Quantum Field Theory in Curved Spacetime  

In General > s.a. effective theories.
* Idea: A theory in which matter fields are quantized but the spacetime metric acts as a fixed background; It is not thought of as a fundamental theory, but is useful in the study of specific effects; Studied by DeWitt, and in the cosmological context by Parker (1969), it received impetus after Hawking's work.
* Difficulties: No canonical definition of > 0 modes and particles (no Poincaré invariance); Solved only for stationary cases or in/out states (S-matrix) in asymptotically flat cases; Can generalize only fields with s 1; Only the linear case is understood; There may be no unitary evolution between states defined on two arbitrary Cauchy surfaces–even in Minkowski.
@ Unitarity of evolution: Friedman et al PRD(92); Torre & Varadarajan CQG(99); > s.a. different backgrounds.
@ Limitations: Parentani gq/97-in [validity]; Giddings PRD(07)ht [black-hole background]; > s.a. semiclassical gravity.
@ Related topics: Singh & Mobed a0902 [breakdown of Casimir invariance]; Yao a0907/FP [arbitrary observer]; Kleinert a0910 [quantization in the neighborhood of a point].

Covariant Quantization
* Idea: Look for a set of field operators (x) satisfying Heisenberg's equations of motion and the commutation relations (e.g. for a scalar field):

[(x), (y)] = – i G(x, y) ,  where G is such that  (y) = Sigma G(y, x) (←a – a→) (x) da ;

If M is globally hyperbolic, G exists and is unique.

Other Approaches > s.a. algebraic and axiomatic approach; geometric quantization; path-integral approach.
@ Canonical: Fulling PRD(73) [ambiguity]; Wyrozumski PRD(90) [fiber bundle formalism]; Furlani JMP(99) [massive vector fields]; Calixto et al IJMPA(00)ht/97 [group quantization]; Corichi et al PRD(02)gq, CQG(03)gq/02, AP(04)ht/02 [scalar field, Schrödinger, Fock, and algebraic]; Moschella & Schaeffer JCAP(09)-a0802, AIP(09)-a0904 [new formulation].
@ Geometric quantization: Woodhouse RPMP(77); Kalinowski & Piechocki IJMPA(99).
@ Hadamard states: Sahlmann & Verch RVMP(01)mp/00; Moyassari a0705 [2D]; Sanders a0903.
@ Other states: Buchholz RVMP(00)mp/98 [from spacetime transformations]; Oeckl PLB(05)ht [on timelike hypersurfaces]; > s.a. thermal states.
@ Correlation dynamics, statistical aspects: Wald in(93) [cosmology, and horizons]; Hu gq/95 [and black-hole information]; > s.a. correlations.
@ Histories approach: Blencowe AP(91); Anastopoulos JMP(00)gq/99 [time-dependent Hilbert space].
@ Operator product expansion: Hollands & Wald GRG(08)-a0805.

Techniques > s.a. complex structure; formalism and techniques; green functions [propagator].
@ Renormalization: Castagnino et al PRD(86) [Hadamard and minimal compared]; Buchbinder et al RNC(89); Hollands & Wald CMP(03)gq/02 [scalar]; Banks & Mannelli PRD(03)ht/02 [in de Sitter space]; Shapiro CQG(08)-a0801 [semiclassical, pedagogical]; Casadio JPCS(09)-a0902 ["gravitational"].
@ Regularization methods: Parker & Fulling PRD(74), PRD(74) [adiabatic]; Moretti JMP(99)gq/98 [comparison], gq/99-in, Elizalde G&C(02)ht/01 [-function].
@ Related topics: Habib & Kandrup AP(89) [density matrix and Wigner functions]; Prugovecki CQG(96) [Hilbert bundles on spacetime]; Antonsen PRD(97)ht [from Wigner function]; Mashkevich gq/98, gq/98 [alternative approach]; Hollands & Wald CMP(01)gq, RVMP(05)gq/04 [conditions on Wick polynomials and Tab conservation]; > s.a. Foldy-Wouthuysen Representation.

Theories > s.a. various backgrounds and effects; klein-gordon fields; non-commutative field theory; quantum gauge theories.
@ Scalar field: Haba JPA(02)ht [4 in scale-invariant quantum metric].
@ Scalar field, renormalization: Tichy & Flanagan PRD(98)gq; Décanini & Folacci PRD(08)gq/05 [Hadamard, arbitrary dimensionality].
@ Scalar field, semiclassical: Camblong & Ordóñez PRD(05) [and black-hole thermodynamics]; Grain & Barrau NPB(06)ht [WKB approach]; Grain & Barrau PRD(07) [propagator, pedagogical].

References > s.a. entropy bound and quantum entropy; particles [creation].
@ Simple: Polarski Rech(90).
@ General: Choquet-Bruhat in(68); Hájícek in(77), PRD(77); Horowitz & Wald PRD(78); Birrell & Taylor JMP(78); Fulling GRG(79); Wald AP(79) [S-matrix]; Ashtekar & Magnon GRG(80); Kibble & Randjbar-Daemi JPA(80); Kay in(82); Martellini NCA(82), CQG(84); Haag et al CMP(84); Brunetti & Fredenhagen a0901-ln; Stoyanovsky a0910 [mathematical definition].
@ Textbooks: Birrell & Davies 84; Kayin 88; Fulling 89; Wald 94; Mukhanov & Winitzki 07; Parker & Toms 09.
@ Reviews: DeWitt RMP(57), PRP(75); Isham in(77); Parker in(77), in(79); Gibbons in(79); Davies in(80); Birrell in(81); Duff in(81); Hu in(82); Wald in(95), gq/95-in, gq/98-in; Ford gq/97-in; Liberati gq/00-PhD [vacuum effects]; Jacobson gq/03-ln; Kay in(06)gq; Wald gq/06-in [history and status], a0907-in.

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