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Position in models of quantum mechanics with a minimal length
1  University of Lethbridge

Published: 22 February 2021 by MDPI in 1st Electronic Conference on Universe session High Energy Physics

Candidate theories of quantum gravity predict the presence of a minimal measurable length at high energies. Such feature is in contrast with the Heisenberg Uncertainty Principle, which does not present any minimal length. Therefore, phenomenological approaches to quantum gravity introduced models spelled as modifications of quantum mechanics including a minimal length. Such models are often described by modifying the commutation relation between position and momentum. The effects of such modification are expected to be relevant at large energies/small lengths. One first consequence is that position eigenstates are not included in such models due to the presence of a minimal uncertainty in position. Furthermore, depending on the particular modification of the position-momentum commutator, when such models are considered from momentum space, the position operator is changed and a measure factor appears to let the position operator be self-adjoint. Although such modifications in momentum space represent small complication, at least formally, the (quasi-)position representation acquires numerous issues, source of misunderstandings. In fact, such representation is formally similar to that in which states are described in terms of Gaussian states in standard quantum mechanics. Consequently, the position operator is no longer a multiplicative operator and the momentum of a free particle does not correspond directly to its wave-number, with a consequent modification of the de Broglie relation.

In this presentation, I will review such issues, clarifying some of the aspects of minimal length models, with particular reference to the representation of the position operator. Furthermore, I will show how such a (quasi-)position description of quantum mechanical models with a minimal length affects results concerning simple systems, resulting in effects not accounted for in the literature.

Keywords: quantum gravity phenomenology; minimal length; generalized uncertainty principle; quantum mechanics