Partial symplectic quantum tomography schemes. Observables, evolution equations, and stationary states equations

TL;DR

This paper develops partial symplectic quantum tomography schemes, deriving operator symbols, evolution, and stationary state equations, with examples using Gaussian distributions to illustrate the formalism.

Contribution

It introduces new partial symplectic quantum tomography representations, deriving their operator symbols and evolution equations, expanding the tools for quantum state analysis.

Findings

Derived dual symbols for key quantum operators.

Formulated evolution and stationary state equations for the new representations.

Provided Gaussian-based examples demonstrating the formalism.

Abstract

Partial symplectic conditional and joint probability representations of quantum mechanics are considered. The correspondence rules for most interesting physical operators are found and the expressions of the dual symbols of operators are derived. Calculations were made by use of general formalism of quantizers and dequantizers determining the star product quantization scheme in these representations. Taking the Gaussian functions as the distributions of the tomographic parameters the examples of joint probability representations were considered. Evolution equations and stationary states equations for partial symplectic conditional and joint probability distributions are obtained.