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AIP conference proceedings

Free particle propagator in polymer quantum mechanics

Ernesto Flores-González, Hugo Aurelio Morales-Técotl, Juan D. Reyes

Published Jan 1, 2012• 2 citations
Publisher

Abstract

We construct the quantum propagator of the free particle in the polymer representation of quantum mechanics, both in position and momentum polarization. We verify the propagator satisfies standard properties such as composition, and that it reduces to the usual Schrödinger propagator in the limit of small parameter μ0, the length scale introduced in the polymer dynamics.

DOI
10.1063/1.4748547

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We construct the quantum propagator of the free particle in the polymer representation of quantum mechanics, both in position and momentum polarization. We verify the propagator satisfies standard properties such as composition, and that it reduces to the usual Schrödinger propagator in the limit of small parameter μ0, the length scale introduced in the polymer dynamics.
In this paper, we provide the representation of the symplectic group Sp(2n,R)Sp(2n,\mathbb{R})Sp(2n,R) in polymer quantum mechanics. We derive the propagator of the polymer free particle and the polymer harmonic oscillator without considering a polymer scale. The polymer scale is then introduced to reconcile our results with those expressions for the polymer free particle. The propagator for the polymer harmonic oscillator implies nonunitary evolution.
"Polymer quantum mechanics" is the name given to a quantization scheme inspired by loop quantum gravity in which the configuration space of the theory is chosen to have a discrete topology. Polymer quantization yields a representation of the canonical commutation relations that is genuinely distinct from the conventional "Schrödinger" representation. In this paper, we summarize the main features of polymer quantum mechanics and investigate in detail the polymer quantization of systems with configuration spaces that are classically compact. We show explicitly how using the standard construction of polymer states leads to a Hilbert space of states defined on a finite graph of points. By way of example, we find the exact energy eigenvalues and eigenfunctions for a particle on a ring and a particle in a box defined on such lattices, and discuss similarities and differences from standard Schrödinger quantum mechanics. We also explore the continuum limit of states in these systems, and demonstrate in detail how the exact eigenfunctions in the position representation approach their continuum counterparts.
A recent proposal to connect the loop quantization with the spin foam model for cosmology via the path integral is hereby adapted to the case of mechanical systems within the framework of the so-called polymer quantum mechanics. The mechanical models we consider are deparametrized and thus the group averaging technique is used to deal with the corresponding constraints. The transition amplitudes are written in a vertex expansion form used in the spin foam models, where here a vertex is actually a jump in position. Polymer propagators previously obtained by spectral methods for a nonrelativistic polymer particle, both free and in a box, are regained with this method and as a new result we obtain the polymer propagator of the relativistic particle. All of them reduce to their standard form in the continuum limit for which the length scale parameter of the polymer quantization is taken to be small. Our results are robust thanks to their analytic and exact character which in turn come from the fact that presented models are solvable. They lend support to the vertex expansion scheme of the polymer path integral explored before in a formal way for cosmological models. Some possible future developments are commented upon in the discussion.
Abstract We analyze the polymer representation of quantum mechanics within the deformation quantization formalism. In particular, we construct the Wigner function and the star-product for the polymer representation as a distributional limit of the Schrödinger representation for the Weyl algebra in a Gaussian weighted measure, and we observe that the quasi-probability distribution limit of this Schrödinger representation agrees with the Wigner function for loop quantum cosmology. Further, the introduced polymer star-product fulfills Bohr’s correspondence principle even though not all the operators are well defined in the polymer representation. Finally, within our framework, we also derive a generalized uncertainty principle which resembles the one appearing in different scenarios, including theories with a minimal length.