A rigorous path integral for quantum spin using flat-space Wiener regularization

Adapting ideas of Daubechies and Klauder [J. Math. Phys. 26, 2239 (1985)] we derive a rigorous continuum path-integral formula for the semigroup generated by a spin Hamiltonian. More precisely, we use spin coherent vectors parametrized by complex numbers to relate the coherent representation of this semigroup to a suitable Schrödinger semigroup on the Hilbert space L2(R2) of Lebesgue square-integrable functions on the Euclidean plane R2. The path-integral formula emerges from the standard Feynman-Kac-Itô formula for the Schrödinger semigroup in the ultradiffusive limit of the underlying Brownian bridge on R2. In a similar vein, a path-integral formula can be constructed for the coherent representation of the unitary time evolution generated by the spin Hamiltonian. PACS numbers: 02.50.Ey, 75.10.Jm Appeared in: J. Math. Phys. 4