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

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 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_ds. Supposing that both Lagrangians describe the same physical system then provides the correlation of L and L_. In the existing literature, the discussion is restricted to only those extended Lagrangians L_ that are homogeneous forms of first order in the velocities. As a new result, it is shown that a class of extended Lagrangians L_ exists that are correlated to corresponding conventional Lagrangians L without being homogeneous functions in the velocities. With these extended Lagrangians, the system's dynamics is described as a motion on a hypersurface within a symplectic extended phase space of even dimension. As a consequence of the formal similarity of conventional and extended Lagrange formalisms, Feynman's non-relativistic path integral approach can be converted into a form appropriate for relativistic quantum physics. To provide an example, the non-homogeneous extended Lagrangian L_ of a classical relativistic point particle in an external electromagnetic field will be presented. This extended Lagrangian has the remarkable property to be a quadratic function in the velocities. With this L_, it is shown that the generalized path integral approach yields the Klein-Gordon equation as the corresponding quantum description. This result can be regarded as the proof of principle of the relativistic generalization of Feynman's path integral approach to quantum physics.