Definition and formulas for harmonic oscillator coherent states and spin coherent states are reviewed in detail. The path integral formalism and its relation with the partition function of a system are also reviewed. The harmonic oscillator coherent state path integral is evaluated exactly at the discrete level and then used to find its continuum limit using various regularizations. The computation of the path integral for a particle of spin s put in a constant magnetic field is carried out using harmonic oscillator coherent states and spin coherent states, with a careful analysis of infinitesimal terms (in 1/N where N is the number of time slices) appearing in the Lagrangian. A mapping of the spin system into a CP1 model is shown explicitly. The theory of a spinless particle in the field of a magnetic monopole and its relation with the spin system are explained. The equivalence of these two models is established up to infinitesimal order by the introduction of an external field correction. This gives a new representation of a coherent state path integral in terms of a more familiar Feynman path integral.
Coherent states can be used for diverse applications in quantum physics including the construction of coherent state path integrals. Most definitions make use of a lattice regularization; however, recent definitions employ a continuous-time regularization that may involve a Wiener measure concentrated on continuous phase space paths. The introduction of constraints is both natural and economical in coherent state path integrals involving only the dynamical and Lagrange multiplier variables. A preliminary indication of how these procedures may possibly be applied to quantum gravity is briefly discussed.