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Hafnian point processes and quasi-free states on the CCR algebra
Maryam Gharamah Ali Alshehri,
Eugene Lytvynov 
Infinite Dimensional Analysis, Quantum Probability and Related Topics, Volume: 25, Issue: 1
Swansea University Author:
Eugene Lytvynov
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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...
| Published in: | Infinite Dimensional Analysis, Quantum Probability and Related Topics |
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| ISSN: | 0219-0257 1793-6306 |
| Published: |
World Scientific Pub Co Pte Ltd
2022
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| Online Access: |
Check full text
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| URI: | https://cronfa.swan.ac.uk/Record/cronfa58782 |
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| 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 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. |
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| Keywords: |
Hafnian point process; Cox process; permanental point process; quasi-free state on CCR algebra |
| College: |
Faculty of Science and Engineering |
| Issue: |
1 |