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Particle-Hole Transformation in the Continuum and Determinantal Point Processes
Communications in Mathematical Physics, Volume: 403, Issue: 2, Pages: 627 - 659
Swansea University Author:
Eugene Lytvynov
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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...
Published in: | Communications in Mathematical Physics |
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ISSN: | 0010-3616 1432-0916 |
Published: |
Springer Science and Business Media LLC
2023
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Online Access: |
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URI: | https://cronfa.swan.ac.uk/Record/cronfa63994 |
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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. 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. |
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College: |
Faculty of Science and Engineering |
Funders: |
Swansea University |
Issue: |
2 |
Start Page: |
627 |
End Page: |
659 |