Approaches to Quantum Field Theory

Canonical Quantization > s.a. canonical quantum mechanics [including group quantization]; geometric quantization.
* Idea: Choose a Hilbert space of states 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 mp/02/FAA [Schrödinger equation as variational differential equation]; Corichi et al CQG(03)gq/02 [curved spacetime ambiguities]; Solomon qp/03 [vs Heisenberg]; > 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: Nikolic 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].
@ Related topics: Banai JMP(87); Danos ht/97 [particles and systems]; Oeckl PRD(06)ht/05 [general boundary].
> Specific theories: see canonical quantum gravity; klein-gordon fields; types of theories [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: [a(f), a*(g)] = (f | g), [a(f), a(g)] = 0; a(f)'s are unbounded (can see from commutation relations), and have to generate the algebra using exp[i ( a + *a*)].
* Fermions: {a(f), a*(g)} = (f | g), {a(f), a(g)} = 0; a(f)'s are bounded.

Other Approaches > s.a. algebraic and axiomatic; path integrals; quantum fields in curved spacetime; stochastic quantization.
@ Relationships: Teleki & Noga ht/06 [operator and path integral].
@ Phase space approach: Zachos & Curtright PTPS(99)ht-in.
@ Loop-based: Rayner CQG(90) [gravity and scalar field]; Solovyov et al ht/04; Manrique et al CQG(06) [loop quantization and 2D Ising field theory].
@ Without infinities / renormalization: Takook IJMPE(05)gq/00 [⁴]; Stefanovich AP(01); Biswas ht/05 [composite particles].
@ Classical non-local model: Morgan qp/01 [for Klein-Gordon], qp/01 [for electromagnetism].
@ Other approaches: Brodsky FdP(02)ht/01 [Heisenberg matrix formulation]; Gurau et al a0807 [based on marked trees].
@ World-line approach: Rylov ht/01; Gies & Hämmerling PRD(05) [spin-gauge field coupling, QED].
@ Related topics: Czachor qp/99 [single oscillator]; > s.a. quantum field theory techniques, Hopf Algebra.
@ Generalizations: Yurtsever CQG(94)gq/93 [quantum field theory on a topology, and small-scale structure].

Main page – Abbreviations – Journals – Comments – Other sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified 27 jun 2008