Path integrals of spin-J systems in the holomorphic representation Original Research Article

A path integral expression for the matrix element and the diagonal representative of the spin-J evolution operator or Boltzmann factor is obtained using Bloch coherent states in the holomorphic representation, yielding the appropriate boundary conditions. Quantum Dyson-Schwinger equations of motion are derived and used as criteria to select the appropriate action. In the case of the diagonal representative the equations of motion satisfy the commutation relations of the spin operators and are the classical limit of the Dyson-Schwinger equations, if we redefine the Lagrangian together with the integration measure. It is shown that in the case of a spin in a time-independent magnetic field the saddle-point approximation for both representatives gives the exact results. We present analytical algorithms to obtain the exponential factor and the prefactor in the saddle-point expansion, without an ad hoc reduction to an effective one-dimensional problem, but keeping instead a 2 × 2 matrix structure. We apply our results to several models.