Star products and quantum groups in quantum mechanics and field theory

Abstract

Hopf algebras and quantum groups have recently been applied to the analysis of the combinatorics of Feynman graphs in relativistic quantum field theory. On the other hand, in accordance with the program of deformation quantization, the relation between star products and the perturbative expansion in field theory has also been the subject of intensive study. In the present work we clarify the relation between these two approaches. We show how these techniques can be applied in a unified way to quantum systems with a finite number of degrees of freedom and to quantum field theories. In particular, we find that the time-ordered product of quantum fields is the Weyl transform of a certain twisted product. We also show that one can pass from systems involving bosons to systems with fermions, essentially just by replacing the symmetric algebra of the relevant vector space by its exterior algebra.