Approaches to Quantum Field Theory  

Canonical Quantization > s.a. canonical quantum mechanics [including group quantization]; geometric quantization.
* Idea: Choose a Hilbert space of states \(\cal H\) which carries a representation of an algebra of observables; The fields are operator-valued distributions, that have to be smeared in order to give well-defined predictions.
* Schrödinger picture: States are functionals on the quantum configuration space, which is not the classical one of C2 functions on a spacelike hypersurface but, e.g., for scalar fields, the space of tempered distributions.
* Caveat: Evolution cannot be unitarily implemented between arbitrary curved Cauchy surfaces!
* Operators: In order to be well-defined, the usual field operators must be smeared with 3D distributions.
@ Schrödinger picture: Stoyanovsky in(04)mp/02 [Schrödinger equation as variational differential equation]; Corichi et al CQG(03)gq/02 [curved spacetime ambiguities]; Solomon qp/03 [vs Heisenberg]; Stoyanovsky a1008; > s.a. representations of quantum mechanics.
@ Arbitrary spacelike slices: Torre & Varadarajan PRD(98)ht/97, CQG(99)ht/98; Helfer ht/99, ht/99.
@ Covariant, based on de Donder-Weyl: Nikolić EPJC(05)ht/04; von Hippel & Wohlfarth EPJC(06)ht/05.
@ Choice of representation: Ashtekar & Isham PLB(92) [inequivalent algebras]; Florig & Summers PLMS(00); Czachor & Wilczewski IJTP(07)qp/05 [and experiment].
@ Precanonical quantization: Kanatchikov ATMP-a1112, ATMP-a1312 [and the Schrödinger wave functional]; > s.a. approaches to quantum gravity.
@ General-boundary formulation: Oeckl PRD(06)ht/05; Oeckl Sigma(12)-a1009 [holomorphic quantization, linear field theory]; Colosi & Dohse JGP(17)-a1011 [S-matrix, curved spacetimes]; Oeckl JGP(12)-a1104 [affine holomorphic quantization]; Oeckl ATMP(15)-a1201 [Schrödinger-Feynman quantization], AIP(12)-a1210 [reverse-engineering quantum field theory]; Banisch et al CQG(13)-a1310 [Unruh-DeWitt detector and vacuum]; > s.a. klein-gordon fields in ads spacetime.
@ Variations, generalizations: Rayner CQG(90) [loop-based, gravity and scalar field]; Solovyov et al ht/04 [loop-based]; Manrique et al CQG(06) [loop quantization and 2D Ising field theory]; Adorno & Klauder a1403 [enhanced quantization, for bosons, fermions, and anyons].
@ Related topics: Banai JMP(87); Danos ht/97 [particles and systems].
> Specific theories: see canonical quantum gravity; klein-gordon fields; types of quantum fields [including polymer variables].
> Related topics: see fock space; green functions; symplectic structures in physics, and types.

Covariant Quantization > s.a. klein-gordon fields; quantum gravity.
* Steps: Define pure frequency solutions, 1-particle Hilbert space, creation and annihilation operators, full Hilbert space with Fock structure.
* Bosons: The commutation relations between annihilation and creation operators are [a(f), a*(g)] = (f | g), [a(f), a(g)] = 0; The a(f)s are unbounded (this can be seen from the commutation relations), and one has to generate the algebra using exp[i (α a + α*a*)].
* Fermions: The annihilation and creation operators satisfy the anticommutation relations {a(f), a*(g)} = (f | g), {a(f), a(g)} = 0; The a(f)s are bounded.

Other Approaches > s.a. algebraic and axiomatic; formalism and techniques; perturbative approach.
@ Relationships: Teleki & Noga ht/06 [operator and path integral].
@ Phase space approach: Zachos & Curtright PTPS(99)ht-conf.
@ Without infinities / renormalization: Takook IJMPE(05)gq/00 [λφ4]; Stefanovich AP(01); Biswas ht/05 [composite particles]; Wang ChPC(11)-a1006 [extended particles]; Klauder MPLA(12)-a1112 [curing the mutually-singular-measures problem]; Czachor a1209; Teufel & Tumulka a1505 [Hamiltonians with new boundary conditions]; Galvan a1607 [interaction Laplacian method].
@ Classical non-local model: Morgan qp/01 [for Klein-Gordon], qp/01 [for electromagnetism].
@ Other approaches: Czachor qp/99 [single oscillator]; Brodsky FdP(02)ht/01 [Heisenberg matrix formulation]; Gurău et al (not JNCG)-a0807 [based on marked trees]; Stoyanovsky in(07)-a0901; Floerchinger a1004 [as a bilocal statistical field theory]; Johnson a1203; Anselmi EPJC(13)-a1205 [general field-covariant approach, and renormalization]; Sulis a1502 [process algebra approach]; Tilloy a1702 [as a Lorentz invariant statistical field theory]; ; Sorkin a1703 [based on the Wightman function]; > s.a. Fermionic Projector.
@ World-line approach: Rylov ht/01; Gies & Hämmerling PRD(05) [spin-gauge field coupling, QED].
@ PT-symmetric scalar field theory: Shalaby & Al-Thoyaib PRD(10)-a0901 [and the cosmological constant]; Bender et PRD(12)-a1201.
@ Generalizations: Yurtsever CQG(94)gq/93 [quantum field theory on a topology, and small-scale structure]; > s.a. generalized quantum field theories.
> Related topics: see Hopf Algebra; path integrals; quantum fields in curved spacetime; stochastic quantization.

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