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.
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)
d
a ;
If M is globally hyperbolic, G exists and is unique.
Other Approaches > s.a. algebraic
and axiomatic;
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:
Schrödinger, Fock, algebraic]; Moschella & Schaeffer a0802 [new formulation].
@ Geometric quantization:
Woodhouse
RPMP(77);
Kalinowski & Piechocki IJMPA(99).
@ Hadamard
states: Sahlmann & Verch RVMP(01)mp/00;
Moyassari a0705 [2D].
@ Other states: Buchholz RVMP(00)mp/98 [from
spacetime transformations]; Oeckl PLB(05)ht [on
timelike hypersurfaces].
@ 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 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]; Shapiro CQG(08)-a0801 [semiclassical,
pedagogical].
@ 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; 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 gq/05 [Hadamard, arbitrary D].
@ 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).
@ Textbooks: Birrell & Davies 84; Kayin 88; Fulling 89; Wald 94;
Mukhanov & Winitzki 07.
@ 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].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
24 jun 2008