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. 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
[*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 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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2017