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];
Barbado et al a1811 [method for computing the evolution];
Colosi & Oeckl a1903 [vacuum as Lagrangian subspace].

@ __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 21 mar 2019