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arXiv:gr-qc/0610072 · Classical and Quantum Gravity

Hamiltonian and physical Hilbert space in polymer quantum mechanics

Alejandro Corichi, Tatjana Vukašinac, José A Zapata

Published Mar 6, 2007• 53 citations
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Abstract

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.

DOI
10.1088/0264-9381/24/6/008

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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.
"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 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.
This thesis addresses the representations of the Weyl algebra of quantum mechanics with an explicitly Born reciprocal structure and the dynamics of physical systems in these representations. In this context, we discuss both the regular modular representations based on Aharonov's modular variables, as well as the irregular modular polymer representations inspired by the spin network representation in loop quantum gravity. \n \nWe introduce the modular space as the quantum configuration space corresponding to the modular representation and treat it as an alternative to the classical configuration space. The modular representations have a built-in Abelian gauge symmetry. We discuss the singular limits in which the modular representation converges to the Schrodinger and momentum representations, and show that these limits require two distinct fixings of the modular gauge symmetry. \n \nIn order to explore the propagation of quantum states in the modular space, we construct a Feynman path integral for the harmonic oscillator explicitly as the transition amplitude between two modular states. The result is a functional integral over the compact modular space, which shows novel features of the dynamics such as winding modes, an Aharonov-Bohm phase, and a new action on the modular space. We compare the stationary trajectories that extremize this modular action to the phase space solution of Hamilton's equations in the Schrodinger representation and find a new translation symmetry. We also identify the other symmetries of the harmonic oscillator in the modular action and show their correspondence to the classical case. \n \nWe generalize the relationship between the classical Hamiltonian function and the modular action by postulating a new modular Legendre transform. For demonstration we apply this transformation on the Kepler problem and reformulate it in terms of modular variables. \n \nWe then switch our focus to the representations of the Weyl algebra that violate the assumptions of the Stone-von Neumann theorem. In a similar fashion to the polymer representations studied as a toy model for quantum gravity, we polymerize the modular representation to obtain an inequivalent representation called the modular polymer (MP) representation. \n \nThe new MP representation lacks both position and momentum operators, therefore the Hamiltonian of the harmonic oscillator is regularized with the Weyl operators and the non-separable MP Hilbert space is split into superselection sectors. The solutions to the Schrodinger equation fall into two categories depending on whether the ratio between the modular scale and the polymeric regularization scale is a rational or irrational number. We find a discrete and finite energy spectrum in each superselection sector in the former case, and a continuous, bounded one in the latter.
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.