Hermitian vector fields and special phase functions. (English) Zbl 1095.53046
Summary: We start by analyzing the Lie algebra of Hermitian vector fields of a Hermitian line bundle. Then, we specify the base space of the above bundle by considering a Galilei, or an Einstein space-time. Namely, in the first case, we consider, a fibred manifold over absolute time equipped with a space-like Riemannian metric, a space-time connection (preserving the time fibring and the space-like metric) and an electromagnetic field. In the second case, we consider a space-time equipped with a Lorentzian metric and an electromagnetic field. In both cases, we exhibit a natural Lie algebra of special phase functions and show that the Lie algebra of Hermitian vector fields turns out to be naturally isomorphic to the Lie algebra of special phase functions. Eventually, we compare the Galilei and Einstein cases.
MSC:
| 53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |
| 17B66 | Lie algebras of vector fields and related (super) algebras |
| 17B81 | Applications of Lie (super)algebras to physics, etc. |
| 53B35 | Local differential geometry of Hermitian and Kählerian structures |
| 53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |
| 55R10 | Fiber bundles in algebraic topology |
| 58A10 | Differential forms in global analysis |
| 81R20 | Covariant wave equations in quantum theory, relativistic quantum mechanics |
| 81S10 | Geometry and quantization, symplectic methods |
| 83C99 | General relativity |
| 83E99 | Unified, higher-dimensional and super field theories |
Keywords:
Hermitian vector fields; quantum bundle; special phase functions; Galilei space-time; Lorentz space-timeReferences:
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