Journal article 212 views 44 downloads
Particle-Hole Transformation in the Continuum and Determinantal Point Processes
Communications in Mathematical Physics
Swansea University Author:
Eugene Lytvynov
-
PDF | Version of Record
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
Download (538.33KB)
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 |
|---|---|
| ISSN: | 0010-3616 1432-0916 |
| Published: |
Springer Science and Business Media LLC
2023
|
| Online Access: |
Check full text
|
| URI: | https://cronfa.swan.ac.uk/Record/cronfa63994 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| 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. |
|---|---|
| College: |
Faculty of Science and Engineering |
| Funders: |
Swansea University |