Journal article 697 views 71 downloads
GaugeInvariant QuasiFree States on the Algebra of the Anyon Commutation Relations / Eugene Lytvynov
Communications in Mathematical Physics, Volume: 351, Issue: 2, Pages: 653  687
Swansea University Author: Eugene Lytvynov

PDF  Accepted Manuscript
Download (454.63KB)
DOI (Published version): 10.1007/s0022001627865
Abstract
Let $X=\mathbb R^2$ and let $q\in\mathbb C$, $q=1$. For $x=(x^1,x^2)$ and $y=(y^1,y^2)$ from $X^2$, we define a function $Q(x,y)$ to be equal to $q$ if $x^1<y^1$, to $\bar q$ if $x^1>y^1$, and to $\Re q$ if $x^1=y^1$. Let $\partial_x^+$, $\partial_x^$ ($x\in X$) be operatorvalued distribut...
Published in:  Communications in Mathematical Physics 

ISSN:  00103616 14320916 
Published: 
2017

Online Access: 
Check full text

URI:  https://cronfa.swan.ac.uk/Record/cronfa29639 
Tags: 
Add Tag
No Tags, Be the first to tag this record!

Abstract: 
Let $X=\mathbb R^2$ and let $q\in\mathbb C$, $q=1$. For $x=(x^1,x^2)$ and $y=(y^1,y^2)$ from $X^2$, we define a function $Q(x,y)$ to be equal to $q$ if $x^1<y^1$, to $\bar q$ if $x^1>y^1$, and to $\Re q$ if $x^1=y^1$. Let $\partial_x^+$, $\partial_x^$ ($x\in X$) be operatorvalued distributions such that $\partial_x^+$ is the adjoint of $\partial_x^$. We say that $\partial_x^+$, $\partial_x^$ satisfy the anyon commutation relations (ACR) if $\partial^+_x\partial_y^+=Q(y,x)\partial_y^+\partial_x^+$ for $x\ne y$ and $\partial^_x\partial_y^+=\delta(xy)+Q(x,y)\partial_y^+\partial^_x$ for $(x,y)\in X^2$. In particular, for $q=1$, the ACR become the canonical commutation relations and for $q=1$, the ACR become the canonical anticommutation relations. We define the ACR algebra as the algebra generated by operatorvalued integrals of $\partial_x^+$, $\partial_x^$. We construct a class of gaugeinvariant quasifree states on the ACR algebra. Each state from this class is completely determined by a positive selfadjoint operator $T$ on the real space $L^2(X,dx)$ which commutes with any operator of multiplication by a bounded function $\psi(x^1)$. In the case $\Re q<0$, the operator $T$ additionally satisfies $0\le T\le 1/\Re q$. Further, for $T=\kappa^2\mathbf 1$ ($\kappa>0$), we discuss the corresponding particle density $\rho(x):=\partial_x^+\partial_x^$. For $\Re q\in(0,1]$, using a renormalization, we rigorously define a vacuum state on the commutative algebra generated by operatorvalued integrals of $\rho(x)$. This state is given by a negative binomial point process. A scaling limit of these states as $\kappa\to\infty$ gives the gamma random measure, depending on parameter $\Re q$. 

College: 
College of Science 
Issue: 
2 
Start Page: 
653 
End Page: 
687 