Evolution Equation for Joint Tomographic Probability Distribution of Spin-1 Particles

TL;DR

This paper develops evolution equations for a comprehensive joint tomographic probability distribution of spin-1 particles, enabling a wave-function-free quantum description that captures full spatial and spin information.

Contribution

It introduces a nine-component vector optical tomogram and extends the approach to symplectic and quasidistribution representations, with evolution equations for arbitrary Hamiltonians.

Findings

Derived evolution equations for vector optical and symplectic tomograms.

Formulated analogs of the Proca equation in tomographic representations.

Demonstrated the possibility of describing spin-1 quantum systems via joint probability distributions.

Abstract

The nine-component positive vector optical tomographic probability portrait of quantum state of spin-1 particles containing full spatial and spin information about the state without redundancy is constructed. Also the suggested approach is expanded to symplectic tomography representation and to representations with quasidistributions. The evolution equations for constructed vector optical and symplectic tomograms and vector quasidistributions for arbitrary Hamiltonian are found. The evolution equations are also obtained in special case of the quantum system of charged spin-1 particle in arbitrary electro-magnetic field, which are analogs of non-relativistic Proca equation in appropriate representations. The generalization of proposed approach to the cases of arbitrary spin is discussed. The possibility of formulation of quantum mechanics of the systems with spins in terms of joint probability distributions without the use of wave functions or density matrixes is explicitly demonstrated.