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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2406.06530 |
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| _version_ | 1866911912330002432 |
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| author | Struckmeier, Jürgen |
| author_facet | Struckmeier, Jürgen |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_06530 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians Struckmeier, Jürgen Quantum Physics 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. |
| title | Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2406.06530 |