"2006 Transtrum, Mark"
Transtrum, Mark Snyder Space and Non-Commutative Quantum Mechanics Faculty Mentor: Jean-Francois Van Huele, Physics and Astronomy Non-commutative quantum mechanics is an active field of research in which the operators corresponding to different spatial dimensions are assumed to not commute, as a consequence of quantization of spacetime. This field is important because it investigates the relationships between fundamental physical ideas such as space, time, momentum, and position on very small scales and will provide the basis for experiments probing these small scales in the future. A well-known result from this field of research maps the simple harmonic oscillator (SHO) system in non-commutative coordinates into a corresponding system in normal quantum mechanics. The corresponding system is still a SHO, but with a constant magnetic field superimposed on top of the potential. Results such as these are fascinating and drive the field of non-commutative quantum mechanics to be extremely active. The commutation algebras used are generally not Lorentz-invariant. Since there is no experimental evidence that Lorentz breaking occurs, the domain of applicability of these results is unknown. By contrast, Hartland Snyder published the first formulation of quantum mechanics in a quantized spacetime in 1947 in a way to guarantee Lorentz-invariance. His motivations were quite different from those investigating the field today, and so his formulation is usually only cited for historical reasons. For these reasons, i.e. historical significance and a unique approach to an active field of research, we studied the simple harmonic oscillator in Snyder Space in a project funded by previous ORCA grant. The goal of the current project was understand more fully the relationship between Snyder space and other noncommutative theories prevalent in the literature. This project began with an extensive literature search in the field of noncommutative physics. The field is very active and it would be impossible for someone to study in any depth every result that is published. Because of the shear volume of literature available, this project was very daunting; however, we found many important results. Most importantly, we discovered that Snyder space is much more closely related to modern noncommutative theories than we had supposed. In particular, we learned about two commutation algebras that are intimately connected with Snyder space. The most common algebras considered in the literature are usually associated with the field of NonCommutative Quantum Mechanics (NCQM). The algebras of NCQM essentially replace the vanishing commutators of the Heisenberg algebra with small nonzero constants. These algebras are related to a low-energy prediction of string theory, and therefore, have been studied extensively. Before beginning this project we were aware of these algebras. While structure of the algebra is very different from that of Snyder space, it should be noted that this algebra is being considered as a viable model of nature that could remedy perceived shortcomings of the current theory. Similarly, Snyder space was originally proposed in the same spirit in 1947, as a workable solution to divergence problems of quantum field theory. By contrast, many theories of the 1950s until the present have been considered only out of mathematical curiosity and were not considered practical models of nature. Beyond this similarity, however, there is little connection between Snyder space and NCQM. Another commutation algebra, of which we were unaware at the outset of this project, is known as the Minimal Length Uncertainty Relations (MLUR). Kempf proposed this commutation algebra in the late 1980s and proposed that it could be useful in describing systems with a fundamental length, such as in string theory or quark physics. It turns out that string theorists predicted the same commutation algebra independently. The commutation algebra of MLUR is very closely related to Snyder space. In fact, Snyder space is recovered in the limit that one of the parameters of MLUR approaches zero. Chang and others studied the simple harmonic oscillator with MLUR and found an energy spectrum. The spectrum matches the spectrum derived by Van Huele and myself for the simple harmonic oscillator in Snyder space. Our derivation method was different from Chang’s, and the agreement in spectral formulae vindicates our method. The final algebra that we compared to Snyder space is known as Dynamical Quantization (DQ). This algebra adds a term to the position-momentum commutator proportional to the Hamiltonian of the system. Thus, in the case of the nonrelativistic free particle, DQ is also identical to Snyder space. DQ was proposed as a model of high-energy interactions, such as quark physics, in the early 1980s by Saavedra and others. Interest has been revived recently in this algebra because of the work being done in other areas of modified commutation relations. Saavedra et al. reported finding energy spectra for the simple harmonic oscillator and the infinite square well potentials. Their work inspired us to study the infinite square well in Snyder space, for which we were able to reproduce the spectrum predicted by Saavedra et al. using the methods that we had developed to study the simple harmonic oscillator. Some of the results mentioned here are also mentioned in a paper submitted for publication in the Journal of Physics A. These results have also been presented at the Spring Research Conference of the College of Physical and Mathematical Sciences at BYU. I have also presented these results at a meeting of the Theoretical and Mathematical Physics research group in the Department of Physics and Astronomy at BYU. The most comprehensive report of my results can be found in the fifth chapter of my thesis and the interested reader is referred to that document for more information. I consider this project to have been very successful. It has given me as a student a valuable experience to learn about how theoretical research is performed in quantum mechanics and high- energy theoretical physics. It has given me a window into the many areas of research being performed by individuals outside of BYU and has related my work to theirs. This will be very helpful as I begin my graduate studies at Cornell University this fall and begin looking for a thesis topic. I am very grateful to ORCA for the generosity of the grant that made this possible.