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arXiv:1309.3183 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Singularity resolution by lattice shifts in discretized quantum mechanics

Jorma Louko, Samuel P. Philpott, Matthew D. Waller

Published Feb 19, 2014• 0 citations
arXivHTMLPDF

Abstract

We investigate the robustness of singularity avoidance mechanisms in nonrelativistic quantum mechanics on the discretized real line when lattice points are allowed to approach a singularity of the classical potential. We consider the attractive Coulomb potential and the attractive scale-invariant potential, on an equispaced parity-noninvariant lattice and on a nonequispaced parity-invariant lattice, and we examine the energy eigenvalues by a combination of analytic and numerical techniques. While the lowest one or two eigenvalues descend to negative infinity in the singular limit, we find that the higher eigenvalues remain finite and form degenerate pairs, close to the eigenvalues of a theory in which a lattice point at the singularity is regularized either by Thiemann's loop quantum gravity singularity-avoidance prescription or by a restriction to the odd-parity sector. The approach to degeneracy can be reproduced from a nonsingular discretized half-line quantum theory by tuning a boundary condition parameter. The results show that Thiemann's singularity-avoidance prescription and the discretized half-line boundary condition reproduce quantitatively correct features of the singular limit spectrum apart from the lowest few eigenvalues.

DOI
10.1103/physrevd.89.044032

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We present a polymer quantization of the $\ensuremath{-}\ensuremath{\lambda}∕{r}^{2}$ potential on the positive real line and compute numerically the bound state eigenenergies in terms of the dimensionless coupling constant $\ensuremath{\lambda}$. The singularity at the origin is handled in two ways: first, by regularizing the potential and adopting either symmetric or antisymmetric boundary conditions; second, by keeping the potential unregularized but allowing the singularity to be balanced by an antisymmetric boundary condition. The results are compared to the semiclassical limit of the polymer theory and to the conventional Schr\"odinger quantization on L2(R+){L}_{2}({\mathbb{R}}_{+})L2​(R+​). The various quantization schemes are in excellent agreement for the highly excited states but differ for the low-lying states, and the polymer spectrum is bounded below even when the Schr\"odinger spectrum is not. We find, as expected, that for the antisymmetric boundary condition the regularization of the potential is redundant: the polymer quantum theory is well defined even with the unregularized potential and the regularization of the potential does not significantly affect the spectrum.
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