Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians

Abstract:In the extended Lagrange formalism of classical point dynamics, the system's dynamics is parametrized along a system evolution parameter $s$, and the physical time $t$ is treated as a \emph{dependent} variable $t(s)$ on equal footing with all other configuration space variables $q^{i}(s)$. In the action principle, the conventional classical action $L dt$ is then replaced by the generalized action $L_{e}ds$. A direct reparametrization of the conventional action yields an extended Lagrangian that is homogeneous of degree one in the velocities. A regular extended Lagrangian that is not homogeneous of degree one can also reproduce the conventional dynamics on a suitably selected invariant hypersurface. With this representation, the physical motion takes place on a hypersurface within a symplectic extended phase space of even dimension. The formal similarity of conventional and extended Lagrange formalisms provides a direct connection with the proper-time path integral approach to relativistic quantum physics. To provide an example, the non-homogeneous extended Lagrangian $L_{e}$ of a classical relativistic spinless point particle in an external electromagnetic field will be presented. This extended Lagrangian is a quadratic function in the velocities, and its selected zero-energy hypersurface is the classical mass shell. With this $L_{e}$, the generalized path integral first yields an evolution equation in $s$. Imposing the corresponding quantum mass-shell condition then yields the Klein-Gordon equation as the physical wave equation. This establishes the relation between the classical extended Lagrangian description and proper-time quantization for a charged spin-zero particle.

6 pages. arXiv admin note: substantial text overlap with arXiv:0811.0496

arXiv:2406.06530 [quant-ph]

(or arXiv:2406.06530v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2406.06530 Focus to learn more arXiv-issued DOI via DataCite

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.