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;
Barbado et al a1811 [method for computing the evolution].

@ __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; states in quantum field theory [including
non-equilibrium]; temperature.

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