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__: 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 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 & Panigrahi 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].

@ __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 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.

@ __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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send feedback and suggestions to bombelli at olemiss.edu – modified 17
oct 2017