Fock
Space| 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 
1,
the corresponding Fock space is the direct sum of n-particle spaces
given by 
():= 
n=0infty n,
where n is
the (anti)symmetrized
tensor product of n 1-particle spaces; In this basis,
= (
0, 1,
...), where n 
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:= ak
ak
.
* 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 
0
| T(A1(t1)
A2(t2))
| 0
,
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].
@ Other topics: Peres PRL(95) [non-local measurement effects]; Anastopoulos PRD(97)ht/96 [n-particle
sector]; Yang & Jing
MPLA(01)
[parasupersymmetric]; Dragan & Zin qp/07 [interference
in a single measurement]; > s.a. bogoliubov
transformation,
Time-Ordered Product.
References > s.a. QED.
@ General: in Emch 72 [original refs]; Streater & Wightman 64.
@ 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 
-Poincaré
symmetries].
@ Generalized: Marcinek m.QA/04-in
[non-commutative]; Garidi et al JPA(05)gq/04 [over
a Krein space]; Silva et al PhyA(08) [for stochastic spin lattices].
@ 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:
Mishra & Rajasekaran in(00)ht/01;
Yuri mp/06 [adelic
model]; > s.a. annihilation operator
@ 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].
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2009