On Multiparameter CAR Flows

Abstract

Let $P$ be a pointed, closed convex cone in ${\mathbb{R}}^d$. We prove that for two pure isometric representations $V^{(1)}$ and $V^{(2)}$ of $P$, the associated CAR flows ${\beta }^{V^{(1)}}$ and ${\beta }^{V^{(2)}}$ are cocycle conjugate if and only if $V^{(1)}$ and $V^{(2)}$ are unitarily equivalent. We also give a complete description of pure isometric representations of $P$ with commuting range projections that give rise to type I CAR flows. We show that such an isometric representation is completely reducible with each irreducible component being a pullback of the shift semigroup $\{S_t{\}}_{t\ge 0}$ on $L^2[0,\infty )$. We also compute the index and the gauge group of the associated CAR flows and show that the action of the gauge group on the set of normalised units need not be transitive.