A unifying perspective on the Moyal and Voros products and their physical meanings

Abstract

The Moyal and Voros formulations of non-commutative quantum field theory has been a point of controversy in the recent past. Here we address this issue in the context of non-commutative non-relativistic quantum mechanics. In particular we show that the two formulations simply correspond to two different representations associated with two different choices of basis on the quantum Hilbert space. From a mathematical perspective the two formulations are therefore completely equivalent, but we also argue that only the Voros formulaton admits a consistent physical interpretation. These considerations are elucidated by considering the free particle transition amplitude in the two representations.

arXiv comment10 pages

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• Non-commutative Quantum Mechanics in Three Dimensions and Rotational Symmetry
• Aspects of finite electrodynamics in D=3 dimensions
• Spectral triplets, statistical mechanics and emergent geometry in non-commutative quantum mechanics
• On the role of Schwinger's SU(2) generators for Simple Harmonic Oscillator in 2D Moyal plane
• Fingerprints of the quantum space-time in time-dependent quantum mechanics: An emergent geometric phase
• Spectral Distance on Lorentzian Moyal Plane
• Revisiting Connes' Finite Spectral Distance on Non-commutative Spaces: Moyal Plane and Fuzzy Sphere
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