Approaches to Quantum Field Theory| 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];
Colosi & Oeckl a2009 [fully local description];
Colosi & Oeckl a2104 [quantization of the evanescent sector];
> s.a. quantization of gauge theories;
klein-gordon fields in AdS spacetime.
@ Light-front quantization:
Collins a1801 [non-triviality of the vacuum];
Mannheim et al a2005,
Polyzou a2102
[comparison with instant-time quantization].
@ 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 IJMPA(14)-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 affine quantization, 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
[including non-perturbative approach]; 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];
Prabhu a1905
[using statistical mechanics, finite vacuum energy density].
@ 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 AHP(09)-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 field theory [conceptual]; quantum fields in
curved spacetime; stochastic quantization.
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