Types of Quantum Field Theories

In General > s.a. [quantum field theory]; boundaries in field theory; conformal invariance; quantum gauge theories.
* Free vs interacting: A field is free if the representation describing a many-particle system is the tensor product of the corresponding single-particle representations.
@ General references: Lev qp/98; Ratsimbarison a0706 [construction of probabilistic theories]; > s.a. types of field theories.
@ Exactly solvable: Ushveridze MPLA(98) [quasi-exactly solvable]; Brodsky et al AP(02) [and Pauli-Villars fields].
@ Massless: Lev TMP(04)ht/02 [massless particles]; Aste LMP(07)ht [self-coupling and mass resummation].
@ Scalar: Frasca IJMPA(07)ht/06 [triviality of ⁴]; Loran PLB(07)ht/06 [finite-temperature ⁴ on a circle]; > s.a. klein-gordon quantum fields.
@ Spin, fermion fields: Nolland & Mansfield IJMPA(00) [fermions, Schrödinger representation]; Iliev ht/04 [spin-1/2, momentum picture]; Forte ht/05 [spin-statistics, path integrals, etc]; Qiu et al IJGMP(06)ht [spin-3/2]; > s.a. ising model.
@ Coupled to atoms: Hu & Raval qp/97; Retamal et al PLA(06) [entanglement]; > s.a. Dicke Model, Friedrichs Model.
@ Diffeo-invariant, or background-independent: Fredenhagen & Haag CMP(87); Kuchar in(88); Horowitz CMP(89) [exactly soluble]; Husain PRD(93)gq [scalar, and loop-based observables; Rovelli NPB(93), JMP(95)gq [and model for quantum geometry]; Thiemann gq/93, CQG(95)gq/99; Salehi IJTP(97) [dynamics formalism]; Baez & Krasnov JMP(98) [with fermions]; Conrady et al PRD(04) [vacuum]; Fredenhagen ht/04-in; Varadarajan PRD(04)gq [scalar, path integral]; Dreyer ht/04; Balachandran et al ht/06 [on Groenewold-Moyal plane]; Sahlmann CQG(07)gq/06 [scalar, diffeo-invariant Hilbert space]; Campiglia et al PRD(06)gq [uniform discretizations]; > s.a. approaches, quantum gauge theories, parametrized, quantum gravity.
@ Theories of connections: Ashtekar et al JMP(95)gq; Bojowald & Kastrup CQG(00)ht/99 [symmetry reduction]; Lewandowski et al CMP(06) [uniqueness of representations]; Okolow gq/06 [non-compact G].
@ Polymer variables: Kaminski et al CQG(06)gq/05, CQG(06)gq [scalar]; > s.a. 2D quantum gravity; fock space; frw quantum cosmology; klein-gordon fields; representations of quantum mechanics.
@ Theories with S unbounded below: Greensite & Halpern NPB(84) [Euclidean quantum theory]; Zavialov et al TMP(96).
@ UV-finite: Lemes et al JPA(01) [criterion].
@ Ultralocal: Klauder CMP(70), APA(71) [quantization]; Klauder JPA(01)qp/00 [and reparametrization-invariant quantum field theory].
@ 0+1: Boozer EJP(07) [as toy model].
@ 1+1: Derezinski & Meissner mp/04 [massless]; Schroer ht/05-in [rev], AP(06)ht/05 [as testing ground]; Dorey et al ed-JPA(06) [low-dim].
@ Other specific types: Segal JMP(60), JMP(64) [non-linear]; Grigore JMP(95) [free]; Chalmers JHEP(98)ht/97 [non-polynomial]; Ho et al PRE(98)qp [scalar, open system]; Maiani & Testa AP(98) [unstable]; Helfer ht/99, ht/99 [bosonic]; Brouder ht/03 [degenerate systems]; Harrivel mp/06 [scalar, perturbative expansion]; Leclerc gq/06 [spin-2, Faddeev-Jackiw quantization].
> Specific types: see composite systems; dirac quantum field theory; effective actions; QED; QCD; supersymmetric field theory; topological theories.

Modifications and Generalizations > s.a. canonical quantum gravity; fock space; poincaré group; quantum fields in curved spacetime.
* Motivation, limits of validity: A natural UV cutoff in the validity of quantum field theory is expected from quantum gravity or string theory, and would help solve divergence problems.
* Galilei-invariant: The quantum version of a field theory which is not relativistically invariant, but only invariant under the Galilei transformations; In it, there is no particle creation and annihilation.
* Higher-derivative theories: They are often assumed to have ghosts, but in reality it is the fourth+second-order theory with a mass parameter m that has ghosts, while the pure fourth-order one is a singular limit and doesn't; This arises in the linearization of conformal gravity.
* Non-local: Several, differently motivated attempts at non-local (not generated by pointlike fields) relativistic particle theories have been made, the most recent one being quantum field theory on non-commutative spacetime.
@ Limits to quantum field theory: Cohen et al PRL(99)ht/98 [entropy bounds and large V's]; Carmona & Cortés PRD(02)ht/00 [100 TeV cutoff, and quantum gravity]; > s.a. quantum gravity phenomenology.
@ Quaternionic: Adler CMP(86); Brumby & Joshi FP(96)ht [consequences].
@ Non-Fock Hilbert spaces: Tsirelson ht/99 [fermions].
@ Finite-temperature: Ccapa Ttira et al PRD(08)-a0803 [dual path-integral representations].
@ Generalized background: Kaiser AP(87) [complex spacetime]; Eyink CMP(89), CMP(89) [fractal spacetime]; Birmingham & Rakowski MPLA(94) [simplicial complex, intersection form action].
@ With fundamental length scale: Brüning & Nagamachi JMP(04) [ito ultra-hyperfunctions]; Hossenfelder CQG(08)-a0712.
@ Discrete: Kur'yan in(91) [discrete spacetime]; Norton & Jaroszkiewicz JPA(98) [discrete t]; Häußling AP(02) [and non-commutative geometry]; > s.a. on graphs.
@ Deformed: Gadiyar ht/96; Hurth & Skenderis NPB(99)ht/98, LNP(00)ht/98 [with symmetries]; García-Compeán et al IJMPA(01)ht/99 [scalar and abelian gauge theory], JPA(02)ht/01 [second quantization of Schrödinger equation]; Kosinski et al ht/00-in, ht/00-in; Iorio et al AP(01)ht [deformation and curved spacetime]; Bezerra et al PRD(02), PRD(02) [q-deformed, perturbative]; Dito m.QA/02-in [covariant field theory]; Sardanashvily ht/02 [polysymplectic]; Hirshfeld & Henselder AP(02)ht [star products]; Matsuo & Shibusa MPLA(06)ht/05 [based on gup]; > s.a. non-commutative field theory [including braided].
@ Higher-derivative theories: Weldon AP(03); Nguyen a0709 [self-interacting scalar field].
@ Non-local: Cornish IJMPA(92); Breckenridge et al CQG(95)ht [in quantised spacetime]; Barci et al IJMPA(96)ht/95; Amorim & Barcelos-Neto JMP(99) [non-local massive s = 1]; Piacitelli JHEP(04) [diagram rules]; Schroer AP(05)ht/04 [rev]; Wang JMP(08); > s.a. causality, quantum systems, types of field theories.
@ Other types: Anco & Wald PRD(89) [algebra-valued fields]; Haag CMP(93) [characterizing models]; Ribaric & Sustersic ht/97 [transport-theoretic], FizB(02)ht/01 [finite alternative theory]; Yang ht/98-in, ht/98 [as effective theory from finite one]; Lev ht/02 [and spin-statistics], ht/02 [supersymmetry], ht/04 [over Galois field]; Dürr et al JPA(05)qp/04 [Bell-type Markov processes/trajectories]; Balakov et al CMP(07) [bilocal, scalar]; > s.a. analysis [fractional derivatives], perfect fluids, non-standard analysis, states [including non-equilibrium].

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