 Quote by tom.stoer
I think my example with Sł was missleading; I'll try it again:
The dimension of the vector space of C-functions on Sł is infinite-dimensional. But if we look at the Lie group SU(2) we can write its elements as
[tex]U[f] = \exp\left\{i t^a f^a(x)\right\}[/tex]
The functions fa are members of an infinite dimensional vector space, but nevertheless we talk about a finite dimensional group b/c we have finitly many generators ta.
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Diffeomorphisms which are infinitesimally close to the identity look like gauge transformations. The problem is that infinitesimal diffeomorphisms do not generate the whole group.
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Looking at the Ashtekar formulation of gravity there are the Gauss-law generators Ga(x) and the spacelike diffeomorphism generators = vector-constraints Va(x). The transformations are generated via functionals like
[tex]G[f] = \int d^3x\,G^a(x)\,f(x);\quad V[f] = \int d^3x\,V^a(x)\,f(x);\quad[/tex]
Again we have finitly many generators even if the functions f belong to an infinite dimensional vector space.
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If there are a finite number of generators, we are not talking about the full diffeomorphism group, only the infinitesimal diffeomorphisms.