E-Thesis 405 views 109 downloads
Quasi-free states on algebras of multicomponent commutation relations / Nedal Othman
Swansea University Author: Nedal Othman
DOI (Published version): 10.23889/SUthesis.62993
Abstract
Let X = R² and let V be a finite-dimensional complex inner product space. Let C : X² → L (V ⊗²) be a continuous function such that, for each (x, y) ∈ X², C(x, y) is a unitary operator in V ⊗², C∗(x, y) = C(y, x), and the functional Yang-Baxter equa-tion is satisfied. The dissertation deals with the...
| Published: |
Swansea University
2023
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| Institution: | Swansea University |
| Degree level: | Doctoral |
| Degree name: | Ph.D |
| Supervisor: | Lytvynov, Eugene, Finkelshtein, Dmitri. |
| URI: | https://cronfa.swan.ac.uk/Record/cronfa62993 |
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| Abstract: |
Let X = R² and let V be a finite-dimensional complex inner product space. Let C : X² → L (V ⊗²) be a continuous function such that, for each (x, y) ∈ X², C(x, y) is a unitary operator in V ⊗², C∗(x, y) = C(y, x), and the functional Yang-Baxter equa-tion is satisfied. The dissertation deals with the multicomponent commutation relations governed by the function C, see [A. Liguori, M. Mintchev, Comm. Math. Phys. 169 (1995) 635–652]. We introduce the ∗-algebra of the C-multicomponent commutation re-lations (C-MCR algebra). We propose definitions of a gauge-invariant quasi-free state andof a strongly quasi-free state on the C-MCR algebra, A. Under restrictive assumptions on the function C, we construct a class of gauge-invariant quasi-free states on A, which, for some functions C, are also strongly quasi-free. We show that, when dim V = 1 (i.e., when we deal with the anyon commutation relations), among all gauge-invariant quasi- free states on A, only the Fock state is strongly quasi-free. In the case dim V = 2 (i.e., when we deal with two-component systems), we present a non-trivial class of examples of function C to which our theory is applicable, and hence, we can construct gauge-invariant quasi-free states, or even strongly quasi-free states on A. |
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| Item Description: |
Copyright: The author, Nedal Othman, 2023 |
| Keywords: |
Fock space, deformed commutation relations, multicomponent quantum system, anyon, plekton |
| College: |
Faculty of Science and Engineering |