0811.0496 (Jürgen Struckmeier)

Extended Hamilton-Lagrange formalism and its application to Feynman's
path integral for relativistic quantum physics    [PDF]

Jürgen Struckmeier

We present a consistent and comprehensive treatise on the foundations of the {\em extended Hamilton-Lagrange formalism}--where the dynamical system is parameterized along a general system evolution parameter $s$, and the time $t$ is treated as a {\em 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 $L_{\e}ds$, with $L$ and $L_{\e}$ denoting the conventional and the extended Lagrangian, respectively. It is shown that a unique correlation of $L_{\e}$ and $L$ exists if we refrain from performing simultaneously a transformation of the dynamical variables. With the appropriate correlation of $L_{\e}$ and $L$ in place, the extension of the formalism preserves its {\em canonical form}. In the extended formalism, the dynamical system is described as a {\em constrained} motion within an {\em extended} space. We show that the value of the constraint and the parameter $s$ constitutes an {\em additional pair} of canonically conjugate variables. In the corresponding quantum system, we thus encounter an additional {\em uncertainty relation}. We derive the extended Lagrangian $L_{\e}$ of a classical relativistic point particle in an external electromagnetic field and show that the generalized path integral approach yields the Klein-Gordon equation as the corresponding quantum description. We furthermore derive the space-time propagator for a free relativistic particle from its extended Lagrangian $L_{\e}$. These results can be regarded as the proof of principle of the {\em relativistic generalization} of Feynman's path integral approach to quantum physics.

View original: http://arxiv.org/abs/0811.0496