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A coherent-state-based path integral for quantum mechanics on the Moyal plane. (English) Zbl 1106.81050

Summary: Inspired by a recent work that proposes using coherent states to evaluate the Feynman kernel in noncommutative space, we provide an independent formulation of the path-integral approach for quantum mechanics on the Moyal plane, with the transition amplitude defined between two coherent states of mean position coordinates. In our approach, we invoke solely a representation of the noncommutative algebra in terms of commutative variables. The kernel expression for a general Hamiltonian was found to contain Gaussian-like damping terms, and it is non-perturbative in the sense that it does not reduce to the commutative theory in the limit of vanishing \(\theta\) – the noncommutative parameter. As an example, we studied the free particle’s propagator which turned out to be oscillating with period being the product of its mass and \(\theta\). Further, it satisfies the Pauli equation for a charged particle with its spin aligned to a constant, orthogonal B field in the ordinary Landau problem, thus providing an interesting evidence of how noncommutativity can induce spin-like effects at the quantum mechanical level.

MSC:

81R30 Coherent states
81S40 Path integrals in quantum mechanics
81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
81R60 Noncommutative geometry in quantum theory