Further Representations of the Canonical Commutation Relations

Abstract

We construct a new class of representations of the canonical commutation relations, which generalizes previously known classes. We perturb the infinitesimal generator of the initial Fock representation (i.e. the free quantum field) by a function of the field which is square-integrable with respect to the associated Gaussian measure. We characterize which such perturbations lead to representations of the canonical commutation relations. We provide conditions entailing the irreducibility of such representations, show explicitly that our class of representations subsumes previously studied classes, and give necessary and sufficient conditions for our representations to be unitarily equivalent, resp. quasi-equivalent, with Fock, coherent or quasifree representations

A Fock Sheaf For Givental Quantization

“…However, since H is infinite-dimensional, the quantization has many difficulties. For example, it is known that there are uncountably many irreducible representaions of the infinite-dimensional Heisenberg algebra [54,55], and so the argument in Example 3.2 fails in our situation. The following heuristic discussion will be only used as a motivation 6 .…”

Inequivalent Vacuum States in Algebraic Quantum Theory

“…A remarkable result about CCR for finite degrees of freedom is the Stone-von Neumann uniqueness theorem which states that all irreducible representations of these CCR are unitarily equivalent [40]. On the contrary, CCR of infinite degrees of freedom admit infinitely many inequivalent irreducible representations [21].…”

On the unitary representations of the affine $ax+b$-group, $\widehat{sl}(2,\mathbb{R})$ and their relatives

“…To author's knowledge the classification of the unitary representations of the loop Heisenberg group is not yet known (see e.g. [7] for review of the subject). The same argument as in Theorem 3.4, shows that if λ(u) = 0 for u ∈ [a, b] the representation ρ l λ,k is not irreducible.…”