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arXiv:1207.0858 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Violation of the Robertson-Schrödinger uncertainty principle and noncommutative quantum mechanics

Catarina Bastos, Orfeu Bertolami A, Nuno Costa Dias, João Nuno Prata

Published Nov 19, 2012• 28 citations
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Abstract

We show that a possible violation of the Robertson-Schr\"odinger uncertainty principle may signal the existence of a deformation of the Heisenberg-Weyl algebra. More precisely, we prove that any Gaussian in phase space (even if it violates the Robertson-Schr\"odinger uncertainty principle) will always be a quantum state of an appropriate noncommutative extension of quantum mechanics. Conversely, all canonical noncommutative extensions of quantum mechanics display states that violate the Robertson-Schr\"odinger uncertainty principle.

DOI
10.1103/physrevd.86.105030

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We show that a possible violation of the Robertson-Schr"odinger uncertainty principle may signal the existence of a deformation of the Heisenberg-Weyl algebra. More precisely, we prove that any Gaussian in phase space (even if it violates the Robertson-Schr"odinger uncertainty principle) will always be a quantum state of an appropriate noncommutative extension of quantum mechanics. Conversely, all canonical noncommutative extensions of quantum mechanics display states that violate the Robertson-Schr"odinger uncertainty principle.
Abstract We consider a non-canonical phase-space deformation of the Heisenberg–Weyl algebra that was recently introduced in the context of quantum cosmology. We prove the existence of minimal uncertainties for all pairs of non-commuting variables. We also show that the states which minimize each uncertainty inequality are ground states of certain positive operators. The algebra is shown to be stable and to violate the usual Heisenberg–Pauli–Weyl inequality for position and momentum. The techniques used are potentially interesting in the context of time-frequency analysis.
We consider a non-canonical phase-space deformation of the Heisenberg– Weyl algebra that was recently introduced in the context of quantum cosmology. We prove the existence of minimal uncertainties for all pairs of non-commuting variables. We also show that the states which minimize each uncertainty inequality are ground states of certain positive operators. The algebra is shown to be stable and to violate the usual Heisenberg–Pauli–Weyl inequality for position and momentum. The techniques used are potentially interesting in the context of time-frequency analysis
Abstract We investigate properties of the covariance matrix in the framework of non-commutative quantum mechanics for an one-parameter family of transformations between the familiar Heisenberg–Weyl algebra and a particular extension of it. Employing as a measure of the Robertson–Schrödinger uncertainty principle the linear symplectic capacity of the Weyl ellipsoid (and its dual), we determine its corresponding bounds. Inequalities between the capacities for non-commutative phase-spaces are established. We also present a constructive example based on a simple model to justify our theoretical predictions.
We revisit Ozawa's uncertainty principle (OUP) in the framework of noncommutative (NC) quantum mechanics. We derive a matrix version of OUP accommodating any NC structure in the phase space, and compute NC corrections to lowest order for two measurement interactions, namely the backaction evading quadrature amplifier and noiseless quadrature transducers. These NC corrections alter the nature of the measurement interaction, as a noiseless interaction may acquire noise, and an interaction of independent intervention may become dependent on the object system. However the most striking result is that noncommutativity may lead to a violation of the OUP itself. The NC corrections for the backaction evading quadrature amplifier reveal a new term which may potentially be amplified in such a way that the violation of the OUP becomes experimentally testable. On the other hand, the NC corrections to the noiseless quadrature transducer shows an incompatibility of this model with NC quantum mechanics. We discuss the implications of this incompatibility for NC quantum mechanics and for Ozawa's uncertainty principle.
The Heisenberg uncertainty principle is foundational to quantum mechanics, yet its standard formulation is limited to Hilbert space operator commutators. Recent advances in noncommutative geometry (NCG) allow a reformulation of quantum observables and spacetime itself using operator algebras, providing a deeper framework for uncertainty relations. In this paper, we develop a generalized uncertainty relation using spectral triples, extending the Robertson– Schrödinger inequality into the noncommutative regime. Explicit derivations are given for operator-valued distances, modified commutators, and position–momentum operators in a noncommutative configuration space. Our results reveal the emergence of a minimal measurable length scale, consistent with predictions from quantum gravity, and demonstrate that uncertainty is fundamentally geometric in origin.