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Hafnian point processes and quasi-free states on the CCR algebra

Maryam Gharamah Ali Alshehri, Eugene Lytvynov Orcid Logo

Infinite Dimensional Analysis, Quantum Probability and Related Topics, Volume: 25, Issue: 1

Swansea University Author: Eugene Lytvynov Orcid Logo

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DOI (Published version): 10.1142/s0219025722500023

Abstract

Let X be a locally compact Polish space and σ a nonatomic reference measure on X (typically X=Rd and σ is the Lebesgue measure). Let X2∋(x,y)↦K(x,y)∈C2×2 be a 2×2-matrix-valued kernel that satisfies KT(x,y)=K(y,x). We say that a point process μ in X is hafnian with correlation kernel K(x,y) if, for...

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Published in: Infinite Dimensional Analysis, Quantum Probability and Related Topics
ISSN: 0219-0257 1793-6306
Published: World Scientific Pub Co Pte Ltd 2022
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URI: https://cronfa.swan.ac.uk/Record/cronfa58782
first_indexed 2021-11-25T11:50:25Z
last_indexed 2025-04-25T04:51:24Z
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spelling 2025-04-24T15:02:50.8485285 v2 58782 2021-11-25 Hafnian point processes and quasi-free states on the CCR algebra e5b4fef159d90a480b1961cef89a17b7 0000-0001-9685-7727 Eugene Lytvynov Eugene Lytvynov true false 2021-11-25 MACS Let X be a locally compact Polish space and σ a nonatomic reference measure on X (typically X=Rd and σ is the Lebesgue measure). Let X2∋(x,y)↦K(x,y)∈C2×2 be a 2×2-matrix-valued kernel that satisfies KT(x,y)=K(y,x). We say that a point process μ in X is hafnian with correlation kernel K(x,y) if, for each n∈N, the nth correlation function of μ (with respect to σ⊗n) exists and is given by k(n)(x1,…,xn)=haf[K(xi,xj)]i,j=1,…,n. Here haf(C) denotes the hafnian of a symmetric matrix C. Hafnian point processes include permanental and 2-permanental point processes as special cases. A Cox process ΠR is a Poisson point process in X with random intensity R(x). Let G(x) be a complex Gaussian field on X satisfying ∫ΔE(∣∣G(x)∣∣2)σ(dx)<∞ for each compact Δ⊂X. Then the Cox process ΠR with R(x)=|G(x)|2 is a hafnian point process. The main result of the paper is that each such process ΠR is the joint spectral measure of a rigorously defined particle density of a representation of the canonical commutation relations (CCRs), in a symmetric Fock space, for which the corresponding vacuum state on the CCR algebra is quasi-free. Journal Article Infinite Dimensional Analysis, Quantum Probability and Related Topics 25 1 World Scientific Pub Co Pte Ltd 0219-0257 1793-6306 Hafnian point process; Cox process; permanental point process; quasi-free state on CCR algebra 19 1 2022 2022-01-19 10.1142/s0219025722500023 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University 2025-04-24T15:02:50.8485285 2021-11-25T11:42:03.4379856 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Maryam Gharamah Ali Alshehri 1 Eugene Lytvynov 0000-0001-9685-7727 2 58782__21686__19f8a1c1dcfd41b797aa53161d50436f.pdf hafnian.pdf 2021-11-25T11:46:30.9356399 Output 407694 application/pdf Accepted Manuscript true 2023-01-19T00:00:00.0000000 true eng https://v2.sherpa.ac.uk/id/publication/9673
title Hafnian point processes and quasi-free states on the CCR algebra
spellingShingle Hafnian point processes and quasi-free states on the CCR algebra
Eugene Lytvynov
title_short Hafnian point processes and quasi-free states on the CCR algebra
title_full Hafnian point processes and quasi-free states on the CCR algebra
title_fullStr Hafnian point processes and quasi-free states on the CCR algebra
title_full_unstemmed Hafnian point processes and quasi-free states on the CCR algebra
title_sort Hafnian point processes and quasi-free states on the CCR algebra
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 Infinite Dimensional Analysis, Quantum Probability and Related Topics
container_volume 25
container_issue 1
publishDate 2022
institution Swansea University
issn 0219-0257
1793-6306
doi_str_mv 10.1142/s0219025722500023
publisher World Scientific Pub Co Pte Ltd
college_str Faculty of Science and Engineering
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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
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description Let X be a locally compact Polish space and σ a nonatomic reference measure on X (typically X=Rd and σ is the Lebesgue measure). Let X2∋(x,y)↦K(x,y)∈C2×2 be a 2×2-matrix-valued kernel that satisfies KT(x,y)=K(y,x). We say that a point process μ in X is hafnian with correlation kernel K(x,y) if, for each n∈N, the nth correlation function of μ (with respect to σ⊗n) exists and is given by k(n)(x1,…,xn)=haf[K(xi,xj)]i,j=1,…,n. Here haf(C) denotes the hafnian of a symmetric matrix C. Hafnian point processes include permanental and 2-permanental point processes as special cases. A Cox process ΠR is a Poisson point process in X with random intensity R(x). Let G(x) be a complex Gaussian field on X satisfying ∫ΔE(∣∣G(x)∣∣2)σ(dx)<∞ for each compact Δ⊂X. Then the Cox process ΠR with R(x)=|G(x)|2 is a hafnian point process. The main result of the paper is that each such process ΠR is the joint spectral measure of a rigorously defined particle density of a representation of the canonical commutation relations (CCRs), in a symmetric Fock space, for which the corresponding vacuum state on the CCR algebra is quasi-free.
published_date 2022-01-19T04:55:26Z
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