Geometrical Path Integrals and Their Applications

Takayoshi Ootsuka, Erico Tanaka

Osvaldo Civitarese, Manuel Gadella

Masao Matsumoto

We develop the formulation of the spin(SU(2)) coherent state path integrals based on arbitrary fiducial vectors. The resultant action in the path integral expression extensively depends on the vector; It differs from the conventional one in that it has a generalized form having some additional terms. We also study, as physical applications, the geometric phases associated with the coherent state path integrals to find that new effects of the terms may appear in experiments. We see that the formalism gives a clear insight into geometric phases.

Masao Matsumoto

We develop the formulation of the spin(SU(2)) coherent state path integrals based on arbitrary starting vectors. The resultant action in the path integral expression extensively depends on the vector; it differs from the conventional one in that it has a generalized form having some additional terms. We also study, as physical applications, the geometric phases associated with the CS path integrals to find that new effects of the terms may appear in experiments.

Eugene P. Gross

Javier Jiménez-López, V. J. García-Garrido

We formulate Lagrangian descriptors (LDs) in the path integral framework. Averaging the classical LD over fluctuations about extremal trajectories defines a quantum LD that incorporates quantum effects. Invariant manifolds, which sharply organize classical transport, become finite-width phase space structures under quantum fluctuations, and their overlap provides a geometric mechanism consistent with tunneling as fluctuation-induced delocalization of transport barriers. We demonstrate this approach for the Hamiltonian saddle, where path integral sampling reveals manifold broadening and barrier penetration. This establishes a geometric framework for studying phase space transport and tunneling beyond the classical regime, while also providing a natural route toward the application of LDs to field theory.