Polymer quantization versus the Snyder noncommutative space

Abstract

We study a noncanonical Hilbert space representation of the polymer quantum mechanics. It is shown that the Heisenberg algebra gets some modifications in the constructed setup, from which a generalized uncertainty principle will naturally come out. Although the extracted physical results are the same as those obtained from the standard canonical representation, the noncanonical representation may be notable in view of its possible connection with the generalized uncertainty theories suggested by string theory. In this regard, by considering a Snyder-deformed Heisenberg algebra, we show that since the translation group is not deformed, it can be identified with a polymer-modified Heisenberg algebra. In the classical level, it is shown that the noncanonical Poisson brackets are related to their canonical counterparts by means of a Darboux transformation on the corresponding phase space.