Abstract
The deformation of the relativistic dispersion relation caused by noncommutative (NC) Quantum Mechanics (QM) is studied using the extended phase-space formalism. The introduction of the additional commutation relations induces Lorentz invariance violation. It is shown that this deformation does not affect the propagation speed of free massless particles. From the deformation of the dispersion relation for massless particles, gamma ray burst data is used to establish an upper bound on the noncommutative parameter, η, namely η≲10−12eV/c. Additionally, a putative metric structure for the noncommutative phase-space is discussed.
Highlights
The deformation of the relativistic dispersion relation caused by noncommutative (NC) Quantum Mechanics (QM) is studied using the extended phase-space formalism
Quantum mechanics may be viewed in the Wigner-Weyl (WW) phase space formalism, which is equivalent to the standard operator-state formalism [9–11]
The introduction of the additional commutation relations in Phase-space NCQM (PSNCQM) produces a deformation of this symplectic form, which captures all effects of NCQM through a modified Moyal product
Summary
Lorentz symmetry is one of the foundational principles of modern physics and, in particular, of Special and General Relativity. Noncommutative (NC) quantum mechanics breaks this symmetry explicitly by introducing additional commutation relations, which provide an intrinsic momentum and length scale to the theory As a consequence, this would lead to a deformation of the dispersion relation due to such scales. [12], a noncommutative extension of this formalism is possible and, besides being fully consistent, it is equivalent to the deformation of the HW algebra This formalism embeds the quantum effects on the defined ⋆-product, which depends on the symplectic form of the phase-space. The introduction of the additional commutation relations in PSNCQM produces a deformation of this symplectic form, which captures all effects of NCQM through a modified Moyal product This suggests that the effect of PSNCQM is to alter the geometric symplectic structure of the phase-space manifold and this will be used in the present work in order to compute the deformation in the relativistic dispersion relation. Indices for p and k will be denoted as subscripts
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