Variable Planck's constant and scaling properties of states on Weyl algebra

Abstract

We consider the possible quantum effect for infinite systems produced by variations of the Planck's constant. Using the algebraic formulation of quantum theory we study behaviour of states $\omega$ defined as positive, normalized functionals on the canonical commutation relations algebra (CCR-algebra) under the changes of the defining relations of the CCR. These defining relations of the multiplication in the CCR-algebra depend explicitly on the value of the Planck's constant. We analyse to what extend changes of the $\hbar$ preserve the original state space (this gives restrictions on the admissible changes of the Plank's constant) and what properties have original quantum states $\omega$ 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 Planck's constant. The second important class of states we study are the KMS-states, 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 of the $\hbar$, but then one has to restrict the CCR-algebra to a subalgebra.