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

## TL;DR

This paper investigates how variations in Planck's constant affect states on Weyl algebra, revealing invariance conditions, state transformations, and the impact on quantum dynamics and algebraic structures.

## Contribution

It analyzes the effects of changing Planck's constant on CCR-algebra states, especially quasi-free and KMS-states, and explores conditions for invariance and algebraic restrictions.

## Key findings

- Universal invariant states are convex combinations of Fock states with different Planck's constants.
- Rescaling Planck's constant nontrivially alters the dynamics of KMS-states.
- Restrictions on the algebra are necessary to accommodate larger variations in Planck's constant.

## 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.