Phase space spinor amplitudes for spin 1/2 systems

TL;DR

This paper introduces a novel phase space amplitude framework for spin-1/2 systems, providing a more fundamental and geometrically consistent description of quantum states on spheres and lattices, with applications to state superposition and dynamics.

Contribution

It generalizes phase space amplitudes to finite-dimensional spin systems, demonstrating their transformation properties and relation to Wigner and Husimi functions for spin-1/2.

Findings

Amplitudes transform correctly as spinors under rotations.

Wigner function expressed as star product of amplitude and conjugate.

Application to phase space evolution of spin-1/2 magnetic dipole.

Abstract

The concept of phase space amplitudes for systems with continuous degrees of freedom is generalized to finite-dimensional spin systems. Complex amplitudes are obtained on both a sphere and a finite lattice, in each case enabling a more fundamental description of pure spin states than that previously given by Wigner functions. In each case the Wigner function can be expressed as the star product of the amplitude and its conjugate, so providing a generalized Born interpretation of amplitudes that emphasizes their more fundamental status. The ordinary product of the amplitude and its conjugate produces a (generalized) spin Husimi function. The case of spin-$\half$ is treated in detail, and it is shown that phase space amplitudes on the sphere transform correctly as spinors under under rotations, despite their expression in terms of spherical harmonics. Spin amplitudes on a lattice are also found to transform as spinors. Applications are given to the phase space description of state superposition, and to the evolution in phase space of the state of a spin-$\half$ magnetic dipole in a time-dependent magnetic field.