A coherent-state-based path integral for quantum mechanics on the Moyal plane

Abstract

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 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 $θ$ - 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 $θ$. 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.