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 C^{2} 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. quantization of gauge theories;
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];
Collins a1801 [light-front quantization, vacuum].

@ __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 [*λφ*^{4}];
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 IJGMP(17)-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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