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arXiv:2403.16893 · The European Physical Journal Plus

Explanation of the generalizations of uncertainty principle from coordinate and momentum space periodicity

Subir Ghosh

Published Jul 2, 2024• 4 citations
arXivHTMLPDF
DOI
10.1140/epjp/s13360-024-05366-x

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Generalized uncertainty principles are able to serve as useful descriptions of some of the phenomenology of quantum gravity effects, providing an intuitive grasp on nontrivial space-time structures such as a fundamental discreteness of space, a universal band limit or an irreducible extendedness of elementary particles. In this article, uncertainty relations for single-particle quantum mechanics are derived by a moment expansion of states for quantum systems with a discrete coordinate and, correspondingly, a periodic momentum. Corrections to standard uncertainty relations are found, with some similarities but also key differences to what is often assumed in this context. The relations provided can be applied to discrete models of matter or space-time, including loop quantum cosmology.
It is generally thought desirable that quantum theory entail an uncertainty relation for time and energy similar to the one for position and momentum. Nevertheless, the existence of such a relation has still remained problematic. Here, it is shown that the problem is due to a confusion between the position coordinates of a point particle (a material system) and the coordinates of a point in space: The time coordinate should be put on a par with the space coordinates, not with the canonical position coordinates of a material system. Whereas quantum mechanics incorporates a Heisenberg uncertainty relation between the canonical position coordinates and their conjugate momenta, there is no reason why a Heisenberg relation should hold between the space coordinates and the canonical momenta, or between the time coordinate and the energy of the system. However, uncertainty relations of a different kind exist between the space coordinates and the total momentum of the system and between the time coordinate and the total energy. These relations are completely similar and may be taken together to form a relativistically covariant set of uncertainty relations. The relation between the time coordinate and the energy implies the well-known relation between the lifetime of a state and its energy spread.
We employ a dressed-state approach to perform a study on the behavior of the uncertainty principle for a system in a heated cavity. We find, in a small cavity for a given temperature, an oscillatory behavior of the momentum-coordinate product $(\ensuremath{\Delta}\phantom{\rule{0.16em}{0ex}}p)\phantom{\rule{0.16em}{0ex}}(\ensuremath{\Delta}\phantom{\rule{0.16em}{0ex}}q)$, which attains periodically finite absolute minimum (maximum) values, no matter how large the elapsed time is. This behavior is in sharp contrast to what happens in free space, where the product $(\ensuremath{\Delta}\phantom{\rule{0.16em}{0ex}}p)\phantom{\rule{0.16em}{0ex}}(\ensuremath{\Delta}\phantom{\rule{0.16em}{0ex}}q)$ tends asymptotically, for each temperature, to a constant value, independent of time.