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arXiv:math-ph/9811016 · Journal of Mathematical Physics

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

Bernhard Bodmann, Hajo Leschke, Simone Warzel

Published Jun 1, 1999• 8 citations
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Abstract

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.

DOI
10.1063/1.532714

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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.
Adapting ideas of Daubechies and Klauder [J. Math. Phys. 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 Schrödinger semigroup on the Hilbert space L2 (R2) of Lebesgue squareintegrable functions on the Euclidean plane R2. The path-integral formula emerges from the standard Feynman-Kac-Itô formula for the Schrödinger semigroup in the ultra-diffusive 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.
Adapting ideas of Daubechies and Klauder we derive a continuum path-integral formula for the time evolution generated by a spin Hamiltonian. For this purpose we identify the finite-dimensional spin Hilbert space with the ground-state eigenspace of a suitable Schödinger operator on L2(R2)L^2({\mathbb{R}}^2)L2(R2), the Hilbert space of square-integrable functions on the Euclidean plane R2{\mathbb{R}}^2R2, and employ the Feynman-Kac-Itô formula.
A generalized Feynman-Kac formula based on the Wiener measure is presented. Within the setting of a quantum particle in an electromagnetic field it yields the standard Feynman-Kac formula for the corresponding Schrödinger semigroup. In this case rigorous criteria for its validity are compiled. Finally, phase-space path-integral representations for more general quantum Hamiltonians are derived. These representations rely on a generalized Lie-Trotter formula which takes care of the operator-ordering multiplicity, but in general is not related to a path measure.
Weak coherent states share many properties of the usual coherent states, but do not admit a resolution of unity expressed in terms of a local integral. They arise e.g. in the case that a group acts on an inadmissible fiducial vector. Motivated by the recent Affine Quantum Gravity Program, the present work studies the path integral representation of the affine weak coherent state matrix elements of the unitary time-evolution operator. Since weak coherent states do not admit a resolution of unity, it is clear that the standard way of constructing a path integral, by time slicing, is predestined to fail. Instead, a well-defined path integral with Wiener measure, based on a continuous-time regularization, is used to approach this problem. The dynamics is rigorously established for linear Hamiltonians, and the difficulties presented by more general Hamiltonians are addressed.
This paper is a review of path integral representations for semigroups {exp(-t H_r/ħ)}{t≥0} where H_r's are relativistic quantum Hamiltonians. We consider three different cases: in the first one Hr is a relativistic Schrödinger operator, in the second is the Hamiltonian associated to Klein-Gordon equation and in the third is that coming from the Dirac one. The paths are trajectories of diffusion processes and, for the Dirac case, they differ from previous constructions based on jump Markov processes. All formulas describe one relativistic particle in an external static electromagnetic field therefore they are relativistic versions of the Feymann-Kac representation for Schrödinger semigroups.
We construct a representation of the coherent state path integral using the Weyl symbol of the Hamiltonian operator. This representation is very different from the usual path integral forms suggested by Klauder and Skagerstam (1985 Coherent States: Applications in Physics and Mathematical Physics (Singapore: World Scientific)), which involve the normal or the antinormal ordering of the Hamiltonian. These different representations, although equivalent quantum mechanically, lead to different semiclassical limits. We show that the semiclassical limit of the coherent state propagator in the Weyl representation involves classical trajectories that are independent of the width of coherent states. This propagator is also free from the phase corrections found in Baranger et al ( 2001 J. Phys. A: Math. Gen. 34 7227 ) for the two Klauder forms and provides an explicit connection between the Wigner and the Husimi representations of the evolution operator.