Klein-Gordon Quantum Field Theory| 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(x−x') .
* Hamiltonian / momentum densities: From the stress-energy, one gets
Ttt
= \(1\over2\)[(∂tφ)2
+ (∂iφ)
(∂iφ)
+ m2φ2]
, Tti
= ∂tφ
∂iφ
,
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 ak†n 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.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 13 mar 2021