We show how the Feynman path integral representation can be used to map between ray and wave descriptions in classical optics and look at a GRIN system and the Gouy phase anomaly from this perspective.
Classical ray and wave optics share a common structure with quantum mechanics. Much of this results from the representations given to abstract operators. It is possible to build a mathematical bridge that allows one to transform back and forth between the ray and wave picture. This is given by the Feynman path integral representation. The map between the ray and wave description is given in terms of a sum over histories integration. In particular, it allows for the computation of wave phenomena by the summation of multiple ray paths of a phase function connected to the classical optical path length.
We provide a unified description of the Gouy phase anomaly, highlighting the pivotal role of local fluctuations in transverse momentum and quantal phase changes associated with electron trajectories that touch caustics. Using arguments based on wave optics, at the interface between classical and quantum mechanics, we derive the magnitude of the quantized Gouy phase changes for scalar waves. Our analysis unifies disparate descriptions of the Gouy anomaly within a single theoretical framework. In particular, we find that the phase anomaly connects Maslov indices, quantal Berry phases, Heisenberg momentum fluctuations, and confinement-induced L\'evy-Leblond phase shifts.
The definition of Feynman path integrals (Feynman functional integrals) as integrals with respect to a generalized measure, called the Lebesgue–Feynman measure in the paper and being an infinite-dimensional analogue of the classical Lebesgue measure on finite-dimensional Euclidean space, is discussed. This definition, which is a formalization of Feynman’s original definition, is different from those used previously in the mathematical literature. It makes it possible to give a description of the origin of quantum anomaly which is a mathematically correct version of the description given in the book Path Integrals and Quantum Anomalies by K. Fujikawa and H. Suzuki (Oxford, 2004) (and erroneously qualified as wrong in the book Functional Integration: Action and Symmetries by P. Cartier and C. DeWitt-Morette (Cambridge Univ. Press, Cambridge, 2006)).