379:
15376:
15374:
15371:
15369:
15366:
15364:
15361:
15359:
15356:
15354:
15351:
15349:
15346:
15344:
15341:
15339:
15336:
15334:
15331:
15329:
15326:
15324:
15321:
15319:
15316:
15314:
15311:
15309:
15306:
15304:
15301:
15299:
15296:
15294:
15291:
15289:
15286:
15284:
15281:
15279:
15276:
15274:
15271:
15269:
15266:
15264:
15261:
15259:
15256:
15254:
15251:
15249:
15246:
15244:
15241:
15239:
15236:
15234:
15231:
15229:
15226:
15224:
15221:
15219:
15216:
15214:
15211:
15209:
15206:
15204:
15201:
15199:
15196:
15194:
15191:
15189:
15186:
15184:
15181:
15179:
15176:
15174:
15171:
15169:
15166:
15164:
15161:
15159:
15156:
15154:
15151:
15149:
15146:
15144:
15141:
15140:
15133:
15132:
15125:
15122:
15120:
15117:
15115:
15112:
15110:
15109:Supersymmetry
15107:
15105:
15104:String theory
15102:
15101:
15095:
15094:
15087:
15084:
15082:
15079:
15077:
15074:
15072:
15069:
15068:
15065:
15060:
15059:
15052:
15049:
15047:
15044:
15042:
15039:
15037:
15034:
15032:
15029:
15027:
15024:
15022:
15019:
15018:
15012:
15011:
15004:
15001:
14999:
14996:
14994:
14991:
14989:
14986:
14984:
14981:
14979:
14976:
14974:
14971:
14969:
14966:
14964:
14961:
14959:
14956:
14954:
14951:
14949:
14946:
14944:
14941:
14939:
14936:
14934:
14931:
14929:
14926:
14924:
14921:
14919:
14916:
14914:
14911:
14909:
14906:
14905:
14899:
14898:
14891:
14888:
14886:
14883:
14881:
14878:
14876:
14873:
14871:
14868:
14866:
14863:
14861:
14858:
14856:
14853:
14851:
14848:
14846:
14843:
14841:
14838:
14837:
14834:
14829:
14828:
14821:
14818:
14816:
14813:
14811:
14808:
14806:
14803:
14801:
14798:
14796:
14793:
14791:
14788:
14786:
14783:
14781:
14778:
14777:
14771:
14770:
14767:
14764:
14763:
14759:
14754:
14750:
14749:
14746:
14743:
14742:
14734:
14732:
14728:
14724:
14704:
14699:
14696:
14693:
14689:
14685:
14681:
14678:
14675:
14672:
14665:
14664:
14663:
14655:
14628:
14625:
14622:
14606:
14603:
14600:
14594:
14590:
14586:
14580:
14577:
14574:
14562:
14561:
14560:
14556:
14554:
14545:
14537:
14531:
14518:
14503:
14499:
14494:
14490:
14486:
14485:Wick rotation
14482:
14477:
14460:
14400:
14393:
14388:
14369:
14366:
14356:
14347:
14336:
14330:
14316:
14309:
14305:
14299:
14295:
14291:
14226:
14220:
14216:
14213:
14210:
14203:
14202:
14201:
14193:
14189:
14169:
14139:
14136:
14133:
14127:
14118:
14112:
14109:
14103:
14100:
14097:
14091:
14085:
14073:
14066:
14065:
14064:
14047:
14044:
14041:
14035:
13998:
13994:
13990:
13980:
13964:
13960:
13949:
13945:
13942:
13932:
13918:
13915:
13912:
13908:
13904:
13901:
13895:
13892:
13889:
13883:
13876:
13875:
13874:
13872:
13867:
13853:
13850:
13844:
13792:
13788:
13767:
13759:
13755:
13737:
13733:
13724:
13697:
13658:
13655:
13650:
13637:
13621:
13616:
13613:
13608:
13604:
13589:
13584:
13581:
13575:
13563:
13557:
13554:
13545:
13544:
13543:
13540:
13538:
13518:
13515:
13510:
13500:
13486:
13483:
13478:
13465:
13456:
13438:
13435:
13429:
13425:
13422:
13413:
13402:
13359:
13351:
13350:
13349:
13303:
13258:
13242:
13238:
13228:
13218:
13207:
13164:
13160:
13156:
13150:
13147:
13141:
13129:
13123:
13120:
13115:
13109:
13106:
13103:
13097:
13090:
13089:
13088:
13086:
13085:Wick rotation
13064:
13054:
13048:
13045:
13041:
13014:
13004:
12998:
12995:
12992:
12988:
12967:
12964:
12961:
12941:
12927:
12924:
12923:Wick rotation
12891:
12885:
12877:
12871:
12864:
12851:
12841:
12835:
12827:
12821:
12814:
12808:
12802:
12789:
12783:
12777:
12767:
12766:
12765:
12739:
12730:
12727:
12721:
12715:
12709:
12703:
12700:
12697:
12694:
12686:
12680:
12668:
12664:
12657:
12649:
12645:
12633:
12629:
12622:
12614:
12610:
12603:
12586:
12582:
12578:
12572:
12566:
12559:
12555:
12551:
12545:
12539:
12535:
12531:
12523:
12519:
12503:
12499:
12486:
12475:
12470:
12466:
12453:
12442:
12438:
12427:
12423:
12419:
12414:
12410:
12397:
12386:
12381:
12377:
12364:
12353:
12349:
12338:
12334:
12330:
12327:
12315:
12304:
12299:
12295:
12284:
12280:
12269:
12268:
12267:
12241:
12232:
12229:
12223:
12217:
12211:
12205:
12202:
12199:
12196:
12188:
12182:
12171:
12168:
12161:
12155:
12149:
12132:
12128:
12124:
12118:
12112:
12105:
12101:
12097:
12091:
12085:
12081:
12073:
12072:
12071:
12052:
12048:
12033:
12030:
12017:
12006:
12001:
11997:
11984:
11975:
11966:
11963:
11959:
11956:
11944:
11933:
11928:
11924:
11913:
11909:
11891:
11888:
11877:
11854:
11851:
11848:
11843:
11836:
11830:
11827:
11822:
11815:
11812:
11803:
11796:
11793:
11784:
11781:
11777:
11771:
11768:
11762:
11758:
11755:
11752:
11745:
11744:
11743:
11740:
11723:
11720:
11704:
11698:
11695:
11691:
11684:
11678:
11675:
11668:
11667:
11666:
11658:
11656:
11652:
11648:
11624:
11621:
11615:
11612:
11609:
11599:
11598:
11597:
11580:
11577:
11573:
11563:
11560:
11549:
11540:
11537:
11533:
11525:
11524:
11523:
11520:
11518:
11514:
11495:
11492:
11489:
11483:
11480:
11475:
11472:
11466:
11460:
11457:
11452:
11449:
11443:
11440:
11431:
11428:
11422:
11419:
11409:
11408:
11407:
11405:
11400:
11370:
11364:
11356:
11353:
11347:
11341:
11335:
11321:
11320:
11319:
11315:
11311:
11289:
11283:
11268:
11262:
11259:
11253:
11250:
11247:
11241:
11228:
11222:
11216:
11209:
11208:
11207:
11188:
11185:
11180:
11175:
11169:
11154:
11148:
11145:
11139:
11136:
11133:
11127:
11114:
11108:
11102:
11096:
11086:
11085:
11084:
11061:
11054:
11048:
11045:
11039:
11036:
11033:
11027:
11018:
11015:
11012:
11006:
11003:
10997:
10994:
10989:
10986:
10980:
10977:
10971:
10961:
10945:
10938:
10932:
10929:
10923:
10920:
10917:
10911:
10902:
10896:
10893:
10887:
10884:
10879:
10876:
10870:
10867:
10861:
10851:
10850:
10849:
10844:The quantity
10842:
10823:
10818:
10812:
10806:
10803:
10797:
10794:
10791:
10785:
10778:
10777:
10776:
10771:
10746:
10739:
10733:
10730:
10724:
10721:
10718:
10712:
10706:
10700:
10697:
10692:
10689:
10679:
10678:
10677:
10673:
10669:
10665:The quantity
10649:
10646:
10640:
10635:
10629:
10626:
10621:
10618:
10612:
10607:
10604:
10601:
10594:
10593:
10592:
10585:
10569:
10559:
10545:
10534:
10517:
10514:
10510:
10504:
10500:
10495:
10488:
10479:
10476:
10471:
10468:
10461:
10455:
10451:
10446:
10438:
10437:
10436:
10432:
10428:
10407:
10404:
10398:
10395:
10391:
10386:
10382:
10379:
10372:
10366:
10362:
10355:
10347:
10338:
10329:
10326:
10315:
10303:
10300:
10296:
10290:
10286:
10282:
10278:
10273:
10270:
10264:
10258:
10254:
10247:
10239:
10235:
10231:
10224:
10223:
10222:
10200:
10197:
10191:
10188:
10184:
10179:
10175:
10172:
10165:
10162:
10150:
10147:
10136:
10127:
10124:
10118:
10110:
10101:
10097:
10091:
10088:
10082:
10076:
10072:
10065:
10057:
10053:
10049:
10042:
10041:
10040:
10018:
10015:
10003:
10000:
9994:
9988:
9985:
9979:
9976:
9973:
9970:
9964:
9961:
9957:
9951:
9948:
9942:
9936:
9932:
9925:
9917:
9913:
9909:
9902:
9901:
9900:
9896:
9892:
9888:
9881:
9877:
9870:
9866:
9862:
9858:
9854:
9850:
9843:
9839:
9818:
9815:
9803:
9800:
9794:
9791:
9785:
9782:
9778:
9772:
9769:
9763:
9757:
9753:
9746:
9738:
9734:
9730:
9723:
9722:
9721:
9718:
9702:
9683:
9679:
9672:
9666:
9663:
9658:
9647:
9644:
9637:
9633:
9630:
9627:
9621:
9613:
9600:
9599:
9598:
9577:
9573:
9570:
9562:
9559:
9552:
9544:
9541:
9538:
9532:
9526:
9517:
9511:
9508:
9505:
9502:
9498:
9491:
9488:
9485:
9479:
9476:
9473:
9467:
9464:
9461:
9458:
9455:
9449:
9442:
9441:
9440:
9428:
9424:
9402:
9397:
9392:
9385:
9382:
9373:
9369:
9360:
9354:
9351:
9348:
9345:
9341:
9333:
9332:
9331:
9329:
9295:
9288:
9285:
9278:
9272:
9269:
9264:
9261:
9248:
9242:
9239:
9234:
9227:
9224:
9215:
9212:
9198:
9195:
9192:
9187:
9183:
9179:
9175:
9169:
9166:
9160:
9157:
9154:
9148:
9143:
9140:
9134:
9128:
9124:
9117:
9114:
9111:
9105:
9092:
9088:
9084:
9078:
9075:
9072:
9069:
9066:
9060:
9053:
9052:
9051:
9046:
9036:
9034:
9030:
9026:
9001:
8995:
8990:
8987:
8981:
8978:
8975:
8968:
8967:
8966:
8955:
8947:
8939:
8915:
8909:
8902:
8895:
8892:
8886:
8883:
8879:
8875:
8870:
8866:
8858:
8857:
8856:
8839:
8836:
8833:
8830:
8827:
8824:
8821:
8818:
8815:
8812:
8800:
8796:
8793:
8788:
8785:
8779:
8775:
8772:
8769:
8766:
8762:
8758:
8753:
8750:
8747:
8744:
8741:
8737:
8730:
8726:
8723:
8720:
8717:
8711:
8703:
8702:
8701:
8698:
8692:
8685:
8681:
8661:
8655:
8649:
8646:
8640:
8636:
8633:
8630:
8627:
8621:
8614:
8611:
8605:
8596:
8591:
8588:
8582:
8577:
8569:
8565:
8561:
8556:
8552:
8548:
8543:
8539:
8535:
8530:
8526:
8519:
8512:
8511:
8510:
8506:
8502:
8496:
8476:
8471:
8466:
8457:
8454:
8451:
8448:
8445:
8441:
8437:
8434:
8427:
8424:
8421:
8418:
8414:
8408:
8404:
8398:
8394:
8390:
8384:
8376:
8373:
8370:
8367:
8364:
8360:
8356:
8353:
8346:
8343:
8340:
8337:
8334:
8330:
8326:
8323:
8316:
8310:
8305:
8301:
8297:
8292:
8287:
8283:
8278:
8273:
8263:
8258:
8255:
8249:
8245:
8240:
8237:
8230:
8227:
8222:
8217:
8211:
8208:
8205:
8202:
8199:
8195:
8191:
8188:
8184:
8176:
8172:
8169:
8166:
8163:
8157:
8150:
8147:
8141:
8132:
8127:
8124:
8118:
8113:
8105:
8101:
8097:
8092:
8088:
8084:
8079:
8075:
8071:
8066:
8062:
8055:
8048:
8047:
8046:
8042:
8038:
8019:
8011:
8007:
7988:
7984:
7960:
7952:
7942:
7938:
7929:
7925:
7916:
7912:
7903:
7899:
7893:
7883:
7878:
7874:
7870:
7864:
7860:
7855:
7852:
7842:
7839:
7836:
7832:
7828:
7826:
7816:
7809:
7806:
7803:
7800:
7793:
7789:
7783:
7779:
7775:
7772:
7769:
7766:
7763:
7760:
7752:
7747:
7743:
7739:
7734:
7729:
7725:
7715:
7712:
7706:
7703:
7692:
7686:
7681:
7678:
7672:
7669:
7663:
7657:
7654:
7651:
7648:
7642:
7639:
7636:
7631:
7628:
7622:
7617:
7615:
7605:
7601:
7597:
7592:
7588:
7584:
7579:
7575:
7571:
7566:
7562:
7555:
7544:
7543:
7542:
7539:
7532:
7525:
7505:
7493:
7489:
7485:
7480:
7476:
7469:
7466:
7463:
7457:
7454:
7451:
7446:
7443:
7427:
7423:
7419:
7413:
7409:
7397:
7393:
7389:
7384:
7380:
7373:
7370:
7367:
7357:
7353:
7349:
7344:
7340:
7333:
7313:
7309:
7303:
7299:
7296:
7288:
7284:
7280:
7275:
7271:
7267:
7262:
7258:
7254:
7249:
7245:
7238:
7231:
7230:
7229:
7212:
7206:
7203:
7198:
7195:
7192:
7189:
7183:
7179:
7171:
7167:
7161:
7157:
7151:
7148:
7144:
7133:
7130:
7127:
7123:
7115:
7114:
7113:
7111:
7110:sinc function
7092:
7077:
7073:
7069:
7064:
7060:
7050:
7047:
7044:
7040:
7034:
7031:
7024:
7023:
7022:
6997:
6994:
6989:
6984:
6978:
6973:
6967:
6964:
6954:
6950:
6946:
6941:
6937:
6930:
6924:
6919:
6916:
6912:
6900:
6897:
6894:
6890:
6886:
6870:
6866:
6860:
6856:
6854:
6844:
6839:
6830:
6826:
6822:
6817:
6813:
6804:
6800:
6796:
6788:
6784:
6780:
6775:
6771:
6763:
6755:
6752:
6742:
6736:
6733:
6726:
6721:
6717:
6707:
6703:
6697:
6692:
6689:
6684:
6680:
6676:
6673:
6667:
6662:
6659:
6646:
6643:
6640:
6636:
6619:
6615:
6609:
6605:
6602:
6600:
6590:
6586:
6582:
6577:
6573:
6569:
6564:
6560:
6556:
6551:
6547:
6540:
6529:
6528:
6527:
6510:
6506:
6497:
6493:
6489:
6484:
6480:
6471:
6467:
6463:
6455:
6451:
6447:
6442:
6438:
6430:
6422:
6419:
6409:
6403:
6400:
6393:
6388:
6384:
6377:
6374:
6361:
6358:
6355:
6351:
6347:
6338:
6334:
6331:
6324:
6323:
6322:
6297:
6290:
6279:
6275:
6271:
6266:
6262:
6255:
6252:
6249:
6242:
6238:
6232:
6228:
6224:
6221:
6213:
6209:
6205:
6200:
6196:
6189:
6186:
6183:
6175:
6170:
6166:
6162:
6157:
6152:
6148:
6138:
6134:
6131:
6126:
6123:
6118:
6116:
6108:
6105:
6100:
6094:
6090:
6084:
6080:
6076:
6070:
6067:
6061:
6056:
6049:
6046:
6039:
6033:
6030:
6023:
6015:
6011:
6003:
5999:
5994:
5990:
5987:
5984:
5969:
5965:
5957:
5953:
5948:
5944:
5942:
5931:
5919:
5918:
5917:
5900:
5889:
5885:
5881:
5876:
5872:
5865:
5862:
5859:
5849:
5845:
5841:
5838:
5832:
5829:
5826:
5818:
5814:
5810:
5799:
5795:
5791:
5786:
5782:
5775:
5772:
5769:
5761:
5758:
5753:
5749:
5742:
5739:
5736:
5728:
5724:
5720:
5714:
5702:
5694:
5693:
5692:
5689:
5682:
5678:
5671:
5667:
5663:
5656:
5652:
5648:
5641:
5637:
5617:
5612:
5608:
5602:
5598:
5594:
5588:
5585:
5579:
5574:
5567:
5564:
5557:
5551:
5548:
5542:
5528:
5527:
5526:
5522:
5521:Mehler kernel
5518:
5494:
5489:
5485:
5479:
5474:
5464:
5461:
5456:
5452:
5445:
5433:
5426:
5425:
5424:
5396:
5393:
5380:
5377:
5373:
5367:
5364:
5358:
5352:
5347:
5344:
5338:
5332:
5328:
5321:
5313:
5309:
5305:
5302:
5299:
5296:
5289:
5286:
5283:
5280:
5277:
5271:
5265:
5257:
5253:
5249:
5246:
5240:
5232:
5228:
5220:
5219:
5218:
5197:
5194:
5189:
5184:
5174:
5171:
5165:
5162:
5159:
5153:
5147:
5144:
5140:
5131:
5130:
5129:
5112:
5105:
5102:
5095:
5087:
5084:
5081:
5075:
5069:
5065:
5059:
5056:
5053:
5050:
5047:
5041:
5034:
5033:
5032:
5030:
5029:Wick rotation
5002:
4999:
4994:
4989:
4979:
4973:
4970:
4967:
4961:
4955:
4952:
4948:
4939:
4938:
4937:
4920:
4917:
4914:
4911:
4904:
4901:
4898:
4895:
4892:
4886:
4883:
4876:
4875:
4874:
4871:
4869:
4863:
4845:
4837:
4834:
4827:
4819:
4816:
4813:
4804:
4800:
4796:
4790:
4787:
4784:
4781:
4778:
4772:
4765:
4764:
4763:
4742:
4735:
4729:
4725:
4721:
4715:
4711:
4707:
4701:
4698:
4695:
4683:
4673:
4672:
4671:
4654:
4647:
4641:
4637:
4633:
4627:
4623:
4619:
4613:
4605:
4595:
4584:
4583:
4582:
4561:
4556:
4552:
4548:
4540:
4532:
4522:
4515:
4509:
4506:
4503:
4491:
4481:
4480:
4479:
4472:
4464:
4440:
4435:
4431:
4427:
4424:
4421:
4416:
4412:
4408:
4403:
4399:
4395:
4389:
4386:
4383:
4380:
4377:
4371:
4364:
4363:
4362:
4353:
4343:
4339:
4332:
4328:
4324:
4296:
4293:
4281:
4277:
4272:
4267:
4262:
4255:
4249:
4246:
4240:
4237:
4234:
4228:
4222:
4214:
4211:
4205:
4201:
4197:
4194:
4189:
4185:
4179:
4176:
4170:
4164:
4159:
4156:
4150:
4144:
4140:
4136:
4130:
4127:
4124:
4121:
4118:
4112:
4105:
4104:
4103:
4086:
4083:
4071:
4067:
4056:
4051:
4044:
4041:
4030:
4025:
4021:
4017:
4013:
4009:
4006:
4001:
3998:
3992:
3986:
3981:
3978:
3972:
3966:
3962:
3958:
3952:
3949:
3946:
3943:
3940:
3934:
3927:
3926:
3925:
3918:
3901:
3898:
3887:
3882:
3875:
3872:
3863:
3860:
3857:
3850:
3849:
3848:
3844:
3837:
3821:Free particle
3815:
3777:
3768:
3757:
3751:
3737:
3733:
3730:
3727:
3724:
3715:
3704:
3693:
3686:
3685:
3684:
3670:
3650:
3630:
3627:
3621:
3532:
3516:
3512:
3500:
3489:
3478:
3475:
3471:
3452:
3449:
3443:
3431:
3425:
3422:
3417:
3411:
3408:
3405:
3399:
3392:
3391:
3390:
3385:Path integral
3382:
3379:
3364:
3360:
3356:
3332:
3322:
3306:
3301:
3281:
3276:
3272:
3269:
3264:
3258:
3255:
3252:
3248:
3244:
3239:
3235:
3228:
3223:
3213:
3205:
3201:
3196:
3193:
3190:
3185:
3182:
3179:
3175:
3171:
3163:
3157:
3153:
3150:
3143:
3142:
3141:
3121:
3115:
3112:
3107:
3094:
3091:
3080:
3075:
3072:
3067:
3064:
3061:
3058:
3055:
3046:
3043:
3037:
3034:
3028:
3021:
3020:
3019:
3017:
3008:
2975:
2968:
2961:
2954:
2944:
2938:
2934:In the limit
2932:
2913:
2909:
2905:
2901:and velocity
2898:
2894:
2887:
2883:
2879:
2859:
2854:
2850:
2846:
2842:
2836:
2832:
2828:
2823:
2819:
2816:
2802:
2796:
2793:
2787:
2781:
2771:
2764:
2760:
2752:
2748:
2743:
2734:
2728:
2724:
2721:
2713:
2705:
2701:
2697:
2689:
2681:
2677:
2669:
2668:
2667:
2664:
2662:
2643:
2637:
2634:
2631:
2624:
2620:
2616:
2611:
2607:
2600:
2597:
2591:
2588:
2581:
2580:
2579:
2575:
2571:
2563:
2559:
2555:
2545:
2523:
2519:
2515:
2510:
2507:
2504:
2500:
2496:
2491:
2487:
2483:
2478:
2475:
2472:
2468:
2464:
2461:
2458:
2453:
2449:
2445:
2440:
2436:
2432:
2427:
2423:
2415:
2414:
2413:
2383:
2378:
2376:
2370:
2355:
2353:
2349:
2345:
2341:
2336:
2334:
2330:
2326:
2321:
2320:(see below).
2319:
2315:
2311:
2307:
2303:
2302:path integral
2299:
2294:
2290:
2282:
2278:
2277:time integral
2275:given by the
2274:
2265:
2260:
2257:
2253:
2250:
2246:
2242:
2241:
2240:
2236:
2234:
2224:
2222:
2215:
2203:
2192:
2185:
2130:
2125:
2121:
2096:
2092:
2087:
2083:
2076:
2069:
2064:
2060:
2050:
2043:
2035:
2031:
2027:
2020:
2005:
2001:
1977:
1972:
1969:
1966:
1962:
1954:
1953:
1952:
1945:
1943:
1918:
1903:
1895:
1878:
1870:
1867:
1864:
1858:
1855:
1852:
1848:
1840:
1839:
1838:
1835:
1831:
1814:
1806:
1803:
1800:
1794:
1791:
1788:
1785:
1781:
1773:
1772:
1771:
1768:
1757:
1753:
1749:
1742:
1738:
1733:
1727:
1723:
1703:
1695:
1689:
1686:
1683:
1680:
1676:
1668:
1667:
1666:
1664:
1645:
1630:
1627:
1624:
1618:
1615:
1612:
1599:
1593:
1590:
1584:
1581:
1578:
1572:
1562:
1552:
1548:
1540:
1539:
1538:
1512:
1507:
1482:
1476:
1468:
1459:
1453:
1450:
1440:
1439:
1438:
1421:
1418:
1415:
1412:
1406:
1403:
1397:
1394:
1391:
1388:
1385:
1378:
1377:
1376:
1372:
1368:
1364:
1344:
1341:
1315:
1306:
1303:
1292:
1283:
1280:
1273:
1272:
1271:
1254:
1251:
1248:
1235:
1229:
1226:
1220:
1217:
1214:
1208:
1195:
1189:
1186:
1183:
1180:
1173:
1172:
1171:
1169:
1164:
1162:
1158:
1154:
1148:
1146:
1142:
1138:
1134:
1130:
1125:
1123:
1119:
1114:
1108:
1103:
1089:
1085:
1083:
1079:
1075:
1071:
1067:
1063:
1059:
1055:
1051:
1047:
1042:
1040:
1036:
1032:
1028:
1024:
1023:string theory
1020:
1015:
1013:
1009:
1005:
1001:
997:
993:
989:
985:
980:
978:
974:
970:
966:
962:
958:
954:
950:
946:
942:
938:
934:
929:
927:
923:
919:
915:
911:
907:
895:
890:
888:
883:
881:
876:
875:
873:
872:
864:
861:
859:
856:
854:
851:
849:
846:
844:
841:
839:
836:
834:
831:
829:
826:
824:
821:
819:
816:
814:
811:
809:
806:
804:
801:
799:
796:
794:
791:
789:
786:
784:
781:
779:
776:
774:
771:
769:
766:
764:
761:
759:
756:
754:
751:
749:
746:
744:
741:
739:
736:
734:
731:
729:
726:
724:
721:
719:
716:
714:
711:
709:
706:
704:
701:
699:
696:
694:
691:
689:
686:
684:
681:
679:
676:
674:
671:
669:
666:
664:
661:
659:
656:
654:
651:
649:
646:
644:
641:
639:
636:
635:
628:
627:
620:
617:
615:
612:
610:
607:
605:
602:
600:
597:
595:
594:Quantum chaos
592:
590:
587:
585:
582:
580:
577:
575:
572:
571:
565:
564:
556:
553:
551:
550:Transactional
548:
546:
543:
541:
540:Quantum logic
538:
536:
533:
531:
528:
522:
519:
518:
517:
514:
513:
512:
509:
507:
504:
502:
499:
497:
494:
492:
489:
487:
484:
483:
479:
474:
473:
465:
462:
460:
457:
455:
452:
450:
447:
445:
442:
441:
434:
433:
425:
422:
420:
417:
415:
412:
410:
407:
405:
402:
400:
397:
396:
392:
389:
388:
382:
381:
373:
370:
368:
365:
363:
360:
359:
353:
350:
349:
348:
345:
344:
340:
337:
335:
332:
330:
327:
325:
322:
320:
317:
315:
312:
310:
307:
305:
302:
300:
297:
295:
292:
291:
284:
283:
273:
270:
269:
268:
267:Wave function
265:
263:
260:
258:
255:
253:
250:
248:
247:Superposition
245:
243:
240:
238:
235:
233:
230:
228:
225:
223:
220:
218:
215:
213:
210:
208:
205:
204:
197:
196:
188:
185:
183:
180:
179:
175:
172:
170:
167:
165:
162:
161:
155:
154:
149:
146:
144:
141:
139:
136:
135:
134:
133:
129:
96:
90:
73:
70:
66:
58:
51:
50:
47:
44:
43:
39:
38:
33:
32:line integral
19:
26381:
26370:
26360:
26347:
26340:
26302:Joan Feynman
26279:
26268:
26257:
26246:
26235:
26224:
26213:
26202:
26191:
26136:Parton model
26130:
25759:Quantum mind
25671:Franck–Hertz
25513:Klein–Gordon
25491:
25467:Formulations
25460:Formulations
25389:Interference
25379:Entanglement
25357:Ground state
25352:Energy level
25325:Fundamentals
25289:Introduction
25196:
25174:
25149:
25120:
25115:Weinberg, S.
25070:
25066:
25039:
25003:
24999:
24973:
24951:
24929:
24908:
24899:
24880:
24845:
24822:
24800:
24781:
24762:
24719:
24685:
24664:
24643:
24621:
24618:Styer, D. F.
24614:Hibbs, A. R.
24589:
24562:
24558:
24530:10.1142/5852
24512:
24482:. Retrieved
24462:
24458:
24435:
24431:
24383:(1): 47–67.
24380:
24376:
24354:
24307:
24303:
24275:
24271:
24245:
24236:
24225:Bibliography
24211:
24200:
24189:
24178:. Retrieved
24173:
24163:
24136:
24133:Scholarpedia
24132:
24122:
24110:
24098:
24087:
24075:
24063:
24057:Feynman 1948
24052:
24040:
24031:
24018:
24007:
24000:Feynman 1948
23995:
23974:
23963:
23952:
23941:
23930:. Retrieved
23926:
23913:
23902:. Retrieved
23896:
23886:
23851:. Retrieved
23844:the original
23830:
23825:, Chapter 9.
23818:
23794:
23781:
23706:
23697:
23694:
23598:
23574:
23562:
23547:
23074:
23058:
22926:
22871:
22868:
22865:
22539:
22369:
22286:
22280:
22276:
22269:
22142:
22065:
21956:
21894:
21891:
21788:
21704:
21697:
21691:
21627:
21600:
21597:
21582:
21494:
21483:
21467:
21461:
21458:Localization
21451:
21445:
21376:
21366:
21362:target space
21356:
21343:
21337:
21334:
21254:
21119:
21013:
20738:
20549:
20531:
20508:
20504:
20497:
20493:
20473:
20470:
20464:
20265:
20118:
19930:
19820:
19798:source field
19796:(called the
19791:
19784:
19772:
19678:
19534:
19489:
19423:
19414:
19403:
19337:
19333:
19309:
19306:
19028:
19018:
19014:
19005:
19001:
18993:
18989:
18971:
18969:
18957:
18950:
18809:
18792:time-ordered
18774:
18748:
18738:
18731:
18726:
18725:is called a
18716:
18712:
18707:
18701:
18697:
18691:
18678:
18676:
18667:
18659:
18653:
18649:
18642:
18554:
18551:
18367:
18360:
18353:
18350:
18346:
18339:
18217:
18210:
18200:
18193:
18190:
18013:
17910:
17900:
17897:
17752:
17748:
17741:This is the
17740:
17606:
17603:
17484:
17351:
17214:
17210:
17206:
17202:
17198:
17032:
17017:
16892:
16883:
16864:
16857:
16739:
16735:
16731:
16724:
16638:
16634:
16630:
16526:
16451:
16447:
16439:
16435:
16431:
16428:
16231:
16224:
16213:
16043:
16039:
16026:
16014:
16010:
16002:
15989:
15979:
15881:
15842:
15688:Stueckelberg
15418:Jona-Lasinio
14993:Vacuum state
14978:Quantization
14815:Gauge theory
14795:Strong force
14780:Field theory
14730:
14720:
14662:is given by
14653:
14646:
14557:
14543:
14535:
14529:
14516:
14492:
14488:
14478:
14398:
14391:
14384:
14199:
14190:
14158:
14027:
13868:
13689:
13541:
13536:
13534:
13294:
12933:
12920:
12763:
12265:
11897:
11886:
11875:
11869:
11741:
11738:
11664:
11644:
11595:
11521:
11510:
11403:
11401:
11385:
11313:
11309:
11306:
11205:
11078:
10843:
10840:
10769:
10763:
10671:
10667:
10664:
10586:
10575:
10551:
10535:
10532:
10430:
10426:
10423:
10216:
10034:
9894:
9890:
9886:
9879:
9875:
9868:
9864:
9860:
9856:
9852:
9848:
9841:
9837:
9834:
9719:
9708:
9699:
9592:
9426:
9422:
9419:
9313:
9048:
9022:
8953:
8945:
8935:
8854:
8696:
8690:
8683:
8679:
8676:
8504:
8500:
8494:
8491:
8040:
8036:
8017:
8009:
8005:
7986:
7982:
7979:
7534:
7527:
7523:
7520:
7227:
7107:
7020:
6525:
6316:
5915:
5687:
5680:
5676:
5669:
5665:
5661:
5654:
5650:
5646:
5639:
5635:
5632:
5524:
5411:
5212:
5127:
5017:
4935:
4872:
4864:
4860:
4757:
4669:
4576:
4470:
4462:
4455:
4341:
4337:
4335:centered at
4330:
4326:
4322:
4311:
4101:
3919:
3916:
3842:
3835:
3824:
3814:Cornu Spiral
3554:
3388:
3380:
3362:
3358:
3351:
3327:
3320:
3304:
3297:
3139:
3009:
2970:
2963:
2956:
2949:
2936:
2933:
2911:
2907:
2903:
2896:
2892:
2885:
2881:
2877:
2874:
2665:
2661:time-slicing
2660:
2658:
2573:
2569:
2561:
2557:
2550:
2543:
2540:
2379:
2372:
2352:perturbative
2337:
2322:
2318:interference
2308:but varying
2306:equal weight
2305:
2301:
2292:
2286:
2263:
2237:
2230:
2211:
2190:
2183:
2128:
2123:
2119:
2094:
2090:
2085:
2081:
2074:
2067:
2062:
2058:
2048:
2045:
2041:
2029:
2025:
2018:
2003:
1999:
1992:
1946:
1919:
1904:
1894:which means
1893:
1836:
1829:
1769:
1751:
1747:
1740:
1736:
1725:
1721:
1718:
1662:
1660:
1508:
1497:
1436:
1370:
1366:
1362:
1330:
1269:
1165:
1149:
1126:
1112:
1099:
1043:
1016:
1012:random walks
981:
976:
963:), than the
960:
956:
930:
905:
903:
449:Klein–Gordon
423:
385:Formulations
222:Energy level
217:Entanglement
200:Fundamentals
187:Interference
138:Introduction
26385:(2013 film)
26375:(2001 play)
26364:(1996 film)
26017:EPR paradox
25797:Quantum bus
25666:Double-slit
25644:Experiments
25610:Many-worlds
25548:Schrödinger
25497:Phase space
25487:Schrödinger
25477:Interaction
25434:Uncertainty
25404:Nonlocality
25399:Measurement
25394:Decoherence
25384:Hamiltonian
25121:Foundations
25036:Tomé, W. A.
24493:Etingof, P.
24444:Duru, İ. H.
24139:(2). 8674.
23751:Propagators
23565:decoherence
23558:nonlocality
21503:, and that
18951:The symbol
18801:functional
16458:convolution
15828:Zinn-Justin
15678:Sommerfield
15603:Pomeranchuk
15568:Osterwalder
15563:Oppenheimer
15493:Łopuszański
15313:Fredenhagen
15114:Technicolor
14270:where
10775:, so that:
4352:convolution
3300:Riemann sum
2333:Hamiltonian
2245:probability
1770:Next comes
1145:four-vector
1102:Hamiltonian
1082:Hamiltonian
965:Hamiltonian
945:coordinates
838:von Neumann
823:Schrödinger
599:EPR paradox
530:Many-worlds
464:Schrödinger
419:Schrödinger
414:Phase-space
404:Interaction
309:Double-slit
287:Experiments
262:Uncertainty
232:Nonlocality
227:Measurement
212:Decoherence
182:Hamiltonian
26401:Categories
25945:Extensions
25779:Technology
25625:Relational
25576:Copenhagen
25472:Heisenberg
25419:Tunnelling
25282:Background
24484:2007-11-25
24180:2021-07-16
24092:Simon 1979
24023:Hilke, M.
24012:Dirac 1933
23957:Dirac 1933
23932:2024-06-21
23904:2024-06-21
23877:2003.12683
23853:2018-12-15
23810:References
23061:regulators
21611:derivation
21501:Lagrangian
19931:Note that
19802:dual space
19545:polynomial
19541:reciprocal
18785:functional
18687:functional
16216:propagator
16036:Propagator
15823:Zimmermann
15718:Vainshtein
15593:Polchinski
15453:Kontsevich
15398:Iliopoulos
15373:Heisenberg
15198:Bogoliubov
15136:Scientists
14973:Propagator
14855:T-symmetry
14850:P-symmetry
14845:C-symmetry
14833:Symmetries
14790:Weak force
14774:Background
14028:where now
13725:, denoted
11653:explained
5515:See also:
4312:where the
3643:and where
3016:Lagrangian
2572:= 1, ...,
2296:elaborate
2289:integrates
2281:Lagrangian
1942:Paul Dirac
1129:Lagrangian
1078:Lagrangian
1062:Paul Dirac
1058:Lagrangian
984:stochastic
953:Lagrangian
833:Sommerfeld
748:Heisenberg
743:Gutzwiller
683:de Broglie
631:Scientists
545:Relational
496:Copenhagen
399:Heisenberg
257:Tunnelling
158:Background
26437:Integrals
26331:Namesakes
25651:Bell test
25506:Equations
25332:Born rule
25117:(2002) ,
25028:121370426
24970:Simon, B.
24754:117837329
24413:119669964
24342:119581543
24176:. 02-1020
24080:Hall 2013
24068:Hall 2013
23986:1004.3578
23800:Hall 2013
23721:to model
23673:ℏ
23650:−
23642:
23621:Γ
23522:^
23510:^
23495:^
23483:^
23441:^
23429:^
23400:^
23388:^
23336:^
23324:^
23295:^
23283:^
23228:ℏ
23187:ℏ
23178:−
23146:^
23134:^
23119:^
23107:^
23005:δ
23001:δ
22993:−
22973:φ
22938:∫
22822:δ
22818:δ
22810:−
22790:φ
22743:δ
22739:δ
22731:−
22712:μ
22702:μ
22698:∂
22668:δ
22664:δ
22656:−
22636:φ
22589:δ
22585:δ
22577:−
22503:μ
22493:μ
22489:∂
22471:⟩
22447:⟨
22441:⟩
22414:⟨
22405:⟩
22381:⟨
22346:φ
22318:φ
22308:∫
22251:φ
22220:φ
22215:μ
22211:∂
22204:∂
22200:∂
22195:−
22181:μ
22159:μ
22116:μ
22106:μ
22102:∂
22066:so that\
22038:φ
22023:−
22002:δ
21983:φ
21916:∫
21864:μ
21850:μ
21837:∂
21833:∫
21815:⟩
21800:⟨
21762:φ
21726:φ
21716:∫
21668:φ
21649:φ
21639:∫
21557:μ
21547:μ
21543:∂
21425:
21416:−
21400:^
21292:∂
21286:−
21181:δ
21177:δ
21169:−
21149:φ
21146:δ
21134:δ
21070:φ
21052:φ
21049:δ
21037:δ
20962:δ
20958:δ
20928:∂
20911:∂
20905:⋯
20880:δ
20876:δ
20846:∂
20829:∂
20810:−
20778:δ
20774:δ
20766:−
20705:φ
20677:∂
20660:∂
20654:⋯
20635:φ
20607:∂
20590:∂
20578:φ
20514:)⟩
20437:⟩
20434:φ
20425:⟨
20416:φ
20390:φ
20380:∫
20368:⟩
20365:φ
20356:⟨
20347:φ
20318:φ
20309:φ
20299:∫
20284:⟩
20277:⟨
20224:φ
20220:⋯
20204:φ
20148:⋯
20074:φ
20071:⋯
20052:φ
19985:δ
19982:⋯
19960:δ
19946:δ
19906:⟩
19903:φ
19894:⟨
19885:φ
19859:φ
19849:∫
19722:−
19683:. In the
19656:φ
19653:δ
19645:φ
19632:δ
19623:φ
19606:−
19593:φ
19590:δ
19582:φ
19573:δ
19535:for some
19515:φ
19506:−
19467:φ
19380:φ
19377:δ
19369:φ
19356:δ
19272:φ
19258:φ
19234:φ
19218:φ
19208:∫
19194:⊂
19187:∑
19167:φ
19153:φ
19129:φ
19113:φ
19103:∫
19089:∩
19083:⊂
19076:∑
19060:∣
19051:
18925:φ
18904:φ
18894:∫
18884:φ
18860:φ
18851:φ
18841:∫
18832:⟩
18826:⟨
18779:, if the
18723:spacetime
18622:ε
18603:−
18531:ε
18525:−
18498:→
18486:−
18461:ε
18428:→
18416:−
18300:→
18288:−
18279:−
18146:→
18085:→
18073:−
17983:−
17971:→
17961:−
17877:α
17841:α
17833:−
17815:−
17805:∞
17796:∫
17707:α
17704:−
17680:−
17668:−
17658:∞
17649:∫
17636:−
17549:−
17535:∞
17526:∫
17513:−
17444:−
17432:−
17417:α
17414:−
17389:−
17329:−
17278:∫
17267:α
17264:−
17239:−
17179:τ
17172:α
17169:−
17155:˙
17130:∫
17075:∫
17054:−
16997:ε
16968:→
16952:−
16946:−
16938:−
16885:Gaussian
16837:ε
16809:→
16797:−
16786:−
16588:−
16566:∝
16494:∗
16485:ψ
16472:ψ
16358:ψ
16333:∫
16280:ψ
16270:∫
16248:ψ
16126:∫
16119:⟩
16107:∣
16095:⟨
16049:to point
15959:−
15944:δ
15922:φ
15913:∂
15897:φ
15758:Wetterich
15743:Weisskopf
15693:Sudarshan
15643:Schwinger
15553:Nishijima
15518:Maldacena
15483:Leutwyler
15443:Kinoshita
15343:Goldstone
15333:Gell-Mann
15248:Doplicher
15015:Equations
14731:classical
14700:β
14694:−
14682:
14632:⟩
14623:α
14611:ℏ
14595:−
14584:⟩
14575:α
14464:ℏ
14461:≫
14348:˙
14296:∫
14249:ℏ
14217:∫
14173:^
14128:ψ
14122:^
14113:−
14092:ψ
14083:∂
14079:∂
14074:ℏ
14036:ψ
13995:μ
13961:ψ
13954:ℏ
13916:∫
13913:−
13905:∫
13884:ψ
13789:μ
13734:μ
13638:˙
13622:∫
13609:−
13564:∫
13537:Euclidean
13457:˙
13426:∫
13414:˙
13239:ψ
13233:ℏ
13219:˙
13161:−
13130:∫
13098:ψ
13069:ℏ
13058:^
13046:−
13019:ℏ
13008:^
12993:−
12962:−
12892:ϕ
12881:ℏ
12865:ϕ
12852:∫
12842:ϕ
12831:ℏ
12815:ϕ
12803:ϕ
12790:∫
12781:⟩
12775:⟨
12731:˙
12695:∫
12690:ℏ
12536:∫
12529:⟩
12490:^
12479:ℏ
12471:−
12457:^
12420:−
12401:^
12390:ℏ
12382:−
12368:^
12331:−
12319:^
12308:ℏ
12300:−
12277:⟨
12233:˙
12197:∫
12192:ℏ
12082:∫
12058:⟩
12021:^
12010:ℏ
12002:−
11988:^
11960:−
11948:^
11937:ℏ
11929:−
11906:⟨
11828:−
11816:˙
11797:˙
11759:∫
11679:μ
11676:∫
11564:˙
11555:∂
11547:∂
11513:Itō lemma
11467:−
11432:˙
11354:−
11290:ε
11260:−
11254:ε
11189:ε
11186:ε
11181:≈
11176:ε
11146:−
11140:ε
11103:−
11062:ε
11046:−
11040:ε
11019:ε
10946:ε
10930:−
10924:ε
10824:ε
10819:≈
10804:−
10798:ε
10747:ε
10731:−
10725:ε
10608:∫
10501:ψ
10477:δ
10469:δ
10452:ψ
10367:ε
10353:∂
10345:∂
10339:−
10330:˙
10321:∂
10313:∂
10287:∫
10279:−
10255:∫
10236:ψ
10232:∫
10221:, gives:
10166:˙
10163:ε
10151:˙
10142:∂
10134:∂
10125:ε
10116:∂
10108:∂
10102:∫
10073:∫
10054:ψ
10050:∫
10004:˙
10001:ε
9989:˙
9977:ε
9933:∫
9914:ψ
9910:∫
9804:˙
9754:∫
9735:ψ
9731:∫
9684:ψ
9664:−
9655:∇
9634:⋅
9619:∂
9614:ψ
9611:∂
9563:ε
9542:−
9509:ε
9503:−
9480:ψ
9477:∫
9474:≈
9468:ε
9450:ψ
9403:ε
9386:˙
9352:ε
9346:−
9322:close to
9240:−
9228:˙
9199:ε
9184:∫
9161:ε
9125:∫
9106:ψ
9101:∞
9096:∞
9093:−
9089:∫
9079:ε
9061:ψ
8982:∫
8910:ω
8907:ℏ
8837:…
8809:for
8773:ω
8767:−
8751:ω
8742:−
8724:ω
8718:−
8647:⋅
8634:ω
8628:−
8600:ℏ
8597:π
8592:ω
8455:ω
8446:−
8438:−
8425:ω
8419:−
8385:−
8374:ω
8365:−
8357:−
8344:ω
8335:−
8267:ℏ
8259:ω
8250:−
8241:
8223:−
8209:ω
8200:−
8192:−
8170:ω
8164:−
8136:ℏ
8133:π
8128:ω
7953:∗
7926:ψ
7900:ψ
7888:ℏ
7865:−
7856:
7848:∞
7833:∑
7807:ω
7804:
7773:−
7767:ω
7764:
7716:ω
7696:ℏ
7682:
7655:ω
7652:
7646:ℏ
7640:π
7632:ω
7486:−
7470:ω
7467:
7461:ℏ
7455:π
7447:ω
7434:ℏ
7390:−
7374:ω
7371:
7350:−
7334:ω
7324:ℏ
7204:π
7196:π
7193:
7152:−
7139:∞
7124:∏
7070:−
7054:ℏ
7048:π
6990:−
6968:π
6947:−
6931:ω
6920:−
6906:∞
6891:∏
6881:ℏ
6823:−
6801:ω
6797:−
6781:−
6756:π
6711:ℏ
6693:
6674:∫
6663:π
6652:∞
6637:∏
6630:ℏ
6490:−
6468:ω
6464:−
6448:−
6423:π
6367:∞
6352:∑
6272:−
6256:ω
6253:
6222:−
6206:−
6190:ω
6187:
6135:ω
6081:ω
6062:−
6050:˙
5995:∫
5949:∫
5882:−
5866:ω
5863:
5842:−
5833:ω
5830:
5792:−
5776:ω
5773:
5759:−
5743:ω
5740:
5599:ω
5580:−
5568:˙
5486:ψ
5471:∇
5465:−
5453:ψ
5443:∂
5439:∂
5329:∫
5310:ψ
5306:∫
5281:−
5254:ψ
5250:∫
5229:ψ
5181:∇
5085:−
5066:∝
5051:−
4986:∇
4896:−
4884:∫
4817:−
4805:−
4797:∝
4782:−
4716:−
4687:~
4634:ε
4628:−
4606:ε
4599:~
4557:ε
4533:ε
4526:~
4495:~
4436:ε
4428:∗
4425:⋯
4422:∗
4417:ε
4409:∗
4404:ε
4381:−
4278:ε
4263:ε
4247:−
4241:ε
4206:−
4198:
4186:∏
4141:∫
4122:−
4045:˙
4022:∫
4018:−
4010:
3963:∫
3944:−
3876:˙
3864:∫
3829:to point
3769:˙
3728:∫
3716:˙
3513:ψ
3501:˙
3432:∫
3400:ψ
3265:ε
3256:−
3245:−
3217:~
3176:∑
3172:ε
3167:ℏ
3154:
3113:−
3095:˙
3062:−
3047:˙
3010:Actually
2843:⋯
2744:∫
2738:ℏ
2725:
2717:∞
2709:∞
2706:−
2702:∫
2698:⋯
2693:∞
2685:∞
2682:−
2678:∫
2617:−
2595:Δ
2589:ε
2476:−
2462:⋯
2298:curlicues
1997:and time
1970:ε
1871:ε
1792:ε
1786:−
1681:−
1616:ε
1613:−
1591:−
1585:ε
1502:at fixed
1474:∂
1466:∂
1454:˙
1416:ε
1413:−
1407:˙
1395:ε
1386:ε
1345:˙
1307:˙
1298:∂
1290:∂
1252:ε
1249:−
1227:−
1221:ε
1181:ε
1027:cosmology
990:with the
969:unitarity
949:canonical
863:Zeilinger
708:Ehrenfest
437:Equations
114:⟩
111:Ψ
100:^
88:⟩
85:Ψ
62:ℏ
26361:Infinity
26315:(nephew)
26306:(sister)
26040:Category
25834:Timeline
25586:Ensemble
25566:Bayesian
25528:Majorana
25444:Collapse
25316:Glossary
25299:Timeline
25195:(2023).
25173:(2004).
25107:16577107
25061:(1928).
25038:(1998).
24972:(1979).
24843:(2004).
24821:(2010).
24620:(2010).
24495:(2002).
24450:(1979).
24438:: 64–72.
24423:(1933).
24235:(1971).
23729:See also
23591:models.
23589:spinfoam
22905:⟩
22888:⟨
22518:⟩
22481:⟨
21870:⟩
21825:⟨
21486:on shell
21347:, where
21092:⟩
21030:⟨
20542:operator
20517:are its
20492:⟨
20241:⟩
20199:⟨
20094:⟩
20048:⟨
19775:on-shell
19754:⟩
19729:⟨
19715:⟩
19699:⟨
19663:⟩
19613:⟨
19599:⟩
19567:⟨
19318:Big Bang
19027: P(
18812:⟩
18808:⟨
16053:in time
15773:Wightman
15738:Weinberg
15728:Virasoro
15708:Tomonaga
15703:Thirring
15698:Symanzik
15658:Semenoff
15633:Schrader
15598:Polyakov
15513:Majorana
15448:Klebanov
15403:Ivanenko
15393:'t Hooft
15363:Guralnik
15308:Fröhlich
15303:Fritzsch
15298:Frampton
15213:Buchholz
15158:Bargmann
15148:Anderson
14928:Crossing
12172:′
12034:′
11967:′
10511:⟩
10447:⟨
2976:⟩
2948:⟨
2567:, where
2405:at time
2391:at time
2331:for the
2267:, where
2205:—
2052:, where
1762:at time
1076:using a
977:deriving
973:S-matrix
788:Millikan
713:Einstein
698:Davisson
653:Blackett
638:Aharonov
506:Ensemble
486:Bayesian
391:Overview
272:Collapse
252:Symmetry
143:Glossary
26324:Related
26185:"
25993:Related
25972:History
25711:Science
25543:Rydberg
25294:History
25144:Zee, A.
25098:1085402
25075:Bibcode
25008:Bibcode
24724:Bibcode
24567:Bibcode
24518:Bibcode
24467:Bibcode
24405:0309456
24385:Bibcode
24322:Bibcode
24280:Bibcode
24141:Bibcode
23773:Remarks
23581:Hawking
23548:In one
23050:Caveats
20747:, then
20525:is its
20519:moments
19821:defined
19543:of any
19415:If the
19034:
19021:)
19010:
16725:and in
15783:Wilczek
15748:Wentzel
15723:Veltman
15668:Shirkov
15663:Shifman
15653:Seiberg
15638:Schwarz
15618:Rubakov
15538:Naimark
15488:Lipatov
15478:Lehmann
15458:Kreimer
15438:Kendall
15328:Gelfand
15323:Glashow
15283:Feynman
15263:Faddeev
15258:Englert
15228:Coleman
15218:Cachazo
15203:Brodsky
15188:Bjorken
15178:Berezin
15168:Belavin
14908:Anomaly
14766:History
14548:
14525:
14521:
14507:
14385:is the
11873:√
10767:√
8958:
8942:
8034:
8022:
8003:
7991:
4475:
4459:
3375:
3348:
3298:in the
2993:and at
2931:terms.
2279:of the
2271:is the
2107:s, say
1917:basis.
828:Simmons
818:Rydberg
783:Moseley
763:Kramers
753:Hilbert
738:Glauber
733:Feynman
718:Everett
688:Compton
459:Rydberg
148:History
26373:
26313:
26304:
26295:Family
26286:(2005)
26284:
26275:(1999)
26273:
26264:(1999)
26262:
26253:(1997)
26251:
26242:(1988)
26240:
26231:(1985)
26229:
26220:(1985)
26218:
26209:(1965)
26207:
26198:(1964)
26196:
26187:(1959)
26089:Career
25686:Popper
25203:
25181:
25158:
25133:
25105:
25095:
25046:
25026:
24980:
24958:
24938:
24915:
24887:
24853:
24829:
24807:
24788:
24769:
24752:
24742:
24692:
24673:
24650:
24628:
24598:
24536:
24411:
24403:
24361:
24340:
24252:
22143:where
21957:where
21705:Then,
21587:where
21450:about
20521:, and
20503:) ...
20266:where
19823:to be
18781:action
18749:called
18679:direct
17604:where
15838:Zumino
15803:Yukawa
15793:Witten
15788:Wilson
15778:Wigner
15713:Tyutin
15673:Skyrme
15623:Ruelle
15588:Plefka
15583:Peskin
15573:Parisi
15533:Møller
15523:Migdal
15508:Maiani
15503:Lüders
15468:Landau
15463:Kuraev
15433:Källén
15423:Jordan
15408:Jackiw
15348:Gribov
15238:DeWitt
15233:Dashen
15223:Callan
15193:Bleuer
15163:Becchi
15153:Anselm
14406:, and
14387:action
13690:where
13295:where
9889:(0) =
9873:where
8913:
7958:
7090:
6295:
5519:, and
5423:does:
3555:where
2962:|
2875:where
2382:zigzag
2273:action
1663:matrix
1437:where
1360:holds
1161:action
998:. The
858:Zeeman
853:Wigner
803:Planck
773:Landau
758:Jordan
409:Matrix
339:Popper
26175:Works
25596:Local
25538:Pauli
25518:Dirac
25024:S2CID
24996:(PDF)
24866:arXiv
24750:S2CID
24705:arXiv
24555:(PDF)
24455:(PDF)
24428:(PDF)
24409:S2CID
24338:S2CID
24312:arXiv
24028:(PDF)
23981:arXiv
23872:arXiv
23847:(PDF)
23840:(PDF)
23456:, or
23246:, or
21609:is a
18683:field
15833:Zuber
15683:Stora
15648:Segal
15628:Salam
15613:Proca
15608:Popov
15578:Pauli
15558:Oehme
15548:Neveu
15543:Nambu
15528:Mills
15413:Jaffe
15388:Hagen
15383:Higgs
15358:Gupta
15353:Gross
15338:Glimm
15318:Furry
15288:Fierz
15278:Fermi
15273:Fayet
15268:Fadin
15253:Dyson
15243:Dirac
15208:Brout
15183:Bethe
15143:Adler
14902:Tools
13829:with
11307:Then
9897:) = 0
9713:and
8498:into
8039:(1 +
8008:(1 −
5412:then
3606:with
2310:phase
1002:is a
813:Raman
798:Pauli
793:Onnes
728:Fermi
703:Debye
693:Dirac
658:Bloch
648:Bethe
516:Local
454:Pauli
444:Dirac
242:State
25523:Weyl
25201:ISBN
25179:ISBN
25156:ISBN
25131:ISBN
25103:PMID
25044:ISBN
24978:ISBN
24956:ISBN
24936:ISBN
24913:ISBN
24885:ISBN
24851:ISBN
24827:ISBN
24805:ISBN
24786:ISBN
24767:ISBN
24740:ISBN
24690:ISBN
24671:ISBN
24648:ISBN
24626:ISBN
24596:ISBN
24534:ISBN
24359:ISBN
24250:ISBN
23587:and
21621:and
21619:BRST
20739:and
20557:for
19008:) =
18363:→ +∞
18356:→ −∞
17022:and
16030:fail
15984:and
15798:Yang
15768:Wick
15763:Weyl
15753:Wess
15733:Ward
15428:Jost
15378:Hepp
15368:Haag
15293:Fock
15173:Bell
11655:here
11206:Let
10580:and
10540:and
9863:) +
9855:) =
9835:Now
9027:and
8016:cos
8014:and
7985:sin
7521:Let
5664:) +
5653:) =
2497:<
2484:<
2465:<
2459:<
2446:<
2254:The
2243:The
2191:L dt
2080:, …
1924:and
1525:and
1270:and
1033:and
1025:and
904:The
848:Wien
843:Weyl
808:Rabi
778:Laue
768:Lamb
723:Fock
678:Bose
673:Born
668:Bohr
663:Bohm
643:Bell
26371:QED
25093:PMC
25083:doi
25016:doi
24732:doi
24575:doi
24526:doi
24475:doi
24463:84B
24393:doi
24330:doi
24288:doi
24276:149
24149:doi
23701:eff
23639:exp
22874:= 0
22287:not
21603:= 0
21255:or
20532:If
20479:QFT
20119:or
19792:If
19342:as
18797:of
18775:In
18743:in
18721:of
18708:One
17609:(Τ)
17489:to
15990:are
15818:Zee
15498:Low
15473:Lee
14656:= 0
14401:= t
14394:= 0
12954:by
11657:).
11515:in
8238:exp
7853:exp
7801:sin
7761:cos
7679:exp
7649:sin
7464:sin
7368:sin
7190:sin
6690:exp
6250:sin
6184:cos
5860:sin
5827:sin
5770:sin
5737:sin
4195:exp
4007:exp
3845:= 1
3838:= 1
3326:...
3311:to
3151:exp
2939:→ ∞
2722:exp
2576:+ 1
2564:− 1
2546:+ 1
2398:to
2293:all
2021:(0)
2002:+ 2
1932:to
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959:or
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21625:.
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13866:.
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10846:xẋ
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10039::
9035:.
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7989:=
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6319:δS
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5686:+
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5666:δx
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18854:F
18846:D
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18803:F
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18715:(
18713:ϕ
18704:)
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17909:−
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17359:Τ
17354:Τ
17337:.
17332:L
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17309:T
17305:(
17302:x
17297:x
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17102:(
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17091:=
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17085:0
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17071:=
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17048:(
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17024:ε
17020:E
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16965:p
16958:i
16949:E
16941:i
16932:=
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16926:E
16923:,
16920:p
16917:(
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16834:i
16831:+
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16771:,
16768:p
16765:(
16760:F
16756:G
16742:)
16740:T
16736:p
16734:(
16732:K
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16710:,
16703:m
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16633:(
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16609:T
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16506:T
16503:;
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16440:y
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16434:(
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16411:x
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16393:S
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16382:)
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16376:0
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16340:(
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16326:x
16323:d
16319:)
16316:T
16313:;
16310:y
16307:,
16304:x
16301:(
16298:K
16295:)
16292:x
16289:(
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16274:x
16266:=
16263:)
16260:y
16257:(
16252:T
16234:)
16232:x
16230:(
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16225:ψ
16220:x
16199:.
16196:x
16193:D
16187:]
16184:x
16181:[
16178:S
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16162:=
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16153:(
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16142:=
16139:)
16136:0
16133:(
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16104:T
16101:;
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16092:=
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16083:;
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16077:,
16074:x
16071:(
16068:K
16055:T
16051:y
16047:x
15986:y
15982:x
15965:)
15962:y
15956:x
15953:(
15948:3
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15937:=
15934:]
15931:)
15928:y
15925:(
15917:t
15909:,
15906:)
15903:x
15900:(
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15871:e
15864:t
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14705:]
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14629:0
14626:;
14619:|
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14604:H
14601:i
14591:e
14587:=
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14578:;
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14544:ħ
14540:/
14536:t
14533:B
14530:k
14528:i
14517:T
14513:/
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14360:)
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14320:[
14317:L
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14285:x
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14276:S
14264:x
14258:D
14245:]
14241:x
14237:[
14232:S
14227:i
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14214:=
14211:Z
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14143:)
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14134:x
14131:(
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14110:=
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14095:(
14086:t
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14048:t
14045:,
14042:x
14039:(
14024:,
14012:)
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13984:)
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13970:(
13965:0
13950:/
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13939:)
13936:)
13933:t
13930:(
13926:x
13922:(
13919:V
13909:e
13902:=
13899:)
13896:t
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13887:(
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13851:=
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13845:0
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13816:x
13793:x
13768:x
13738:x
13709:)
13705:x
13701:(
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13617:2
13614:m
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13601:)
13597:x
13593:(
13590:f
13585:x
13582:=
13579:)
13576:0
13573:(
13569:x
13558:Z
13555:1
13531:.
13519:t
13516:d
13511:]
13507:)
13504:)
13501:t
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13490:(
13487:V
13484:+
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13469:)
13466:t
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13453:x
13445:|
13439:2
13436:m
13430:[
13423:=
13420:)
13410:x
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13389:n
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13380:d
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13243:0
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13225:)
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13200:(
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13165:S
13157:e
13151:x
13148:=
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13142:0
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13135:x
13124:Z
13121:1
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13110:t
13107:,
13104:x
13101:(
13065:/
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13049:t
13042:e
13015:/
13005:H
12999:t
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12965:i
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12912:.
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12886:S
12878:i
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12868:]
12862:[
12857:D
12845:]
12839:[
12836:S
12828:i
12822:e
12818:)
12812:(
12809:F
12806:]
12800:[
12795:D
12784:=
12778:F
12760:,
12746:)
12743:)
12740:t
12737:(
12728:x
12722:,
12719:)
12716:t
12713:(
12710:x
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12701:t
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12650:2
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12623:x
12620:(
12615:1
12611:F
12607:]
12604:x
12601:[
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12587:f
12583:x
12579:=
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12573:t
12570:(
12567:x
12560:i
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12552:=
12549:)
12546:0
12543:(
12540:x
12532:=
12524:i
12520:x
12515:|
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12325:(
12316:H
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12156:F
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12147:[
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12125:=
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12098:=
12095:)
12092:0
12089:(
12086:x
12053:i
12049:x
12044:|
12038:)
12031:t
12027:(
12018:H
12007:i
11998:e
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11985:x
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11976:F
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11945:H
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11813:x
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11785:j
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11769:m
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11724:.
11721:x
11716:D
11708:]
11705:x
11702:[
11699:S
11696:i
11692:e
11688:]
11685:x
11682:[
11625:i
11622:=
11619:]
11616:p
11613:,
11610:x
11607:[
11581:1
11578:=
11574:]
11561:x
11550:S
11541:,
11538:x
11534:[
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11490:x
11484:t
11481:d
11476:x
11473:d
11461:t
11458:d
11453:x
11450:d
11444:x
11441:=
11438:]
11429:x
11423:,
11420:x
11417:[
11396:L
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11388:f
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11365:2
11361:)
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11336:=
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11316:)
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11312:(
11310:f
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11278:)
11272:)
11269:t
11266:(
11263:x
11257:)
11251:+
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11242:x
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11229:=
11226:)
11223:t
11220:(
11217:f
11170:2
11164:)
11158:)
11155:t
11152:(
11149:x
11143:)
11137:+
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11131:(
11128:x
11123:(
11115:=
11112:]
11109:1
11106:[
11100:]
11097:2
11094:[
11081:ε
11058:)
11055:t
11052:(
11049:x
11043:)
11037:+
11034:t
11031:(
11028:x
11022:)
11016:+
11013:t
11010:(
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10981:x
10978:=
10975:]
10972:2
10969:[
10942:)
10939:t
10936:(
10933:x
10927:)
10921:+
10918:t
10915:(
10912:x
10906:)
10903:t
10900:(
10897:x
10894:=
10888:t
10885:d
10880:x
10877:d
10871:x
10868:=
10865:]
10862:1
10859:[
10816:)
10813:t
10810:(
10807:x
10801:)
10795:+
10792:t
10789:(
10786:x
10770:t
10743:)
10740:t
10737:(
10734:x
10728:)
10722:+
10719:t
10716:(
10713:x
10707:=
10701:t
10698:d
10693:x
10690:d
10674:)
10672:t
10670:(
10668:x
10650:t
10647:d
10641:2
10636:)
10630:t
10627:d
10622:x
10619:d
10613:(
10605:=
10602:S
10589:i
10582:p
10578:x
10566:ħ
10562:ħ
10554:ħ
10542:x
10538:ẋ
10518:0
10515:=
10505:0
10496:|
10492:)
10489:t
10486:(
10480:x
10472:S
10462:|
10456:0
10433:)
10431:t
10429:(
10427:ε
10408:u
10405:D
10399:S
10396:i
10392:e
10387:)
10383:t
10380:d
10376:)
10373:t
10370:(
10363:)
10356:u
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10327:u
10316:S
10304:t
10301:d
10297:d
10291:(
10283:(
10274:x
10271:=
10268:)
10265:0
10262:(
10259:u
10251:)
10248:x
10245:(
10240:0
10219:t
10201:u
10198:D
10192:S
10189:i
10185:e
10180:)
10176:t
10173:d
10148:u
10137:S
10128:+
10119:u
10111:S
10098:(
10092:x
10089:=
10086:)
10083:0
10080:(
10077:u
10069:)
10066:x
10063:(
10058:0
10037:ε
10019:u
10016:D
10010:)
9995:+
9986:u
9980:,
9974:+
9971:u
9968:(
9965:S
9962:i
9958:e
9952:x
9949:=
9946:)
9943:0
9940:(
9937:u
9929:)
9926:x
9923:(
9918:0
9895:T
9893:(
9891:ε
9887:ε
9882:)
9880:t
9878:(
9876:ε
9871:)
9869:t
9867:(
9865:ε
9861:t
9859:(
9857:u
9853:t
9851:(
9849:x
9844:)
9842:t
9840:(
9838:x
9819:x
9816:D
9810:)
9801:x
9795:,
9792:x
9789:(
9786:S
9783:i
9779:e
9773:x
9770:=
9767:)
9764:0
9761:(
9758:x
9750:)
9747:x
9744:(
9739:0
9715:ẋ
9711:x
9680:)
9676:)
9673:x
9670:(
9667:V
9659:2
9648:2
9645:1
9638:(
9631:i
9628:=
9622:t
9595:ψ
9578:.
9574:x
9571:d
9560:2
9553:2
9549:)
9545:y
9539:x
9536:(
9533:i
9527:e
9521:)
9518:x
9515:(
9512:V
9506:i
9499:e
9495:)
9492:t
9489:;
9486:x
9483:(
9471:)
9465:+
9462:t
9459:;
9456:y
9453:(
9437:ε
9433:i
9429:)
9427:x
9425:(
9423:ψ
9398:2
9393:2
9383:x
9374:i
9370:e
9364:)
9361:x
9358:(
9355:V
9349:i
9342:e
9324:x
9320:y
9316:ẋ
9299:)
9296:1
9293:(
9289:x
9286:d
9282:)
9279:t
9276:(
9273:x
9270:D
9265:t
9262:d
9257:)
9252:)
9249:x
9246:(
9243:V
9235:2
9225:x
9216:2
9213:1
9206:(
9196:+
9193:t
9188:t
9180:i
9176:e
9170:y
9167:=
9164:)
9158:+
9155:t
9152:(
9149:x
9144:x
9141:=
9138:)
9135:t
9132:(
9129:x
9121:)
9118:t
9115:;
9112:x
9109:(
9085:=
9082:)
9076:+
9073:t
9070:;
9067:y
9064:(
9005:)
9002:t
8999:(
8996:r
8991:t
8988:d
8979:=
8976:s
8963:t
8954:r
8950:/
8946:e
8916:.
8903:)
8896:2
8893:1
8887:+
8884:n
8880:(
8876:=
8871:n
8867:E
8840:.
8834:,
8831:2
8828:,
8825:1
8822:,
8819:0
8816:=
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8797:n
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8731:2
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8686:)
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8637:T
8631:i
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8615:2
8612:1
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8589:m
8583:(
8578:=
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8566:t
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8557:i
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7904:n
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7840:=
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7790:x
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7618:=
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7473:(
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7428:c
7424:S
7420:i
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7410:=
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7377:(
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7300:Q
7297:=
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7213:,
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7184:=
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7172:2
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7162:2
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7131:=
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7093:.
7083:)
7078:i
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7065:f
7061:t
7057:(
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7035:=
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6985:)
6979:2
6974:)
6965:j
6960:)
6955:i
6951:t
6942:f
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6934:(
6925:(
6917:1
6913:(
6901:1
6898:=
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6875:c
6871:S
6867:i
6861:e
6857:=
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6840:)
6836:)
6831:i
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6810:(
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6789:i
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6620:S
6616:i
6610:e
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6603:=
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6591:i
6587:t
6583:,
6578:i
6574:x
6570:;
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6561:t
6557:,
6552:f
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6544:(
6541:K
6511:.
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6503:)
6498:i
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6359:=
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6335:=
6332:S
6298:.
6291:)
6285:)
6280:i
6276:t
6267:f
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6145:(
6139:(
6132:m
6127:2
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6119:=
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6106:d
6101:)
6095:2
6091:x
6085:2
6077:m
6071:2
6068:1
6057:2
6047:x
6040:m
6034:2
6031:1
6024:(
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6012:t
6004:i
6000:t
5991:=
5988:t
5985:d
5979:L
5970:f
5966:t
5958:i
5954:t
5945:=
5936:c
5932:S
5901:.
5895:)
5890:i
5886:t
5877:f
5873:t
5869:(
5855:)
5850:i
5846:t
5839:t
5836:(
5819:f
5815:x
5811:+
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5800:i
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5787:f
5783:t
5779:(
5765:)
5762:t
5754:f
5750:t
5746:(
5729:i
5725:x
5721:=
5718:)
5715:t
5712:(
5707:c
5703:x
5684:c
5681:S
5677:S
5672:)
5670:t
5668:(
5662:t
5660:(
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5651:t
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5642:)
5640:t
5638:(
5636:x
5618:.
5613:2
5609:x
5603:2
5595:m
5589:2
5586:1
5575:2
5565:x
5558:m
5552:2
5549:1
5543:=
5538:L
5495:.
5490:t
5480:2
5475:2
5462:=
5457:t
5446:t
5434:i
5421:K
5416:t
5414:ψ
5397:,
5394:x
5389:D
5381:S
5378:i
5374:e
5368:y
5365:=
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5359:t
5356:(
5353:x
5348:x
5345:=
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5339:0
5336:(
5333:x
5325:)
5322:x
5319:(
5314:0
5303:=
5300:x
5297:d
5293:)
5290:t
5287:;
5284:y
5278:x
5275:(
5272:K
5269:)
5266:x
5263:(
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5247:=
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5241:y
5238:(
5233:t
5215:K
5198:.
5195:K
5190:2
5185:2
5175:i
5172:=
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5166:T
5163:;
5160:x
5157:(
5154:K
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5141:d
5113:,
5106:T
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5096:2
5092:)
5088:y
5082:x
5079:(
5076:i
5070:e
5063:)
5060:T
5057:;
5054:y
5048:x
5045:(
5042:K
5025:ε
5020:i
5003:.
5000:K
4995:2
4990:2
4980:=
4977:)
4974:T
4971:;
4968:x
4965:(
4962:K
4956:t
4953:d
4949:d
4918:=
4915:y
4912:d
4908:)
4905:T
4902:;
4899:y
4893:x
4890:(
4887:K
4846:.
4838:T
4835:2
4828:2
4824:)
4820:y
4814:x
4811:(
4801:e
4794:)
4791:T
4788:;
4785:y
4779:x
4776:(
4773:K
4760:K
4743:.
4736:2
4730:2
4726:p
4722:T
4712:e
4708:=
4705:)
4702:T
4699:;
4696:p
4693:(
4684:K
4655:,
4648:2
4642:2
4638:p
4624:e
4620:=
4617:)
4614:p
4611:(
4596:G
4579:G
4562:.
4553:/
4549:T
4545:)
4541:p
4538:(
4523:G
4516:=
4513:)
4510:T
4507:;
4504:p
4501:(
4492:K
4471:ε
4467:/
4463:T
4441:,
4432:G
4413:G
4400:G
4396:=
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4390:T
4387:;
4384:y
4378:x
4375:(
4372:K
4358:ε
4356:G
4348:ε
4344:)
4342:t
4340:(
4338:x
4333:)
4331:ε
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4325:(
4323:x
4318:ε
4314:D
4297:,
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4273:2
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4232:(
4229:x
4223:(
4215:2
4212:1
4202:(
4190:t
4180:y
4177:=
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4171:T
4168:(
4165:x
4160:x
4157:=
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4151:0
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4137:=
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4131:T
4128:;
4125:y
4119:x
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4113:K
4087:.
4084:x
4079:D
4072:)
4068:t
4064:d
4057:2
4052:2
4042:x
4031:T
4026:0
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3999:=
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3993:T
3990:(
3987:x
3982:x
3979:=
3976:)
3973:0
3970:(
3967:x
3959:=
3956:)
3953:T
3950:;
3947:y
3941:x
3938:(
3935:K
3922:i
3902:,
3899:t
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3888:2
3883:2
3873:x
3861:=
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3843:ħ
3836:m
3831:y
3827:x
3816:.
3784:)
3781:)
3778:t
3775:(
3765:x
3758:,
3755:)
3752:t
3749:(
3745:x
3741:(
3738:L
3734:t
3731:d
3725:=
3722:]
3712:x
3705:,
3701:x
3697:[
3694:S
3671:S
3651:Z
3631:x
3628:=
3625:)
3622:0
3619:(
3615:x
3593:x
3571:x
3565:D
3539:)
3536:)
3533:t
3530:(
3526:x
3522:(
3517:0
3507:]
3497:x
3490:,
3486:x
3482:[
3479:S
3476:i
3472:e
3466:x
3460:D
3453:x
3450:=
3447:)
3444:0
3441:(
3437:x
3426:Z
3423:1
3418:=
3415:)
3412:t
3409:,
3406:x
3403:(
3372:2
3369:/
3363:j
3359:x
3354:j
3352:x
3343:j
3338:j
3330:n
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3315:n
3313:x
3308:1
3305:x
3282:)
3277:)
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3249:x
3240:j
3236:x
3229:,
3224:j
3214:x
3206:(
3202:L
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3194:+
3191:n
3186:1
3183:=
3180:j
3164:i
3158:(
3125:)
3122:x
3119:(
3116:V
3108:2
3103:|
3092:x
3085:|
3081:m
3076:2
3073:1
3068:=
3065:V
3059:T
3056:=
3053:)
3044:x
3038:,
3035:x
3032:(
3029:L
3012:L
3004:b
3002:x
2997:b
2995:t
2990:a
2988:x
2983:a
2981:t
2973:a
2971:t
2966:a
2964:x
2959:b
2957:t
2952:b
2950:x
2937:n
2929:n
2925:j
2920:j
2914:)
2912:t
2910:(
2908:ẋ
2904:v
2899:)
2897:t
2895:(
2893:x
2888:)
2886:v
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2880:(
2878:L
2860:,
2855:n
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2847:d
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2833:x
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2820:t
2817:d
2811:)
2806:)
2803:t
2800:(
2797:v
2794:,
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2788:t
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2777:(
2772:L
2765:b
2761:t
2753:a
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2735:i
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2714:+
2690:+
2644:.
2638:1
2635:+
2632:n
2625:a
2621:t
2612:b
2608:t
2601:=
2598:t
2592:=
2574:n
2570:j
2562:j
2558:t
2553:j
2551:t
2544:n
2524:b
2520:t
2516:=
2511:1
2508:+
2505:n
2501:t
2492:n
2488:t
2479:1
2473:n
2469:t
2454:1
2450:t
2441:0
2437:t
2433:=
2428:a
2424:t
2409:b
2407:t
2402:b
2400:x
2395:a
2393:t
2388:a
2386:x
2269:S
2264:e
2217:ħ
2200:h
2196:q
2184:T
2177:∫
2171:F
2167:q
2163:F
2158:k
2156:q
2152:F
2148:F
2143:k
2141:q
2136:k
2134:q
2129:e
2124:h
2122:/
2120:F
2116:h
2111:k
2109:q
2105:q
2101:h
2095:t
2091:q
2086:m
2082:q
2078:2
2075:q
2071:1
2068:q
2063:T
2059:q
2054:F
2049:e
2032:)
2030:t
2028:(
2026:q
2019:q
2013:S
2009:H
2004:ε
2000:t
1995:t
1978:,
1973:S
1967:i
1963:e
1949:S
1938:H
1934:q
1930:p
1926:q
1922:p
1915:p
1911:q
1907:H
1898:q
1879:,
1874:)
1868:+
1865:t
1862:(
1859:q
1856:p
1853:i
1849:e
1815:,
1810:)
1807:q
1804:,
1801:p
1798:(
1795:H
1789:i
1782:e
1764:t
1760:p
1754:)
1752:t
1750:(
1748:p
1743:)
1741:t
1739:(
1737:q
1728:)
1726:t
1724:(
1722:q
1704:,
1699:)
1696:t
1693:(
1690:q
1687:p
1684:i
1677:e
1646:.
1639:]
1634:)
1631:q
1628:,
1625:p
1622:(
1619:H
1608:)
1603:)
1600:t
1597:(
1594:q
1588:)
1582:+
1579:t
1576:(
1573:q
1568:(
1563:p
1558:[
1553:i
1549:e
1535:q
1531:p
1527:q
1523:p
1519:p
1515:q
1504:q
1500:p
1483:,
1477:p
1469:H
1460:=
1451:q
1422:,
1419:H
1404:q
1398:p
1392:=
1389:L
1373:)
1371:ε
1367:t
1365:(
1363:q
1342:q
1316:,
1304:q
1293:L
1284:=
1281:p
1255:L
1244:)
1239:)
1236:t
1233:(
1230:q
1224:)
1218:+
1215:t
1212:(
1209:q
1204:(
1199:)
1196:t
1193:(
1190:p
1187:=
1184:H
1113:i
1111:−
893:e
886:t
879:v
107:|
97:H
91:=
81:|
74:t
71:d
67:d
59:i
34:.
20:)
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