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* = (*α*_{0},
*α*_{1}, ...), where
*α*_{n} ∈
\({\cal H}_n\) 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}
*N*_{k} ,
where *N*_{k}:=
*a*_{k}^{†}*a*_{k}
.

* __Properties__: For a free
scalar field, [*N*_{k},
*H*] = [*N*_{k},
*P*_{i}] = 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(*A*_{1}(*t*_{1})
*A*_{2}(*t*_{2}))
− :*A*_{1}(*t*_{1})
*A*_{2}(*t*_{2}):.

* __Relationships__: This contraction turns out to be the c-number
\(\langle\)0 | T(*A*_{1}(*t*_{1})
*A*_{2}(*t*_{2})) | 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].

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