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arXiv:2206.15422 · Classical and Quantum Gravity

Space and time transformations with a minimal length

Pasquale Bosso

Published Jan 19, 2023• 8 citations
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Abstract

Abstract Phenomenological studies of quantum gravity have proposed a modification of the commutator between position and momentum in quantum mechanics to introduce a minimal uncertainty in position in quantum mechanics. In the present work, we show the influence of space and time transformations in shaping quantities such as momentum, energy, and their relations with the generators of transformations. Thus, such an influence determines, among other aspects, the time evolution of a quantum system. In the exemplary case of Galilean transformations, the Schrödinger equation is identical to the ordinary case.

DOI
10.1088/1361-6382/acb4d5

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Abstract Phenomenological studies of quantum gravity have proposed a modification of the commutator between position and momentum in quantum mechanics to introduce a minimal uncertainty in position in quantum mechanics. In the present work, we show the influence of space and time transformations in shaping quantities such as momentum, energy, and their relations with the generators of transformations. Thus, such an influence determines, among other aspects, the time evolution of a quantum system. In the exemplary case of Galilean transformations, the Schrödinger equation is identical to the ordinary case.
A proper deformation of the underlying coordinate and momentum commutation relations in quantum mechanics provides a phenomenological approach to account for the influence of gravity on small scales. Introducing the squared momentum term results in a generalized uncertainty principle, which limits the minimum uncertainty in particle position to the Planck length. However, such a deformation of the commutator significantly changes the formalism, making it separate from the canonical formalism of quantum mechanics. In this study, it is shown that the deformed algebra of position and momentum operators can be incorporated into the framework of ordinary quantum mechanics.
Abstract The problem of time in quantum mechanics (QM) concerns the fact that in the Schrödinger equation time is a parameter, not an operator. Pauli's objection to a time–energy uncertainty relation analogue to the position–momentum one, conjectured by Heisenberg early on, seemed to exclude the existence of such an operator. However Dirac's formulation of an electron's relativistic QM does allow the introduction of a dynamical time operator that is self-adjoint. Consequently, it can be considered as the generator of a unitary transformation of the system, as well as an additional system observable subject to uncertainty. In the present paper these aspects are examined within the standard framework of relativistic QM.
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. Therefore, phenomenological approaches to quantum gravity introduced models spelled as modifications of quantum mechanics including a minimal length. 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. As a consequence, the (quasi-)position representation acquires numerous issues. For example, the position operator is no longer a multiplicative operator, and the momentum of a free particle does not correspond directly to its wave number. Here, we will review such issues, clarifying aspects of minimal length models, with particular reference to the representation of the position operator. Furthermore, we will show how such a (quasi-)position description of quantum mechanical models with a minimal length affects results concerning simple systems.
Abstract Many theories of quantum gravity suggest that there should exist a minimal scale with which it is possible to measure distances and time. We present a formalism of quantum mechanics exchibiting a minimal length and time scale. The work is based on the Page-Wootters formalism which gives a possible solution to the problem of time in quantum mechanics. This formalism is an example of a relational approach to quantum dynamics. It is based on the idea that reference frames with respect to which motion is described are themselves physical systems interacting with the degrees of freedom they wish to describe. Quantum dynamics is described in terms of quantum correlations between a clock and a system. In our work we modify the commutation relations between the time and frequency operators leading to a minimal uncertainty of time measurement. This results in a modified version of the Schrödinger equation. A minimal time scale also allows us to introduce a discrete Schrödinger equation describing time evolution on a lattice. We show that both descriptions of time evolution are equivalent. We further investigate the e ects of such modi cation on a couple simple quantum systems.
Several approaches to quantum gravity imply the presence of a minimal measurable length at high energies. This is in tension with the Heisenberg Uncertainty Principle. Such a contrast is then considered in phenomenological approaches to quantum gravity by introducing a minimal length in quantum mechanics via the Generalized Uncertainty Principle. Several features of the standard theory are affected by such a modification. For example, position eigenstates are no longer included in models of quantum mechanics with a minimal length. Furthermore, while the momentum-space description can still be realized in a relatively straightforward way, the (quasi-)position representation acquires numerous issues. Here, we will review such issues, clarifying aspects regarding models with a minimal length. Finally, we will consider the effects of such models on simple quantum mechanical systems.
The existence of a minimum measurable length scale was suggested by various theories of quantum gravity, string theory and black hole physics. Motivated by this, we examine a quantum theory exhibiting a minimum measurable time scale. We use the Page–Wootters formalism to describe time evolution of a quantum system with the modified commutation relations between the time and frequency operator. Such modification leads to a minimal uncertainty in the measurement of time. This causes breaking of the time-translation symmetry and results in a modified version of the Schrödinger equation. A minimal time scale also allows us to introduce a discrete Schrödinger equation describing time evolution on a lattice. We show that both descriptions of time evolution are equivalent. We demonstrate the developed theory on a couple simple quantum systems.