Coherent states and N dimensional coordinate noncommutativity

Abstract

Considering coordinates as operators whose measured values are expectations between generalized coherent states based on the group SO(N, 1) leads to coordinate noncommutativity together with full N dimensional rotation invariance. Through the introduction of a gauge potential this theory can additionally be made invariant under N dimensional translations. Fluctuations in coordinate measurements are determined by two scales. For small distances these fluctuations are fixed at the noncommutativity parameter while for larger distances they are proportional to the distance itself divided by a very large number. Limits on this number will lbe available from LIGO measurements.

Noncommutative coordinates invariant under rotations and Lorentz transformations

“…This product does respect a ''twisted'' Poincaré invariance [2,3]; implications of such invariance for field theories have been discussed [4] in great detail recently. A different approach, in which the position coordinates are replaced by operators that have nontrivial commutation relations among each other and under rotations transform into each other preserving these commutation relations, has been pursued by this author [5]. In that paper the space coordinates are represented by operators acting on coherent states based (in three space dimensions) on the group SO1; 3.…”

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

“…In [12], noncommutative quantum mechanics has been formulated on the premise that measurement of position operators, or functions of such operators is determined by their expectation values between generalized coherent states [23] based on the group SO(N, 1).…”

“…What is also interesting is that it provides an avenue for noncommutativity on other types of spaces -including compact ones -to be realised quantum mechanically via coherent states. For example, one may be able to analyse the quantum mechanics of the noncommutative fuzzy sphere via generalised coherent states of SU(2) following [12]. Thus, a possible further extension of our work would be to perform coherent state-based path integration in higher dimensions and on certain topologies, for example, on hyperspheres and compare results with the commutative versions.…”

Hydrogen atom in rotationally invariant noncommutative space

“…In this case the author considered an isotropic harmonic oscillator. The way to maintain the N dimensional rotation invariance was considered in [29]. In this article the coordinates are represented by operators.…”