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arXiv:1905.01406 · Journal of Physics A Mathematical and Theoretical

Uncertainty relations for a non-canonical phase-space noncommutative algebra

Nuno C Dias, João Nuno Prata

Published Apr 15, 2019• 2 citations
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Abstract

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.

DOI
10.1088/1751-8121/ab194b

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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
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.
The commutation relations, uncertainty relations, and spectra of position and momentum operators were studied within the framework of quantum group symmetric Heisenberg algebras and their (Bargmann) Fock representations. As an effect of the underlying noncommutative geometry, a length and a momentum scale appear, leading to the existence of nonzero minimal uncertainties in the positions and momenta. The usual quantum mechanical behavior is recovered as a limiting case for not too small and not too large distances and momenta.
Our main focus is to explore different models in noncommutative spaces in higher dimensions. We provide a procedure to relate a three dimensional q-deformed oscillator algebra to the corresponding algebra satisfied by canonical variables describing non-commutative spaces. The representations for the corresponding operators obey algebras whose uncertainty relations lead to minimal length, areas and volumes in phase space, which are in principle natural candidates of many different approaches of quantum gravity. We study some explicit models on these types of noncommutative spaces, first by utilising the perturbation theory, later in an exact manner. In many cases the operators are not Hermitian, therefore we use PT -symmetry and pseudo-Hermiticity property, wherever applicable, to make them self-consistent. Apart from building mathematical models, we focus on the physical implications of noncommutative theories too. We construct Klauder coherent states for the perturbative and nonperturbative noncommutative harmonic oscillator associated with uncertainty relations implying minimal lengths. In both cases, the uncertainty relations for the constructed states are shown to be saturated and thus imply to the squeezed coherent states. They are also shown to satisfy the Ehrenfest theorem dictating the classical like nature of the coherent wavepacket. The quality of those states are further underpinned by the fractional revival structure. More investigations into the comparison are carried out by a qualitative comparison between the dynamics of the classical particle and that of the coherent states based on numerical techniques. The qualitative behaviour is found to be governed by the Mandel parameter determining the regime in which the wavefunctions evolve as soliton like structures.
Abstract Realizing the canonical commutation relations (CCR) [N, Θ] = - i as N = - i d/dϑ and Θ to be the multiplication by ϑ on the Hilbert space of square integrable functions on [0, 2π], in the physical literature there seems to be some contradictions concerning the Heisenberg uncertainty principle ⟨ΔN⟨ ⟨ΔΘ⟨ ≥ 1/4. The difficulties may be overcome by a rigorous mathematical analysis of the domain of state vectors, for which Heisenberg's inequality is valid. It is shown that the exponentials exp {i t N} and exp{i sΘ} satisfy some commutation relations, which are not the Weyl relations. Finally, the present work aims at a better understanding of the phase and number operators in non-Fock representations.
The structure of the state-vector space of identical bosons in noncommutative spaces is investigated. To maintain Bose–Einstein statistics the commutation relations of phase space variables should simultaneously include coordinate-coordinate non-commutativity and momentum-momentum non-commutativity, which leads to a kind of deformed Heisenberg-Weyl algebra. Although there is no ordinary number representation in this state-vector space, several set of orthogonal and complete state-vectors can be derived which are common eigenvectors of corresponding pairs of commuting Hermitian operators. As a simple application of this state-vector space, an explicit form of two-dimensional canonical coherent state is constructed and its properties are discussed.