On the Path Integral Representation for Spin Systems

TL;DR

This paper develops a classical constrained Hamiltonian framework for spin systems, demonstrating that Dirac brackets encode spin commutation relations and deriving the partition function consistent with coherent state methods.

Contribution

It introduces a novel classical Hamiltonian approach for spins, linking Dirac brackets to quantum commutation relations and deriving the partition function via a new method.

Findings

Dirac brackets represent spin commutation relations

Partition function matches coherent state results

Explicit evaluation for a spin in a magnetic field

Abstract

We propose a classical constrained Hamiltonian theory for the spin. After the Dirac treatment we show that due to the existence of second class constraints the Dirac brackets of the proposed theory represent the commutation relations for the spin. We show that the corresponding partition function, obtained via the Fadeev-Senjanovic procedure, coincides with the one obtained using coherent states. We also evaluate this partition function for the case of a single spin in a magnetic field.