## Abstract

We apply a method analogous to the eikonal approximation to the Maxwell wave equations in an inhomogeneous anisotropic medium and geodesic motion in a three dimensional Riemannian manifold, using a method which identifies the symplectic structure of the corresponding mechanics, to the five dimensional generalization of Maxwell theory required by the gauge invariance of Stueckelberg's covariant classical and quantum dynamics. In this way, we demonstrate, in the eikonal approximation, the existence of geodesic motion for the flow of mass in a four dimensional pseudo-Riemannian manifold. These results provide a foundation for the geometrical optics of the five dimensional radiation theory and establish a model in which there is mass flow along geodesics. We then discuss the interesting case of relativistic quantum theory in an anisotropic medium as well. In this case the eikonal approximation to the relativistic quantum mechanical current coincides with the geodesic flow governed by the pseudo-Riemannian metric obtained from the eikonal approximation to solutions of the Stueckelberg-Schrödinger equation. The locally symplectic structure which emerges is that of a generally covariant form of Stueckelberg's mechanics on this manifold.

Original language | English |
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Pages (from-to) | 1323-1338 |

Number of pages | 16 |

Journal | Foundations of Physics |

Volume | 33 |

Issue number | 9 |

DOIs | |

State | Published - Sep 2003 |

### Bibliographical note

Funding Information:We wish to thank C. Piron and F. W. Hehl for discussions at an early stage of this work, and H. Goldenberg and N. Erez for helpful discussions of propagation of waves in a medium. One of us (L.H.) wishes to thank S. L. Adler and the Institute for Advanced Study in Princeton for hospitality in the Spring Semester 2003, and, in particular, for a grant in aid from the Funds for Natural Sciences.

### Funding

We wish to thank C. Piron and F. W. Hehl for discussions at an early stage of this work, and H. Goldenberg and N. Erez for helpful discussions of propagation of waves in a medium. One of us (L.H.) wishes to thank S. L. Adler and the Institute for Advanced Study in Princeton for hospitality in the Spring Semester 2003, and, in particular, for a grant in aid from the Funds for Natural Sciences.

Funders | Funder number |
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Funds for Natural Sciences |

## Keywords

- Eikonal approximation
- Relativistic quantum theory
- Space-time metric