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Extended Hamilton-Lagrange formalism and...
Struckmeier, Jürgen...

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Extended Hamilton-Lagrange formalism and its application to Feynman's path integral for relativistic quantum physics by Struckmeier, Jürgen ( Author )
Lemonjar Open Access
N.A
04-11-2008
With this paper, a consistent and comprehensive treatise on the foundations of the extended Hamilton-Lagrange formalism will be presented. In this formalism, 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 qi(s). In the action principle, the conventional classical action Ldt is then replaced by the generalized action Leds, with L and Le denoting the conventional and the extended Lagrangian, respectively. It is shown that a class of extended Lagrangians Le exists that are correlated to corresponding conventional Lagrangians L without being homogeneous functions in the velocities. Then the Legendre transformation of Le to an extended Hamiltonian He exists. With this class of extended Hamiltonians, an extended canonical formalism is presented that is completely analogous to the conventional Hamiltonian formalism. The physical time t and the negative value of the conventional Hamiltonian then constitute and an additional pair of conjugate canonical variables. The extended formalism also includes a theory of extended canonical transformations, where the time variable t(s) is also subject to transformation. In the extended formalism, the system's dynamics is described as a motion on a hypersurface within an extended phase space of even dimension. With the extended Lagrangian Le, it is shown that the generalized path integral approach yields the Klein-Gordon equation as the corresponding quantum description. Moreover, the space-time propagator for a free relativistic particle will be derived. These results can be regarded as the proof of principle of the relativistic generalization of Feynman's path integral approach to quantum physics.
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English
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MYR 0.00
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https://arxiv.org/abs/0811.0496
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