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.
* History: Precursor work was done by Schrödinger in the 1930s, but the field was really started by Bryce DeWitt, and in the cosmological context by a 1969-1971 series of papers by Leonard Parker on particle creation in an expanding universe; It received a bigger impetus after Hawking's work on black-hole radiation.
* Approaches: Quantum field theory in curved spacetime may be defined either through a manifestly unitary canonical approach, or via the manifestly covariant path integral formalism; For gauge theories, these two approaches have produced conflicting results.
* Difficulty with unitarity: In general there may be no unitary evolution between states defined on two arbitrary Cauchy surfaces, even in Minkowski spacetime–but one must remember that a general di eomorphism is not a symmetry of the theory; However, > see QED in curved spacetime.
* Other difficulties: There is no canonical definition of ω > 0 modes and particles (no Poincaré invariance); This issue has been 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.
@ Unitarity of evolution: Friedman et al PRD(92); Torre & Varadarajan CQG(99); Colosi & Oeckl ONPPJ(11)-a0912; Agulló & Ashtekar PRD(15)-a1503 [generalized notion]; > s.a. different backgrounds.
@ Limitations: Parentani gq/97-proc [validity]; Giddings PRD(07)ht [black-hole background]; > s.a. semiclassical gravity [including corrections to gravity].
@ Related topics: Singh & Mobed AdP(10)-a0902 [breakdown of Casimir invariance]; Yao a0907/FP [arbitrary observer]; Kleinert EJTP(09)-a0910 [quantization in the neighborhood of a point]; Fredenhagen & Hack a1308 [and phenomenological applications to cosmology].

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

[φ(x), φ(y)] = – i\(\hbar\) G(x, y) ,  where G is such that  φ(y) = Σ G(y, x) (←∂a – ∂a→) φ(x) dσa ;

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]; 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]; Cortez et al JCAP(10)-a1004 [unique Fock quantization]; > s.a. Proca Theory.
@ Geometric quantization: Woodhouse RPMP(77); Kalinowski & Piechocki IJMPA(99).
@ 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]; Hadamard States; regularization.
@ 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"]; Barceló et al PRD(12) [equivalence between two different renormalized stress-energy tensors]; Viet Dang a1312 [causal approach].
@ Related topics: Habib & Kandrup AP(89) [density matrix and Wigner functions]; Prugovečki 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]; Doukas et al CQG(15)-a1306 [discriminating quantum field theories in curved spacetime]; > s.a. Foldy-Wouthuysen Representation.

Theories > s.a. various backgrounds and effects; non-commutative field theory.
@ Scalar field: Haba JPA(02)ht [λφ4 in scale-invariant quantum metric]; > s.a. klein-gordon fields.
@ 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].
@ Spin-1 field: Buchbinder et al a1703 [non-minimally coupled massive, effective action]; > s.a. generalized theories [non-local]; Proca Theory.
@ Other theories: Pahlavan & Bahari IJTP(09), Takook et al EPJC(12)-a1206 [spin-3/2 fields, in de Sitter space]; Bilić et al PLB(12)-a1110 [supersymmetric model]; de Medeiros & Hollands CQG(13)-a1305 [superconformal]; > s.a. dirac fields; quantum gauge theories.

References > s.a. entropy bound and quantum entropy; particles [creation].
@ Simple: Polarski Rech(90).
@ General: Choquet-Bruhat in(68); Hájíček 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 LNP(09)-a0901; Stoyanovsky a0910 [mathematical definition]; Baer & Ginoux SPM(12)-a1104; Fewster & Liberati GRG(14)-a1402 [GR20 report]; Benini & Dappiaggi a1505 [three explicit examples].
@ Textbooks: Birrell & Davies 84; Kay in(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-GR14, gq/98-in; Ford gq/97-proc; Liberati PhD(00)gq [vacuum effects]; Jacobson gq/03-ln; Kay in(06)gq; Wald gq/06-conf [history and status], a0907-proc; Haro a1011-ln; Benini et al IJMPA(13)-a1306 [primer]; Hollands & Wald PRP(15)-a1401; Fredenhagen & Rejzner JMP(16)-a1412 [framework and examples]; Miao et al a1505-in [non-technical].

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