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 2*s* (spinor) internal indices, symmetric under any
index permutation; if *τ*^{a}
= Pauli matrices and **p**:= *p*_{a}
*τ*^{a} (positive definite),

(*φ*,*ψ*):= ∫
dΩ *φ**^{A... B}(*p*)
(**p**/*m*)_{A}^{M} ...
(**p**/*m*)_{B}^{N}
*ψ*_{M... 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, *φ*^{4}).

* __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].

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