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arXiv:0704.0007 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Polymer quantum mechanics and its continuum limit

Alejandro Corichi, Tatjana Vukašinac, José Antonio Zapata

Published Aug 21, 2007• 179 citations
arXivHTMLPDF

Abstract

A rather nonstandard quantum representation of the canonical commutation relations of quantum mechanics systems, known as the polymer representation, has gained some attention in recent years, due to its possible relation with Planck scale physics. In particular, this approach has been followed in a symmetric sector of loop quantum gravity known as loop quantum cosmology. Here we explore different aspects of the relation between the ordinary Schr\"odinger theory and the polymer description. The paper has two parts. In the first one, we derive the polymer quantum mechanics starting from the ordinary Schr\"odinger theory and show that the polymer description arises as an appropriate limit. In the second part we consider the continuum limit of this theory, namely, the reverse process in which one starts from the discrete theory and tries to recover back the ordinary Schr\"odinger quantum mechanics. We consider several examples of interest, including the harmonic oscillator, the free particle, and a simple cosmological model.

DOI
10.1103/physrevd.76.044016

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A rather nonstandard quantum representation of the canonical commutation relations of quantum mechanics systems, known as the polymer representation, has gained some attention in recent years, due to its possible relation with Planck scale physics. In particular, this approach has been followed in a symmetric sector of loop quantum gravity known as loop quantum cosmology. Here we explore different aspects of the relation between the ordinary Schr"odinger theory and the polymer description. The paper has two parts. In the first one, we derive the polymer quantum mechanics starting from the ordinary Schr"odinger theory and show that the polymer description arises as an appropriate limit. In the second part we consider the continuum limit of this theory, namely, the reverse process in which one starts from the discrete theory and tries to recover back the ordinary Schr"odinger quantum mechanics. We consider several examples of interest, including the harmonic oscillator, the free particle, and a simple cosmological model.
"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.
In this paper, a version of polymer quantum mechanics, which is inspired by loop quantum gravity, is considered and shown to be equivalent, in a precise sense, to the standard, experimentally tested, Schroedinger quantum mechanics. The kinematical cornerstone of our framework is the so called polymer representation of the Heisenberg-Weyl (H-W) algebra, which is the starting point of the construction. The dynamics is constructed as a continuum limit of effective theories characterized by a scale, and requires a renormalization of the inner product. The result is a physical Hilbert space in which the continuum Hamiltonian can be represented and that is unitarily equivalent to the Schroedinger representation of quantum mechanics. As a concrete implementation of our formalism, the simple harmonic oscillator is fully developed.
In this paper, a version of polymer quantum mechanics, which is inspired by loop quantum gravity, is considered and shown to be equivalent, in a precise sense, to the standard, experimentally tested, Schroedinger quantum mechanics. The kinematical cornerstone of our framework is the so called polymer representation of the Heisenberg-Weyl (H-W) algebra, which is the starting point of the construction. The dynamics is constructed as a continuum limit of effective theories characterized by a scale, and requires a renormalization of the inner product. The result is a physical Hilbert space in which the continuum Hamiltonian can be represented and that is unitarily equivalent to the Schroedinger representation of quantum mechanics. As a concrete implementation of our formalism, the simple harmonic oscillator is fully developed.
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.
One of the key conceptual challenges in quantum gravity is to understand how quantum theory should modify the very notion of spacetime. One way to investigate this question is to study the alternatives to Schrödinger quantum mechanics. The polymer representation, inspired by loop quantum gravity, can be understood as capturing features of discrete spatial geometry. The modular representation, on the other hand, has a built-in unification of position and momentum polarizations via a length scale. In this paper, we introduce the modular polymer representations of the Weyl algebra, in which neither position nor momentum exists as a well-defined operator. As inequivalent representations, they are candidates for describing new physics. We illustrate this by studying the dynamics of the harmonic oscillator as an example, with the prospect of eventually applying this representation to quantum cosmology.