Fock Space

In General > s.a. klein-gordon quantum field theory; photon; quantum field theory in curved spacetime; types of quantum field theories.
* Idea: The particle number representation Hilbert space for a canonically formulated quantum theory, which comes equipped with a preferred basis of n-particle states, eigenstates of the number operators, and coherent states, eigenstates of the annihilation operators.
* Limitations: It can be used for linear field theories, but is not applicable in principle for interacting theories, although it is sometimes used in that situation too; The problem is that one usually doesn't have consistent representations of the relevant observable algebras in general.
$Def: Given a 1-particle Hilbert space $${\cal H}_1$$, the corresponding Fock space is the direct sum of $$n$$-particle spaces given by $${\cal F}({\cal H}):= \bigoplus_{n = 0}^\infty {\cal H}_n$$, where $${\cal H}_n$$ is the (anti)symmetrized tensor product of $$n$$ one-particle spaces; In this basis, ψ = (α0, α1, ...), where αn ∈ $${\cal H}_n$$ is interpreted as the n-particle component. Number Operator > s.a. annihilation and creation operator.$ Def: For a system with multiple degrees of freedom, the operator

N:= ∑k Nk ,    where   Nk:= akak .

* Properties: For a free scalar field, [Nk, H] = [Nk, Pi] = 0.
@ References: Besnard LMP(01)mp/00, mp/01 [algebras admitting N, types of particles]; Bueler mp/01 [on Riemannian manifolds]; Gour qp/01, FP(02)qp/01 [conjugate phase operator]; Dumitru qp/02 [number-phase problem].

Dyson-Wick Contraction or Chronological Pairing
\$ Def: For two operators in the algebra of fermion and boson operators on Fock space, depending on a parameter t (time), their contraction is defined by T(A1(t1) A2(t2)) − :A1(t1) A2(t2):.
* Relationships: This contraction turns out to be the c-number $$\langle$$0 | T(A1(t1) A2(t2)) | 0$$\rangle$$, and is often called also a propagator or a Green's function.

Related Topics, Systems and States > s.a. Hopf Algebra and lie algebra representations.
* Oscillator: The lifetime of a Fock state with excitation number n scales as 1/n.
@ QED: Baseia & Dantas PLA(99), Gerry & Benmoussa PLA(02) [with holes in photon number distribution]; Valverde et al PLA(03) [truncated states]; Nayak qp/03; Rohde et al NJP(07)qp/06 [Fock states vs multiphoton states]; > s.a. QED.
@ Oscillator: Brune et al PRL(08), Wang et al PRL(08), Blais & Gambetta Phy(08) [preparing states and watching them evolve].
@ States: Anastopoulos PRD(97)ht/96 [n-particle sector]; > s.a. types of coherent states [fermionic, photon-added].
@ Phenomenology, experiments: Rossetti et al PRA(14)-a1409 [engineering interactions confined to subspaces of the Fock space]; Wang et al PRL(17)-a1703 [generating arbitrary Fock-state superpositions in a superconducting cavity].
@ Other topics: Peres PRL(95) [non-local measurement effects]; Yang & Jing MPLA(01) [parasupersymmetric]; Dragan & Zin PRA(07)qp [interference in a single measurement]; Cortez et al PRD(11)-a1101 [in cosmological spacetimes]; D'Amico et al PRL(11) [metric space structure, and particle densities]; > s.a. bogoliubov transformation; Time-Ordered Product.

References > s.a. formulations of quantum theory [metric on state space]; QED.
@ General: in Emch 72 [original refs]; Streater & Wightman 64; Shchesnovich a1308-ln.
@ Exponential Hilbert space: Friedrichs 53; Klauder JMP(70).
@ Deformed Fock space: Roknizadeh & Tavassoly JPA(04)mp [coherent states]; Jing et al CTP(06)ht/05; Meng & Wang IJTP(07) [squeezed states]; Arzano & Marcianò PRD(07)-a0707 [scalar field with κ-Poincaré symmetries]; Tavassoly & Lake ChPC-a1204 [coherent and squeezed states]; Bożejko et al JMP(17)-a1603 [Q-deformed commutation relations]; > s.a. modified coherent states; momentum space; particle models; quantum particles [3D relativistic quantum particles with curved momentum space].
@ And polymer representation: Varadarajan & Zapata CQG(00), Ashtekar & Lewandowski CQG(01), et al CQG(03) [scalar field].
@ And Schrödinger representation: Corichi et al AP(04)ht/02, CQG(03)gq/02; Beggi et al EJP(18)-a1805 [Fock space and Hilbert space].
@ Generalized: Marcinek in(03)m.QA/04 [non-commutative]; Garidi et al JPA(05)gq/04 [over a Krein space]; Mishra & Rajasekaran in(00)ht/01; Yuri mp/06 [adelic model]; Silva et al PhyA(08) [for stochastic spin lattices]; Antipin et al PPNL(15)-a1301 [anti-Fock representations, realized on a Krein space]; Chen & Lin IJTP(17)-a1602 [categorical fermionic Fock space]; Alpay & Porat a1804; > s.a. annihilation operator; brownian motion; types of coherent states [Fock-Krein spaces].
@ Related topics: Howe JPA(97) [decomposition]; Gough mp/03 [transformation fermionic/bosonic]; Hiroshima & Ito mp/03 [canonical transformations]; Laloë EPJD(05)qp/04 ["hidden phase"]; Kupsch & Banerjee mp/04 [ultracoherence and canonical transformations]; Gudder IJTP(04) [computational logic]; Lieb et al PRL(05) [k = 0 mode as c-number in bosonic Hamiltonian]; Rodríguez-Vázquez et al AP(14)-a1403 [for local quanta, of a 1D Klein-Gordon field].