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arXiv:quant-ph/0610077

Displacement deformed quantum fields

Morgan, Peter

Published Oct 10, 2006• 0 citations
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Abstract

A displacement operator d_ζis introduced, verifying commutation relations [d_ζ, a_f^\dagger]=[d_ζ, a_f]=ζ(f)d_ζwith field creation and annihilation operators that verify [a_f,a_g]=0, [a_f,a_g^\dagger]=(g,f), as usual. f and g are test functions, ζis a Poincare invariant real-valued function on the test function space, and (g,f) is a Poincare invariant Hermitian inner product. The *-algebra generated by all these operators, and a state defined on it, nontrivially extends the *-algebra of creation and annihilation operators and its Fock space representation. If the usual requirement for linearity is weakened, as suggested in quant-ph/0512190, we obtain a deformation of the free quantum field.

DOI
10.48550/arxiv.quant-ph/0610077

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A displacement operator d_ζis introduced, verifying commutation relations [d_ζ, a_f^\dagger]=[d_ζ, a_f]=ζ(f)d_ζwith field creation and annihilation operators that verify [a_f,a_g]=0, [a_f,a_g^\dagger]=(g,f), as usual. f and g are test functions, ζis a Poincare invariant real-valued function on the test function space, and (g,f) is a Poincare invariant Hermitian inner product. The *-algebra generated by all these operators, and a state defined on it, nontrivially extends the *-algebra of creation and annihilation operators and its Fock space representation. If the usual requirement for linearity is weakened, as suggested in quant-ph/0512190, we obtain a deformation of the free quantum field.
We consider Fock representations of the Q-deformed commutation relations ∂s∂t†=Q(s,t)∂t†∂s+δ(s,t) for s,t∈T. Here T:=Rd (or more generally T is a locally compact Polish space), the function Q:T2→C satisfies |Q(s,t)|≤1 and Q(s,t)=Q(t,s)¯, and ∫T2h(s)g(t)δ(s,t) σ(ds)σ(dt):=∫Th(t)g(t) σ(dt), σ being a fixed reference measure on T. In the case, where |Q(s,t)|≡1, the Q-deformed commutation relations describe a generalized statistics studied by Liguori and Mintchev. These generalized statistics contain anyon statistics as a special case (with T=R2 and a special choice of the function Q). The related Q-deformed Fock space F(H) over H:=L2(T→C,σ) is constructed. An explicit form of the orthogonal projection of H⊗n onto the n-particle space Fn(H) is derived. A scalar product in Fn(H) is given by an operator Pn≥0 in H⊗n which is strictly positive on Fn(H). We realize the smeared operators ∂t† and ∂t as creation and annihilation operators in F(H), respectively. Additional Q-commutation relations are obtained between the creation operators and between the annihilation operators. They are of the form ∂s†∂t†=Q(t,s)∂t†∂s†, ∂s∂t=Q(t,s)∂t∂s, valid for those s,t∈T for which |Q(s, t)| = 1.
Let [Formula: see text] be a separable Hilbert space and [Formula: see text] be a self-adjoint bounded linear operator on [Formula: see text] with norm [Formula: see text], satisfying the Yang–Baxter equation. Bożejko and Speicher ([10]) proved that the operator [Formula: see text] determines a [Formula: see text]-deformed Fock space [Formula: see text]. We start with reviewing and extending the known results about the structure of the [Formula: see text]-particle spaces [Formula: see text] and the commutation relations satisfied by the corresponding creation and annihilation operators acting on [Formula: see text]. We then choose [Formula: see text], the [Formula: see text]-space of [Formula: see text]-valued functions on [Formula: see text]. Here [Formula: see text] and [Formula: see text] with [Formula: see text]. Furthermore, we assume that the operator [Formula: see text] acting on [Formula: see text] is given by [Formula: see text]. Here, for a.a. [Formula: see text], [Formula: see text] is a linear operator on [Formula: see text] with norm [Formula: see text] that satisfies [Formula: see text] and the spectral quantum Yang–Baxter equation. The corresponding creation and annihilation operators describe a multicomponent quantum system. A special choice of the operator-valued function [Formula: see text] in the case [Formula: see text] determines non-Abelian anyons (also called plektons). For a multicomponent system, we describe its [Formula: see text]-deformed Fock space and the available commutation relations satisfied by the corresponding creation and annihilation operators. Finally, we consider several examples of multicomponent quantum systems.
This work addresses the study of the oscillator algebra, defined by four parameters ppp, qqq, ααα, and ννν. The time-independent Schrödinger equation for the induced deformed harmonic oscillator is solved; explicit analytic expressions of the energy spectrum are given. Deformed states are built and discussed with respect to the criteria of coherent state construction. Various commutators involving annihilation and creation operators and their combinatorics are computed and analyzed. Finally, the correlation functions of matrix elements of main normal and antinormal forms, pertinent for quantum optics analysis, are computed.
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Let HHH be a separable Hilbert space and TTT be a self-adjoint bounded linear operator on H⊗2H^{\otimes 2}H⊗2 with norm ≤1\le1≤1, satisfying the Yang--Baxter equation. Bo\.zejko and Speicher (1994) proved that the operator TTT determines a TTT-deformed Fock space F(H)=⨁n=0∞Fn(H)\mathcal F(H)=\bigoplus_{n=0}^\infty\mathcal F_n(H)F(H)=⨁n=0∞​Fn​(H). We start with reviewing and extending the known results about the structure of the nnn-particle spaces Fn(H)\mathcal F_n(H)Fn​(H) and the commutation relations satisfied by the corresponding creation and annihilation operators acting on F(H)\mathcal F(H)F(H). We then choose H=L2(X→V)H=L^2(X\to V)H=L2(X→V), the L2L^2L2-space of VVV-valued functions on XXX. Here X:=RdX:=\mathbb R^dX:=Rd and V:=CmV:=\mathbb C^mV:=Cm with m≥2m\ge2m≥2. Furthermore, we assume that the operator TTT acting on H⊗2=L2(X2→V⊗2)H^{\otimes 2}=L^2(X^2\to V^{\otimes 2})H⊗2=L2(X2→V⊗2) is given by (Tf(2))(x,y)=Cx,yf(2)(y,x)(Tf^{(2)})(x,y)=C_{x,y}f^{(2)}(y,x)(Tf(2))(x,y)=Cx,y​f(2)(y,x). Here, for a.a.\ (x,y)∈X2(x,y)\in X^2(x,y)∈X2, Cx,yC_{x,y}Cx,y​ is a linear operator on V⊗2V^{\otimes 2}V⊗2 with norm ≤1\le1≤1 that satisfies Cx,y∗=Cy,xC_{x,y}^*=C_{y,x}Cx,y∗​=Cy,x​ and the spectral quantum Yang--Baxter equation. The corresponding creation and annihilation operators describe a multicomponent quantum system. A special choice of the operator-valued function CxyC_{xy}Cxy​ in the case d=2d=2d=2 determines non-Abelian anyons (also called plektons). For a multicomponent system, we describe its TTT-deformed Fock space and the available commutation relations satisfied by the corresponding creation and annihilation operators. Finally, we consider several examples of multicomponent quantum systems.