Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

Constructing measures for spin-variable path integrals

John R. Klauder

Published Oct 1, 1982• 22 citations
Publisher

Abstract

By exploiting the overcompleteness of the spin-coherent states we derive expressions for spin-kinematics path integrals (specifically for spin 1/2 and spin 1) in terms of genuine (Wiener) measures on continuous paths lying on the unit sphere and for certain dynamical systems which when projected onto the subspace spanned by the proper spin-coherent-state matrix elements yield the appropriate quantum-mechanical propagator.

DOI
10.1063/1.525233

Related papers

Measures for Path Integrals

John R. Klauder, Ingrid Daubechies

By exploitation of the overcompleteness of coherent states expressions are presented for path integrals in terms of genuine (Wiener) path-space measures for driven harmonic oscillators which when projected onto the subspace spanned by coherent-state matrix elements yield the appropriate quantum mechanical propagator.

Physical Review Letters•Jan 18, 1982•24 citations

Quantum-mechanical path integrals with Wiener measure for all polynomial Hamiltonians. II

Ingrid C. Daubechies, John R. Klauder

The coherent-state representation of quantum-mechanical propagators as well-defined phase-space path integrals involving Wiener measure on continuous phase-space paths in the limit that the diffusion constant diverges is formulated and proved. This construction covers a wide class of self-adjoint Hamiltonians, including all those which are polynomials in the Heisenberg operators; in fact, this method also applies to maximal symmetric Hamiltonians that do not possess a self-adjoint extension. This construction also leads to a natural covariance of the path integral under canonical transformations. An entirely parallel discussion for spin variables leads to the representation of the propagator for an arbitrary spin-operator Hamiltonian as well-defined path integrals involving Wiener measure on the unit sphere, again in the limit that the diffusion constant diverges.

Journal of Mathematical Physics•Sep 1, 1985•101 citations

Constructing measures for path integrals

Ingrid C. Daubechies, John R. Klauder

The overcompleteness of the coherent states for the Heisenberg–Weyl group implies that many different integral kernels can be used to represent the same operator. Within such an equivalence class we construct an integral kernel to represent the quantum-mechanical evolution operator for certain dynamical systems in the form of a path integral that involves genuine (Wiener) measures on continuous phase-space paths. To achieve this goal it is necessary to employ an expression for the classical action different from the usual one.

Journal of Mathematical Physics•Oct 1, 1982•15 citations

Wiener measures for path integrals with affine kinematic variables

Ingrid C. Daubechies, John R. Klauder, Thierry Paul

The results obtained earlier have been generalized to show that the path integral for the affine coherent state matrix element of a unitary evolution operator exp(−iTH) can be written as a well-defined Wiener integral, involving Wiener measure on the Lobachevsky half-plane, in the limit that the diffusion constant diverges. This approach works for a wide class of Hamiltonians, including, e.g., −d2/dx2+V(x) on L2(R+), with V sufficiently singular at x=0.

Journal of Mathematical Physics•Jan 1, 1987•45 citations

Semiclassical dynamics of a spin-½ in an arbitrary magnetic field

Adrian Alscher, Hermann Grabert

The spin coherent state path integral describing the dynamics of a spin-½ system in a magnetic field of arbitrary time dependence is considered. Defining the path integral as the limit of a Wiener regularized expression, the semiclassical approximation leads to a continuous minimal action path with jumps at the endpoints. The resulting semiclassical propagator is shown to coincide with the exact quantum mechanical propagator. A nonlinear transformation of the angle variables allows for a determination of the semiclassical path and the jumps without solving a boundary-value problem. The semiclassical spin dynamics is thus readily amenable to numerical methods.

arXiv:quant-ph/9904102•Journal of Physics A Mathematical and General•Jan 1, 1999•17 citations

Coherent State Path Integrals at (Nearly) 40

John R. Klauder

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.

arXiv:quant-ph/9810043•Oct 14, 1998•4 citations
By exploiting the overcompleteness of the spin-coherent states we derive expressions for spin-kinematics path integrals (specifically for spin 1/2 and spin 1) in terms of genuine (Wiener) measures on continuous paths lying on the unit sphere and for certain dynamical systems which when projected onto the subspace spanned by the proper spin-coherent-state matrix elements yield the appropriate quantum-mechanical propagator.
The coherent-state representation of quantum-mechanical propagators as well-defined phase-space path integrals involving Wiener measure on continuous phase-space paths in the limit that the diffusion constant diverges is formulated and proved. This construction covers a wide class of self-adjoint Hamiltonians, including all those which are polynomials in the Heisenberg operators; in fact, this method also applies to maximal symmetric Hamiltonians that do not possess a self-adjoint extension. This construction also leads to a natural covariance of the path integral under canonical transformations. An entirely parallel discussion for spin variables leads to the representation of the propagator for an arbitrary spin-operator Hamiltonian as well-defined path integrals involving Wiener measure on the unit sphere, again in the limit that the diffusion constant diverges.
The overcompleteness of the coherent states for the Heisenberg–Weyl group implies that many different integral kernels can be used to represent the same operator. Within such an equivalence class we construct an integral kernel to represent the quantum-mechanical evolution operator for certain dynamical systems in the form of a path integral that involves genuine (Wiener) measures on continuous phase-space paths. To achieve this goal it is necessary to employ an expression for the classical action different from the usual one.
The spin coherent state path integral describing the dynamics of a spin-½ system in a magnetic field of arbitrary time dependence is considered. Defining the path integral as the limit of a Wiener regularized expression, the semiclassical approximation leads to a continuous minimal action path with jumps at the endpoints. The resulting semiclassical propagator is shown to coincide with the exact quantum mechanical propagator. A nonlinear transformation of the angle variables allows for a determination of the semiclassical path and the jumps without solving a boundary-value problem. The semiclassical spin dynamics is thus readily amenable to numerical methods.
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.