Connes spectral distance and nonlocality of generalized noncommutative phase spaces

Abstract

We study the Connes spectral distance of quantum states and analyse the nonlocality of a 4D generalized noncommutative phase space. By virtue of the Hilbert-Schmidt operatorial formulation, we obtain the Dirac operator and construct a spectral triple corresponding to the noncommutative phase space. Based on the ball condition, we obtain some constraint relations about the optimal elements, and then calculate the Connes spectral distance between two Fock states. Due to the noncommutativity, the spectral distances between Fock states in generalized noncommutative phase spaces are shorter than those in normal phase spaces. This shortening of distances implies some type of nonlocality caused by the noncommutativity. These spectral distances in the 4D generalized noncommutative phase space are additive and satisfy the normal Pythagoras theorem. When the noncommutative parameters go to zero, the results return to those in normal quantum phase spaces.