Dmitry Treschev

Quantum observables: algebra and calculus

We consider quantum observables as series in non-commuting generators $\hat x$, $\hat p$. The space of such series turns out to be an infinite-dimensional associative and Lie algebra. We present the concept of convergence for such series. In this language quantum objects turn out to be non-commutative analogs of classical ones. We prove quantum analogs for several basic theorems of classical mechanics.