Random walk on quantum blobs

TL;DR

This paper explores the symplectic group's action on quantum states called squeezed states, extending it to a semigroup, and investigates implications for quantum monitoring and fractal pattern formation.

Contribution

It extends the metaplectic representation to the symplectic semigroup and analyzes the geometry of squeezing in quantum states.

Findings

Extended the metaplectic representation to the semigroup.

Proposed use of noncommuting semigroup elements for quantum monitoring.

Analyzed shape geometry of squeezing and potential fractal patterns.

Abstract

We describe the action of the symplectic group on the homogeneous space of squeezed states (quantum blobs) and extend this action to the semigroup. We then extend the metaplectic representation to the metaplectic (or oscillator) semigroup and study the properties of such an extension using Bargmann-Fock space. The shape geometry of squeezing is analyzed and noncommuting elements from the symplectic semigroup are proposed to be used in simultaneous monitoring of noncommuting quantum variables - which should lead to fractal patterns on the manifold of squeezed states.