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 \(\cal F\)(\(\cal H\)):= ⊕_{n=0}^{∞}\(\cal H\)_{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].

@ __Phenomenology, experiments__: Rossetti et al PRA(14)-a1409 [engineering interactions confined to subspaces of the Fock space]; Wang et al 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 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.

@ __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 a1602 [categorical fermionic Fock space]; > s.a. annihilation operator;
brownian motion.

@ __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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