Unitary equivalence in Generalized Uncertainty Principle theories
Sebastiano Segreto, Matteo Bruno
TL;DR
The work addresses unitary equivalence in one-dimensional Generalized Uncertainty Principle (GUP) theories by formalizing a GUP theory as a triple $(\mathcal{H}, q, p)$ with a deformed commutator $[q,p]=i\hbar f(p)$. It constructs an explicit unitary map between the Kempf–Mangano–Mann representation and the standard $(Q,P)$ realization, proving dynamical and spectral equivalence for typical Hamiltonians (e.g., the harmonic oscillator) and for free fall, using Gel'fand triples for rigor. It also demonstrates inequivalence with polymer-like PQM representations, arguing that a universal Stone–von Neumann generalization may not hold in GUP theories. Together, these results clarify when different GUP realizations describe the same physics and highlight the limitations of equivalence in more exotic representations, paving the way for well-defined quantum-GUP canonical transformations.
Abstract
We analyze the issue of unitary equivalence within Generalized Uncertainty Principle (GUP) theories in the one-dimensional case. For a deformed Heisenberg algebra, its representation in terms of Hilbert space and conjugate operators is not uniquely determined, raising the question of whether different realizations of the same algebra are equivalent and describe the same physics. After proposing a definition of a quantum GUP theory, we establish conditions for unitary equivalence. Using this framework, we rigorously prove that two commonly used representations are unitarily equivalent, specifying the conditions under which this equivalence holds. We demonstrate this equivalence explicitly by providing a unitary map and showing how both GUP formulations yield the same physical results in two examples: the quantum harmonic oscillator and a free-falling particle. Finally, we discuss a case in which equivalence fails, suggesting that a generalization of the Stone-von Neumann theorem may not be possible within the GUP framework under our definition of unitary equivalence.