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 dieomorphism 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*)
d*σ** ^{a}* ;

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 *T*_{ab} 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 [*λφ*^{4} 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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send feedback and suggestions to bombelli at olemiss.edu – modified 18 jun 2017