Abstract
We present the path integral techniques in a non-commutative phase space and illustrate the calculation in the case of an exact problem of the coupled oscillator in two dimensions. The non-commutativity, with respect to Poisson (classical) and Heisenberg (quantum) brackets, in this phase space, is governed by two small constant parameters. They characterize the geometric deformation of this space. The path integral is formulated in a mixed representation due to the non-commutativity of the coordinates on one hand and those of the momentum on the other hand. Using a canonical linear transformation in this non-commutative phase space, we retrieve the commutative phase space properties by which the study becomes more suitable. The case of the non-commutative coupled harmonic oscillator in two dimensions is treated and the thermodynamics properties of the assembly of such oscillators are derived too.
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