Noncommutative quantum mechanics: uniqueness of the functional description

Abstract

The generalized Weyl transform of index $\alpha$ is used to implement the time-slice definition of the phase space path integral yielding the Feynman kernel in the case of noncommutative quantum mechanics. As expected, this representation for the Feynman kernel is not unique but labeled by the real parameter $\alpha$. We succeed in proving that the $\alpha$-dependent contributions disappear at the limit where the time slice goes to zero. This proof of consistency turns out to be intricate because the Hamiltonian involves products of noncommuting operators originating from the non-commutativity. The antisymmetry of the matrix parameterizing the non-commutativity plays a key role in the cancelation mechanism of the $\alpha$-dependent terms.

13 pages

References in corpus (7)

• Non-Commutative Quantum Mechanics
• Comments on the Hydrogen Atom Spectrum in the Noncommutative Space
• Noncommutative Quantum Mechanics: The Two-Dimensional Central Field
• Noncommutative Quantum Field Theories
• The noncommutative degenerate electron gas
• Born series and unitarity in noncommutative quantum mechanics
• On the quantum dynamics of non-commutative systems

Cited by in corpus (4)

• A comparative review of four formulations of noncommutative quantum mechanics
• Twist Deformation of Rotationally Invariant Quantum Mechanics
• Noncommutative quantum mechanics as a gauge theory
• From Noncommutative Sphere to Nonrelativistic Spin