Towards a tomographic representation of quantum mechanics on the plane

TL;DR

This paper introduces a new tomographic representation of quantum mechanics on the plane, using symplectic quantum tomograms to map observables and states to probability distributions, enabling calculations via plane integrations.

Contribution

It defines a novel probability distribution on the plane based on symplectic quantum tomograms and establishes a dual map for polynomial observables, expanding the mathematical framework of quantum mechanics.

Findings

Probability distribution on the plane derived from symplectic quantum tomograms

Dual map connects polynomial observables to functions of two variables

Average values computed through integration over the plane

Abstract

On the base of symplectic quantum tomogram we define a probability distribution on the plane. The dual map transfers all observables which are polynomials of the position and momentum operators to the set of polynomials of two variables. In this representation the average values of observables can be calculated by means of integration over all the plane.