- #1

- 2,984

- 1,531

With an infinitesimal symmetry on your space-time M one can look at the corresponding transformation of the canonical variables in phase-space PS. This can be done by a phase space function H[]:M --> PS. The generators of these symmetries give you global charges.

If I generate my symmetry with a vector field [itex]\xi[/itex] or [itex]\alpha [/itex], the statement which is often made is that

[tex]

\{H[\xi],H[\alpha] \} = H [[\xi,\alpha]]

[/tex]

In words: the Poisson algebra {} of the charges (LHS) is isomorphic (=) to the Lie algebra of the corresponding symmetries on your space-time (RHS).

My question is: why is this so natural to assume? Henneaux and Brown wrote an article about the details and adjustments of this assumption (central charges in the canonical realization...), but I don't see why this should be "natural" in the first place. I'm missing something here.

Thanks in forward :)