4.2 Review

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

Journal

REVIEWS IN MATHEMATICAL PHYSICS
Volume 32, Issue 5, Pages -

Publisher

WORLD SCIENTIFIC PUBL CO PTE LTD
DOI: 10.1142/S0129055X20300046

Keywords

Deformed commutation relations; deformed Fock space; multicomponent quantum system; non-Abelian anyons (plektons)

Funding

  1. London Mathematical Society

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Let H be a separable Hilbert space and T be a self-adjoint bounded linear operator on H-circle times 2 with norm <= 1, satisfying the Yang-Baxter equation. Bozejko and Speicher ([10]) proved that the operator T determines a T-deformed Fock space F(H) = circle plus F-infinity(n=8)n(H). We start with reviewing and extending the known results about the structure of the n-particle spaces .F-n (H) and the commutation relations satisfied by the corresponding creation and annihilation operators acting on F(H). We then choose H = L-2 (X -> V), the L-2-space of V-valued functions on X. Here X := R-d and V := C-m with m >= 2. Furthermore, we assume that the operator T acting on H-circle times 2 = L-2 (X-2 -> V-circle times 2) is given by (Tf-(2))(x, y) = C-x,C-y f((2))(y, x). Here, for a.a. (x, y) is an element of X-2, C-x,C-y is a linear operator on V-circle times 2 with norm <= 1 that satisfies C*(x,y) = C-y(,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 C-xy in the case d = 2 determines non-Abelian anyons (also called plektons). For a multicomponent system, we describe its T-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.

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