A pseudodifferential calculus on non-standard symplectic space

Abstract

We introduce a class of pseudodifferential operators e A ! acting on functions defined on an arbitrary symplectic space (R 2n , !). These operators arise naturally when one considers the generalized commutation relations from non-commutative quantum mechanics. The connection with the usual Weyl operator bA with symbol a is made using a family of intertwiners W g defined in terms of the cross-Wigner transform W( f, g). We show that if a belongs to some adequate Shubin symbol classes there is a simple relation between the eigenvalues of e A and those of b A.

Noncommutative mapping from the symplectic formalism

“…Further, it was also extended to induce noncommutativity into commutative systems 24,34,35 . There are a lot of ways to introduce NC 5,15,[36][37][38][39][40][41][42][43] into a system, however, a brief presentation of the NC symplectic induction formalism 24,34,35 will be necessary, since the Boop's shifts will be mathematically generalized through the symplectic framework.…”