Superfluids in Polymer Quantum Mechanics

Abstract

The polymer quantization stands for a non-regular representation of quantum mechanics that results non unitarily equivalent to the standard Schrödinger representation. This quantization is obtained by enforcing the unitary implementation of diffeomorphisms by means of non-regular representation techniques to models with a finite number of degrees of freedom. The non-regular nature of the polymer representation implies that only unitary Weyl operators are well-defined and, in consequence, in a particular polarization, the quantum configuration space results discrete by the appearance of a minimal-length parameter (which in the context of loop quantum gravity is usually associated with Planck's length). In this work, we analyze the corrections obtained on a homogeneous one-dimensional Bose gas within the high densities limit by means of the polymer quantization scheme. Thus, starting from the Bogoliubov formalism, we analyze the ground expectation value of the polymer momentum operator in terms of semiclassical states, in order to obtain an analytic expression for the ground state energy of the N-body system, which allows us to solve the pathological behavior commonly associated with the one-dimensional Bose-Einstein condensation through the introduction of finite size effects characterized by the contribution of the polymer corrections. We also discuss the speed of sound in our polymer version of the Bose gas and the corresponding relative shift induced by the introduction of a minimum length parameter. Finally, by considering the idea that the Bose-Einstein condensation phenomenon is closely related to that of superfluidity, we investigate the emergent superfluid behavior in our polymer model by implementing an appropriate Landau's criterion. In this case, we are able to consequently analyze the changes in the critical velocity which defines the limit between the superfluid-condensate regions, thus deducing that the polymer length acts as a kind of pseudo-potential which induces a dissipationless flow associated with the superfluid phase even in the absence of self-interactions.