📝 SummarXiv

Abstract

We analyze the fundamental issue of unitary equivalence within the framework of Generalized Uncertainty Principle (GUP) theories in the one-dimensional case. Given a deformed Heisenberg algebra, its representation in terms of the Hilbert space and quantum conjugate operators is not uniquely determined. This raises the question of whether two different realizations of the same algebra are equivalent and thus whether they describe the same physics. Having agreed on the formal definition of a quantum GUP theory, we establish the conditions that must be satisfied to discuss unitary equivalence. We use this framework to rigorously prove that two of the main representations used in the literature are actually unitarily equivalent, and we specify the conditions that must be met for this equivalence to hold. We explicitly demonstrate unitary equivalence by providing the unitary map and then discuss two relevant physical examples, the quantum harmonic oscillator and the quantum particle in free fall, showing, for each of them, how both GUP formulations yield the same physical results. Finally, we discuss the relationship between these standard GUP representations and the polymer-like representation, inspired by Polymer Quantum Mechanics and related to Loop Quantum Cosmology. We show that, whenever a GUP theory leads to the emergence of a minimal length, these formulations are not unitarily equivalent and therefore describe different quantum physics. This is particularly relevant, as it may indicate that a generalization of the Stone-von Neumann theorem, which holds in ordinary Quantum Mechanics, may not be possible in the GUP framework, at least within our definition of unitary equivalence.