Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2003-06-09
Class.Quant.Grav. 20 (2003) 3279-3294
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
22 pages, version to be published in Classical & Quantum Gravity
Scientific paper
10.1088/0264-9381/20/15/301
Relying on known results of the Noether theory of symmetries extended to constrained systems, it is shown that there exists an obstruction that prevents certain tangent-space diffeomorphisms to be projectable to phase-space, for generally covariant theories. This main result throws new light on the old fact that the algebra of gauge generators in the phase space of General Relativity, or other generally covariant theories, only closes as a soft algebra and not a a Lie algebra. The deep relationship between these two issues is clarified. In particular, we see that the second one may be understood as a side effect of the procedure to solve the first. It is explicitly shown how the adoption of specific metric-dependent diffeomorphisms, as a way to achieve projectability, causes the algebra of gauge generators (constraints) in phase space not to be a Lie algebra --with structure constants-- but a soft algebra --with structure {\it functions}.
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