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 fluid).
* 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 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 a = Pauli matrices and p:= paa (positive definite),

(,):= d *A...B(p) (p/m)AM ... (p/m)BN M...N(p) .

Concepts and Techniques > s.a. hamiltonian, jacobi, and lagrangian dynamics; path integrals; topology; waves.
* Classical dynamics: 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: Adler 81 [random fields]; 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-in [smoothness].
@ Current algebra: Adler & Dashen 68; Treiman et al 72; de Alfaro et al 73; Cardenas et al MPLA(98)ht [path integral approach].
@ 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.
@ Path integral approach: in Gozzi et al IJMPA(05); Mauro ht/05-in [anomaly cancellation].
@ Without self-energy: de Haan AP(04) [Lee model], AP(04)qp [scalar + atom], AP(06) [electromagnetism].
@ Formulations: Sniatycki RPMP(84) [covariant Cauchy]; Vasiliev IJGMP(06)ht/05 [unfolded dynamics, Yang-Mills and general relativity examples].
@ Non-equilibrium: Blagoev et al PRD(01) [Schwinger-Dyson approach].
> Related concepts: see Configuration Space; Coupling Constant; energy-momentum; interactions; multipoles; renormalization; symmetries.

Features, Effects > s.a. boundaries; chaotic systems; KAM Theory; mass; quantum field theory effects; scattering; velocity.
* Conformal invariance: Possible (some say desirable) for massless free fields, and some interacting ones (em, 4).
* Linearity: There is no classical experimental evidence yet for non-linearity [@ Jackson], but 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 ht/04-in [anticipatory aspects].
@ Non-relativistic physics: Deser AJP(05)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; particle types; Phonon; Quasiparticles; Skyrmion; soliton; Sphaleron.
@ Related topics: Mashhoon AdP(03)ht [non-local: accelerated frames]; Holdom JPA(06) [quantumlike behavior]; > s.a. diffraction; duality; modified lorentz symmetry; particle models; Self-Organization.

General References > s.a. causality; electromagnetism, electromagnetism in curved spacetime.
@ Classical: Weisskopf yr(58); Hagedorn 64 [and dispersion relations]; Schwinger 70, 73; Landau & Lifshitz 75; Balian & Zinn-Justin ed-76; Das 93 [with special relativity]; Low 97 [III; electromagnetism and gravity]; Thirring 97; Siegel ht/99-text; Burgess 02.
@ Statistical: Amit & Martín-Mayor 05.
@ Effective: Jaffe & Mende NPB(92); Barceló et al CQG(01)gq, IJMPD(01)gq [from linearization in a background].
@ Geometric: Binz, Fischer & Sniatycki 86; Aldrovandi & Pereira 95; de León et al mp/02 [including singular field theory], mp/02 [rev]; Sánchez a0803.
@ Conceptual: Tian 96; BenDaniel phy/98/CSF; Madore et al EPJC(01)ht/00 [geometry vs field]; Harpaz EJP(02) [nature of fields]; > s.a. philosophy of physics.
@ 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].

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