Klein-Gordon Quantum Field Theory  

Canonical Quantization
* Kinematical setup: Choose a foliation of spacetime generated by a (Killing) vector field t, with hypersurfaces diffeomorphic to some .
* Phase space: Classically, the set of pairs (, ) on that are sufficiently smooth and rapidly vanishing at infinity; The quantum one includes distributional fields.
* 1-particle Hilbert space: The space of smooth pairs of functions (, ) on , with finite Klein-Gordon norm.
* Equal time commutation relations: If the canonical momentum is (t, x):= /(t) = t,

[(t, x), (t, x')] = [(t, x), (t, x')] = 0 ,   and   [(t, x), (t, x')] = i n–1(x–x') .

* Hamiltonian/momentum densities: From the stress-energy, one gets

Ttt = [(t)2 + (i)(i) + m22] ,   Tti = t i ,
H = Sigma Ttt dn–1x = k (Nk + ) ,   Pi = Sigma Tti dn–1x = k Nk ki .

@ General references: Corichi et al PRD(02)gq [Schrödinger representation in curved spacetime], CQG(03)gq/02 [Fock vs algebraic], AP(04)ht/02 [Fock vs Schrödinger]; Comay Ap(05)qp/04 [Hamiltonian operator].
@ Inner product: Mostafazadeh & Zamani qp/03, AP(06)qp; Kleefeld CzJP(06)qp.
@ Polymer representation: Ashtekar et al CQG(03)gq/02 [and Fock]; Kaminski et al CQG(06)gq/05; Laddha & Varadarajan a0805 [2D].
@ Related topics: Arageorgis et al SHPMP(02) [non-unitary implementability of dynamics]; Engle CQG(06)gq/05 [symmetry reduction]; Mostafazadeh IJMPA(06) [PC, C, CPT, and position operators].

Covariant Fock Space Quantization [@ in Wald 84] > s.a. fock space.
* Positive frequency solutions: A complete set of field modes uk(t, x) defined by the choice of a timelike vector field /t.
* Creation/annihilation operators: Obtained as coefficients of the field expansion

(t, x) = k [ak uk(t, x) + ak uk*(t, x)] .

* 1-particle Hilbert space : The completion of the (Klein-Gordon) inner product space of smooth, rapidly falling positive-frequency solutions of the Klein-Gordon equation; It is isomorphic to L2(positive mass shell in Fourier transform space).
* Full Hilbert space: S():= C [n = 1infty (Sn )], where "s" means symmetric; It has a natural Fock space structure, in which the particle number basis elements are (these change under a Bogoliubov transformation)

|nk, n'k', ..., n''k'' = (n! n'! ... n''!)–1/2 akn ak'n'... ak''n'' |0 .

In Curved Spacetime > s.a. quantum field theory in curved spacetime [representations]; quantum cosmology; renormalization; schwarzschild.
@ General references: Hájícek & Isham JMP(96)gq/95 [group quantization]; Helfer CQG(96)gq [stress-energy operator], ht/99, ht/99 [existence]; Strohmaier LMP(00)mp; Agnew & Dray GRG(01)gq/00 [distributional modes]; Iorio et al AP(01)ht [and deformed algebra].
@ Robertson-Walker: Zecca IJTP(97); Trucks CMP(98)gq/97 [m 0 Hadamard state]; > s.a. FRW spacetime.
@ Related topics: Hortaçsu & Özdemir MPLA(98) [cosmic strings]; Accioly & Blas PRD(02)gq [massive scalar, Foldy-Wouthuysen representation]; Haba JPA(03)ht [static quantum metric]; Camblong & Ordóñez PRD(05)ht/04 [semiclassical, and black hole thermodyamics]; Redmount PRD(06)gq/05 [massive in dS space, 1-particle + coherent states].
> Specific spacetimes: see gowdy spacetime.

Topics and References > s.a. path integral; quantum field theory [deformation, interpretation, ...].
@ General references: Pauli & Weisskopf HPA(34); in Birrell & Davies 82; in Ryder 96; Mostafazadeh AP(04)gq/03 [inner products, observables].
@ 2D: Fewster CQG(99)gq/98, CQG(99)gq/98 [cylinder]; Faber & Ivanov ht/02 [different approaches], ht/02 [ground state].
@ Modifications: Namsrai IJTP(98) [sqrt Klein-Gordon operator]; Oeckl PRD(06)ht/05 [general boundary formulation]; > s.a. lorentz symmetry violations.
@ Related topics: Weaver m.OA/02 [operator algebras]; Mostafazadeh IJMPA(06)qp/03 [C, P, T]; Morgan PLA(05)qp/04 [and classical random field]; > s.a. coherent states; feynman propagator; geometric phase.

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