Classical Field Theory

In General > s.a. partial differential equations / states in quantum field theory [semiclassical]; types of field theories.
* Idea, motivation: An approach to interactions that assigns "substance" to them (for example, the electric and magnetic fields); In the usual formulations, fields incorporate relativistic invariance; They transfer perturbations at finite speed, without the need for an action at a distance.
* Remark: The only classical field theories are electromagnetism and gravitation, there can be no classical theory of a half-integer spin field and even the other field theories are usually constructed only as a step towards their quantization and obtaining information about particles; One can consider hydrodynamics as a classical field theory, but it does not have any unconstrained action principle (> see fluids).
* As collection of oscillators: For a smooth field, we can decompose it into Fourier components, and we get a precise statement of the equivalence of a field theory in a finite box and a mechanical system with infinitely-many degrees of freedom.
* Mathematical description: A massive field of spin s (= 0, 1/2, 1, 3/2, 2, ...) is described by a tensor field with 2s (spinor) internal indices, symmetric under any index permutation; if τa = Pauli matrices and p:= pa τa (positive definite),

(φ,ψ):= dΩ φ*A... B(p) (p/m)AM ... (p/m)BN ψM... N(p) .

Concepts and Techniques > s.a. hamiltonian, jacobi, and lagrangian dynamics; interactions; waves.
* Classical dynamics: It can be treated in Lagrangian or Hamiltonian form.
* Current: A quadratic or higher-order combination of fields appearing in the Lagrangian or Hamiltonian of a field theory.
@ General references: Visscher 88 [and computers]; Klishevich TMP(98)ht/97 [field redefinitions and degrees of freedom]; García Pérez et al NPB(99) [smoothing procedure]; BenDaniel mp/99-talk [smoothness]; Pons AIP(10)-a0909 [field redefinitions]; Brunetti et al CMP(19)-a1209 [algebraic structure, functorial approach]; Bekenstein & Majhi NPB(15)-a1411 [field equations from the action without variation].
@ Current algebra: Adler & Dashen 68; Treiman et al 72; de Alfaro et al 73; Cardenas et al MPLA(98)ht [path-integral approach].
@ Currents, other: Vyas et al IJMPA(19)-a1411 [topological charges and symmetry transformations].
@ Discretizations: Di Bartolo et al JMP(05)gq/04 [consistent & mimetic], JMP(05)gq/04 [constrained]; > s.a. computational physics; lattice field theory; numerical general relativity; types of field theories.
@ Path-integral approach: in Gozzi et al IJMPA(05); Mauro ht/05-proc [anomaly cancellation].
@ Without self-energy: de Haan AP(04) [Lee model], AP(04)qp [scalar + atom], AP(06) [electromagnetism].
@ Formulations: Śniatycki RPMP(84) [covariant Cauchy]; Vasiliev IJGMP(06)ht/05 [unfolded dynamics, Yang-Mills and general relativity examples]; Öttinger a1902 [based on the energy-momentum tensor].
@ Non-equilibrium: Blagoev et al PRD(01) [Schwinger-Dyson approach].
> Related concepts: see Configuration Space; Coupling Constant; energy-momentum tensor; multipoles; symmetries; random processes [random fields].
> Techniques: see Newton's Laws; path integrals; renormalization; symplectic structures in physics; topology in physics.
> Online resources: see E Tonti's Algebraic Formulation of Physical Fields site.

Features, Effects > s.a. boundaries; chaotic systems; KAM Theory; mass; quantum field theory effects; scattering; velocity.
* Conformal invariance: It is possible (some say desirable) for massless free fields, and for some interacting ones (electromagnetism, φ4).
* Linearity: The only classical experimental evidence we have so far for non-linearity is in gravitational theory, but verious other models have been proposed (> see modified electromagnetism, sigma-models), and quantum effects do lead to non-linearity.
@ Linearity: Deser GRG(70)gq/04 [need for non-linearity]; Audretsch & Lämmerzahl JMP(91) [reason; Ø].
@ Conservation laws: Anderson & Torre PRL(96)ht; Anco JPA(03)mp [scale invariant].
@ Causality: Lusanna AIP(04)ht [anticipatory aspects].
@ Non-relativistic physics: Deser AJP(05)aug-gq/04 [potentials]; Holland & Brown SHPMP(03) [non-relativistic limit of electromagnetism and Dirac].
@ Particles, localized configurations: Sen 68; Buniy & Kephart PRD(03) [conditions for existence of lumps]; > s.a. geon; instanton; Meron; monopole; non-linear electromagnetism; particle models; particle types; Phonon; Quasiparticles; Skyrmion; soliton; Sphaleron.
@ Related topics: Mashhoon AdP(03)ht [non-local: accelerated frames]; Holdom JPA(06) [quantumlike behavior]; Benioff QIP(16)-a1508 [spacetime-dependent number scaling and effect on physical and geometric quantities]; > s.a. diffraction; duality; modified lorentz symmetry; particle models; Self-Organization.

References > s.a. causality; Continuous Media; electromagnetic theory; electromagnetism in curved spacetime.
@ General: Weisskopf yr(58); Hagedorn 64 [and dispersion relations]; Schwinger 70, 73; Landau & Lifshitz v2; Balian & Zinn-Justin ed-76; Das 93 [with special relativity]; Low 97 [III; electromagnetism and gravity]; Thirring 97; Siegel ht/99-text; Burgess 02; Giachetta et al 09 [mathematical, geometrical]; Popławski a0911-ln [and spacetime]; in Franklin 10 [IIb, including spin-2 fields]; Scheck 12 [electromagnetism, gauge theory and gravity]; Setlur 13 [and quantum fields]; Cortés & Haupt book(17)-a1612 [lecture notes, mathematical]; Franklin 17; Susskind & Friedman 17; Năstase 19.
@ Statistical: Amit & Martín-Mayor 05; Brézin 10.
@ Effective: Jaffe & Mende NPB(92); Barceló et al CQG(01)gq, IJMPD(01)gq [from linearization in a background].
@ Geometric: Binz, Fischer & Śniatycki 88; de León et al mp/02 [including singular field theory], mp/02 [rev]; Sánchez a0803; Sardanashvily IJGMP(08)-a0811 [in terms of fibre bundles, graded manifolds, jet manifolds]; de León et al 15; Aldrovandi & Pereira 16.
@ Conceptual: Tian 96; BenDaniel CSF(99)phy/98; Madore et al EPJC(01)ht/00 [geometry vs field]; Harpaz EJP(02) [nature of fields]; Brody & Hughston TN(00)-a0910 [classical fields as statistical states]; Samaroo PhSc(11) [background structures]; > s.a. Hilbert's Program [6th problem]; philosophy of physics; Trajectory [constructing field evolution].
@ Other references: Moon & Spencer 88 [handbook]; Graneau & Graneau 93 [cranks against field theory]; Dmitriyev NCA(98)phy/99, Ap(00)phy/99, phy/99 [mechanical models].