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Generally covariant theories: the Noether obstruction for realizing certain space–time diffeomorphisms in phase space

Josep M Pons

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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 being 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 as 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 functions.


PACS

04.20.Fy Canonical formalism, Lagrangians, and variational principles

04.20.Gz Spacetime topology, causal structure, spinor structure

MSC

83Cxx General relativity

Subjects

Gravitation and cosmology

Dates

Issue 15 (7 August 2003)

Received 3 March 2003, in final form 15 May 2003

Published 8 July 2003



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