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Particle-Hole Transformation in the Continuum and Determinantal Point Processes

Maryam Gharamah Ali Alshehri, Eugene Lytvynov Orcid Logo

Communications in Mathematical Physics

Swansea University Author: Eugene Lytvynov Orcid Logo

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DOI (Published version): 10.1007/s00220-023-04803-9

Abstract

Let X be an underlying space with a reference measure σ. Let K be an integral operator in L2(X,σ) with integral kernel K(x, y). A point process μ on X is called determinantal with the correlation operator K if the correlation functions of μ are given by k(n) (x1,..., xn) = det[K(xi, x j)]i,j=1,...,n...

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Published in: Communications in Mathematical Physics
ISSN: 0010-3616 1432-0916
Published: Springer Science and Business Media LLC 2023
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URI: https://cronfa.swan.ac.uk/Record/cronfa63994
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spelling v2 63994 2023-07-28 Particle-Hole Transformation in the Continuum and Determinantal Point Processes e5b4fef159d90a480b1961cef89a17b7 0000-0001-9685-7727 Eugene Lytvynov Eugene Lytvynov true false 2023-07-28 SMA Let X be an underlying space with a reference measure σ. Let K be an integral operator in L2(X,σ) with integral kernel K(x, y). A point process μ on X is called determinantal with the correlation operator K if the correlation functions of μ are given by k(n) (x1,..., xn) = det[K(xi, x j)]i,j=1,...,n. It is known that each determinantal point process with a self-adjoint correlation operator K is the joint spectral measure of the particle density ρ(x) = A+(x)A−(x) (x ∈ X), where the operator-valued distributions A+(x), A−(x) come from a gauge-invariant quasi-free representation of the canonical anticommutation relations (CAR). If the space X is discrete and divided into two disjoint parts, X1 and X2, by exchanging particles and holes on the X2 part of the space, one obtains from a determinantal point process with a self-adjoint correlation operator K the determinantal point process with the J -self-adjoint correlation operator K = K P1 + (1 − K)P2. Here Pi is the orthogonal projection of L2(X,σ) onto L2(Xi,σ). In the case where the space X is continuous, the exchange of particles and holes makes no sense. Instead, we apply a Bogoliubov transformation to a gauge-invariant quasi-free representation of the CAR. This transformation acts identically on the X1 part of the space and exchanges the creation operators A+(x) and the annihilation operators A−(x) for x ∈ X2. This leads to a quasi-free representation of the CAR, which is not anymore gauge-invariant. We prove that the joint spectral measure of the corresponding particle density is the determinantal point process with the correlation operator K. Journal Article Communications in Mathematical Physics Springer Science and Business Media LLC 0010-3616 1432-0916 5 8 2023 2023-08-05 10.1007/s00220-023-04803-9 http://dx.doi.org/10.1007/s00220-023-04803-9 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University SU Library paid the OA fee (TA Institutional Deal) Swansea University 2023-09-07T13:30:04.7649617 2023-07-28T10:26:13.4617789 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Maryam Gharamah Ali Alshehri 1 Eugene Lytvynov 0000-0001-9685-7727 2 63994__28318__a9ab6b559f8d4a2f9e75221ceb6c7250.pdf 63994.VOR.pdf 2023-08-18T12:13:45.1376189 Output 551253 application/pdf Version of Record true This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made true eng http://creativecommons.org/licenses/by/4.0/
title Particle-Hole Transformation in the Continuum and Determinantal Point Processes
spellingShingle Particle-Hole Transformation in the Continuum and Determinantal Point Processes
Eugene Lytvynov
title_short Particle-Hole Transformation in the Continuum and Determinantal Point Processes
title_full Particle-Hole Transformation in the Continuum and Determinantal Point Processes
title_fullStr Particle-Hole Transformation in the Continuum and Determinantal Point Processes
title_full_unstemmed Particle-Hole Transformation in the Continuum and Determinantal Point Processes
title_sort Particle-Hole Transformation in the Continuum and Determinantal Point Processes
author_id_str_mv e5b4fef159d90a480b1961cef89a17b7
author_id_fullname_str_mv e5b4fef159d90a480b1961cef89a17b7_***_Eugene Lytvynov
author Eugene Lytvynov
author2 Maryam Gharamah Ali Alshehri
Eugene Lytvynov
format Journal article
container_title Communications in Mathematical Physics
publishDate 2023
institution Swansea University
issn 0010-3616
1432-0916
doi_str_mv 10.1007/s00220-023-04803-9
publisher Springer Science and Business Media LLC
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
url http://dx.doi.org/10.1007/s00220-023-04803-9
document_store_str 1
active_str 0
description Let X be an underlying space with a reference measure σ. Let K be an integral operator in L2(X,σ) with integral kernel K(x, y). A point process μ on X is called determinantal with the correlation operator K if the correlation functions of μ are given by k(n) (x1,..., xn) = det[K(xi, x j)]i,j=1,...,n. It is known that each determinantal point process with a self-adjoint correlation operator K is the joint spectral measure of the particle density ρ(x) = A+(x)A−(x) (x ∈ X), where the operator-valued distributions A+(x), A−(x) come from a gauge-invariant quasi-free representation of the canonical anticommutation relations (CAR). If the space X is discrete and divided into two disjoint parts, X1 and X2, by exchanging particles and holes on the X2 part of the space, one obtains from a determinantal point process with a self-adjoint correlation operator K the determinantal point process with the J -self-adjoint correlation operator K = K P1 + (1 − K)P2. Here Pi is the orthogonal projection of L2(X,σ) onto L2(Xi,σ). In the case where the space X is continuous, the exchange of particles and holes makes no sense. Instead, we apply a Bogoliubov transformation to a gauge-invariant quasi-free representation of the CAR. This transformation acts identically on the X1 part of the space and exchanges the creation operators A+(x) and the annihilation operators A−(x) for x ∈ X2. This leads to a quasi-free representation of the CAR, which is not anymore gauge-invariant. We prove that the joint spectral measure of the corresponding particle density is the determinantal point process with the correlation operator K.
published_date 2023-08-05T13:30:06Z
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