Gauge-invriant 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 $\Re q$ 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 $\Re q<0$, the operator $T$ additionally satisfies $0\le T\le -1/\Re q$. Further, for $T={\kappa }^{2}1$ ($\kappa >0$), 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 $\Re q$.

References in corpus (4)

• Anyons and the quantum Hall effect - a pedagogical review
• On the Fock space for nonrelativistic anyon fields and braided tensor products
• Non-commutative Lévy processes for generalized (particularly anyon) statistics
• L2 dimensions of spaces of braid-invariant harmonic forms

Cited by in corpus (3)

• Fock representations of $Q$-deformed commutation relations
• Fock representations of multicomponent (particularly non-Abelian anyon) commutation relations
• Quasi-free states on a class of algebras of multicomponent commutation relations