My advisor and I have stumbled upon a very strange claim in Rovelli's book. There he defines the P_diff map from S0 onto its algebraic dual S0* as(adsbygoogle = window.adsbygoogle || []).push({});

P_diff(Psi) Psi' = sum (Psi'' = phi Psi) (Psi'',Psi')

This is indeed a well-defined map that yields diff-invariant states. However, Rovelli claims, further, that the image of this map is indeed the space of _all_ diff-invariant states -- that is, that all diff-invariant states are of the form P_diff(Psi) for some Psi that is a _finite_ sum of spin network states.

However, as I see it, a diff-invariant state f can be seen as an arbitrary mapping of each s-knot state (diffeomorphism equivalence class of spin network states) k to a certain number f(k), whereas if f is of the form P_diff(Psi), with Psi = sum_s Psi_s, then f(k) can only be non-zero for those k which have the same knot as one of these s. Indeed, a simple example such as the state f(k) = 1 for all k is clearly _not_ of the form P_diff Psi. Is this reasoning correct? If so, what is it that one can state about the space K_diff?

Thanks in advance,

-- Rafael Kaufmann

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# The space of diff-invariant states

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