Lagrangian (field theory)

Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom.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 partial differential equations.It enables various theorems to be provided, ranging from proofs of existence to the uniform convergence of formal series to the general settings of potential theory.In addition, insight and clarity is obtained by generalizations to Riemannian manifolds and fiber bundles, allowing the geometric structure to be clearly discerned and disentangled from the corresponding equations of motion.In field theory, the independent variable is replaced by an event in spacetime (x, y, z, t), or more generally still by a point s on a Riemannian manifold.Abraham and Marsden's textbook[1] provided the first comprehensive description of classical mechanics in terms of modern geometrical ideas, i.e., in terms of tangent manifolds, symplectic manifolds and contact geometry.Bleecker's textbook[2] provided a comprehensive presentation of field theories in physics in terms of gauge invariant fiber bundles.Jost[3] continues with a geometric presentation, clarifying the relation between Hamiltonian and Lagrangian forms, describing spin manifolds from first principles, etc.In field theory, the independent variable t is replaced by an event in spacetime (x, y, z, t) or still more generally by a point s on a manifold.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 notationDo not be misled: the volume form is implicitly present in the integral above, even if it is not explicitly written.A large variety of physical systems have been formulated in terms of Lagrangians over fields.The sigma model describes the motion of a scalar point particle constrained to move on a Riemannian manifold, such as a circle or a sphere.The most famous and well-studied of these is the Skyrmion, which serves as a model of the nucleon that has withstood the test of time.So the Lagrange density for electromagnetism in special relativity written in terms of Lorentz vectors and tensors isBy the equivalence principle, it becomes simple to extend the notion of electromagnetism to curved spacetime.[5][6] Using differential forms, the electromagnetic action S in vacuum on a (pseudo-) Riemannian manifoldThis 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.That is, classical electrodynamics, all of its effects and equations, can be completely understood in terms of a circle bundle over Minkowski spacetime.The Weyl spinors provide a more general foundation; they can be constructed directly from the Clifford algebra of spacetime; the construction works in any number of dimensions,[3] and the Dirac spinors appear as a special case.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.The gravitational field itself was historically ascribed to the metric tensor; the modern view is that the connection is "more fundamental".(This last statement must be qualified: there is no "force field" per se; moving bodies follow geodesics on the manifold described by the connection.The Lagrangian for general relativity can also be written in a form that makes it manifestly similar to the Yang–Mills equations.This is done by noting that most of differential geometry works "just fine" on bundles with an affine connection and arbitrary Lie group.So the tracelessness of the energy momentum tensor implies that the curvature scalar in an electromagnetic field vanishes.Solving both Einstein and Maxwell's equations around a spherically symmetric mass distribution in free space leads to the Reissner–Nordström charged black hole, with the defining line element (written in natural units and with charge Q):[5][2] 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.
classical field theoryLagrangian mechanicsdegrees of freedomfieldsquantum field theorypartial differential equationsSobolev spacesuniform convergencepotential theoryRiemannian manifoldsfiber bundlesChern–Gauss–Bonnet theoremRiemann–Roch theoremAtiyah–Singer index theoremChern–Simons theoryspacetimeRiemannian manifoldequations of motionactionfunctionalindependent variablesdensityvolume formfiber bundlegeodesicsclassical mechanicstangent manifoldssymplectic manifoldscontact geometryspin manifoldsnon-rigidtensor algebrasquantum groupsaffine Lie algebrasLie groupsVirasoro algebrageneralized coordinatescoordinatessectionsjet bundlevector fieldstensor fieldsspinor fieldsfermionsBosonsscalar fieldsvector fieldisomorphictime integralvolume integralwedge productmetric tensorMinkowski spacetimeHodge starEuler–Lagrange equationsgeodesic flowvariationboundary conditionsgravitational potentialgravitational constantNordström's theory of gravitationGauss's law for gravityScalar field theoryMexican hat potentialHiggs fieldssigma modeldifferentialRiemannian metriclocal coordinatescoordinate chartLie groupsymmetric spaceKilling formMaurer–Cartan formtopological solitonSkyrmionnucleonCovariant formulation of classical electromagnetismelectromagnetic fieldGauss' lawAmpère's lawtensor notationfour-vectorselectromagnetic tensorMinkowski metricLevi-Civita tensorequivalence principledifferential formsnatural unitsexact formaffine connectioncircle bundleYang–Mills equationsStandard modelChern–Simons functionalgrand unified theoryGinzburg–Landau theoryYang–Mills actionsectionvector bundlesuperconductorHiggs field"Sombrero hat" potentialYang–Mills fieldHodge star operatorYang–Mills–Higgs equationsSeiberg–Witten theoryDirac equationDirac fieldDirac spinorDirac adjointFeynman slash notationWeyl spinorsClifford algebravielbeinspin structureQuantum electrodynamicsgauge covariant derivativeFeynman notationelectromagnetic four-potentialquantum chromodynamicsDirac spinorsgluon field strength tensorEinstein–Hilbert actioncosmological constantcurvature scalarRicci tensorRiemann tensorKronecker deltatidal forceChristoffel symbolsmetric connectiontorsionstraight lineframe fieldsEinstein field equationsenergy momentum tensorJacobian determinantMaxwell's equations in curved spacetimecovariant derivativeReissner–Nordström charged black holeKaluza–Klein theoryfactorizationstheory of everythingBF modelsolitonsinstantonstopological field theoriesCalculus of variationsCovariant classical field theoryEuler–Lagrange equationFunctional derivativeFunctional integralHamiltonian mechanicsHamiltonian field theoryKinetic termLagrangian and Eulerian coordinatesLagrangian pointLagrangian systemNoether's theoremOnsager–Machlup functionPrinciple of least actionfour-gradientJost, Jürgen