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
\({\cal H}_1\), 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:= 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
\(\langle\)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].
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