Lagrangian (field theory)

8646: 8961: 8249: 49:, which is infamously beset by formal difficulties that make it unacceptable as a mathematical theory. The Lagrangians presented here are identical to their quantum equivalents, but, in treating the fields as classical fields, instead of being quantized, one can provide definitions and obtain solutions with properties compatible with the conventional formal approach to the mathematics of 2342: 4463: 8368: 3427: 7787:. These alter the metric without altering the geometry one bit. As to the actual "direction in which gravity points" (e.g. on the surface of the Earth, it points down), this comes from the Riemann tensor: it is the thing that describes the "gravitational force field" that moving bodies feel and react to. (This last statement must be qualified: there is no "force field" 8692: 993: 10029: 8009: 5274: 9648: 9284: 2987: 490: 7459: 2174: 4225: 1419: 7665: 3223: 6098: 2457: 829: 647:. Bleecker's textbook provided a comprehensive presentation of field theories in physics in terms of gauge invariant fiber bundles. Such formulations were known or suspected long before. Jost continues with a geometric presentation, clarifying the relation between Hamiltonian and Lagrangian forms, describing 9881: 5399: 4722: 5720: 9877:

is an index which takes values 0 (for the time coordinate), and 1, 2, 3 (for the spatial coordinates), so strictly only one derivative or coordinate would be present. In general, all the spatial and time derivatives will appear in the Lagrangian density, for example in Cartesian coordinates, the

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operator. This is exactly the same Lagrangian as in the section above, except that the treatment here is coordinate-free; expanding the integrand into a basis yields the identical, lengthy expression. Note that with forms, an additional integration measure is not necessary because forms have

1843:

and so it is commonly omitted, when discussing field theory in flat spacetime. Likewise, the use of the wedge-product symbols offers no additional insight over the ordinary concept of a volume in multivariate calculus, and so these are likewise dropped. Some older textbooks, e.g., Landau and

8641:{\displaystyle {\begin{aligned}{\mathcal {L}}(x)&=j^{\mu }(x)A_{\mu }(x)-{1 \over 4\mu _{0}}F_{\mu \nu }(x)F_{\rho \sigma }(x)g^{\mu \rho }(x)g^{\nu \sigma }(x)+{\frac {c^{4}}{16\pi G}}R(x)\\&={\mathcal {L}}_{\text{Maxwell}}+{\mathcal {L}}_{\text{Einstein–Hilbert}}.\end{aligned}}} 1871:

for the volume form, since the minus sign is appropriate for metric tensors with signature (+−−−) or (−+++) (since the determinant is negative, in either case). When discussing field theory on general Riemannian manifolds, the volume form is usually written in the abbreviated notation

7024: 8956:{\displaystyle T^{\mu \nu }(x)={\frac {2}{\sqrt {-g(x)}}}{\frac {\delta }{\delta g_{\mu \nu }(x)}}{\mathcal {S}}_{\text{Maxwell}}={\frac {1}{\mu _{0}}}\left(F_{{\text{ }}\lambda }^{\mu }(x)F^{\nu \lambda }(x)-{\frac {1}{4}}g^{\mu \nu }(x)F_{\rho \sigma }(x)F^{\rho \sigma }(x)\right)} 6265: 2618: 9075: 3791: 2811: 317: 7267: 8244:{\displaystyle T_{\mu \nu }\equiv {\frac {-2}{\sqrt {-g}}}{\frac {\delta ({\mathcal {L}}_{\mathrm {matter} }{\sqrt {-g}})}{\delta g^{\mu \nu }}}=-2{\frac {\delta {\mathcal {L}}_{\mathrm {matter} }}{\delta g^{\mu \nu }}}+g_{\mu \nu }{\mathcal {L}}_{\mathrm {matter} }\,.} 1294: 7538: 73:, allowing the geometric structure to be clearly discerned and disentangled from the corresponding equations of motion. A clearer view of the geometric structure has in turn allowed highly abstract theorems from geometry to be used to gain insight, ranging from the 5982: 7971: 822: 2357: 5546: 9681:

Lagrangian, short for "Background Field", describes a system with trivial dynamics, when written on a flat spacetime manifold. On a topologically non-trivial spacetime, the system will have non-trivial classical solutions, which may be interpreted as

4198: 2337:{\displaystyle 0={\frac {\delta {\mathcal {S}}}{\delta \varphi }}=\int _{M}*(1)\left(-\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\varphi )}}\right)+{\frac {\partial {\mathcal {L}}}{\partial \varphi }}\right).} 4458:{\displaystyle {\mathcal {L}}(\mathbf {x} ,t)=-\rho (\mathbf {x} ,t)\phi (\mathbf {x} ,t)+\mathbf {j} (\mathbf {x} ,t)\cdot \mathbf {A} (\mathbf {x} ,t)+{\epsilon _{0} \over 2}{E}^{2}(\mathbf {x} ,t)-{1 \over {2\mu _{0}}}{B}^{2}(\mathbf {x} ,t).} 5962:

although the general case is of general interest. In all cases, there is no need for any quantization to be performed. Although the Yang–Mills equations are historically rooted in quantum field theory, the above equations are purely classical.

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on spacetime. The gravitational field itself was historically ascribed to the metric tensor; the modern view is that the connection is "more fundamental". This is due to the understanding that one can write connections with non-zero

3422:{\displaystyle {\mathcal {L}}={\frac {1}{2}}\partial ^{\mu }\phi \partial _{\mu }\phi -V(\phi )={\frac {1}{2}}\partial ^{\mu }\phi \partial _{\mu }\phi -{\frac {1}{2}}m^{2}\phi ^{2}-\sum _{n=3}^{\infty }{\frac {1}{n!}}g_{n}\phi ^{n}} 7521:. As for the electrodynamics case above, the appearance of the word "quantum" above only acknowledges its historical development. The Lagrangian and its gauge invariance can be formulated and treated in a purely classical fashion. 3623: 6683: 3935: 6873: 6138: 2479: 3654: 988:{\displaystyle {\mathcal {L}}(\varphi _{1},{\boldsymbol {\nabla }}\varphi _{1},\partial \varphi _{1}/\partial t,\ldots ,\varphi _{n},{\boldsymbol {\nabla }}\varphi _{n},\partial \varphi _{n}/\partial t,\ldots ,\mathbf {x} ,t)} 5057: 5960: 10024:{\displaystyle {\mathcal {L}}\left(\varphi ,{\frac {\partial \varphi }{\partial x}},{\frac {\partial \varphi }{\partial y}},{\frac {\partial \varphi }{\partial z}},{\frac {\partial \varphi }{\partial t}},x,y,z,t\right)} 7230:. Although the word "quantum" appears in the above, this is a historical artifact. The definition of the Dirac field requires no quantization whatsoever, it can be written as a purely classical field of anti-commuting 3171: 9660:, such as the fact that the 7-sphere can be written as a product of the 4-sphere and the 3-sphere, or that the 11-sphere is a product of the 4-sphere and the 7-sphere, accounted for much of the early excitement that a 6542: 4948: 9865: 4792: 238: 1480: 5787: 5269:{\displaystyle {\epsilon _{0} \over 2}{E}^{2}-{1 \over {2\mu _{0}}}{B}^{2}=-{\frac {1}{4\mu _{0}}}F_{\mu \nu }F^{\mu \nu }=-{\frac {1}{4\mu _{0}}}F_{\mu \nu }F_{\rho \sigma }\eta ^{\mu \rho }\eta ^{\nu \sigma }} 3564:, such as a circle or a sphere. It generalizes the case of scalar and vector fields, that is, fields constrained to move on a flat manifold. The Lagrangian is commonly written in one of three equivalent forms: 2007: 9643:{\displaystyle \mathrm {d} s^{2}=\left(1-{\frac {2M}{r}}+{\frac {Q^{2}}{r^{2}}}\right)\mathrm {d} t^{2}-\left(1-{\frac {2M}{r}}+{\frac {Q^{2}}{r^{2}}}\right)^{-1}\mathrm {d} r^{2}-r^{2}\mathrm {d} \Omega ^{2}} 7803: 7197: 7853: 749: 6759: 2063: 4048:

in hiding; the Killing form provides a quadratic form on the field manifold, the lagrangian is then just the pullback of this form. Alternately, the Lagrangian can also be seen as the pullback of the

1258: 9350: 9279:{\displaystyle R^{\mu \nu }={\frac {8\pi G}{c^{4}}}{\frac {1}{\mu _{0}}}\left({F^{\mu }}_{\lambda }(x)F^{\nu \lambda }(x)-{\frac {1}{4}}g^{\mu \nu }(x)F_{\rho \sigma }(x)F^{\rho \sigma }(x)\right)} 2109: 687:

is replaced by a Lagrangian density, a function of the fields in the system and their derivatives, and possibly the space and time coordinates themselves. In field theory, the independent variable

9656:. Effectively, one constructs an affine bundle, just as for the Yang–Mills equations given earlier, and then considers the action separately on the 4-dimensional and the 1-dimensional parts. Such 6834: 5408: 2982:{\displaystyle \delta {\mathcal {L}}(\mathbf {x} ,t)=-\rho (\mathbf {x} ,t)\delta \Phi (\mathbf {x} ,t)-{2 \over 8\pi G}(\nabla \Phi (\mathbf {x} ,t))\cdot (\nabla \delta \Phi (\mathbf {x} ,t)).} 8373: 485:{\displaystyle {\mathcal {S}}\left=\int {{\mathcal {L}}\left(\varphi _{i}(s),\left\{{\frac {\partial \varphi _{i}(s)}{\partial s^{\alpha }}}\right\},\{s^{\alpha }\}\right)\,\mathrm {d} ^{n}s},} 4088: 3991: 1115: 7454:{\displaystyle {\mathcal {L}}_{\mathrm {QCD} }=\sum _{n}{\bar {\psi }}_{n}\left(i\hbar c{D}\!\!\!\!/\ -m_{n}c^{2}\right)\psi _{n}-{1 \over 4}G^{\alpha }{}_{\mu \nu }G_{\alpha }{}^{\mu \nu }} 2466:

A large variety of physical systems have been formulated in terms of Lagrangians over fields. Below is a sampling of some of the most common ones found in physics textbooks on field theory.

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The Lagrange density of electromagnetism in general relativity also contains the Einstein–Hilbert action from above. The pure electromagnetic Lagrangian is precisely a matter Lagrangian

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of spacetime; the construction works in any number of dimensions, and the Dirac spinors appear as a special case. Weyl spinors have the additional advantage that they can be used in a

1414:{\displaystyle {\mathcal {S}}=\int {\mathcal {L}}(\varphi ,{\boldsymbol {\nabla }}\varphi ,\partial \varphi /\partial t,\mathbf {x} ,t)\,\mathrm {d} ^{3}\mathbf {x} \,\mathrm {d} t.} 4474: 7660:{\displaystyle {\mathcal {L}}_{\text{GR}}={\mathcal {L}}_{\text{EH}}+{\mathcal {L}}_{\text{matter}}={\frac {c^{4}}{16\pi G}}\left(R-2\Lambda \right)+{\mathcal {L}}_{\text{matter}}} 4031: 6410: 6793: 2707: 671:

are, in a sense "rigid", as they are determined by their Lie algebra. When reformulated on a tensor algebra, they become "floppy", having infinite degrees of freedom; see e.g.,

8310: 614: 173: 6322: 6093:{\displaystyle {\mathcal {S}}=\int _{\mathcal {M}}\mathrm {tr} \left(\mathbf {A} \wedge d\mathbf {A} +{\frac {2}{3}}\mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \right).} 3567: 1181: 1144: 8687: 306: 6599: 5590: 3865: 2664: 1518: 1285: 578: 266: 8000: 7845: 7097: 7054: 4981: 4585: 4220: 3645: 2795: 1841: 7224: 9414: 3503: 2452:{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial \varphi }}=\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\varphi )}}\right).} 1724: 3852: 1869: 9377: 7685: 6564: 3532: 3218: 2169: 2145: 1542: 744: 1686: 3821: 6703: 6342: 6285: 4986: 3459: 1899: 8271:

is the determinant of the metric tensor when regarded as a matrix. Generally, in general relativity, the integration measure of the action of Lagrange density is

3097: 1754: 6489: 4888: 9808: 8269: 7709: 6398: 6374: 1919: 1797: 1777: 1565: 1062: 5394:{\displaystyle \partial _{\mu }F^{\mu \nu }=-\mu _{0}j^{\nu }\quad {\text{and}}\quad \epsilon ^{\mu \nu \lambda \sigma }\partial _{\nu }F_{\lambda \sigma }=0} 4717:{\displaystyle 0=\mathbf {j} (\mathbf {x} ,t)+\epsilon _{0}{\dot {\mathbf {E} }}(\mathbf {x} ,t)-{1 \over \mu _{0}}\nabla \times \mathbf {B} (\mathbf {x} ,t)} 186: 1430: 9072:

So the tracelessness of the energy momentum tensor implies that the curvature scalar in an electromagnetic field vanishes. The Einstein equations are then

5715:{\displaystyle {\mathcal {S}}=-\int _{\mathcal {M}}\left({\frac {1}{2}}\,\mathbf {F} \wedge \ast \mathbf {F} -\mathbf {A} \wedge \ast \mathbf {J} \right).} 5744: 4076: 1928: 7802:

The Lagrangian for general relativity can also be written in a form that makes it manifestly similar to the Yang–Mills equations. This is called the

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This Lagrangian is obtained by simply replacing the Minkowski metric in the above flat Lagrangian with a more general (possibly curved) metric

7019:{\displaystyle {\mathcal {L}}_{\mathrm {QED} }={\bar {\psi }}(i\hbar c{D}\!\!\!\!/\ -mc^{2})\psi -{1 \over 4\mu _{0}}F_{\mu \nu }F^{\mu \nu }} 6260:{\displaystyle {\mathcal {L}}(\psi ,A)=\vert F\vert ^{2}+\vert D\psi \vert ^{2}+{\frac {1}{4}}\left(\sigma -\vert \psi \vert ^{2}\right)^{2}} 2613:{\displaystyle {\mathcal {L}}(\mathbf {x} ,t)=-{1 \over 8\pi G}(\nabla \Phi (\mathbf {x} ,t))^{2}-\rho (\mathbf {x} ,t)\Phi (\mathbf {x} ,t)} 1544:. This ensures that the action is invariant under general coordinate transformations. In mathematical literature, spacetime is taken to be a 1217: 9289: 3786:{\displaystyle {\mathcal {L}}={\frac {1}{2}}\sum _{i=1}^{n}\sum _{j=1}^{n}g_{ij}(\phi )\;\partial ^{\mu }\phi _{i}\partial _{\mu }\phi _{j}} 2070: 2111:

is frequently seen. Do not be misled: the volume form is implicitly present in the integral above, even if it is not explicitly written.

8329: 31: 3940: 8689:. We can generate the Einstein Field Equations in the presence of an EM field using this lagrangian. The energy-momentum tensor is 7143: 4798:. We package the charge density into the current 4-vector and the potential into the potential 4-vector. These two new vectors are 2801:

for how this could be modified to deal with changes over time. This form is reprised in the next example of a scalar field theory.

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combines the Lagrangian for the Dirac field together with the Lagrangian for electrodynamics in a gauge-invariant way. It is:

10123: 8316:. The minus sign is a consequence of the metric signature (the determinant by itself is negative). This is an example of the 9416:. Solving both Einstein and Maxwell's equations around a spherically symmetric mass distribution in free space leads to the 9805:

It is a standard abuse of notation to abbreviate all the derivatives and coordinates in the Lagrangian density as follows:

6105:

was deeply explored in physics, as a toy model for a broad range of geometric phenomena that one might expect to find in a

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One possible way of unifying the electromagnetic and gravitational Lagrangians (by using a fifth dimension) is given by

5405:. So the Lagrange density for electromagnetism in special relativity written in terms of Lorentz vectors and tensors is 6571: 74: 1074: 10222: 10186: 10161: 8337: 7966:{\displaystyle R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu }+g_{\mu \nu }\Lambda ={\frac {8\pi G}{c^{4}}}T_{\mu \nu }\,.} 817:{\displaystyle {\mathcal {L}}(\varphi ,{\boldsymbol {\nabla }}\varphi ,\partial \varphi /\partial t,\mathbf {x} ,t)} 30:. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of 9710: 9691: 7264:, which describes the dynamics of a gauge field; the combined Lagrangian is gauge invariant. It may be written as: 7106: 2712: 10271: 10266: 82: 7738: 7480: 655:

affine structures, (sometimes called "quantum structures") wherein one replaces occurrences of vector spaces by

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in kg·m. This is necessary because using a point source for a field would result in mathematical difficulties.

10276: 10261: 8312:. This makes the integral coordinate independent, as the root of the metric determinant is equivalent to the 5541:{\displaystyle {\mathcal {L}}(x)=j^{\mu }(x)A_{\mu }(x)-{\frac {1}{4\mu _{0}}}F_{\mu \nu }(x)F^{\mu \nu }(x)} 50: 9775: 3996: 2351: 9715: 7767: 7530: 7518: 6401: 4193:{\displaystyle -q\phi (\mathbf {x} (t),t)+q{\dot {\mathbf {x} }}(t)\cdot \mathbf {A} (\mathbf {x} (t),t)} 2120: 6768: 2676: 7065: 6124: 6118: 3648: 3174: 587: 131: 8274: 6298: 5548:

In this notation it is apparent that classical electromagnetism is a Lorentz-invariant theory. By the

1157: 1120: 9780: 8653: 6575: 3505:. The scalar theory is the field-theory generalization of a particle moving in a potential. When the 3087:{\displaystyle 0=-\rho (\mathbf {x} ,t)+{\frac {1}{4\pi G}}\nabla \cdot \nabla \Phi (\mathbf {x} ,t)} 275: 5571: 3429:

It is not at all an accident that the scalar theory resembles the undergraduate textbook Lagrangian

2645: 1659:{\displaystyle {\mathcal {S}}=\int _{M}{\sqrt {|g|}}dx^{1}\wedge \cdots \wedge dx^{m}{\mathcal {L}}} 1499: 1266: 631:

on the fiber bundle. Abraham and Marsden's textbook provided the first comprehensive description of

559: 247: 10214: 9740: 7848: 6405: 5877: 4878:{\displaystyle j^{\mu }=(\rho ,\mathbf {j} )\quad {\text{and}}\quad A_{\mu }=(-\phi ,\mathbf {A} )} 1485: 269: 78: 10115: 7975: 7820: 7071: 7029: 4956: 4568: 4203: 3628: 2771: 53:. This enables the formulation of solutions on spaces with well-characterized properties, such as 41:

One motivation for the development of the Lagrangian formalism on fields, and more generally, for

9730: 9653: 7806:. This is done by noting that most of differential geometry works "just fine" on bundles with an 7202: 6867: 6861: 6102: 2625: 1806: 1496:

In the presence of gravity or when using general curvilinear coordinates, the Lagrangian density

684: 86: 9386: 3464: 10153: 9705: 8003: 7253: 7247: 7100: 7057: 6852:, which, roughly speaking, is a way of formulating spinors consistently in a curved spacetime. 6796: 6288: 4951: 4551:{\displaystyle 0=-\rho (\mathbf {x} ,t)+\epsilon _{0}\nabla \cdot \mathbf {E} (\mathbf {x} ,t)} 4049: 3830: 2639: 1695: 1006: 652: 42: 23: 9355: 7670: 6547: 3508: 3194: 2154: 2130: 1847: 729: 9735: 9720: 7688: 7261: 6479:{\displaystyle D{\star }D\psi ={\frac {1}{2}}\left(\sigma -\vert \psi \vert ^{2}\right)\psi } 6353: 6132: 5976: 5549: 4082: 3535: 2148: 1671: 1523: 581: 10206: 10145: 10107: 5971:

In the same vein as the above, one can consider the action in one dimension less, i.e. in a

3796: 9770: 9755: 9661: 9380: 8313: 7817:

Substituting this Lagrangian into the Euler–Lagrange equation and taking the metric tensor

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are replaced by terms involving a continuous charge density ρ in A·s·m and current density

3432: 545: 46: 27: 9664:

had been found. Unfortunately, the 7-sphere proved not large enough to enclose all of the

1875: 128:. The dependent variables are replaced by the value of a field at that point in spacetime 8: 10207: 10108: 9785: 9725: 7810:

and arbitrary Lie group. Then, plugging in SO(3,1) for that symmetry group, i.e. for the

7775: 6567: 6128: 5870: 4056: 3618:{\displaystyle {\mathcal {L}}={\frac {1}{2}}\mathrm {d} \phi \wedge {*\mathrm {d} \phi }} 3561: 3186: 1800: 1729: 1545: 1041:

are described by tensor fields, which include scalar and vector fields as special cases.

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In mathematical formulations, it is common to express the Lagrangian as a function on a

8254: 7796: 7694: 6678:{\displaystyle {\mathcal {L}}={\bar {\psi }}(i\hbar c{\partial }\!\!\!/\ -mc^{2})\psi } 6383: 6359: 5561: 5402: 3930:{\displaystyle {\mathcal {L}}={\frac {1}{2}}\mathrm {tr} \left(L_{\mu }L^{\mu }\right)} 2347: 1904: 1782: 1762: 1550: 1047: 664: 6836:. There is no particular need to focus on Dirac spinors in the classical theory. The 4725: 57:. It enables various theorems to be provided, ranging from proofs of existence to the 10218: 10182: 10157: 10146: 10119: 9765: 7807: 7779: 5850: 3855: 3824: 538: 9760: 7712: 7235: 6841: 6570:, i.e. the fully antisymmetric tensor. These equations are closely related to the 5972: 5277: 3859: 2067:

Not infrequently, the notation above is considered to be entirely superfluous, and

1196: 998: 672: 644: 636: 180: 62: 5552:, it becomes simple to extend the notion of electromagnetism to curved spacetime. 5280:

to raise the indices on the EMF tensor. In this notation, Maxwell's equations are

7732: 4732: 4041: 1424: 35: 7535:

The Lagrange density for general relativity in the presence of matter fields is

10076:

David Bleecker, (1981) "Gauge Theory and Variational Principles" Addison-Wesley

9665: 9657: 7784: 7728: 6849: 6587: 6345: 5888: 5861:. That is, classical electrodynamics, all of its effects and equations, can be 2991:

After integrating by parts, discarding the total integral, and dividing out by

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has units of J·m. Here the interaction term involves a continuous mass density

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If we take the trace of both sides of the Einstein Field Equations, we obtain

5789:

These are Maxwell's equations for the electromagnetic potential. Substituting

5052:{\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }} 620:

of the field function, i.e., the measure of the domain of the field function.

10255: 9868: 9421: 7720: 6840:

provide a more general foundation; they can be constructed directly from the

6762: 6292: 5955:{\displaystyle \mathrm {SU} (3)\times \mathrm {SU} (2)\times \mathrm {U} (1)} 5866: 5593: 4559: 2124: 1757: 1689: 1192: 660: 648: 54: 3166:{\displaystyle 4\pi G\rho (\mathbf {x} ,t)=\nabla ^{2}\Phi (\mathbf {x} ,t)} 627:, wherein the Euler–Lagrange equations can be interpreted as specifying the 9745: 7724: 7716: 7257: 6706: 6537:{\displaystyle D{\star }F=-\operatorname {Re} \langle D\psi ,\psi \rangle } 5858: 4943:{\displaystyle -\rho \phi +\mathbf {j} \cdot \mathbf {A} =j^{\mu }A_{\mu }} 4222:

in A·m. The resulting Lagrangian density for the electromagnetic field is:

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Jurgen Jost, (1995) "Riemannian Geometry and Geometric Analysis", Springer

9860:{\displaystyle {\mathcal {L}}(\varphi ,\partial _{\mu }\varphi ,x_{\mu })} 4950:

Additionally, we can package the E and B fields into what is known as the

4787:{\displaystyle -\rho \phi (\mathbf {x} ,t)+\mathbf {j} \cdot \mathbf {A} } 8963:

It can be shown that this energy momentum tensor is traceless, i.e. that

8317: 7811: 7771: 7231: 6837: 6593: 6349: 4795: 3560:

describes the motion of a scalar point particle constrained to move on a

3557: 3551: 3539: 1568: 1064: 617: 4040:. This group can be replaced by any Lie group, or, more generally, by a 10056:

Ralph Abraham and Jerrold E. Marsden, (1967) "Foundations of Mechanics"

7238:. The full gauge-invariant classical formulation is given in Bleecker. 6848:

for the metric on a Riemannian manifold; this enables the concept of a

5839: 5737: 1922: 1151: 1010: 718:

Often, a "Lagrangian density" is simply referred to as a "Lagrangian".

233:{\displaystyle {\frac {\delta {\mathcal {S}}}{\delta \varphi _{i}}}=0,} 1475:{\displaystyle L=\int {\mathcal {L}}\,\mathrm {d} ^{3}\mathbf {x} \,.} 9687: 7792: 5881: 4034: 1288: 997:

In mathematical formulations, the scalar fields are understood to be

668: 98: 97:

In field theory, the independent variable is replaced by an event in

65:. In addition, insight and clarity is obtained by generalizations to 10246:

Claude Itykson and Jean-Bernard Zuber, (1980) "Quantum Field Theory"

5782:{\displaystyle \mathrm {d} {\ast }\mathbf {F} ={\ast }\mathbf {J} .} 5741:

coordinate differentials built in. Variation of the action leads to

4081:

Consider a point particle, a charged particle, interacting with the

9678: 6845: 5880:

can be written in exactly the same form as above, by replacing the

4060: 2002:{\displaystyle *(1)={\sqrt {|g|}}dx^{1}\wedge \cdots \wedge dx^{m}} 1488:", in that it is a function of the fields (and their derivatives). 628: 1205:. In field theory, a distinction is occasionally made between the 659:. This research is motivated by the breakthrough understanding of 9683: 8320:, previously discussed, becoming manifest in non-flat spacetime. 4064: 1034: 5555: 9383:. For free space, we can set the current tensor equal to zero, 1016: 548:

of the system, including the time variable, and is indexed by

9690:. A variety of extensions exist, forming the foundations for 7795:

on the manifold described by the connection. They move in a "

7474: 7192:{\displaystyle D_{\sigma }=\partial _{\sigma }-ieA_{\sigma }} 4037: 1038: 4059:

solutions. The most famous and well-studied of these is the

683:

In Lagrangian field theory, the Lagrangian as a function of

5884: 5854: 6754:{\displaystyle {\bar {\psi }}=\psi ^{\dagger }\gamma ^{0}} 2058:{\displaystyle {\mathcal {S}}=\int _{M}*(1){\mathcal {L}}} 8323: 7778:

and derivatives of Christoffel symbols, which define the

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for the kinetic term of a free point particle written as

651:

from first principles, etc. Current research focuses on

3191:

The Lagrangian for a scalar field moving in a potential

1005:, and the derivatives of the field are understood to be 635:

in terms of modern geometrical ideas, i.e., in terms of

10106:

Mandl, F.; Shaw, G. (2010). "Lagrangian Field Theory".

1253:{\displaystyle {\mathcal {S}}=\int L\,\mathrm {d} t\,,} 38:, which have an infinite number of degrees of freedom. 9345:{\displaystyle D_{\mu }F^{\mu \nu }=-\mu _{0}j^{\nu }} 8277: 5887:

of electromagnetism by an arbitrary Lie group. In the

5736:

is the field strength 2-form and the star denotes the

2104:{\displaystyle {\mathcal {S}}=\int _{M}{\mathcal {L}}} 1850: 1809: 1698: 1526: 9884: 9811: 9434: 9389: 9358: 9292: 9078: 9024: 8969: 8695: 8656: 8371: 8340: 8257: 8012: 7978: 7856: 7823: 7741: 7697: 7673: 7541: 7483: 7270: 7205: 7146: 7109: 7074: 7032: 6876: 6805: 6771: 6715: 6691: 6602: 6550: 6492: 6413: 6386: 6362: 6330: 6301: 6273: 6141: 5985: 5897: 5805: 5747: 5616: 5574: 5411: 5286: 5065: 4989: 4959: 4891: 4804: 4741: 4593: 4571: 4477: 4228: 4206: 4091: 3999: 3943: 3868: 3833: 3799: 3657: 3631: 3570: 3511: 3467: 3435: 3226: 3197: 3100: 3004: 2814: 2774: 2715: 2679: 2648: 2482: 2360: 2177: 2157: 2133: 2073: 2015: 1931: 1907: 1878: 1785: 1765: 1732: 1674: 1576: 1553: 1502: 1433: 1297: 1269: 1220: 1160: 1123: 1077: 1050: 832: 752: 732: 590: 562: 500: 320: 278: 250: 189: 134: 16:

Application of Lagrangian mechanics to field theories

6855: 6829:{\displaystyle \gamma ^{\sigma }\partial _{\sigma }} 1427:

of the Lagrangian density is the Lagrangian; in 3D,

45:, is to provide a clear mathematical foundation for 7241: 4077:

Covariant formulation of classical electromagnetism

10152:. Princeton: Princeton University Press. pp. 10037:to abbreviate all spatial derivatives as a vector. 10023: 9859: 9642: 9408: 9371: 9344: 9278: 9064: 9010: 8955: 8681: 8640: 8357: 8304: 8263: 8243: 7994: 7965: 7839: 7758: 7703: 7679: 7659: 7509: 7453: 7218: 7191: 7132: 7091: 7048: 7018: 6828: 6787: 6753: 6697: 6677: 6558: 6536: 6478: 6392: 6368: 6352:, after noting that the second term is the famous 6336: 6316: 6279: 6259: 6092: 5954: 5824: 5781: 5714: 5584: 5540: 5393: 5268: 5051: 4975: 4942: 4877: 4786: 4716: 4579: 4550: 4457: 4214: 4192: 4025: 3985: 3929: 3846: 3815: 3785: 3639: 3617: 3526: 3497: 3453: 3421: 3212: 3165: 3086: 2981: 2789: 2760: 2701: 2658: 2612: 2451: 2336: 2163: 2139: 2103: 2057: 2001: 1913: 1893: 1863: 1835: 1791: 1771: 1748: 1718: 1680: 1658: 1559: 1536: 1512: 1474: 1413: 1279: 1252: 1175: 1138: 1109: 1056: 987: 816: 738: 608: 572: 521: 484: 300: 260: 232: 167: 34:. Lagrangian field theory applies to continua and 7506: 7347: 7346: 7345: 7344: 7083: 7082: 7081: 7080: 6934: 6933: 6932: 6931: 6779: 6778: 6777: 6644: 6643: 6642: 6574:. Another closely related Lagrangian is found in 5726:stands for the electromagnetic potential 1-form, 4735:, we can write all this more compactly. The term 2474:The Lagrangian density for Newtonian gravity is: 10253: 7256:combines the Lagrangian for one or more massive 5799:immediately yields the equation for the fields, 5059:The term we are looking out for turns out to be 3986:{\displaystyle L_{\mu }=U^{-1}\partial _{\mu }U} 1110:{\displaystyle \varphi _{1},\dots ,\varphi _{m}} 1484:The action is often referred to as the "action 8358:{\displaystyle {\mathcal {L}}_{\text{matter}}} 3827:on the manifold of the field; i.e. the fields 2804:The variation of the integral with respect to 2768:providing a kinetic term, and the interaction 2673:This Lagrangian can be written in the form of 9420:, with the defining line element (written in 6112: 5568:in vacuum on a (pseudo-) Riemannian manifold 5556:Electromagnetism and the Yang–Mills equations 746:, the Lagrangian density will take the form: 10213:(Third ed.). Springer-Verlag. pp. 9011:{\displaystyle T=g_{\mu \nu }T^{\mu \nu }=0} 7133:{\displaystyle \gamma ^{\sigma }D_{\sigma }} 6531: 6516: 6459: 6452: 6237: 6230: 6196: 6186: 6174: 6167: 2761:{\displaystyle T=-(\nabla \Phi )^{2}/8\pi G} 516: 501: 454: 441: 61:of formal series to the general settings of 9065:{\displaystyle R=-{\frac {8\pi G}{c^{4}}}T} 6404:for the Ginzburg–Landau functional are the 6376:is the (non-Abelian) gauge field, i.e. the 2114: 1214:, of which the time integral is the action 1017:Vector fields, tensor fields, spinor fields 10209:Riemannian Geometry and Geometric Analysis 10205:(2002). "The Ginzburg–Landau Functional". 10072: 10070: 10068: 10066: 10064: 10062: 7759:{\displaystyle {\mathcal {L}}_{\text{EH}}} 7510:{\displaystyle G^{\alpha }{}_{\mu \nu }\!} 5966: 5825:{\displaystyle \mathrm {d} \mathbf {F} =0} 4885:We can then write the interaction term as 3742: 522:{\displaystyle \{\cdot ~\forall \alpha \}} 10181:. Cambridge: Cambridge University Press. 10105: 8288: 8237: 7959: 7234:constructed from first principles from a 6304: 5668: 3545: 1468: 1450: 1399: 1381: 1246: 1237: 1163: 1126: 462: 10031:Here we write the same thing, but using 6344:corresponds to the order parameter in a 3862:of the manifold. A third common form is 26:. It is the field-theoretic analogue of 10059: 8330:Maxwell's equations in curved spacetime 10254: 10176: 10092: 10090: 10088: 10086: 10084: 10082: 9878:Lagrangian density has the full form: 9671: 9286:Additionally, Maxwell's equations are 8324:Electromagnetism in general relativity 6348:; equivalently, it corresponds to the 4071:Electromagnetism in special relativity 3538:, the resulting fields are termed the 3180: 1726:is the square root of the determinant 10139: 10137: 10135: 9418:Reissner–Nordström charged black hole 4794:is actually the inner product of two 4067:that has withstood the test of time. 4026:{\displaystyle U\in \mathrm {SU} (N)} 691:is replaced by an event in spacetime 120:, or more generally still by a point 10201: 10099: 7804:Einstein–Yang–Mills action principle 2469: 10143: 10079: 9751:Lagrangian and Eulerian coordinates 7814:, one obtains the equations above. 7524: 6581: 5891:, it is conventionally taken to be 2147:as a function of time. Taking the 711:or still more generally by a point 13: 10132: 9983: 9975: 9960: 9952: 9937: 9929: 9914: 9906: 9887: 9829: 9814: 9631: 9626: 9598: 9512: 9436: 8782: 8620: 8603: 8378: 8344: 8231: 8228: 8225: 8222: 8219: 8216: 8209: 8163: 8160: 8157: 8154: 8151: 8148: 8141: 8083: 8080: 8077: 8074: 8071: 8068: 8061: 7915: 7774:tensor, and is constructed out of 7745: 7674: 7646: 7632: 7579: 7562: 7545: 7287: 7284: 7281: 7274: 7161: 6893: 6890: 6887: 6880: 6817: 6788:{\displaystyle {\partial }\!\!\!/} 6773: 6638: 6605: 6144: 6023: 6020: 6013: 5988: 5939: 5922: 5919: 5902: 5899: 5849:field can be understood to be the 5807: 5749: 5647: 5619: 5577: 5414: 5363: 5288: 5030: 5007: 4686: 4520: 4231: 4010: 4007: 3971: 3893: 3890: 3871: 3764: 3744: 3660: 3633: 3607: 3592: 3573: 3379: 3315: 3302: 3261: 3248: 3229: 3143: 3134: 3064: 3061: 3055: 2953: 2947: 2918: 2915: 2871: 2820: 2784: 2731: 2728: 2702:{\displaystyle {\mathcal {L}}=T-V} 2682: 2651: 2590: 2537: 2534: 2485: 2424: 2417: 2410: 2405: 2389: 2376: 2369: 2364: 2317: 2310: 2305: 2277: 2270: 2263: 2258: 2242: 2192: 2096: 2076: 2050: 2018: 1651: 1579: 1567:and the integral then becomes the 1505: 1453: 1445: 1401: 1384: 1358: 1347: 1322: 1300: 1272: 1239: 1223: 956: 938: 892: 874: 835: 791: 780: 755: 593: 565: 510: 465: 418: 394: 355: 323: 253: 198: 14: 10288: 8305:{\textstyle {\sqrt {-g}}\,d^{4}x} 7333: 6920: 6856:Quantum electrodynamic Lagrangian 6631: 4063:, which serves as a model of the 4055:In general, sigma models exhibit 2799:Nordström's theory of gravitation 1021:The above can be generalized for 609:{\displaystyle \mathrm {d} ^{n}s} 168:{\displaystyle \varphi (x,y,z,t)} 9711:Covariant classical field theory 7242:Quantum chromodynamic Lagrangian 6317:{\displaystyle \mathbb {C} ^{n}} 6127:combines the Lagrangian for the 6078: 6070: 6062: 6044: 6033: 5997: 5812: 5772: 5759: 5700: 5689: 5681: 5670: 5628: 4913: 4905: 4868: 4828: 4780: 4772: 4755: 4701: 4693: 4653: 4639: 4609: 4601: 4573: 4565:Varying instead with respect to 4535: 4527: 4494: 4439: 4385: 4336: 4328: 4311: 4303: 4286: 4266: 4240: 4208: 4168: 4160: 4137: 4105: 3150: 3117: 3071: 3021: 2960: 2925: 2878: 2855: 2829: 2597: 2577: 2544: 2494: 1803:), the unit volume is one, i.e. 1464: 1395: 1368: 1337: 1287:, which one integrates over all 1195:of the Lagrangian is called the 1176:{\displaystyle \mathbb {R} ^{n}} 1139:{\displaystyle \mathbb {R} ^{m}} 1037:are described by spinor fields. 972: 921: 857: 801: 770: 721: 10240: 10114:(2nd ed.). Wiley. p. 8682:{\displaystyle g_{\mu \nu }(x)} 5342: 5336: 4841: 4835: 301:{\displaystyle \varphi _{i}(s)} 10231: 10195: 10170: 10148:Einstein gravity in a nutshell 10050: 9854: 9819: 9799: 9268: 9262: 9246: 9240: 9224: 9218: 9189: 9183: 9167: 9161: 8945: 8939: 8923: 8917: 8901: 8895: 8866: 8860: 8844: 8838: 8773: 8767: 8739: 8733: 8715: 8709: 8676: 8670: 8587: 8581: 8547: 8541: 8525: 8519: 8503: 8497: 8481: 8475: 8434: 8428: 8415: 8409: 8389: 8383: 8099: 8055: 7313: 7228:electromagnetic four-potential 6959: 6914: 6908: 6722: 6669: 6625: 6619: 6161: 6149: 6001: 5993: 5949: 5943: 5932: 5926: 5912: 5906: 5632: 5624: 5585:{\displaystyle {\mathcal {M}}} 5535: 5529: 5513: 5507: 5466: 5460: 5447: 5441: 5425: 5419: 4872: 4855: 4832: 4818: 4765: 4751: 4711: 4697: 4663: 4649: 4619: 4605: 4545: 4531: 4504: 4490: 4449: 4435: 4395: 4381: 4346: 4332: 4321: 4307: 4296: 4282: 4276: 4262: 4250: 4236: 4187: 4178: 4172: 4164: 4153: 4147: 4124: 4115: 4109: 4101: 4020: 4014: 3739: 3733: 3651:. An equivalent expression is 3521: 3515: 3285: 3279: 3207: 3201: 3160: 3146: 3127: 3113: 3081: 3067: 3031: 3017: 2973: 2970: 2956: 2944: 2938: 2935: 2921: 2912: 2888: 2874: 2865: 2851: 2839: 2825: 2735: 2725: 2659:{\displaystyle {\mathcal {L}}} 2607: 2593: 2587: 2573: 2558: 2554: 2540: 2531: 2504: 2490: 2436: 2420: 2289: 2273: 2230: 2224: 2045: 2039: 1958: 1950: 1941: 1935: 1888: 1882: 1821: 1813: 1742: 1734: 1710: 1702: 1608: 1600: 1513:{\displaystyle {\mathcal {L}}} 1491: 1378: 1327: 1311: 1305: 1280:{\displaystyle {\mathcal {L}}} 982: 840: 811: 760: 678: 573:{\displaystyle {\mathcal {L}}} 413: 407: 381: 375: 295: 289: 261:{\displaystyle {\mathcal {S}}} 162: 138: 51:partial differential equations 1: 6592:The Lagrangian density for a 5564:, the electromagnetic action 4467:Varying this with respect to 2797:the potential term. See also 2346:Solving, with respect to the 1150:, then the field manifold is 1117:, then the field manifold is 556:. The calligraphic typeface, 10044: 7995:{\displaystyle T_{\mu \nu }} 7847:as the field, we obtain the 7840:{\displaystyle g_{\mu \nu }} 7770:. The Riemann tensor is the 7260:with the Lagrangian for the 7092:{\displaystyle {D}\!\!\!\!/} 7049:{\displaystyle F^{\mu \nu }} 6131:with the Lagrangian for the 4976:{\displaystyle F_{\mu \nu }} 4580:{\displaystyle \mathbf {A} } 4215:{\displaystyle \mathbf {j} } 3640:{\displaystyle \mathrm {d} } 2790:{\displaystyle V=\rho \Phi } 1836:{\textstyle {\sqrt {|g|}}=1} 1799:. For flat spacetime (e.g., 179:are obtained by means of an 7: 9698: 7519:gluon field strength tensor 7252:The Lagrangian density for 7219:{\displaystyle A_{\sigma }} 6866:The Lagrangian density for 6400:is its field-strength. The 6123:The Lagrangian density for 4983:. We define this tensor as 2461: 272:of the dependent variables 92: 83:Atiyah–Singer index theorem 10: 10293: 9692:topological field theories 9409:{\displaystyle j^{\mu }=0} 8327: 7528: 7467:gauge covariant derivative 7245: 7066:gauge covariant derivative 6859: 6585: 6572:Yang–Mills–Higgs equations 6116: 6113:Ginzburg–Landau Lagrangian 4074: 3549: 3498:{\displaystyle T=mv^{2}/2} 3184: 1719:{\textstyle {\sqrt {|g|}}} 1044:For example, if there are 494:where the brackets denote 75:Chern–Gauss–Bonnet theorem 10237:Itzykson-Zuber, eq. 3-152 9781:Principle of least action 5865:understood in terms of a 3847:{\displaystyle \phi _{i}} 2632:is the mass density, and 1864:{\textstyle {\sqrt {-g}}} 1520:will include a factor of 1186: 1146:. If the field is a real 9792: 9776:Onsager–Machlup function 9741:Hamiltonian field theory 9372:{\displaystyle D_{\mu }} 7849:Einstein field equations 7680:{\displaystyle \Lambda } 7473:= 1, 2, ...6 counts the 6559:{\displaystyle {\star }} 6402:Euler–Lagrange equations 6354:"Sombrero hat" potential 6135:. It may be written as: 5975:setting. This gives the 5276:We have made use of the 4085:. The interaction terms 4044:. The trace is just the 3527:{\displaystyle V(\phi )} 3213:{\displaystyle V(\phi )} 3094:which is equivalent to: 2352:Euler–Lagrange equations 2164:{\displaystyle \varphi } 2140:{\displaystyle \varphi } 2121:Euler–Lagrange equations 2115:Euler–Lagrange equations 1537:{\textstyle {\sqrt {g}}} 739:{\displaystyle \varphi } 580:, is used to denote the 308:, their derivatives and 9731:Generalized coordinates 9716:Euler–Lagrange equation 9668:, dashing these hopes. 8627:Einstein–Hilbert 7791:; moving bodies follow 7768:Einstein–Hilbert action 7531:Einstein–Hilbert action 6862:Quantum electrodynamics 5977:Chern–Simons functional 5967:Chern–Simons functional 5730:is the current 1-form, 4052:to the base spacetime. 3175:Gauss's law for gravity 2626:gravitational potential 1681:{\displaystyle \wedge } 826:For many scalar fields 685:generalized coordinates 183:principle, written as: 20:Lagrangian field theory 10272:Calculus of variations 10267:Classical field theory 10177:Cahill, Kevin (2013). 10025: 9861: 9706:Calculus of variations 9644: 9410: 9373: 9346: 9280: 9066: 9012: 8957: 8683: 8642: 8359: 8306: 8265: 8245: 8004:energy momentum tensor 7996: 7967: 7841: 7760: 7705: 7681: 7661: 7511: 7455: 7254:quantum chromodynamics 7248:quantum chromodynamics 7220: 7193: 7134: 7093: 7058:electromagnetic tensor 7050: 7020: 6830: 6797:Feynman slash notation 6789: 6755: 6699: 6679: 6560: 6538: 6480: 6394: 6370: 6338: 6318: 6281: 6261: 6125:Ginzburg–Landau theory 6119:Ginzburg–Landau theory 6094: 5956: 5826: 5783: 5716: 5592:can be written (using 5586: 5542: 5395: 5270: 5053: 4977: 4952:electromagnetic tensor 4944: 4879: 4788: 4718: 4581: 4552: 4459: 4216: 4194: 4027: 3987: 3931: 3848: 3817: 3816:{\displaystyle g_{ij}} 3787: 3719: 3698: 3641: 3619: 3546:Sigma model Lagrangian 3528: 3499: 3455: 3423: 3383: 3214: 3167: 3088: 2983: 2791: 2762: 2703: 2660: 2640:gravitational constant 2614: 2453: 2338: 2165: 2141: 2105: 2059: 2003: 1915: 1895: 1865: 1837: 1793: 1773: 1750: 1720: 1682: 1660: 1561: 1538: 1514: 1476: 1415: 1281: 1254: 1177: 1140: 1111: 1058: 989: 818: 740: 610: 574: 523: 486: 302: 262: 234: 169: 43:classical field theory 24:classical field theory 10144:Zee, Anthony (2013). 10026: 9862: 9736:Hamiltonian mechanics 9721:Functional derivative 9645: 9411: 9374: 9347: 9281: 9067: 9013: 8958: 8684: 8643: 8360: 8307: 8266: 8246: 7997: 7968: 7842: 7761: 7706: 7689:cosmological constant 7682: 7662: 7529:Further information: 7512: 7456: 7221: 7194: 7135: 7094: 7051: 7021: 6831: 6790: 6756: 6700: 6698:{\displaystyle \psi } 6680: 6576:Seiberg–Witten theory 6561: 6539: 6481: 6395: 6371: 6339: 6337:{\displaystyle \psi } 6319: 6282: 6280:{\displaystyle \psi } 6262: 6095: 5957: 5827: 5784: 5717: 5587: 5550:equivalence principle 5543: 5396: 5271: 5054: 4978: 4945: 4880: 4789: 4719: 4582: 4553: 4460: 4217: 4195: 4083:electromagnetic field 4028: 3988: 3932: 3849: 3818: 3788: 3699: 3678: 3642: 3620: 3536:Mexican hat potential 3529: 3500: 3456: 3454:{\displaystyle L=T-V} 3424: 3363: 3215: 3168: 3089: 2998:the formula becomes: 2984: 2792: 2763: 2704: 2661: 2615: 2454: 2339: 2166: 2142: 2106: 2060: 2004: 1916: 1896: 1866: 1838: 1794: 1774: 1751: 1721: 1683: 1661: 1562: 1539: 1515: 1477: 1416: 1282: 1255: 1178: 1141: 1112: 1059: 990: 819: 741: 726:For one scalar field 611: 575: 546:independent variables 524: 487: 303: 263: 235: 170: 10277:Quantum field theory 10262:Mathematical physics 10179:Physical mathematics 10110:Quantum Field Theory 9882: 9809: 9756:Lagrangian mechanics 9662:theory of everything 9432: 9387: 9381:covariant derivative 9356: 9290: 9076: 9022: 8967: 8693: 8654: 8369: 8365:. The Lagrangian is 8338: 8314:Jacobian determinant 8275: 8255: 8010: 7976: 7854: 7821: 7739: 7719:contracted with the 7695: 7671: 7539: 7481: 7268: 7203: 7144: 7107: 7072: 7030: 6874: 6803: 6769: 6713: 6689: 6600: 6548: 6490: 6411: 6406:Yang–Mills equations 6384: 6360: 6328: 6299: 6271: 6139: 6107:grand unified theory 5983: 5895: 5878:Yang–Mills equations 5803: 5745: 5614: 5572: 5409: 5284: 5063: 4987: 4957: 4889: 4802: 4739: 4591: 4569: 4475: 4226: 4204: 4089: 3997: 3941: 3866: 3831: 3797: 3655: 3629: 3568: 3509: 3465: 3433: 3224: 3195: 3098: 3002: 2812: 2772: 2713: 2677: 2646: 2480: 2358: 2175: 2155: 2131: 2071: 2013: 1929: 1905: 1894:{\displaystyle *(1)} 1876: 1848: 1807: 1783: 1763: 1730: 1696: 1672: 1574: 1551: 1524: 1500: 1431: 1295: 1267: 1218: 1158: 1121: 1075: 1048: 830: 750: 730: 641:symplectic manifolds 588: 560: 498: 318: 276: 248: 187: 132: 79:Riemann–Roch theorem 67:Riemannian manifolds 47:quantum field theory 28:Lagrangian mechanics 9786:Scalar field theory 9726:Functional integral 9672:Additional examples 9654:Kaluza–Klein theory 8837: 7776:Christoffel symbols 6568:Hodge star operator 6129:scalar field theory 6103:Chern–Simons theory 5979:. It is written as 5871:Minkowski spacetime 4057:topological soliton 3562:Riemannian manifold 3187:Scalar field theory 3181:Scalar field theory 2348:boundary conditions 1801:Minkowski spacetime 1749:{\displaystyle |g|} 1546:Riemannian manifold 1291:to get the action: 665:affine Lie algebras 633:classical mechanics 177:equations of motion 126:Riemannian manifold 87:Chern–Simons theory 59:uniform convergence 10021: 9857: 9640: 9406: 9369: 9342: 9276: 9062: 9008: 8953: 8818: 8679: 8638: 8636: 8355: 8302: 8261: 8241: 8006:and is defined by 7992: 7963: 7837: 7756: 7735:. The integral of 7731:contracted with a 7701: 7677: 7657: 7507: 7451: 7305: 7216: 7189: 7130: 7089: 7046: 7016: 6826: 6785: 6751: 6695: 6675: 6556: 6534: 6476: 6390: 6366: 6334: 6314: 6277: 6257: 6090: 5952: 5822: 5779: 5712: 5582: 5562:differential forms 5538: 5403:Levi-Civita tensor 5391: 5266: 5049: 4973: 4940: 4875: 4784: 4714: 4577: 4548: 4455: 4212: 4190: 4050:Maurer–Cartan form 4023: 3983: 3927: 3844: 3813: 3783: 3637: 3615: 3524: 3495: 3451: 3419: 3220:can be written as 3210: 3163: 3084: 2979: 2787: 2758: 2699: 2656: 2610: 2449: 2350:, one obtains the 2334: 2161: 2137: 2101: 2055: 1999: 1911: 1891: 1861: 1833: 1789: 1769: 1746: 1716: 1678: 1656: 1557: 1534: 1510: 1472: 1411: 1277: 1262:Lagrangian density 1250: 1173: 1136: 1107: 1054: 985: 814: 736: 606: 570: 519: 482: 298: 258: 230: 165: 32:degrees of freedom 22:is a formalism in 10125:978-0-471-49684-7 9990: 9967: 9944: 9921: 9771:Noether's theorem 9766:Lagrangian system 9581: 9554: 9504: 9477: 9203: 9135: 9118: 9057: 8880: 8826: 8811: 8790: 8777: 8743: 8742: 8628: 8611: 8576: 8460: 8352: 8286: 8264:{\displaystyle g} 8188: 8121: 8097: 8047: 8046: 7944: 7881: 7808:affine connection 7780:metric connection 7753: 7704:{\displaystyle R} 7654: 7616: 7587: 7570: 7553: 7405: 7355: 7316: 7296: 7262:Yang–Mills action 6988: 6942: 6911: 6725: 6652: 6622: 6439: 6393:{\displaystyle F} 6369:{\displaystyle A} 6216: 6133:Yang–Mills action 6059: 5851:affine connection 5666: 5492: 5340: 5212: 5158: 5118: 5081: 4839: 4684: 4646: 4421: 4367: 4144: 3887: 3856:local coordinates 3825:Riemannian metric 3676: 3589: 3397: 3338: 3299: 3245: 3053: 2910: 2638:in m·kg·s is the 2529: 2470:Newtonian gravity 2440: 2383: 2324: 2293: 2206: 1962: 1914:{\displaystyle *} 1859: 1825: 1792:{\displaystyle M} 1772:{\displaystyle g} 1714: 1612: 1560:{\displaystyle M} 1532: 1057:{\displaystyle m} 637:tangent manifolds 509: 432: 219: 10284: 10247: 10244: 10238: 10235: 10229: 10228: 10212: 10199: 10193: 10192: 10174: 10168: 10167: 10151: 10141: 10130: 10129: 10113: 10103: 10097: 10094: 10077: 10074: 10057: 10054: 10038: 10036: 10030: 10028: 10027: 10022: 10020: 10016: 9991: 9989: 9981: 9973: 9968: 9966: 9958: 9950: 9945: 9943: 9935: 9927: 9922: 9920: 9912: 9904: 9891: 9890: 9876: 9866: 9864: 9863: 9858: 9853: 9852: 9837: 9836: 9818: 9817: 9803: 9761:Lagrangian point 9649: 9647: 9646: 9641: 9639: 9638: 9629: 9624: 9623: 9611: 9610: 9601: 9596: 9595: 9587: 9583: 9582: 9580: 9579: 9570: 9569: 9560: 9555: 9550: 9542: 9525: 9524: 9515: 9510: 9506: 9505: 9503: 9502: 9493: 9492: 9483: 9478: 9473: 9465: 9449: 9448: 9439: 9427: 9424:and with charge 9415: 9413: 9412: 9407: 9399: 9398: 9378: 9376: 9375: 9370: 9368: 9367: 9351: 9349: 9348: 9343: 9341: 9340: 9331: 9330: 9315: 9314: 9302: 9301: 9285: 9283: 9282: 9277: 9275: 9271: 9261: 9260: 9239: 9238: 9217: 9216: 9204: 9196: 9182: 9181: 9160: 9159: 9154: 9153: 9152: 9136: 9134: 9133: 9121: 9119: 9117: 9116: 9107: 9096: 9091: 9090: 9071: 9069: 9068: 9063: 9058: 9056: 9055: 9046: 9035: 9017: 9015: 9014: 9009: 9001: 9000: 8988: 8987: 8962: 8960: 8959: 8954: 8952: 8948: 8938: 8937: 8916: 8915: 8894: 8893: 8881: 8873: 8859: 8858: 8836: 8831: 8827: 8824: 8812: 8810: 8809: 8797: 8792: 8791: 8788: 8786: 8785: 8778: 8776: 8766: 8765: 8746: 8744: 8726: 8722: 8708: 8707: 8688: 8686: 8685: 8680: 8669: 8668: 8647: 8645: 8644: 8639: 8637: 8630: 8629: 8626: 8624: 8623: 8613: 8612: 8609: 8607: 8606: 8593: 8577: 8575: 8564: 8563: 8554: 8540: 8539: 8518: 8517: 8496: 8495: 8474: 8473: 8461: 8459: 8458: 8457: 8441: 8427: 8426: 8408: 8407: 8382: 8381: 8364: 8362: 8361: 8356: 8354: 8353: 8350: 8348: 8347: 8311: 8309: 8308: 8303: 8298: 8297: 8287: 8279: 8270: 8268: 8267: 8262: 8250: 8248: 8247: 8242: 8236: 8235: 8234: 8213: 8212: 8205: 8204: 8189: 8187: 8186: 8185: 8169: 8168: 8167: 8166: 8145: 8144: 8133: 8122: 8120: 8119: 8118: 8102: 8098: 8090: 8088: 8087: 8086: 8065: 8064: 8050: 8048: 8039: 8038: 8030: 8025: 8024: 8001: 7999: 7998: 7993: 7991: 7990: 7972: 7970: 7969: 7964: 7958: 7957: 7945: 7943: 7942: 7933: 7922: 7914: 7913: 7898: 7897: 7882: 7874: 7869: 7868: 7846: 7844: 7843: 7838: 7836: 7835: 7766:is known as the 7765: 7763: 7762: 7757: 7755: 7754: 7751: 7749: 7748: 7713:curvature scalar 7710: 7708: 7707: 7702: 7686: 7684: 7683: 7678: 7666: 7664: 7663: 7658: 7656: 7655: 7652: 7650: 7649: 7639: 7635: 7617: 7615: 7604: 7603: 7594: 7589: 7588: 7585: 7583: 7582: 7572: 7571: 7568: 7566: 7565: 7555: 7554: 7551: 7549: 7548: 7525:Einstein gravity 7516: 7514: 7513: 7508: 7505: 7504: 7496: 7493: 7492: 7460: 7458: 7457: 7452: 7450: 7449: 7441: 7438: 7437: 7428: 7427: 7419: 7416: 7415: 7406: 7398: 7393: 7392: 7383: 7379: 7378: 7377: 7368: 7367: 7353: 7352: 7343: 7324: 7323: 7318: 7317: 7309: 7304: 7292: 7291: 7290: 7278: 7277: 7236:Clifford algebra 7225: 7223: 7222: 7217: 7215: 7214: 7198: 7196: 7195: 7190: 7188: 7187: 7169: 7168: 7156: 7155: 7139: 7137: 7136: 7131: 7129: 7128: 7119: 7118: 7101:Feynman notation 7098: 7096: 7095: 7090: 7088: 7079: 7055: 7053: 7052: 7047: 7045: 7044: 7025: 7023: 7022: 7017: 7015: 7014: 7002: 7001: 6989: 6987: 6986: 6985: 6969: 6958: 6957: 6940: 6939: 6930: 6913: 6912: 6904: 6898: 6897: 6896: 6884: 6883: 6842:Clifford algebra 6835: 6833: 6832: 6827: 6825: 6824: 6815: 6814: 6794: 6792: 6791: 6786: 6784: 6776: 6760: 6758: 6757: 6752: 6750: 6749: 6740: 6739: 6727: 6726: 6718: 6704: 6702: 6701: 6696: 6684: 6682: 6681: 6676: 6668: 6667: 6650: 6649: 6641: 6624: 6623: 6615: 6609: 6608: 6582:Dirac Lagrangian 6565: 6563: 6562: 6557: 6555: 6543: 6541: 6540: 6535: 6500: 6485: 6483: 6482: 6477: 6472: 6468: 6467: 6466: 6440: 6432: 6421: 6399: 6397: 6396: 6391: 6378:Yang–Mills field 6375: 6373: 6372: 6367: 6343: 6341: 6340: 6335: 6323: 6321: 6320: 6315: 6313: 6312: 6307: 6286: 6284: 6283: 6278: 6266: 6264: 6263: 6258: 6256: 6255: 6250: 6246: 6245: 6244: 6217: 6209: 6204: 6203: 6182: 6181: 6148: 6147: 6099: 6097: 6096: 6091: 6086: 6082: 6081: 6073: 6065: 6060: 6052: 6047: 6036: 6026: 6018: 6017: 6016: 6000: 5992: 5991: 5973:contact geometry 5961: 5959: 5958: 5953: 5942: 5925: 5905: 5837: 5831: 5829: 5828: 5823: 5815: 5810: 5798: 5788: 5786: 5785: 5780: 5775: 5770: 5762: 5757: 5752: 5735: 5721: 5719: 5718: 5713: 5708: 5704: 5703: 5692: 5684: 5673: 5667: 5659: 5652: 5651: 5650: 5631: 5623: 5622: 5609: 5591: 5589: 5588: 5583: 5581: 5580: 5547: 5545: 5544: 5539: 5528: 5527: 5506: 5505: 5493: 5491: 5490: 5489: 5473: 5459: 5458: 5440: 5439: 5418: 5417: 5400: 5398: 5397: 5392: 5384: 5383: 5371: 5370: 5361: 5360: 5341: 5338: 5335: 5334: 5325: 5324: 5309: 5308: 5296: 5295: 5278:Minkowski metric 5275: 5273: 5272: 5267: 5265: 5264: 5252: 5251: 5239: 5238: 5226: 5225: 5213: 5211: 5210: 5209: 5193: 5185: 5184: 5172: 5171: 5159: 5157: 5156: 5155: 5139: 5131: 5130: 5125: 5119: 5117: 5116: 5115: 5099: 5094: 5093: 5088: 5082: 5077: 5076: 5067: 5058: 5056: 5055: 5050: 5048: 5047: 5038: 5037: 5025: 5024: 5015: 5014: 5002: 5001: 4982: 4980: 4979: 4974: 4972: 4971: 4949: 4947: 4946: 4941: 4939: 4938: 4929: 4928: 4916: 4908: 4884: 4882: 4881: 4876: 4871: 4851: 4850: 4840: 4837: 4831: 4814: 4813: 4793: 4791: 4790: 4785: 4783: 4775: 4758: 4723: 4721: 4720: 4715: 4704: 4696: 4685: 4683: 4682: 4670: 4656: 4648: 4647: 4642: 4637: 4634: 4633: 4612: 4604: 4586: 4584: 4583: 4578: 4576: 4557: 4555: 4554: 4549: 4538: 4530: 4519: 4518: 4497: 4470: 4464: 4462: 4461: 4456: 4442: 4434: 4433: 4428: 4422: 4420: 4419: 4418: 4402: 4388: 4380: 4379: 4374: 4368: 4363: 4362: 4353: 4339: 4331: 4314: 4306: 4289: 4269: 4243: 4235: 4234: 4221: 4219: 4218: 4213: 4211: 4199: 4197: 4196: 4191: 4171: 4163: 4146: 4145: 4140: 4135: 4108: 4032: 4030: 4029: 4024: 4013: 3992: 3990: 3989: 3984: 3979: 3978: 3969: 3968: 3953: 3952: 3936: 3934: 3933: 3928: 3926: 3922: 3921: 3920: 3911: 3910: 3896: 3888: 3880: 3875: 3874: 3860:coordinate chart 3853: 3851: 3850: 3845: 3843: 3842: 3822: 3820: 3819: 3814: 3812: 3811: 3792: 3790: 3789: 3784: 3782: 3781: 3772: 3771: 3762: 3761: 3752: 3751: 3732: 3731: 3718: 3713: 3697: 3692: 3677: 3669: 3664: 3663: 3646: 3644: 3643: 3638: 3636: 3624: 3622: 3621: 3616: 3614: 3610: 3595: 3590: 3582: 3577: 3576: 3533: 3531: 3530: 3525: 3504: 3502: 3501: 3496: 3491: 3486: 3485: 3460: 3458: 3457: 3452: 3428: 3426: 3425: 3420: 3418: 3417: 3408: 3407: 3398: 3396: 3385: 3382: 3377: 3359: 3358: 3349: 3348: 3339: 3331: 3323: 3322: 3310: 3309: 3300: 3292: 3269: 3268: 3256: 3255: 3246: 3238: 3233: 3232: 3219: 3217: 3216: 3211: 3172: 3170: 3169: 3164: 3153: 3142: 3141: 3120: 3093: 3091: 3090: 3085: 3074: 3054: 3052: 3038: 3024: 2997: 2988: 2986: 2985: 2980: 2963: 2928: 2911: 2909: 2895: 2881: 2858: 2832: 2824: 2823: 2807: 2796: 2794: 2793: 2788: 2767: 2765: 2764: 2759: 2748: 2743: 2742: 2708: 2706: 2705: 2700: 2686: 2685: 2665: 2663: 2662: 2657: 2655: 2654: 2637: 2631: 2623: 2619: 2617: 2616: 2611: 2600: 2580: 2566: 2565: 2547: 2530: 2528: 2514: 2497: 2489: 2488: 2458: 2456: 2455: 2450: 2445: 2441: 2439: 2432: 2431: 2415: 2414: 2413: 2403: 2397: 2396: 2384: 2382: 2374: 2373: 2372: 2362: 2343: 2341: 2340: 2335: 2330: 2326: 2325: 2323: 2315: 2314: 2313: 2303: 2298: 2294: 2292: 2285: 2284: 2268: 2267: 2266: 2256: 2250: 2249: 2220: 2219: 2207: 2205: 2197: 2196: 2195: 2185: 2170: 2168: 2167: 2162: 2151:with respect to 2146: 2144: 2143: 2138: 2110: 2108: 2107: 2102: 2100: 2099: 2093: 2092: 2080: 2079: 2064: 2062: 2061: 2056: 2054: 2053: 2035: 2034: 2022: 2021: 2008: 2006: 2005: 2000: 1998: 1997: 1976: 1975: 1963: 1961: 1953: 1948: 1920: 1918: 1917: 1912: 1900: 1898: 1897: 1892: 1870: 1868: 1867: 1862: 1860: 1852: 1844:Lifschitz write 1842: 1840: 1839: 1834: 1826: 1824: 1816: 1811: 1798: 1796: 1795: 1790: 1778: 1776: 1775: 1770: 1755: 1753: 1752: 1747: 1745: 1737: 1725: 1723: 1722: 1717: 1715: 1713: 1705: 1700: 1687: 1685: 1684: 1679: 1665: 1663: 1662: 1657: 1655: 1654: 1648: 1647: 1626: 1625: 1613: 1611: 1603: 1598: 1596: 1595: 1583: 1582: 1566: 1564: 1563: 1558: 1543: 1541: 1540: 1535: 1533: 1528: 1519: 1517: 1516: 1511: 1509: 1508: 1481: 1479: 1478: 1473: 1467: 1462: 1461: 1456: 1449: 1448: 1420: 1418: 1417: 1412: 1404: 1398: 1393: 1392: 1387: 1371: 1357: 1340: 1326: 1325: 1304: 1303: 1286: 1284: 1283: 1278: 1276: 1275: 1259: 1257: 1256: 1251: 1242: 1227: 1226: 1213: 1204: 1182: 1180: 1179: 1174: 1172: 1171: 1166: 1145: 1143: 1142: 1137: 1135: 1134: 1129: 1116: 1114: 1113: 1108: 1106: 1105: 1087: 1086: 1063: 1061: 1060: 1055: 994: 992: 991: 986: 975: 955: 950: 949: 934: 933: 924: 916: 915: 891: 886: 885: 870: 869: 860: 852: 851: 839: 838: 823: 821: 820: 815: 804: 790: 773: 759: 758: 745: 743: 742: 737: 710: 673:Virasoro algebra 645:contact geometry 615: 613: 612: 607: 602: 601: 596: 579: 577: 576: 571: 569: 568: 552:= 1, 2, 3, ..., 528: 526: 525: 520: 507: 491: 489: 488: 483: 478: 474: 473: 468: 461: 457: 453: 452: 437: 433: 431: 430: 429: 416: 406: 405: 392: 374: 373: 359: 358: 345: 341: 340: 327: 326: 307: 305: 304: 299: 288: 287: 267: 265: 264: 259: 257: 256: 239: 237: 236: 231: 220: 218: 217: 216: 203: 202: 201: 191: 174: 172: 171: 166: 119: 63:potential theory 10292: 10291: 10287: 10286: 10285: 10283: 10282: 10281: 10252: 10251: 10250: 10245: 10241: 10236: 10232: 10225: 10200: 10196: 10189: 10175: 10171: 10164: 10142: 10133: 10126: 10104: 10100: 10095: 10080: 10075: 10060: 10055: 10051: 10047: 10042: 10041: 10032: 9982: 9974: 9972: 9959: 9951: 9949: 9936: 9928: 9926: 9913: 9905: 9903: 9896: 9892: 9886: 9885: 9883: 9880: 9879: 9872: 9848: 9844: 9832: 9828: 9813: 9812: 9810: 9807: 9806: 9804: 9800: 9795: 9790: 9701: 9674: 9634: 9630: 9625: 9619: 9615: 9606: 9602: 9597: 9588: 9575: 9571: 9565: 9561: 9559: 9543: 9541: 9534: 9530: 9529: 9520: 9516: 9511: 9498: 9494: 9488: 9484: 9482: 9466: 9464: 9457: 9453: 9444: 9440: 9435: 9433: 9430: 9429: 9425: 9394: 9390: 9388: 9385: 9384: 9363: 9359: 9357: 9354: 9353: 9336: 9332: 9326: 9322: 9307: 9303: 9297: 9293: 9291: 9288: 9287: 9253: 9249: 9231: 9227: 9209: 9205: 9195: 9174: 9170: 9155: 9148: 9144: 9143: 9142: 9141: 9137: 9129: 9125: 9120: 9112: 9108: 9097: 9095: 9083: 9079: 9077: 9074: 9073: 9051: 9047: 9036: 9034: 9023: 9020: 9019: 8993: 8989: 8980: 8976: 8968: 8965: 8964: 8930: 8926: 8908: 8904: 8886: 8882: 8872: 8851: 8847: 8832: 8823: 8822: 8817: 8813: 8805: 8801: 8796: 8787: 8781: 8780: 8779: 8758: 8754: 8750: 8745: 8721: 8700: 8696: 8694: 8691: 8690: 8661: 8657: 8655: 8652: 8651: 8635: 8634: 8625: 8619: 8618: 8617: 8608: 8602: 8601: 8600: 8591: 8590: 8565: 8559: 8555: 8553: 8532: 8528: 8510: 8506: 8488: 8484: 8466: 8462: 8453: 8449: 8445: 8440: 8422: 8418: 8403: 8399: 8392: 8377: 8376: 8372: 8370: 8367: 8366: 8349: 8343: 8342: 8341: 8339: 8336: 8335: 8332: 8326: 8293: 8289: 8278: 8276: 8273: 8272: 8256: 8253: 8252: 8215: 8214: 8208: 8207: 8206: 8197: 8193: 8178: 8174: 8170: 8147: 8146: 8140: 8139: 8138: 8134: 8132: 8111: 8107: 8103: 8089: 8067: 8066: 8060: 8059: 8058: 8051: 8049: 8031: 8029: 8017: 8013: 8011: 8008: 8007: 7983: 7979: 7977: 7974: 7973: 7950: 7946: 7938: 7934: 7923: 7921: 7906: 7902: 7890: 7886: 7873: 7861: 7857: 7855: 7852: 7851: 7828: 7824: 7822: 7819: 7818: 7750: 7744: 7743: 7742: 7740: 7737: 7736: 7733:Kronecker delta 7715:, which is the 7696: 7693: 7692: 7672: 7669: 7668: 7651: 7645: 7644: 7643: 7622: 7618: 7605: 7599: 7595: 7593: 7584: 7578: 7577: 7576: 7567: 7561: 7560: 7559: 7550: 7544: 7543: 7542: 7540: 7537: 7536: 7533: 7527: 7497: 7495: 7494: 7488: 7484: 7482: 7479: 7478: 7442: 7440: 7439: 7433: 7429: 7420: 7418: 7417: 7411: 7407: 7397: 7388: 7384: 7373: 7369: 7363: 7359: 7348: 7339: 7329: 7325: 7319: 7308: 7307: 7306: 7300: 7280: 7279: 7273: 7272: 7271: 7269: 7266: 7265: 7250: 7244: 7210: 7206: 7204: 7201: 7200: 7183: 7179: 7164: 7160: 7151: 7147: 7145: 7142: 7141: 7124: 7120: 7114: 7110: 7108: 7105: 7104: 7084: 7075: 7073: 7070: 7069: 7037: 7033: 7031: 7028: 7027: 7007: 7003: 6994: 6990: 6981: 6977: 6973: 6968: 6953: 6949: 6935: 6926: 6903: 6902: 6886: 6885: 6879: 6878: 6877: 6875: 6872: 6871: 6864: 6858: 6820: 6816: 6810: 6806: 6804: 6801: 6800: 6780: 6772: 6770: 6767: 6766: 6745: 6741: 6735: 6731: 6717: 6716: 6714: 6711: 6710: 6690: 6687: 6686: 6663: 6659: 6645: 6637: 6614: 6613: 6604: 6603: 6601: 6598: 6597: 6590: 6584: 6551: 6549: 6546: 6545: 6496: 6491: 6488: 6487: 6462: 6458: 6445: 6441: 6431: 6417: 6412: 6409: 6408: 6385: 6382: 6381: 6361: 6358: 6357: 6329: 6326: 6325: 6308: 6303: 6302: 6300: 6297: 6296: 6272: 6269: 6268: 6251: 6240: 6236: 6223: 6219: 6218: 6208: 6199: 6195: 6177: 6173: 6143: 6142: 6140: 6137: 6136: 6121: 6115: 6077: 6069: 6061: 6051: 6043: 6032: 6031: 6027: 6019: 6012: 6011: 6007: 5996: 5987: 5986: 5984: 5981: 5980: 5969: 5938: 5918: 5898: 5896: 5893: 5892: 5833: 5811: 5806: 5804: 5801: 5800: 5790: 5771: 5766: 5758: 5753: 5748: 5746: 5743: 5742: 5731: 5699: 5688: 5680: 5669: 5658: 5657: 5653: 5646: 5645: 5641: 5627: 5618: 5617: 5615: 5612: 5611: 5607: 5597: 5576: 5575: 5573: 5570: 5569: 5558: 5520: 5516: 5498: 5494: 5485: 5481: 5477: 5472: 5454: 5450: 5435: 5431: 5413: 5412: 5410: 5407: 5406: 5401:where ε is the 5376: 5372: 5366: 5362: 5347: 5343: 5337: 5330: 5326: 5320: 5316: 5301: 5297: 5291: 5287: 5285: 5282: 5281: 5257: 5253: 5244: 5240: 5231: 5227: 5218: 5214: 5205: 5201: 5197: 5192: 5177: 5173: 5164: 5160: 5151: 5147: 5143: 5138: 5126: 5121: 5120: 5111: 5107: 5103: 5098: 5089: 5084: 5083: 5072: 5068: 5066: 5064: 5061: 5060: 5043: 5039: 5033: 5029: 5020: 5016: 5010: 5006: 4994: 4990: 4988: 4985: 4984: 4964: 4960: 4958: 4955: 4954: 4934: 4930: 4924: 4920: 4912: 4904: 4890: 4887: 4886: 4867: 4846: 4842: 4836: 4827: 4809: 4805: 4803: 4800: 4799: 4779: 4771: 4754: 4740: 4737: 4736: 4733:tensor notation 4700: 4692: 4678: 4674: 4669: 4652: 4638: 4636: 4635: 4629: 4625: 4608: 4600: 4592: 4589: 4588: 4572: 4570: 4567: 4566: 4534: 4526: 4514: 4510: 4493: 4476: 4473: 4472: 4468: 4438: 4429: 4424: 4423: 4414: 4410: 4406: 4401: 4384: 4375: 4370: 4369: 4358: 4354: 4352: 4335: 4327: 4310: 4302: 4285: 4265: 4239: 4230: 4229: 4227: 4224: 4223: 4207: 4205: 4202: 4201: 4167: 4159: 4136: 4134: 4133: 4104: 4090: 4087: 4086: 4079: 4073: 4042:symmetric space 4006: 3998: 3995: 3994: 3974: 3970: 3961: 3957: 3948: 3944: 3942: 3939: 3938: 3916: 3912: 3906: 3902: 3901: 3897: 3889: 3879: 3870: 3869: 3867: 3864: 3863: 3838: 3834: 3832: 3829: 3828: 3804: 3800: 3798: 3795: 3794: 3777: 3773: 3767: 3763: 3757: 3753: 3747: 3743: 3724: 3720: 3714: 3703: 3693: 3682: 3668: 3659: 3658: 3656: 3653: 3652: 3632: 3630: 3627: 3626: 3606: 3602: 3591: 3581: 3572: 3571: 3569: 3566: 3565: 3554: 3548: 3510: 3507: 3506: 3487: 3481: 3477: 3466: 3463: 3462: 3434: 3431: 3430: 3413: 3409: 3403: 3399: 3389: 3384: 3378: 3367: 3354: 3350: 3344: 3340: 3330: 3318: 3314: 3305: 3301: 3291: 3264: 3260: 3251: 3247: 3237: 3228: 3227: 3225: 3222: 3221: 3196: 3193: 3192: 3189: 3183: 3149: 3137: 3133: 3116: 3099: 3096: 3095: 3070: 3042: 3037: 3020: 3003: 3000: 2999: 2992: 2959: 2924: 2899: 2894: 2877: 2854: 2828: 2819: 2818: 2813: 2810: 2809: 2805: 2773: 2770: 2769: 2744: 2738: 2734: 2714: 2711: 2710: 2681: 2680: 2678: 2675: 2674: 2650: 2649: 2647: 2644: 2643: 2633: 2629: 2621: 2596: 2576: 2561: 2557: 2543: 2518: 2513: 2493: 2484: 2483: 2481: 2478: 2477: 2472: 2464: 2427: 2423: 2416: 2409: 2408: 2404: 2402: 2398: 2392: 2388: 2375: 2368: 2367: 2363: 2361: 2359: 2356: 2355: 2316: 2309: 2308: 2304: 2302: 2280: 2276: 2269: 2262: 2261: 2257: 2255: 2251: 2245: 2241: 2237: 2233: 2215: 2211: 2198: 2191: 2190: 2186: 2184: 2176: 2173: 2172: 2156: 2153: 2152: 2132: 2129: 2128: 2117: 2095: 2094: 2088: 2084: 2075: 2074: 2072: 2069: 2068: 2049: 2048: 2030: 2026: 2017: 2016: 2014: 2011: 2010: 1993: 1989: 1971: 1967: 1957: 1949: 1947: 1930: 1927: 1926: 1906: 1903: 1902: 1877: 1874: 1873: 1851: 1849: 1846: 1845: 1820: 1812: 1810: 1808: 1805: 1804: 1784: 1781: 1780: 1764: 1761: 1760: 1741: 1733: 1731: 1728: 1727: 1709: 1701: 1699: 1697: 1694: 1693: 1673: 1670: 1669: 1650: 1649: 1643: 1639: 1621: 1617: 1607: 1599: 1597: 1591: 1587: 1578: 1577: 1575: 1572: 1571: 1552: 1549: 1548: 1527: 1525: 1522: 1521: 1504: 1503: 1501: 1498: 1497: 1494: 1463: 1457: 1452: 1451: 1444: 1443: 1432: 1429: 1428: 1425:volume integral 1400: 1394: 1388: 1383: 1382: 1367: 1353: 1336: 1321: 1320: 1299: 1298: 1296: 1293: 1292: 1271: 1270: 1268: 1265: 1264: 1238: 1222: 1221: 1219: 1216: 1215: 1209: 1200: 1189: 1167: 1162: 1161: 1159: 1156: 1155: 1130: 1125: 1124: 1122: 1119: 1118: 1101: 1097: 1082: 1078: 1076: 1073: 1072: 1049: 1046: 1045: 1019: 971: 951: 945: 941: 929: 925: 920: 911: 907: 887: 881: 877: 865: 861: 856: 847: 843: 834: 833: 831: 828: 827: 800: 786: 769: 754: 753: 751: 748: 747: 731: 728: 727: 724: 715:on a manifold. 692: 681: 657:tensor algebras 597: 592: 591: 589: 586: 585: 564: 563: 561: 558: 557: 499: 496: 495: 469: 464: 463: 448: 444: 425: 421: 417: 401: 397: 393: 391: 387: 369: 365: 364: 360: 354: 353: 352: 336: 332: 328: 322: 321: 319: 316: 315: 283: 279: 277: 274: 273: 252: 251: 249: 246: 245: 212: 208: 204: 197: 196: 192: 190: 188: 185: 184: 133: 130: 129: 101: 95: 17: 12: 11: 5: 10290: 10280: 10279: 10274: 10269: 10264: 10249: 10248: 10239: 10230: 10223: 10194: 10187: 10169: 10162: 10131: 10124: 10098: 10078: 10058: 10048: 10046: 10043: 10040: 10039: 10019: 10015: 10012: 10009: 10006: 10003: 10000: 9997: 9994: 9988: 9985: 9980: 9977: 9971: 9965: 9962: 9957: 9954: 9948: 9942: 9939: 9934: 9931: 9925: 9919: 9916: 9911: 9908: 9902: 9899: 9895: 9889: 9856: 9851: 9847: 9843: 9840: 9835: 9831: 9827: 9824: 9821: 9816: 9797: 9796: 9794: 9791: 9789: 9788: 9783: 9778: 9773: 9768: 9763: 9758: 9753: 9748: 9743: 9738: 9733: 9728: 9723: 9718: 9713: 9708: 9702: 9700: 9697: 9696: 9695: 9673: 9670: 9666:Standard model 9658:factorizations 9637: 9633: 9628: 9622: 9618: 9614: 9609: 9605: 9600: 9594: 9591: 9586: 9578: 9574: 9568: 9564: 9558: 9553: 9549: 9546: 9540: 9537: 9533: 9528: 9523: 9519: 9514: 9509: 9501: 9497: 9491: 9487: 9481: 9476: 9472: 9469: 9463: 9460: 9456: 9452: 9447: 9443: 9438: 9405: 9402: 9397: 9393: 9366: 9362: 9339: 9335: 9329: 9325: 9321: 9318: 9313: 9310: 9306: 9300: 9296: 9274: 9270: 9267: 9264: 9259: 9256: 9252: 9248: 9245: 9242: 9237: 9234: 9230: 9226: 9223: 9220: 9215: 9212: 9208: 9202: 9199: 9194: 9191: 9188: 9185: 9180: 9177: 9173: 9169: 9166: 9163: 9158: 9151: 9147: 9140: 9132: 9128: 9124: 9115: 9111: 9106: 9103: 9100: 9094: 9089: 9086: 9082: 9061: 9054: 9050: 9045: 9042: 9039: 9033: 9030: 9027: 9007: 9004: 8999: 8996: 8992: 8986: 8983: 8979: 8975: 8972: 8951: 8947: 8944: 8941: 8936: 8933: 8929: 8925: 8922: 8919: 8914: 8911: 8907: 8903: 8900: 8897: 8892: 8889: 8885: 8879: 8876: 8871: 8868: 8865: 8862: 8857: 8854: 8850: 8846: 8843: 8840: 8835: 8830: 8821: 8816: 8808: 8804: 8800: 8795: 8784: 8775: 8772: 8769: 8764: 8761: 8757: 8753: 8749: 8741: 8738: 8735: 8732: 8729: 8725: 8720: 8717: 8714: 8711: 8706: 8703: 8699: 8678: 8675: 8672: 8667: 8664: 8660: 8633: 8622: 8616: 8605: 8599: 8596: 8594: 8592: 8589: 8586: 8583: 8580: 8574: 8571: 8568: 8562: 8558: 8552: 8549: 8546: 8543: 8538: 8535: 8531: 8527: 8524: 8521: 8516: 8513: 8509: 8505: 8502: 8499: 8494: 8491: 8487: 8483: 8480: 8477: 8472: 8469: 8465: 8456: 8452: 8448: 8444: 8439: 8436: 8433: 8430: 8425: 8421: 8417: 8414: 8411: 8406: 8402: 8398: 8395: 8393: 8391: 8388: 8385: 8380: 8375: 8374: 8346: 8328:Main article: 8325: 8322: 8301: 8296: 8292: 8285: 8282: 8260: 8240: 8233: 8230: 8227: 8224: 8221: 8218: 8211: 8203: 8200: 8196: 8192: 8184: 8181: 8177: 8173: 8165: 8162: 8159: 8156: 8153: 8150: 8143: 8137: 8131: 8128: 8125: 8117: 8114: 8110: 8106: 8101: 8096: 8093: 8085: 8082: 8079: 8076: 8073: 8070: 8063: 8057: 8054: 8045: 8042: 8037: 8034: 8028: 8023: 8020: 8016: 7989: 7986: 7982: 7962: 7956: 7953: 7949: 7941: 7937: 7932: 7929: 7926: 7920: 7917: 7912: 7909: 7905: 7901: 7896: 7893: 7889: 7885: 7880: 7877: 7872: 7867: 7864: 7860: 7834: 7831: 7827: 7747: 7729:Riemann tensor 7700: 7676: 7648: 7642: 7638: 7634: 7631: 7628: 7625: 7621: 7614: 7611: 7608: 7602: 7598: 7592: 7581: 7575: 7564: 7558: 7547: 7526: 7523: 7503: 7500: 7491: 7487: 7448: 7445: 7436: 7432: 7426: 7423: 7414: 7410: 7404: 7401: 7396: 7391: 7387: 7382: 7376: 7372: 7366: 7362: 7358: 7351: 7342: 7338: 7335: 7332: 7328: 7322: 7315: 7312: 7303: 7299: 7295: 7289: 7286: 7283: 7276: 7246:Main article: 7243: 7240: 7213: 7209: 7186: 7182: 7178: 7175: 7172: 7167: 7163: 7159: 7154: 7150: 7127: 7123: 7117: 7113: 7087: 7078: 7043: 7040: 7036: 7013: 7010: 7006: 7000: 6997: 6993: 6984: 6980: 6976: 6972: 6967: 6964: 6961: 6956: 6952: 6948: 6945: 6938: 6929: 6925: 6922: 6919: 6916: 6910: 6907: 6901: 6895: 6892: 6889: 6882: 6860:Main article: 6857: 6854: 6850:spin structure 6823: 6819: 6813: 6809: 6783: 6775: 6748: 6744: 6738: 6734: 6730: 6724: 6721: 6694: 6674: 6671: 6666: 6662: 6658: 6655: 6648: 6640: 6636: 6633: 6630: 6627: 6621: 6618: 6612: 6607: 6588:Dirac equation 6586:Main article: 6583: 6580: 6554: 6533: 6530: 6527: 6524: 6521: 6518: 6515: 6512: 6509: 6506: 6503: 6499: 6495: 6475: 6471: 6465: 6461: 6457: 6454: 6451: 6448: 6444: 6438: 6435: 6430: 6427: 6424: 6420: 6416: 6389: 6365: 6346:superconductor 6333: 6311: 6306: 6276: 6254: 6249: 6243: 6239: 6235: 6232: 6229: 6226: 6222: 6215: 6212: 6207: 6202: 6198: 6194: 6191: 6188: 6185: 6180: 6176: 6172: 6169: 6166: 6163: 6160: 6157: 6154: 6151: 6146: 6117:Main article: 6114: 6111: 6089: 6085: 6080: 6076: 6072: 6068: 6064: 6058: 6055: 6050: 6046: 6042: 6039: 6035: 6030: 6025: 6022: 6015: 6010: 6006: 6003: 5999: 5995: 5990: 5968: 5965: 5951: 5948: 5945: 5941: 5937: 5934: 5931: 5928: 5924: 5921: 5917: 5914: 5911: 5908: 5904: 5901: 5889:Standard model 5821: 5818: 5814: 5809: 5778: 5774: 5769: 5765: 5761: 5756: 5751: 5711: 5707: 5702: 5698: 5695: 5691: 5687: 5683: 5679: 5676: 5672: 5665: 5662: 5656: 5649: 5644: 5640: 5637: 5634: 5630: 5626: 5621: 5605: 5579: 5557: 5554: 5537: 5534: 5531: 5526: 5523: 5519: 5515: 5512: 5509: 5504: 5501: 5497: 5488: 5484: 5480: 5476: 5471: 5468: 5465: 5462: 5457: 5453: 5449: 5446: 5443: 5438: 5434: 5430: 5427: 5424: 5421: 5416: 5390: 5387: 5382: 5379: 5375: 5369: 5365: 5359: 5356: 5353: 5350: 5346: 5333: 5329: 5323: 5319: 5315: 5312: 5307: 5304: 5300: 5294: 5290: 5263: 5260: 5256: 5250: 5247: 5243: 5237: 5234: 5230: 5224: 5221: 5217: 5208: 5204: 5200: 5196: 5191: 5188: 5183: 5180: 5176: 5170: 5167: 5163: 5154: 5150: 5146: 5142: 5137: 5134: 5129: 5124: 5114: 5110: 5106: 5102: 5097: 5092: 5087: 5080: 5075: 5071: 5046: 5042: 5036: 5032: 5028: 5023: 5019: 5013: 5009: 5005: 5000: 4997: 4993: 4970: 4967: 4963: 4937: 4933: 4927: 4923: 4919: 4915: 4911: 4907: 4903: 4900: 4897: 4894: 4874: 4870: 4866: 4863: 4860: 4857: 4854: 4849: 4845: 4834: 4830: 4826: 4823: 4820: 4817: 4812: 4808: 4782: 4778: 4774: 4770: 4767: 4764: 4761: 4757: 4753: 4750: 4747: 4744: 4713: 4710: 4707: 4703: 4699: 4695: 4691: 4688: 4681: 4677: 4673: 4668: 4665: 4662: 4659: 4655: 4651: 4645: 4641: 4632: 4628: 4624: 4621: 4618: 4615: 4611: 4607: 4603: 4599: 4596: 4575: 4547: 4544: 4541: 4537: 4533: 4529: 4525: 4522: 4517: 4513: 4509: 4506: 4503: 4500: 4496: 4492: 4489: 4486: 4483: 4480: 4454: 4451: 4448: 4445: 4441: 4437: 4432: 4427: 4417: 4413: 4409: 4405: 4400: 4397: 4394: 4391: 4387: 4383: 4378: 4373: 4366: 4361: 4357: 4351: 4348: 4345: 4342: 4338: 4334: 4330: 4326: 4323: 4320: 4317: 4313: 4309: 4305: 4301: 4298: 4295: 4292: 4288: 4284: 4281: 4278: 4275: 4272: 4268: 4264: 4261: 4258: 4255: 4252: 4249: 4246: 4242: 4238: 4233: 4210: 4189: 4186: 4183: 4180: 4177: 4174: 4170: 4166: 4162: 4158: 4155: 4152: 4149: 4143: 4139: 4132: 4129: 4126: 4123: 4120: 4117: 4114: 4111: 4107: 4103: 4100: 4097: 4094: 4075:Main article: 4072: 4069: 4022: 4019: 4016: 4012: 4009: 4005: 4002: 3982: 3977: 3973: 3967: 3964: 3960: 3956: 3951: 3947: 3925: 3919: 3915: 3909: 3905: 3900: 3895: 3892: 3886: 3883: 3878: 3873: 3841: 3837: 3810: 3807: 3803: 3780: 3776: 3770: 3766: 3760: 3756: 3750: 3746: 3741: 3738: 3735: 3730: 3727: 3723: 3717: 3712: 3709: 3706: 3702: 3696: 3691: 3688: 3685: 3681: 3675: 3672: 3667: 3662: 3635: 3613: 3609: 3605: 3601: 3598: 3594: 3588: 3585: 3580: 3575: 3550:Main article: 3547: 3544: 3523: 3520: 3517: 3514: 3494: 3490: 3484: 3480: 3476: 3473: 3470: 3450: 3447: 3444: 3441: 3438: 3416: 3412: 3406: 3402: 3395: 3392: 3388: 3381: 3376: 3373: 3370: 3366: 3362: 3357: 3353: 3347: 3343: 3337: 3334: 3329: 3326: 3321: 3317: 3313: 3308: 3304: 3298: 3295: 3290: 3287: 3284: 3281: 3278: 3275: 3272: 3267: 3263: 3259: 3254: 3250: 3244: 3241: 3236: 3231: 3209: 3206: 3203: 3200: 3185:Main article: 3182: 3179: 3162: 3159: 3156: 3152: 3148: 3145: 3140: 3136: 3132: 3129: 3126: 3123: 3119: 3115: 3112: 3109: 3106: 3103: 3083: 3080: 3077: 3073: 3069: 3066: 3063: 3060: 3057: 3051: 3048: 3045: 3041: 3036: 3033: 3030: 3027: 3023: 3019: 3016: 3013: 3010: 3007: 2978: 2975: 2972: 2969: 2966: 2962: 2958: 2955: 2952: 2949: 2946: 2943: 2940: 2937: 2934: 2931: 2927: 2923: 2920: 2917: 2914: 2908: 2905: 2902: 2898: 2893: 2890: 2887: 2884: 2880: 2876: 2873: 2870: 2867: 2864: 2861: 2857: 2853: 2850: 2847: 2844: 2841: 2838: 2835: 2831: 2827: 2822: 2817: 2786: 2783: 2780: 2777: 2757: 2754: 2751: 2747: 2741: 2737: 2733: 2730: 2727: 2724: 2721: 2718: 2698: 2695: 2692: 2689: 2684: 2653: 2642:. The density 2609: 2606: 2603: 2599: 2595: 2592: 2589: 2586: 2583: 2579: 2575: 2572: 2569: 2564: 2560: 2556: 2553: 2550: 2546: 2542: 2539: 2536: 2533: 2527: 2524: 2521: 2517: 2512: 2509: 2506: 2503: 2500: 2496: 2492: 2487: 2471: 2468: 2463: 2460: 2448: 2444: 2438: 2435: 2430: 2426: 2422: 2419: 2412: 2407: 2401: 2395: 2391: 2387: 2381: 2378: 2371: 2366: 2333: 2329: 2322: 2319: 2312: 2307: 2301: 2297: 2291: 2288: 2283: 2279: 2275: 2272: 2265: 2260: 2254: 2248: 2244: 2240: 2236: 2232: 2229: 2226: 2223: 2218: 2214: 2210: 2204: 2201: 2194: 2189: 2183: 2180: 2171:, one obtains 2160: 2136: 2116: 2113: 2098: 2091: 2087: 2083: 2078: 2052: 2047: 2044: 2041: 2038: 2033: 2029: 2025: 2020: 1996: 1992: 1988: 1985: 1982: 1979: 1974: 1970: 1966: 1960: 1956: 1952: 1946: 1943: 1940: 1937: 1934: 1910: 1890: 1887: 1884: 1881: 1858: 1855: 1832: 1829: 1823: 1819: 1815: 1788: 1768: 1744: 1740: 1736: 1712: 1708: 1704: 1677: 1653: 1646: 1642: 1638: 1635: 1632: 1629: 1624: 1620: 1616: 1610: 1606: 1602: 1594: 1590: 1586: 1581: 1556: 1531: 1507: 1493: 1490: 1471: 1466: 1460: 1455: 1447: 1442: 1439: 1436: 1410: 1407: 1403: 1397: 1391: 1386: 1380: 1377: 1374: 1370: 1366: 1363: 1360: 1356: 1352: 1349: 1346: 1343: 1339: 1335: 1332: 1329: 1324: 1319: 1316: 1313: 1310: 1307: 1302: 1274: 1249: 1245: 1241: 1236: 1233: 1230: 1225: 1188: 1185: 1170: 1165: 1133: 1128: 1104: 1100: 1096: 1093: 1090: 1085: 1081: 1053: 1033:. In physics, 1018: 1015: 984: 981: 978: 974: 970: 967: 964: 961: 958: 954: 948: 944: 940: 937: 932: 928: 923: 919: 914: 910: 906: 903: 900: 897: 894: 890: 884: 880: 876: 873: 868: 864: 859: 855: 850: 846: 842: 837: 813: 810: 807: 803: 799: 796: 793: 789: 785: 782: 779: 776: 772: 768: 765: 762: 757: 735: 723: 720: 680: 677: 661:quantum groups 649:spin manifolds 605: 600: 595: 567: 537:} denotes the 518: 515: 512: 506: 503: 481: 477: 472: 467: 460: 456: 451: 447: 443: 440: 436: 428: 424: 420: 415: 412: 409: 404: 400: 396: 390: 386: 383: 380: 377: 372: 368: 363: 357: 351: 348: 344: 339: 335: 331: 325: 297: 294: 291: 286: 282: 255: 229: 226: 223: 215: 211: 207: 200: 195: 164: 161: 158: 155: 152: 149: 146: 143: 140: 137: 94: 91: 55:Sobolev spaces 15: 9: 6: 4: 3: 2: 10289: 10278: 10275: 10273: 10270: 10268: 10265: 10263: 10260: 10259: 10257: 10243: 10234: 10226: 10224:3-540-42627-2 10220: 10216: 10211: 10210: 10204: 10198: 10190: 10188:9781107005211 10184: 10180: 10173: 10165: 10163:9780691145587 10159: 10155: 10150: 10149: 10140: 10138: 10136: 10127: 10121: 10117: 10112: 10111: 10102: 10093: 10091: 10089: 10087: 10085: 10083: 10073: 10071: 10069: 10067: 10065: 10063: 10053: 10049: 10035: 10017: 10013: 10010: 10007: 10004: 10001: 9998: 9995: 9992: 9986: 9978: 9969: 9963: 9955: 9946: 9940: 9932: 9923: 9917: 9909: 9900: 9897: 9893: 9875: 9870: 9869:four-gradient 9849: 9845: 9841: 9838: 9833: 9825: 9822: 9802: 9798: 9787: 9784: 9782: 9779: 9777: 9774: 9772: 9769: 9767: 9764: 9762: 9759: 9757: 9754: 9752: 9749: 9747: 9744: 9742: 9739: 9737: 9734: 9732: 9729: 9727: 9724: 9722: 9719: 9717: 9714: 9712: 9709: 9707: 9704: 9703: 9693: 9689: 9685: 9680: 9676: 9675: 9669: 9667: 9663: 9659: 9655: 9650: 9635: 9620: 9616: 9612: 9607: 9603: 9592: 9589: 9584: 9576: 9572: 9566: 9562: 9556: 9551: 9547: 9544: 9538: 9535: 9531: 9526: 9521: 9517: 9507: 9499: 9495: 9489: 9485: 9479: 9474: 9470: 9467: 9461: 9458: 9454: 9450: 9445: 9441: 9423: 9422:natural units 9419: 9403: 9400: 9395: 9391: 9382: 9364: 9360: 9337: 9333: 9327: 9323: 9319: 9316: 9311: 9308: 9304: 9298: 9294: 9272: 9265: 9257: 9254: 9250: 9243: 9235: 9232: 9228: 9221: 9213: 9210: 9206: 9200: 9197: 9192: 9186: 9178: 9175: 9171: 9164: 9156: 9149: 9145: 9138: 9130: 9126: 9122: 9113: 9109: 9104: 9101: 9098: 9092: 9087: 9084: 9080: 9059: 9052: 9048: 9043: 9040: 9037: 9031: 9028: 9025: 9005: 9002: 8997: 8994: 8990: 8984: 8981: 8977: 8973: 8970: 8949: 8942: 8934: 8931: 8927: 8920: 8912: 8909: 8905: 8898: 8890: 8887: 8883: 8877: 8874: 8869: 8863: 8855: 8852: 8848: 8841: 8833: 8828: 8819: 8814: 8806: 8802: 8798: 8793: 8770: 8762: 8759: 8755: 8751: 8747: 8736: 8730: 8727: 8723: 8718: 8712: 8704: 8701: 8697: 8673: 8665: 8662: 8658: 8648: 8631: 8614: 8597: 8595: 8584: 8578: 8572: 8569: 8566: 8560: 8556: 8550: 8544: 8536: 8533: 8529: 8522: 8514: 8511: 8507: 8500: 8492: 8489: 8485: 8478: 8470: 8467: 8463: 8454: 8450: 8446: 8442: 8437: 8431: 8423: 8419: 8412: 8404: 8400: 8396: 8394: 8386: 8331: 8321: 8319: 8315: 8299: 8294: 8290: 8283: 8280: 8258: 8238: 8201: 8198: 8194: 8190: 8182: 8179: 8175: 8171: 8135: 8129: 8126: 8123: 8115: 8112: 8108: 8104: 8094: 8091: 8052: 8043: 8040: 8035: 8032: 8026: 8021: 8018: 8014: 8005: 7987: 7984: 7980: 7960: 7954: 7951: 7947: 7939: 7935: 7930: 7927: 7924: 7918: 7910: 7907: 7903: 7899: 7894: 7891: 7887: 7883: 7878: 7875: 7870: 7865: 7862: 7858: 7850: 7832: 7829: 7825: 7815: 7813: 7809: 7805: 7800: 7798: 7797:straight line 7794: 7790: 7786: 7781: 7777: 7773: 7769: 7734: 7730: 7726: 7722: 7721:metric tensor 7718: 7714: 7698: 7690: 7640: 7636: 7629: 7626: 7623: 7619: 7612: 7609: 7606: 7600: 7596: 7590: 7573: 7556: 7532: 7522: 7520: 7501: 7498: 7489: 7485: 7476: 7472: 7468: 7464: 7446: 7443: 7434: 7430: 7424: 7421: 7412: 7408: 7402: 7399: 7394: 7389: 7385: 7380: 7374: 7370: 7364: 7360: 7356: 7349: 7340: 7336: 7330: 7326: 7320: 7310: 7301: 7297: 7293: 7263: 7259: 7258:Dirac spinors 7255: 7249: 7239: 7237: 7233: 7229: 7211: 7207: 7184: 7180: 7176: 7173: 7170: 7165: 7157: 7152: 7148: 7125: 7121: 7115: 7111: 7102: 7085: 7076: 7067: 7063: 7059: 7041: 7038: 7034: 7011: 7008: 7004: 6998: 6995: 6991: 6982: 6978: 6974: 6970: 6965: 6962: 6954: 6950: 6946: 6943: 6936: 6927: 6923: 6917: 6905: 6899: 6869: 6863: 6853: 6851: 6847: 6843: 6839: 6821: 6811: 6807: 6798: 6781: 6764: 6763:Dirac adjoint 6746: 6742: 6736: 6732: 6728: 6719: 6708: 6692: 6672: 6664: 6660: 6656: 6653: 6646: 6634: 6628: 6616: 6610: 6595: 6589: 6579: 6577: 6573: 6569: 6552: 6528: 6525: 6522: 6519: 6513: 6510: 6507: 6504: 6501: 6497: 6493: 6473: 6469: 6463: 6455: 6449: 6446: 6442: 6436: 6433: 6428: 6425: 6422: 6418: 6414: 6407: 6403: 6387: 6379: 6363: 6355: 6351: 6347: 6331: 6309: 6294: 6293:vector bundle 6290: 6274: 6252: 6247: 6241: 6233: 6227: 6224: 6220: 6213: 6210: 6205: 6200: 6192: 6189: 6183: 6178: 6170: 6164: 6158: 6155: 6152: 6134: 6130: 6126: 6120: 6110: 6108: 6104: 6100: 6087: 6083: 6074: 6066: 6056: 6053: 6048: 6040: 6037: 6028: 6008: 6004: 5978: 5974: 5964: 5946: 5935: 5929: 5915: 5909: 5890: 5886: 5883: 5879: 5874: 5872: 5868: 5867:circle bundle 5864: 5860: 5856: 5852: 5848: 5843: 5841: 5836: 5819: 5816: 5797: 5793: 5776: 5767: 5763: 5754: 5739: 5734: 5729: 5725: 5709: 5705: 5696: 5693: 5685: 5677: 5674: 5663: 5660: 5654: 5642: 5638: 5635: 5604: 5600: 5595: 5594:natural units 5567: 5563: 5553: 5551: 5532: 5524: 5521: 5517: 5510: 5502: 5499: 5495: 5486: 5482: 5478: 5474: 5469: 5463: 5455: 5451: 5444: 5436: 5432: 5428: 5422: 5404: 5388: 5385: 5380: 5377: 5373: 5367: 5357: 5354: 5351: 5348: 5344: 5331: 5327: 5321: 5317: 5313: 5310: 5305: 5302: 5298: 5292: 5279: 5261: 5258: 5254: 5248: 5245: 5241: 5235: 5232: 5228: 5222: 5219: 5215: 5206: 5202: 5198: 5194: 5189: 5186: 5181: 5178: 5174: 5168: 5165: 5161: 5152: 5148: 5144: 5140: 5135: 5132: 5127: 5122: 5112: 5108: 5104: 5100: 5095: 5090: 5085: 5078: 5073: 5069: 5044: 5040: 5034: 5026: 5021: 5017: 5011: 5003: 4998: 4995: 4991: 4968: 4965: 4961: 4953: 4935: 4931: 4925: 4921: 4917: 4909: 4901: 4898: 4895: 4892: 4864: 4861: 4858: 4852: 4847: 4843: 4824: 4821: 4815: 4810: 4806: 4797: 4776: 4768: 4762: 4759: 4748: 4745: 4742: 4734: 4729: 4727: 4724:which yields 4708: 4705: 4689: 4679: 4675: 4671: 4666: 4660: 4657: 4643: 4630: 4626: 4622: 4616: 4613: 4597: 4594: 4563: 4561: 4558:which yields 4542: 4539: 4523: 4515: 4511: 4507: 4501: 4498: 4487: 4484: 4481: 4478: 4465: 4452: 4446: 4443: 4430: 4425: 4415: 4411: 4407: 4403: 4398: 4392: 4389: 4376: 4371: 4364: 4359: 4355: 4349: 4343: 4340: 4324: 4318: 4315: 4299: 4293: 4290: 4279: 4273: 4270: 4259: 4256: 4253: 4247: 4244: 4184: 4181: 4175: 4156: 4150: 4141: 4130: 4127: 4121: 4118: 4112: 4098: 4095: 4092: 4084: 4078: 4068: 4066: 4062: 4058: 4053: 4051: 4047: 4043: 4039: 4036: 4017: 4003: 4000: 3980: 3975: 3965: 3962: 3958: 3954: 3949: 3945: 3923: 3917: 3913: 3907: 3903: 3898: 3884: 3881: 3876: 3861: 3857: 3839: 3835: 3826: 3808: 3805: 3801: 3778: 3774: 3768: 3758: 3754: 3748: 3736: 3728: 3725: 3721: 3715: 3710: 3707: 3704: 3700: 3694: 3689: 3686: 3683: 3679: 3673: 3670: 3665: 3650: 3611: 3603: 3599: 3596: 3586: 3583: 3578: 3563: 3559: 3553: 3543: 3541: 3537: 3518: 3512: 3492: 3488: 3482: 3478: 3474: 3471: 3468: 3448: 3445: 3442: 3439: 3436: 3414: 3410: 3404: 3400: 3393: 3390: 3386: 3374: 3371: 3368: 3364: 3360: 3355: 3351: 3345: 3341: 3335: 3332: 3327: 3324: 3319: 3311: 3306: 3296: 3293: 3288: 3282: 3276: 3273: 3270: 3265: 3257: 3252: 3242: 3239: 3234: 3204: 3198: 3188: 3178: 3176: 3173:which yields 3157: 3154: 3138: 3130: 3124: 3121: 3110: 3107: 3104: 3101: 3078: 3075: 3058: 3049: 3046: 3043: 3039: 3034: 3028: 3025: 3014: 3011: 3008: 3005: 2995: 2989: 2976: 2967: 2964: 2950: 2941: 2932: 2929: 2906: 2903: 2900: 2896: 2891: 2885: 2882: 2868: 2862: 2859: 2848: 2845: 2842: 2836: 2833: 2815: 2802: 2800: 2781: 2778: 2775: 2755: 2752: 2749: 2745: 2739: 2722: 2719: 2716: 2696: 2693: 2690: 2687: 2671: 2669: 2641: 2636: 2627: 2604: 2601: 2584: 2581: 2570: 2567: 2562: 2551: 2548: 2525: 2522: 2519: 2515: 2510: 2507: 2501: 2498: 2475: 2467: 2459: 2446: 2442: 2433: 2428: 2399: 2393: 2385: 2379: 2353: 2349: 2344: 2331: 2327: 2320: 2299: 2295: 2286: 2281: 2252: 2246: 2238: 2234: 2227: 2221: 2216: 2212: 2208: 2202: 2199: 2187: 2181: 2178: 2158: 2150: 2134: 2127:of the field 2126: 2125:geodesic flow 2123:describe the 2122: 2112: 2089: 2085: 2081: 2065: 2042: 2036: 2031: 2027: 2023: 1994: 1990: 1986: 1983: 1980: 1977: 1972: 1968: 1964: 1954: 1944: 1938: 1932: 1924: 1908: 1885: 1879: 1856: 1853: 1830: 1827: 1817: 1802: 1786: 1766: 1759: 1758:metric tensor 1738: 1706: 1691: 1690:wedge product 1675: 1666: 1644: 1640: 1636: 1633: 1630: 1627: 1622: 1618: 1614: 1604: 1592: 1588: 1584: 1570: 1554: 1547: 1529: 1489: 1487: 1482: 1469: 1458: 1440: 1437: 1434: 1426: 1421: 1408: 1405: 1389: 1375: 1372: 1364: 1361: 1354: 1350: 1344: 1341: 1333: 1330: 1317: 1314: 1308: 1290: 1263: 1247: 1243: 1234: 1231: 1228: 1212: 1208: 1203: 1198: 1194: 1193:time integral 1184: 1168: 1153: 1149: 1131: 1102: 1098: 1094: 1091: 1088: 1083: 1079: 1070: 1069:scalar fields 1066: 1051: 1042: 1040: 1036: 1032: 1031:spinor fields 1028: 1027:tensor fields 1024: 1023:vector fields 1014: 1012: 1008: 1004: 1000: 995: 979: 976: 968: 965: 962: 959: 952: 946: 942: 935: 930: 926: 917: 912: 908: 904: 901: 898: 895: 888: 882: 878: 871: 866: 862: 853: 848: 844: 824: 808: 805: 797: 794: 787: 783: 777: 774: 766: 763: 733: 722:Scalar fields 719: 716: 714: 708: 704: 700: 696: 690: 686: 676: 674: 670: 666: 662: 658: 654: 650: 646: 642: 638: 634: 630: 626: 621: 619: 603: 598: 583: 555: 551: 547: 544: 540: 536: 532: 513: 504: 492: 479: 475: 470: 458: 449: 445: 438: 434: 426: 422: 410: 402: 398: 388: 384: 378: 370: 366: 361: 349: 346: 342: 337: 333: 329: 313: 311: 292: 284: 280: 271: 243: 227: 224: 221: 213: 209: 205: 193: 182: 178: 159: 156: 153: 150: 147: 144: 141: 135: 127: 123: 117: 113: 109: 105: 100: 90: 88: 84: 80: 76: 72: 71:fiber bundles 68: 64: 60: 56: 52: 48: 44: 39: 37: 33: 29: 25: 21: 10242: 10233: 10208: 10203:Jost, Jürgen 10197: 10178: 10172: 10147: 10109: 10101: 10052: 10033: 9873: 9801: 9746:Kinetic term 9651: 8649: 8333: 7816: 7812:frame fields 7801: 7788: 7725:Ricci tensor 7717:Ricci tensor 7534: 7470: 7462: 7251: 7232:Weyl spinors 7061: 6865: 6838:Weyl spinors 6707:Dirac spinor 6591: 6356:. The field 6122: 6101: 5970: 5875: 5862: 5859:fiber bundle 5846: 5844: 5834: 5795: 5791: 5732: 5727: 5723: 5602: 5598: 5565: 5559: 4796:four-vectors 4730: 4726:Ampère's law 4564: 4466: 4080: 4054: 4046:Killing form 3649:differential 3555: 3540:Higgs fields 3190: 2993: 2990: 2803: 2672: 2667: 2634: 2476: 2473: 2465: 2345: 2118: 2066: 1925:. That is, 1667: 1495: 1483: 1423:The spatial 1422: 1261: 1210: 1206: 1201: 1190: 1148:vector field 1043: 1020: 1003:fiber bundle 996: 825: 725: 717: 712: 706: 702: 698: 694: 688: 682: 625:fiber bundle 622: 553: 549: 542: 534: 530: 493: 314: 309: 241: 175:so that the 121: 115: 111: 107: 103: 96: 40: 19: 18: 8318:volume form 7772:tidal force 7477:types, and 7465:is the QCD 6594:Dirac field 6350:Higgs field 6295:with fiber 3558:sigma model 3552:sigma model 2709:, with the 1569:volume form 1492:Volume form 1199:denoted by 999:coordinates 679:Definitions 618:volume form 10256:Categories 9688:instantons 7723:, and the 5863:completely 5840:exact form 5738:Hodge star 4560:Gauss' law 3625:where the 1923:Hodge star 1668:Here, the 1486:functional 1207:Lagrangian 1152:isomorphic 1011:jet bundle 669:Lie groups 270:functional 240:where the 10045:Citations 9984:∂ 9979:φ 9976:∂ 9961:∂ 9956:φ 9953:∂ 9938:∂ 9933:φ 9930:∂ 9915:∂ 9910:φ 9907:∂ 9898:φ 9850:μ 9839:φ 9834:μ 9830:∂ 9823:φ 9632:Ω 9613:− 9590:− 9539:− 9527:− 9462:− 9396:μ 9365:μ 9338:ν 9324:μ 9320:− 9312:ν 9309:μ 9299:μ 9258:σ 9255:ρ 9236:σ 9233:ρ 9214:ν 9211:μ 9193:− 9179:λ 9176:ν 9157:λ 9150:μ 9127:μ 9102:π 9088:ν 9085:μ 9041:π 9032:− 8998:ν 8995:μ 8985:ν 8982:μ 8935:σ 8932:ρ 8913:σ 8910:ρ 8891:ν 8888:μ 8870:− 8856:λ 8853:ν 8834:μ 8829:λ 8803:μ 8763:ν 8760:μ 8752:δ 8748:δ 8728:− 8705:ν 8702:μ 8666:ν 8663:μ 8570:π 8537:σ 8534:ν 8515:ρ 8512:μ 8493:σ 8490:ρ 8471:ν 8468:μ 8451:μ 8438:− 8424:μ 8405:μ 8281:− 8202:ν 8199:μ 8183:ν 8180:μ 8172:δ 8136:δ 8127:− 8116:ν 8113:μ 8105:δ 8092:− 8053:δ 8041:− 8033:− 8027:≡ 8022:ν 8019:μ 7988:ν 7985:μ 7955:ν 7952:μ 7928:π 7916:Λ 7911:ν 7908:μ 7895:ν 7892:μ 7871:− 7866:ν 7863:μ 7833:ν 7830:μ 7793:geodesics 7675:Λ 7633:Λ 7627:− 7610:π 7502:ν 7499:μ 7490:α 7447:ν 7444:μ 7435:α 7425:ν 7422:μ 7413:α 7395:− 7386:ψ 7357:− 7334:ℏ 7314:¯ 7311:ψ 7298:∑ 7212:σ 7185:σ 7171:− 7166:σ 7162:∂ 7153:σ 7126:σ 7116:σ 7112:γ 7042:ν 7039:μ 7012:ν 7009:μ 6999:ν 6996:μ 6979:μ 6966:− 6963:ψ 6944:− 6921:ℏ 6909:¯ 6906:ψ 6822:σ 6818:∂ 6812:σ 6808:γ 6774:∂ 6743:γ 6737:† 6733:ψ 6723:¯ 6720:ψ 6693:ψ 6673:ψ 6654:− 6639:∂ 6632:ℏ 6620:¯ 6617:ψ 6553:⋆ 6532:⟩ 6529:ψ 6523:ψ 6517:⟨ 6514:⁡ 6508:− 6498:⋆ 6474:ψ 6456:ψ 6450:− 6447:σ 6426:ψ 6419:⋆ 6332:ψ 6275:ψ 6234:ψ 6228:− 6225:σ 6193:ψ 6153:ψ 6075:∧ 6067:∧ 6038:∧ 6009:∫ 5936:× 5916:× 5882:Lie group 5768:∗ 5755:∗ 5697:∗ 5694:∧ 5686:− 5678:∗ 5675:∧ 5643:∫ 5639:− 5525:ν 5522:μ 5503:ν 5500:μ 5483:μ 5470:− 5456:μ 5437:μ 5381:σ 5378:λ 5368:ν 5364:∂ 5358:σ 5355:λ 5352:ν 5349:μ 5345:ϵ 5332:ν 5318:μ 5314:− 5306:ν 5303:μ 5293:μ 5289:∂ 5262:σ 5259:ν 5255:η 5249:ρ 5246:μ 5242:η 5236:σ 5233:ρ 5223:ν 5220:μ 5203:μ 5190:− 5182:ν 5179:μ 5169:ν 5166:μ 5149:μ 5136:− 5109:μ 5096:− 5070:ϵ 5045:μ 5035:ν 5031:∂ 5027:− 5022:ν 5012:μ 5008:∂ 4999:ν 4996:μ 4969:ν 4966:μ 4936:μ 4926:μ 4910:⋅ 4899:ϕ 4896:ρ 4893:− 4862:ϕ 4859:− 4848:μ 4822:ρ 4811:μ 4777:⋅ 4749:ϕ 4746:ρ 4743:− 4690:× 4687:∇ 4676:μ 4667:− 4644:˙ 4627:ϵ 4587:, we get 4524:⋅ 4521:∇ 4512:ϵ 4488:ρ 4485:− 4471:, we get 4412:μ 4399:− 4356:ϵ 4325:⋅ 4280:ϕ 4260:ρ 4257:− 4157:⋅ 4142:˙ 4099:ϕ 4093:− 4035:Lie group 4004:∈ 3976:μ 3972:∂ 3963:− 3950:μ 3918:μ 3908:μ 3854:are just 3836:ϕ 3775:ϕ 3769:μ 3765:∂ 3755:ϕ 3749:μ 3745:∂ 3737:ϕ 3701:∑ 3680:∑ 3612:ϕ 3604:∗ 3600:∧ 3597:ϕ 3519:ϕ 3446:− 3411:ϕ 3380:∞ 3365:∑ 3361:− 3352:ϕ 3328:− 3325:ϕ 3320:μ 3316:∂ 3312:ϕ 3307:μ 3303:∂ 3283:ϕ 3274:− 3271:ϕ 3266:μ 3262:∂ 3258:ϕ 3253:μ 3249:∂ 3205:ϕ 3144:Φ 3135:∇ 3111:ρ 3105:π 3065:Φ 3062:∇ 3059:⋅ 3056:∇ 3047:π 3015:ρ 3012:− 2954:Φ 2951:δ 2948:∇ 2942:⋅ 2919:Φ 2916:∇ 2904:π 2892:− 2872:Φ 2869:δ 2849:ρ 2846:− 2816:δ 2785:Φ 2782:ρ 2753:π 2732:Φ 2729:∇ 2723:− 2694:− 2591:Φ 2571:ρ 2568:− 2538:Φ 2535:∇ 2523:π 2511:− 2434:φ 2429:μ 2425:∂ 2418:∂ 2406:∂ 2394:μ 2390:∂ 2380:φ 2377:∂ 2365:∂ 2321:φ 2318:∂ 2306:∂ 2287:φ 2282:μ 2278:∂ 2271:∂ 2259:∂ 2247:μ 2243:∂ 2239:− 2222:∗ 2213:∫ 2203:φ 2200:δ 2188:δ 2159:φ 2149:variation 2135:φ 2086:∫ 2037:∗ 2028:∫ 1984:∧ 1981:⋯ 1978:∧ 1933:∗ 1909:∗ 1880:∗ 1854:− 1676:∧ 1634:∧ 1631:⋯ 1628:∧ 1589:∫ 1441:∫ 1359:∂ 1351:φ 1348:∂ 1342:φ 1338:∇ 1331:φ 1318:∫ 1309:φ 1289:spacetime 1232:∫ 1099:φ 1092:… 1080:φ 966:… 957:∂ 943:φ 939:∂ 927:φ 922:∇ 909:φ 902:… 893:∂ 879:φ 875:∂ 863:φ 858:∇ 845:φ 792:∂ 784:φ 781:∂ 775:φ 771:∇ 764:φ 734:φ 653:non-rigid 629:geodesics 514:α 511:∀ 505:⋅ 450:α 427:α 419:∂ 399:φ 395:∂ 367:φ 350:∫ 334:φ 281:φ 210:φ 206:δ 194:δ 136:φ 99:spacetime 9699:See also 9684:solitons 9679:BF model 6846:vielbein 5832:because 4061:Skyrmion 2462:Examples 1260:and the 1067:-valued 1035:fermions 1007:sections 93:Overview 77:and the 9379:is the 8789:Maxwell 8610:Maxwell 8002:is the 7785:torsion 7727:is the 7711:is the 7687:is the 7517:is the 7226:is the 7064:is the 7056:is the 6761:is its 6566:is the 6289:section 4065:nucleon 3858:on the 3647:is the 3534:is the 2624:is the 2009:and so 1921:is the 1756:of the 1688:is the 1009:of the 616:is the 582:density 312:itself 268:, is a 81:to the 10221: 10217:–381. 10185: 10160: 10156:–390. 10122: 9871:. The 9352:where 8825: 8351:matter 8251:where 7789:per se 7667:where 7653:matter 7586:matter 7461:where 7354: 7199:where 7068:, and 7026:where 6941: 6765:, and 6685:where 6651: 6544:where 6324:. The 6267:where 5838:is an 5722:Here, 5560:Using 4731:Using 4033:, the 2620:where 1901:where 1197:action 1187:Action 1039:Bosons 1029:, and 584:, and 529:; and 508: 242:action 181:action 36:fields 10118:–38. 9793:Notes 7475:quark 7140:with 6705:is a 6291:of a 6287:is a 5869:over 5853:on a 5610:) as 4038:SU(N) 3937:with 3793:with 1001:on a 124:on a 10219:ISBN 10183:ISBN 10158:ISBN 10120:ISBN 9867:see 9677:The 7799:".) 7103:for 6799:for 6596:is: 6486:and 6380:and 5885:U(1) 5876:The 5855:U(1) 5845:The 3993:and 3823:the 3556:The 2808:is: 2119:The 1692:and 1191:The 1065:real 643:and 85:and 69:and 10215:373 10154:344 9686:or 9428:): 7099:is 6868:QED 6795:is 5794:= d 5608:= 1 5339:and 4838:and 1779:on 1154:to 675:.) 663:as 541:of 539:set 533:= { 10258:: 10134:^ 10116:25 10081:^ 10061:^ 8567:16 7752:EH 7691:, 7607:16 7569:EH 7552:GR 7469:, 7060:, 6709:, 6578:. 6511:Re 6109:. 5873:. 5842:. 5601:= 5596:, 4728:. 4562:. 3542:. 3177:. 2628:, 2354:: 1183:. 1071:, 1025:, 1013:. 705:, 701:, 697:, 639:, 244:, 114:, 110:, 106:, 89:. 10227:. 10191:. 10166:. 10128:. 10034:∇ 10018:) 10014:t 10011:, 10008:z 10005:, 10002:y 9999:, 9996:x 9993:, 9987:t 9970:, 9964:z 9947:, 9941:y 9924:, 9918:x 9901:, 9894:( 9888:L 9874:μ 9855:) 9846:x 9842:, 9826:, 9820:( 9815:L 9694:. 9636:2 9627:d 9621:2 9617:r 9608:2 9604:r 9599:d 9593:1 9585:) 9577:2 9573:r 9567:2 9563:Q 9557:+ 9552:r 9548:M 9545:2 9536:1 9532:( 9522:2 9518:t 9513:d 9508:) 9500:2 9496:r 9490:2 9486:Q 9480:+ 9475:r 9471:M 9468:2 9459:1 9455:( 9451:= 9446:2 9442:s 9437:d 9426:Q 9404:0 9401:= 9392:j 9361:D 9334:j 9328:0 9317:= 9305:F 9295:D 9273:) 9269:) 9266:x 9263:( 9251:F 9247:) 9244:x 9241:( 9229:F 9225:) 9222:x 9219:( 9207:g 9201:4 9198:1 9190:) 9187:x 9184:( 9172:F 9168:) 9165:x 9162:( 9146:F 9139:( 9131:0 9123:1 9114:4 9110:c 9105:G 9099:8 9093:= 9081:R 9060:T 9053:4 9049:c 9044:G 9038:8 9029:= 9026:R 9006:0 9003:= 8991:T 8978:g 8974:= 8971:T 8950:) 8946:) 8943:x 8940:( 8928:F 8924:) 8921:x 8918:( 8906:F 8902:) 8899:x 8896:( 8884:g 8878:4 8875:1 8867:) 8864:x 8861:( 8849:F 8845:) 8842:x 8839:( 8820:F 8815:( 8807:0 8799:1 8794:= 8783:S 8774:) 8771:x 8768:( 8756:g 8740:) 8737:x 8734:( 8731:g 8724:2 8719:= 8716:) 8713:x 8710:( 8698:T 8677:) 8674:x 8671:( 8659:g 8632:. 8621:L 8615:+ 8604:L 8598:= 8588:) 8585:x 8582:( 8579:R 8573:G 8561:4 8557:c 8551:+ 8548:) 8545:x 8542:( 8530:g 8526:) 8523:x 8520:( 8508:g 8504:) 8501:x 8498:( 8486:F 8482:) 8479:x 8476:( 8464:F 8455:0 8447:4 8443:1 8435:) 8432:x 8429:( 8420:A 8416:) 8413:x 8410:( 8401:j 8397:= 8390:) 8387:x 8384:( 8379:L 8345:L 8300:x 8295:4 8291:d 8284:g 8259:g 8239:. 8232:r 8229:e 8226:t 8223:t 8220:a 8217:m 8210:L 8195:g 8191:+ 8176:g 8164:r 8161:e 8158:t 8155:t 8152:a 8149:m 8142:L 8130:2 8124:= 8109:g 8100:) 8095:g 8084:r 8081:e 8078:t 8075:t 8072:a 8069:m 8062:L 8056:( 8044:g 8036:2 8015:T 7981:T 7961:. 7948:T 7940:4 7936:c 7931:G 7925:8 7919:= 7904:g 7900:+ 7888:g 7884:R 7879:2 7876:1 7859:R 7826:g 7746:L 7699:R 7647:L 7641:+ 7637:) 7630:2 7624:R 7620:( 7613:G 7601:4 7597:c 7591:= 7580:L 7574:+ 7563:L 7557:= 7546:L 7486:G 7471:n 7463:D 7431:G 7409:G 7403:4 7400:1 7390:n 7381:) 7375:2 7371:c 7365:n 7361:m 7350:/ 7341:D 7337:c 7331:i 7327:( 7321:n 7302:n 7294:= 7288:D 7285:C 7282:Q 7275:L 7208:A 7181:A 7177:e 7174:i 7158:= 7149:D 7122:D 7086:/ 7077:D 7062:D 7035:F 7005:F 6992:F 6983:0 6975:4 6971:1 6960:) 6955:2 6951:c 6947:m 6937:/ 6928:D 6924:c 6918:i 6915:( 6900:= 6894:D 6891:E 6888:Q 6881:L 6782:/ 6747:0 6729:= 6670:) 6665:2 6661:c 6657:m 6647:/ 6635:c 6629:i 6626:( 6611:= 6606:L 6526:, 6520:D 6505:= 6502:F 6494:D 6470:) 6464:2 6460:| 6453:| 6443:( 6437:2 6434:1 6429:= 6423:D 6415:D 6388:F 6364:A 6310:n 6305:C 6253:2 6248:) 6242:2 6238:| 6231:| 6221:( 6214:4 6211:1 6206:+ 6201:2 6197:| 6190:D 6187:| 6184:+ 6179:2 6175:| 6171:F 6168:| 6165:= 6162:) 6159:A 6156:, 6150:( 6145:L 6088:. 6084:) 6079:A 6071:A 6063:A 6057:3 6054:2 6049:+ 6045:A 6041:d 6034:A 6029:( 6024:r 6021:t 6014:M 6005:= 6002:] 5998:A 5994:[ 5989:S 5950:) 5947:1 5944:( 5940:U 5933:) 5930:2 5927:( 5923:U 5920:S 5913:) 5910:3 5907:( 5903:U 5900:S 5857:- 5847:A 5835:F 5820:0 5817:= 5813:F 5808:d 5796:A 5792:F 5777:. 5773:J 5764:= 5760:F 5750:d 5733:F 5728:J 5724:A 5710:. 5706:) 5701:J 5690:A 5682:F 5671:F 5664:2 5661:1 5655:( 5648:M 5636:= 5633:] 5629:A 5625:[ 5620:S 5606:0 5603:ε 5599:c 5578:M 5566:S 5536:) 5533:x 5530:( 5518:F 5514:) 5511:x 5508:( 5496:F 5487:0 5479:4 5475:1 5467:) 5464:x 5461:( 5452:A 5448:) 5445:x 5442:( 5433:j 5429:= 5426:) 5423:x 5420:( 5415:L 5389:0 5386:= 5374:F 5328:j 5322:0 5311:= 5299:F 5229:F 5216:F 5207:0 5199:4 5195:1 5187:= 5175:F 5162:F 5153:0 5145:4 5141:1 5133:= 5128:2 5123:B 5113:0 5105:2 5101:1 5091:2 5086:E 5079:2 5074:0 5041:A 5018:A 5004:= 4992:F 4962:F 4932:A 4922:j 4918:= 4914:A 4906:j 4902:+ 4873:) 4869:A 4865:, 4856:( 4853:= 4844:A 4833:) 4829:j 4825:, 4819:( 4816:= 4807:j 4781:A 4773:j 4769:+ 4766:) 4763:t 4760:, 4756:x 4752:( 4712:) 4709:t 4706:, 4702:x 4698:( 4694:B 4680:0 4672:1 4664:) 4661:t 4658:, 4654:x 4650:( 4640:E 4631:0 4623:+ 4620:) 4617:t 4614:, 4610:x 4606:( 4602:j 4598:= 4595:0 4574:A 4546:) 4543:t 4540:, 4536:x 4532:( 4528:E 4516:0 4508:+ 4505:) 4502:t 4499:, 4495:x 4491:( 4482:= 4479:0 4469:ϕ 4453:. 4450:) 4447:t 4444:, 4440:x 4436:( 4431:2 4426:B 4416:0 4408:2 4404:1 4396:) 4393:t 4390:, 4386:x 4382:( 4377:2 4372:E 4365:2 4360:0 4350:+ 4347:) 4344:t 4341:, 4337:x 4333:( 4329:A 4322:) 4319:t 4316:, 4312:x 4308:( 4304:j 4300:+ 4297:) 4294:t 4291:, 4287:x 4283:( 4277:) 4274:t 4271:, 4267:x 4263:( 4254:= 4251:) 4248:t 4245:, 4241:x 4237:( 4232:L 4209:j 4188:) 4185:t 4182:, 4179:) 4176:t 4173:( 4169:x 4165:( 4161:A 4154:) 4151:t 4148:( 4138:x 4131:q 4128:+ 4125:) 4122:t 4119:, 4116:) 4113:t 4110:( 4106:x 4102:( 4096:q 4021:) 4018:N 4015:( 4011:U 4008:S 4001:U 3981:U 3966:1 3959:U 3955:= 3946:L 3924:) 3914:L 3904:L 3899:( 3894:r 3891:t 3885:2 3882:1 3877:= 3872:L 3840:i 3809:j 3806:i 3802:g 3779:j 3759:i 3740:) 3734:( 3729:j 3726:i 3722:g 3716:n 3711:1 3708:= 3705:j 3695:n 3690:1 3687:= 3684:i 3674:2 3671:1 3666:= 3661:L 3634:d 3608:d 3593:d 3587:2 3584:1 3579:= 3574:L 3522:) 3516:( 3513:V 3493:2 3489:/ 3483:2 3479:v 3475:m 3472:= 3469:T 3449:V 3443:T 3440:= 3437:L 3415:n 3405:n 3401:g 3394:! 3391:n 3387:1 3375:3 3372:= 3369:n 3356:2 3346:2 3342:m 3336:2 3333:1 3297:2 3294:1 3289:= 3286:) 3280:( 3277:V 3243:2 3240:1 3235:= 3230:L 3208:) 3202:( 3199:V 3161:) 3158:t 3155:, 3151:x 3147:( 3139:2 3131:= 3128:) 3125:t 3122:, 3118:x 3114:( 3108:G 3102:4 3082:) 3079:t 3076:, 3072:x 3068:( 3050:G 3044:4 3040:1 3035:+ 3032:) 3029:t 3026:, 3022:x 3018:( 3009:= 3006:0 2996:Φ 2994:δ 2977:. 2974:) 2971:) 2968:t 2965:, 2961:x 2957:( 2945:( 2939:) 2936:) 2933:t 2930:, 2926:x 2922:( 2913:( 2907:G 2901:8 2897:2 2889:) 2886:t 2883:, 2879:x 2875:( 2866:) 2863:t 2860:, 2856:x 2852:( 2843:= 2840:) 2837:t 2834:, 2830:x 2826:( 2821:L 2806:Φ 2779:= 2776:V 2756:G 2750:8 2746:/ 2740:2 2736:) 2726:( 2720:= 2717:T 2697:V 2691:T 2688:= 2683:L 2668:ρ 2652:L 2635:G 2630:ρ 2622:Φ 2608:) 2605:t 2602:, 2598:x 2594:( 2588:) 2585:t 2582:, 2578:x 2574:( 2563:2 2559:) 2555:) 2552:t 2549:, 2545:x 2541:( 2532:( 2526:G 2520:8 2516:1 2508:= 2505:) 2502:t 2499:, 2495:x 2491:( 2486:L 2447:. 2443:) 2437:) 2421:( 2411:L 2400:( 2386:= 2370:L 2332:. 2328:) 2311:L 2300:+ 2296:) 2290:) 2274:( 2264:L 2253:( 2235:( 2231:) 2228:1 2225:( 2217:M 2209:= 2193:S 2182:= 2179:0 2097:L 2090:M 2082:= 2077:S 2051:L 2046:) 2043:1 2040:( 2032:M 2024:= 2019:S 1995:m 1991:x 1987:d 1973:1 1969:x 1965:d 1959:| 1955:g 1951:| 1945:= 1942:) 1939:1 1936:( 1889:) 1886:1 1883:( 1857:g 1831:1 1828:= 1822:| 1818:g 1814:| 1787:M 1767:g 1743:| 1739:g 1735:| 1711:| 1707:g 1703:| 1652:L 1645:m 1641:x 1637:d 1623:1 1619:x 1615:d 1609:| 1605:g 1601:| 1593:M 1585:= 1580:S 1555:M 1530:g 1506:L 1470:. 1465:x 1459:3 1454:d 1446:L 1438:= 1435:L 1409:. 1406:t 1402:d 1396:x 1390:3 1385:d 1379:) 1376:t 1373:, 1369:x 1365:, 1362:t 1355:/ 1345:, 1334:, 1328:( 1323:L 1315:= 1312:] 1306:[ 1301:S 1273:L 1248:, 1244:t 1240:d 1235:L 1229:= 1224:S 1211:L 1202:S 1169:n 1164:R 1132:m 1127:R 1103:m 1095:, 1089:, 1084:1 1052:m 983:) 980:t 977:, 973:x 969:, 963:, 960:t 953:/ 947:n 936:, 931:n 918:, 913:n 905:, 899:, 896:t 889:/ 883:1 872:, 867:1 854:, 849:1 841:( 836:L 812:) 809:t 806:, 802:x 798:, 795:t 788:/ 778:, 767:, 761:( 756:L 713:s 709:) 707:t 703:z 699:y 695:x 693:( 689:t 667:( 604:s 599:n 594:d 566:L 554:n 550:α 543:n 535:s 531:s 517:} 502:{ 480:, 476:s 471:n 466:d 459:) 455:} 446:s 442:{ 439:, 435:} 423:s 414:) 411:s 408:( 403:i 389:{ 385:, 382:) 379:s 376:( 371:i 362:( 356:L 347:= 343:] 338:i 330:[ 324:S 310:s 296:) 293:s 290:( 285:i 254:S 228:, 225:0 222:= 214:i 199:S 163:) 160:t 157:, 154:z 151:, 148:y 145:, 142:x 139:( 122:s 118:) 116:t 112:z 108:y 104:x 102:(

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Knowledge

Lagrangian (field theory)

Index