Deriving the von Neumann equation from the Majorana-Bloch equation for arbitrary spin in any state

TL;DR

This paper generalizes the connection between classical and quantum spin equations from spin-1/2 to arbitrary spins, establishing a one-to-one mapping between the Majorana-Bloch and von Neumann equations for all spin states.

Contribution

It extends the derivation linking the Majorana-Bloch and von Neumann equations to higher spins and arbitrary states, using polynomial representations and symmetrization techniques.

Findings

Derived an invertible mapping for arbitrary spin states.

Showed expectation values follow classical equations of motion.

Unified classical-quantum spin formalism for all spins.

Abstract

After publishing the derivation from the classical Bloch equation to the quantum von Neumann equation to the Schr\"dinger-Pauli equation for spin-$\tfrac{1}{2}$, we proposed renaming the Bloch equation to the Majorana-Bloch equation because Majorana's work predated Bloch's in the presentation of the Bloch equation by 14 years. Here, we first generalize our previous derivation to higher spins or angular momenta in coherent pure states. Using the polynomial representation of the coherent-state projector, we derive an invertible mapping from the Majorana-Bloch equation to the von Neumann equation, establishing a one-to-one correspondence between these two formalisms. Application of the Ehrenfest theorem also shows that expectation values in these states reproduce the classical equation of motion as expected. Then, we obtain arbitrary spin-$s$ states by symmetrizing tensor products of spin-$\tfrac{1}{2}$ primitives, in accordance with the Majorana construction or the Schur-Weyl duality.