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arXiv:1904.11211 · Reviews in Mathematical Physics

Fock representations of multicomponent (particularly non-Abelian anyon) commutation relations

Alexei Daletskii, Alexander Kalyuzhny, Eugene Lytvynov, Daniil Proskurin

Published Oct 18, 2019• 7 citations
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Abstract

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.

DOI
10.1142/s0129055x20300046

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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.
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.
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.
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.
This is a self-contained and hopefully readable account on the method of creation and annihilation operators (also known as the Fock space representation or the "second quantization" formalism) for non-relativistic quantum mechanics of many particles. Assuming knowledge only on conventional quantum mechanics in the wave function formalism, we define the creation and annihilation operators, discuss their properties, and introduce corresponding representations of states and operators of many-particle systems. As the title of the note suggests, we cover most topics treated in sections 6.7 and 6.8 of Feynman's "Statistical Mechanics: A Set of Lectures". As a preliminary, we also carefully discuss the symmetry of wave functions describing indistinguishable particles. We note that all the contents of the present note are completely standard, and the definitions and the derivations presented here have been known to many. Although the style of the present note may be slightly more mathematical than standard physics literatures, we do not try to achieve full mathematical rigor.(Note to experts: In particular we here DERIVE the (anti)commutation relations of the creation and annihilation operators, rather than simply declaring them. In this sense our approach is quite close to that of Feynman's. But we here focus on the action of creation/annihilation operators on general NNN body wave functions, while Feynman makes a heavy use of Slater-determinant-type states from the beginning. We hope that our presentation provides a better perspective on the formalism.)
CAR on Fock spacesThis chapter is devoted to the study of Fock representations of the canonical anticommutation relations, a basic tool of quantum many-body theory and quantum field theory.It is parallel to Chap. 9, where Fock CCR representations were studied.The basic framework is almost the same as in Chap.9. Throughout this chapter Z is a Hilbert space, called the one-particle space.We will consider the Fock CAR representation acting on the fermionic Fock space Γ a (Z).As in Sect.1.3, we introduce the space Y = Re(Z ⊕ Z) := {(z, z) : z ∈ Z}, which will serve as the dual phase space of our system.Recall that in the bosonic case we equipped Y with the structure of the Kähler space consisting of the anti-involution j, the Euclidean scalar product • and the symplectic form ω: j(z, z) := (iz, iz), (13.1) (z, z) • (w, w) := 2Re(z|w), (13.2) (z, z)•ω(w, w) := 2Im(z|w) = -(z, z) • j(w, w).(13.3)In our presentation of the fermionic case, we will need the Kähler anti-involution j.The symplectic form ω will not be used.Instead of the scalar product (13.2) we will use another scalar product,Again we will avoid identifing Z with Y. CY is identified with Z ⊕ Z by the mapY # , the space dual to Y, is canonically identified with Re(Z ⊕ Z) by using the scalar product (9.2), and CY # is identified with Z ⊕ Z. Fock CAR representationConsider the fermionic Fock space Γ a (Z).Recall that, for z ∈ Z, a * (z), resp.a(z) denote the corresponding creation, resp.annihilation operators defined in Sect.3.4.
A description of the Fock space of a parabose oscillator of order p based on commutation relations bilinear in creation and annihilation operators rather than trilinear ones is developed. A new statement of the well-known Green’s ansatz is introduced to help understand the significance of the bilinear commutation relations. It is also applied to show how all parabose oscillators of even order (respectively, odd order) can be obtained as irreducible constituents of the Fock space associated with a Green’s ansatz of two (three) terms. Analogs of the form in which the parabose Green’s ansatz is presented are provided for one parafermi oscillator and for systems of many parabose and parafermi oscillators of the same order. Methods used in the paper allow various new results for q-deformed parabose oscillators to be derived.