Functional approach to coherent states in non commutative theories

In many high dimensional noncommutative theories, no state saturates simultaneously all the non trivial Heisenberg uncertainty relations. This differs from the usual theory where the squeezed states possess this property. The important role played by these states when recovering classical mechanics as a limit of quantum theory makes necessary the investigation of the possible generalizations in the noncommutative context. We propose an extension based on a variational principle. The action considered is the sum of the squares of the terms associated to the non trivial Heisenberg uncertainty relations. We first verify that our proposal works in the usual theory: we find the known gaussian functions and, besides them, other states which can be expressed as products of gaussians with specific hypergeometrics. We illustrate our construction in three models defined on a four dimensional phase space: two models endowed with a minimal length uncertainty and the popular case in which the commutators of the positions are constants. In these three models, our proposal leads to second order partial differential equations. We find analytical solutions in specific cases. We briefly discuss how our method may be applied to the fuzzy sphere. To emphasize that the recovering of classical behaviours is not a trivial question in the non commutative context, we show how the difference of structure between the Poisson brackets and the commutators in the theories analyzed here generically leads to a loss of periodicity for the harmonic oscillator.