Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians
| Autor: | Struckmeier, Jürgen |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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 $L_{\e}ds$. Supposing that both Lagrangians describe the same physical system then provides the correlation of $L$ and $L_{\e}$. In the existing literature, the discussion is restricted to only those extended Lagrangians $L_{\e}$ that are homogeneous forms of first order in the velocities. As a new result, it is shown that a class of extended Lagrangians $L_{\e}$ exists that are correlated to corresponding conventional Lagrangians $L$ \emph{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 \emph{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 \emph{relativistic} quantum physics. To provide an example, the non-homogeneous extended Lagrangian $L_{\e}$ 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 $L_{\e}$, 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 \emph{relativistic generalization} of Feynman's path integral approach to quantum physics. Comment: 4 pages. arXiv admin note: substantial text overlap with arXiv:0811.0496 |
| Databáze: | arXiv |
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