A path integral expression for the matrix element and the diagonal rep
resentative of the spin-J evolution operator or Boltzmann factor is ob
tained 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 app
ropriate action. In the case of the diagonal representative the equati
ons of motion satisfy the commutation relations of the spin operators
and are the classical limit of the Dyson-Schwinger equations, if we re
define the Lagrangian together with the integration measure. It is sho
wn that in the case of a spin in a time-independent magnetic field the
saddle-point approximation for both representatives gives the exact r
esults. We present analytical algorithms to obtain the exponential fac
tor and the prefactor in the saddle-point expansion, without an ad hoc
reduction to an effective one-dimensional problem, but keeping instea
d a 2 x 2 matrix structure. We apply our results to several models.