Properties of states on Weyl algebra with variable multiplication law

Abstract

We consider possible quantum effects for infinite systems implied by variations of the multiplication law in the algebra of observables. Using the algebraic formulation of quantum theory, we study the behavior of states ω under changes in the defining relations of the canonical commutation relations (CCR-algebra). These defining relations of the multiplication law depend explicitly on the symplectic form σ, which encodes commutation relations of canonical field operators. We consider the change in this form given by simple rescaling of σ by a positive parameter h. We analyze to what extent changes in h preserve the original state space (this gives restrictions on the admissible changes in the scaling parameter h) and which properties have original quantum states ω as states on the new algebra. We answer such questions for the quasi-free states. We show that any universally invariant state can be interpreted as a convex combination of Fock states with different values of constant h. The second important class of states we study is the KMS-state; here, the rescaling alters in a nontrivial way the relevant dynamics. We also show that it is possible to go beyond the limits restricting the changes in h, but then one has to restrict the CCR-algebra to a subalgebra.