Extended phase space in general gauge theories. (English) Zbl 1537.81140
Summary: In a recent paper, it was shown that in diffeomorphism-invariant theories, the symplectic vector fields induced by spacetime diffeomorphisms are integrable if one introduces an extended phase space. In this paper we extend the notion of extended phase space to all gauge theories with arbitrary combinations of internal and spacetime local symmetries. We formulate this in terms of a corresponding Atiyah Lie algebroid, a geometric object derived from a principal bundle which features internal symmetries and diffeomorphisms on an equal footing. In this language, gauge transformations are understood as morphisms between Atiyah Lie algebroids that preserve the geometric structures encoded therein. The extended configuration space of a gauge theory can subsequently be understood as the space of pairs \((\varphi, \Phi)\), where \(\varphi\) is a Lie algebroid morphism and \(\Phi\) is a field configuration in the non-extended sense. Starting from this data, we outline a very powerful, manifestly geometric approach to the extended phase space. Using this approach, we find that the action of the group of gauge transformations and diffeomorphisms on the symplectic geometry of any covariant theory is integrable. We motivate our construction by carefully examining the need for extended phase space in Chern-Simons gauge theories and display its usefulness by re-computing the charge algebra. We also describe the implementation of the configuration algebroid in Einstein-Yang-Mills theories.
MSC:
| 81S30 | Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics |
| 37C05 | Dynamical systems involving smooth mappings and diffeomorphisms |
| 60F17 | Functional limit theorems; invariance principles |
| 53A45 | Differential geometric aspects in vector and tensor analysis |
| 83F05 | Relativistic cosmology |
| 81Q80 | Special quantum systems, such as solvable systems |
| 13B02 | Extension theory of commutative rings |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
| 32L05 | Holomorphic bundles and generalizations |
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