Physics – Mathematical Physics
Scientific paper
2012-04-21
Physics
Mathematical Physics
29 pages
Scientific paper
Noncommutative quantum mechanics can be considered as a first step in the construction of noncommutative quantum field theory of generic form. In this paper we discuss the mathematical framework of the non-relativistic quantum mechanics with coordinate operators satisfying the algebra $[\hat {x}^{i},\hat{x}^{j}] =i\theta\hat{\omega}_{q}^{ij}(\hat {x}) $, where $\hat{\omega}_{q}^{ij}(\hat{x}) $ is some given operator describing the noncommutativity of the space and $\theta$ is the parameter of noncommutativity. First we introduce the momenta operators $\hat{p}_{i}$ conjugated to the corresponding coordinates and construct the complete algebra of commutation relations between these operators as a deformation in $\theta$ of a standard Heisenberg algebra. Then we construct a polydifferential representation of this algebra as a deformation of coordinate representation of the Heisenberg algebra. To fix the arbitrariness in our construction we require that the phase space operators should be self-adjoint with respect to the trace functional defined on the above algebra. As an example we consider a free particle in curved noncommutative space.
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