Gauge-Invariant Quasi-Free States on the Algebra of the Anyon Commutation Relations

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 ${\mathfrak{R}q}$ if ${x^1=y^1}$. Let ${\partial_x^+}$, ${\partial_x^-}$ (${x\in X}$) be operator-valued 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 ${\partial^-_x\partial_y^+=\delta(x-y)+Q(x,y)\partial_y^+\partial^-_x}$ 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 operator-valued integrals of ${\partial_x^+}$, ${\partial_x^-}$. We construct a class of gauge-invariant quasi-free states on the ACR algebra. Each state from this class is completely determined by a positive self-adjoint 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 ${\mathfrak{R}q < 0}$, 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 operator-valued 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 ${\mathfrak{R}q}$.