Klein-Gordon
Quantum
Field Theory |

**Canonical Quantization** > s.a. types
of quantum field theories [including polymer representation].

* __Kinematical setup__:
Choose a foliation of spacetime generated by a (Killing) vector field *t*,
with hypersurfaces diffeomorphic to some Σ.

* __Phase space__: Classically,
the set of pairs (*φ*, *π*) on Σ that are sufficiently smooth
and rapidly vanishing at infinity; The quantum one includes distributional fields.

* __1-particle Hilbert space__:
The space \(\cal H\) of smooth pairs of functions (*φ*, *π*)
on Σ with finite Klein-Gordon norm.

* __Equal time commutation relations__:
If the canonical momentum is *π*(*t*,
*x*):= ∂\(\cal L\)/∂(∂_{t}*φ*)
= ∂_{t}*φ*,

[*φ*(*t*,* x*),
*φ*(*t*,* x'*)]
= [*π*(*t*,* x*), *π*(*t*,* x'*)]
= 0 , and [*φ*(*t*,* x*),
*π*(*t*,* x'*)]
= i δ^{n–1}(*x*–*x'*)
.

* __Hamiltonian / momentum
densities__: From the stress-energy, one gets

*T _{tt}* = \(1\over2\)[(∂

@ __General references__: Corichi
et al PRD(02)gq [Schrödinger
representation in curved spacetime], CQG(03)gq/02 [Fock
vs algebraic], AP(04)ht/02 [Fock
vs Schrödinger]; Comay Ap(05)qp/04 [Hamiltonian
operator].

@ __Inner product__: Mostafazadeh & Zamani qp/03,
AP(06)qp;
Kleefeld CzJP(06)qp.

@ __Related topics__: Arageorgis et al SHPMP(02)
[non-unitary implementability of dynamics]; Engle CQG(06)gq/05 [symmetry
reduction]; Mostafazadeh IJMPA(06)
[PC, C, CPT, and position operators]; Cortez et al a1311-MG13, AP-a1509 [unitary evolution as a uniqueness criterion].

**Covariant Fock Space Quantization** [@ in Wald 84] > s.a. fock space.

* __Positive frequency solutions__:
A complete set of field modes *u** _{k}*(

*

*φ*(*t*,* x*)
= ∑* _{k}* [

* __1-particle Hilbert space__:
The completion of the (Klein-Gordon) inner product space of smooth, rapidly falling positive-frequency solutions of the
Klein-Gordon equation; It is isomorphic to L^{2}(positive mass shell in Fourier transform space).

* __Full Hilbert space__: The space \(\cal F\)_{S}(\(\cal H\)):= \(\mathbb C\) ⊕ [⊕_{n}_{ =
1}^{∞} (⊗_{S}^{n} \(\cal H\))],
where "s" means symmetric; It has a natural Fock-space structure,
in which the particle number basis elements are (these change under a Bogoliubov transformation)

|*n** _{k}*,

**In Curved Spacetime** > s.a. quantum
field theory in curved spacetime [representations];
quantum cosmology; renormalization.

@ __General references__: Hájíček & Isham JMP(96)gq/95 [group
quantization]; Helfer CQG(96)gq [stress-energy
operator], ht/99, ht/99 [existence];
Strohmaier LMP(00)mp;
Agnew & Dray
GRG(01)gq/00 [distributional
modes]; Iorio et al AP(01)ht [and
deformed algebra]; Alhaidari & Jellal PLA(15)-a1106.

@ __Robertson-Walker__: Zecca IJTP(97);
Trucks CMP(98)gq/97 [*m* ≠ 0
Hadamard state]; Kaya & Tarman JCAP(12)-a1111 [cosmological backreaction]; > s.a. FLRW spacetime.

@ __de Sitter space__: Redmount PRD(06)gq/05 [massive,
1-particle + coherent states]; Marolf & Morrison CQG(09)
[group averaging]; Page & Wu JCAP(12) [massless, vacuum].

@ __Related topics__: Hortaçsu & Özdemir MPLA(98)
[cosmic strings];
Accioly & Blas PRD(02)gq [massive
scalar, Foldy-Wouthuysen representation]; Haba
JPA(03)ht [static
quantum metric]; Camblong & Ordóñez PRD(05)ht/04 [semiclassical,
and black-hole thermodyamics]; Colosi a0903 [general
boundary formulation]; Cortez et al PRD(09)-a0903, CQG(11)-a1108 [with
time-dependent mass].

> __Specific spacetimes__:
see bianchi I models; gowdy
spacetime; quantum field theory in curved backgrounds [including anti-de Sitter];
reissner-nordström spacetime; schwarzschild
spacetime.

**Topics and References** > s.a. classical klein-gordon fields; path
integral; quantum field theory [deformation, interpretation, ...].

@ __General references__: Pauli & Weisskopf HPA(34);
in Birrell & Davies 82; in Ryder 96; Mostafazadeh AP(04)gq/03 [inner
products, observables].

@ __2D__: Fewster CQG(99)gq/98, CQG(99)gq/98 [cylinder];
Faber & Ivanov ht/02 [different
approaches], ht/02 [ground
state]; Marolf & Morrison CQG(09) [in de Sitter, group averaging].

@ __Modifications__: Namsrai IJTP(98)
[sqrt Klein-Gordon operator]; Oeckl PRD(06)ht/05 [general
boundary formulation]; Koide & Kodama PTEP(15)-a1306 [stochastic variational method]; > s.a. lorentz
symmetry violations.

@ __Related topics__: Weaver m.OA/02 [operator
algebras]; Mostafazadeh IJMPA(06)qp/03 [*C*,
*P*,
*T*]; Morgan PLA(05)qp/04 [and
classical random field]; > s.a. coherent
states; feynman propagator; geometric
phase; temperature.

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