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):= ∂\(\cal L\)/∂(∂tφ) = ∂tφ,

[φ(t, x), φ(t, x')] = [π(t, x), π(t, x')] = 0 ,   and   [φ(t, x), π(t, x')] = i δn−1(xx') .

* Hamiltonian / momentum densities: From the stress-energy, one gets

Ttt = \(1\over2\)[(∂tφ)2 + (∂iφ) (∂iφ) + m2φ2] ,   Tti = ∂tφiφ ,
H = \(\int_\Sigma\)Ttt dn−1x = ∑k (Nk + \(1\over2\)) ω ,   Pi = Σ Tti dn−1x = ∑k Nk ki .

@ 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 uk(t, x) defined by the choice of a timelike vector field ∂/∂t.
* Creation / annihilation operators: They are obtained as coefficients of the field expansion

φ(t, x) = ∑k [ak uk(t, x) + ak uk*(t, x)] .

* 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 L2(positive mass shell in Fourier transform space).
* Full Hilbert space: The space \(\cal F\)S(\(\cal H\)):= \(\mathbb C\) ⊕ [⊕n = 1 (⊗Sn \(\cal H\))], 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)

|nk, n'k', ..., n''k''\(\rangle\) = (n! n'! ... n''!)−1/2 akn ak'n'... ak''n'' |0\(\rangle\).

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; de sitter space; gowdy spacetime; gravitational collapse; 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]; Alkhateeb & Matzkin a2103 [relativistic spin-0 particle in a box]; > s.a. coherent states; feynman propagator; geometric phase; states in quantum field theory [including non-equilibrium]; temperature.

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