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Mathematics

Modifying the Lower Bound of the Heisenberg Uncertainty Relation with Special Deformed Boson Algebra

Filippo Giraldi

Published Apr 29, 2026• 0 citations
Publisher

Abstract

The Heisenberg uncertainty relation, concerning the product of the standard deviations of position and momentum, is studied for a quantum oscillator characterized by deformed boson algebra. Special forms of the deformed number operator are determined such that the ground state and any deformed coherent state of the oscillator under study exhibit minimum uncertainty, as the canonical boson algebra provides, but the product of the corresponding standard deviations is less than the canonical lower bound, ℏ/2. Above all, each standard deviation and, consequently, the uncertainty become arbitrarily small for the ground state, other states of the Fock basis, and deformed coherent states with arbitrarily small magnitude of the label, if the parameters defining the special deformed number operator are properly chosen.

DOI
10.3390/math14091505

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The Heisenberg uncertainty relation, concerning the product of the standard deviations of position and momentum, is studied for a quantum oscillator characterized by deformed boson algebra. Special forms of the deformed number operator are determined such that the ground state and any deformed coherent state of the oscillator under study exhibit minimum uncertainty, as the canonical boson algebra provides, but the product of the corresponding standard deviations is less than the canonical lower bound, ℏ/2. Above all, each standard deviation and, consequently, the uncertainty become arbitrarily small for the ground state, other states of the Fock basis, and deformed coherent states with arbitrarily small magnitude of the label, if the parameters defining the special deformed number operator are properly chosen.
We introduce the deformed boson operators which satisfy a deformed boson algebra in some special types of generalized noncommutative phase space. Based on the deformed boson algebra, we construct coherent state representations. We calculate the variances of the coordinate operators on the coherent states and investigate the corresponding Heisenberg uncertainty relations. It is found that there are some restriction relations of the noncommutative parameters in these special types of noncommutative phase space.
This work establishes a link between deformed angular momentum operators and modifications of the quantum uncertainty principle. Starting from an algebra in which the standard angular momentum operators are deformed by two parameters, [Formula: see text] and [Formula: see text], we construct the corresponding momentum operators via [Formula: see text] where the radial operator [Formula: see text] is determined by the requirement that the standard momentum operator is recovered in the limit [Formula: see text], [Formula: see text]. The resulting commutation relations take the form [Formula: see text] where [Formula: see text] is a position-dependent deformation matrix. This leads to an uncertainty relation [Formula: see text] which is state-dependent, anisotropic and position-dependent. We use the term Generalized Uncertainty Principle (GUP) in its broad sense referring to any modification of the Heisenberg relation motivated by quantum-gravity ideas. However, the model does not yield a fixed minimal length of the kind familiar from conventional GUP models (e.g. [Formula: see text]). In the isotropic, limit the trace of the deformation matrix simplifies the bound to [Formula: see text] which can be considered as a rescaling of the uncertainty constant. Our work thus provides a geometric origin for modified uncertainty relations from symmetry breaking in the rotation group, and offers a framework in which quantum-gravity effects manifest themselves through state- and direction-dependent measurement limits, without necessarily introducing a universal minimal length.
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Several quantum gravity theories predict a minimal length at the order of magnitude of the Planck length, under which the concepts of space and time lose their physical meaning. In quantum mechanics, the insurgence of such a minimal length can be described by introducing a modified position-momentum commutator, which in turn yields a generalized uncertainty principle, where the uncertainty on position measurements has a lower bound. The value of the minimal length is not predicted by theories and must be estimated experimentally. In this paper, we address the quantum bound to the estimability of the minimal uncertainty length by performing measurements on a harmonic oscillator, which is analytically solvable in the deformed algebra induced by the deformed commutation relations.
Uncertainty relations provide fundamental limits on what can be said about the properties of quantum systems. For a quantum particle, the commutation relation of position and momentum observables entails Heisenberg's uncertainty relation. A third observable is presented which satisfies canonical commutation relations with both position and momentum. The resulting triple of pairwise canonical observables gives rise to a Heisenberg uncertainty relation for the product of three standard deviations. We derive the smallest possible value of this bound and determine the specific squeezed state which saturates the triple uncertainty relation. Quantum optical experiments are proposed to verify our findings.
We consider the time-dependent bi-coherent states that are essentially the Gazeau-Klauder coherent states for the two dimensional noncommutative harmonic oscillator. Starting from some q-deformations of the oscillator algebra for which the entire deformed Fock space can be constructed explicitly, we define the q-deformed bi-coherent states. We verify the generalized Heisenberg’s uncertainty relations projected onto these states. For the initial value in time, the states are shown to satisfy a generalized version of Heisenberg’s uncertainty relations. For the initial value in time and for the parameter of noncommutativity θ = 0, the inequalities are saturated for the simultaneous measurement of the position-momentum observables. When the time evolves, the uncertainty products are different from their values at the initial time and do not always respect the generalized uncertainty relations.