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

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

1998-11-18

arxiv.org/abs/math-ph/9811016v2

J. Math. Phys. {\bf 40} 2549-2559 (1999)

Physics

Mathematical Physics

revised version

Scientific paper

10.1063/1.532714

Adapting ideas of Daubechies and Klauder [J. Math. Phys. {\bf 26} (1985) 2239] 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 Schr\"odinger semigroup on the Hilbert space $L^2(R^2)$ of Lebesgue square-integrable functions on the Euclidean plane $R^2$. The path-integral formula emerges from the standard Feynman-Kac-It\^o formula for the Schr\"odinger semigroup in the ultra-diffusive limit of the underlying Brownian bridge on $R^2$. 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.

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