Mark Transtrum and Dr. 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 chose to investigate the simple harmonic oscillator in Snyder Space.
The commutation algebra of Snyder Space is much more complicated than the algebras usually considered today. One complication arises because the algebra does not close. The effect of quantizing space in Snyder’s formulation is that the commutator of the position and momentum operators is no longer constant, but adds a term proportional to the square of the momentum. Thus, there are significant deviations from normal quantum mechanics in a single dimension. By contrast modern non-commutative algebras do no affect quantum mechanics in a single dimension. Because of these complications, we began our study by considering the SHO in a single dimension.
We used two methods to investigate the SHO in one-dimensional Snyder space. The first method used the techniques of non-commutative quantum mechanics to map the problem from Snyder space to a corresponding problem in normal quantum mechanics. This corresponding problem can be considered a SHO with a small perturbation caused by the effects of quantized spacetime. The energy eigenvalues can then be found by using perturbation theory. We calculated the perturbed energies to fourth order of the square of the fundamental length.
The second method involves solving a differential equation analytically. In normal quantum mechanics one can work in either position space or momentum space. In Snyder space, position space cannot be formulated in terms of differential operators, since the position operator no longer takes on a continuum of values. For this reason we were forced to express the differential equation in momentum space. The solution to the differential equation was straightforward to find and involved the associated Legendre functions. The next step was the most difficult; it was to find the condition that would result in the wave function being square integrable, which in turns defines how the energy levels are quantized. An educated guess for what this condition might be provides us with a tantalizing result that agrees with the result obtained from perturbation theory. Although there are many reasons to believe that this solution is correct, we are still seeking a rigorous demonstration of its veracity.
Having successfully studied the one dimensional SHO, we next turned to the two dimensional SHO. Because of the complex nature of the commutation relation, we were not able to find an analytic solution to the differential equation like we did in a single dimension. So our only method of tackling this problem was converting it to a perturbed SHO in normal quantum mechanics. This technique is complicated in two dimensions because the unperturbed energy levels are degenerate. We successfully found a new basis to remove the degeneracy and computed the perturbed energy levels to second order.
Having found a number of original results, we would like to analyze these results and compare them with those mentioned previously from non-commutative quantum mechanics. The simple nature of the commutation algebra usually considered in the literature leads naturally to a translation from an SHO in non-commutative quantum mechanics to an SHO with a magnetic field in normal quantum mechanics. In our case, it can be shown that this cannot be done. A constant magnetic field or any sort of magnetic field cannot describe the translated problem. In fact, the translated problem involves products of momentum and position in such a way as to make it impossible to equate it with any simple physical system.
An interesting analysis of the one-dimensional result is considering the wavelength of the particle as the energy levels increase. The new formula for the energies requires that they be proportional to the square of the number of the energy level. Consequently, as the energy levels increase, the wavelength in the classically allowed region cannot decrease below the fundamental length. This agrees intuitively with what we mean by quantized spacetime.
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. The experience has been complimentary to my quantum mechanics class required by the physics curriculum. By doing this research in conjunction with the class, I was forced to ask the question with each new topic, “How would this change if space were quantized?” This in turn helped me to successfully solve problems in this project as the research progressed. I am very grateful to ORCA for the generosity of the grant that made this possible.