The space of diff-invariant states

[kaufmann]

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
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

 

[marcus]

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
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...

Hi Rafael, could you please give the page reference? I have the hardcopy book handy and it would make it easier to see what you are talking about.

If you are using the online 2003 edition then a page reference would help there too. but the hardcopy is more readily accessible for me at present.

 

[Careful]

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
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

It is a long while ago that I looked upon a spin network, but it is clear that your mapping f(k) = 1 forall k is not a state (it is not even linear) - it is a diffeo invariant FUNCTION on S0 however. Moreover, it is clear that the P_diff(Psi) span up the space of all algebraic diff-invariant states by definition.

 

[kaufmann]

The reference is page 171 of the online edition, the hardcopy has the same text here. Indeed, as Careful has pointed out, my example was not linear, and thus not in the algebraic dual, but what Rovelli does not spell out -- as I have learned from him (see below) -- is that the space K_diff he intends to use is _not_ simply the subspace of S0* consisting of diff-invariant linear functionals, but the even smaller space of _finite-norm_ diff-invariant functionals, that is, the completion P_diff(S0) with respect to the appropriate inner product. Indeed, what is ``clear" to Careful is not even true unless you take into consideration only finite-norm functionals, and even then, only in the extended sense of limits of Cauchy sequences -- certainly not in the sense that ''K_diff is the image of P_diff" as originally claimed in the book. As Rovelli wrote to me:

Dear Rafael Kaufmann Nedal,
thanks for your interest and your remark. In fact, in part you are right. The correct statement is not that K_diff is the image of P_diff, but rather that K_diff is the *closure in the Hilbert norm* of the image of P_diff.
However, notice that K_diff is not the space of all diff-invariant functions of k, but only the finite norm ones. The function f(k) that you consider cannot be arbitrary: in particular the example f(k) = 1 for all k that you mention is not in the Hilbert space of the diff-invariant states of the quantum theory because its norm is infinite. Indeed, the state f(k) = 1 can be written as |f>=\Sum_k |k>, where |k> is a s-knot state and the sum runs over *all* knots; roughly speaking (that it, ignoring the complications due to the knots with symmetries etc), each s-knot state |k> has norm 1, and therefore |f> has infinite norm.
Now, S0 is formed by *finite* linear combinations of spin network states. Hence the image of P_diff is formed by *finite* linear combinations of s-knots |f>=\Sum_{n=1...N} |k_n>. These have clearly finite norm, and they do span K_diff. However, it is true that the image of P_diff is not a Hilbert space, because it does not include the
*infinite* linear combinations of knot states that converge in norm: |f>=\Sum_k c_k |k>, where \Sum_k |c_k|^2 <k|k> < infinity. To obtain the Hilbert space K_diff we have to take the closure in the Hilbert norm of the image of P_diff. In the pages of the book your refer to, I have been imprecise, following the common physicist attitude of disregarding norm-closure issues. This is incorrect since elsewhere in the book I have given a bit more attention to these isseus. Thanks for pointing this out to me.

Carlo Rovelli

Regards,
-- R. Kaufmann

 

[Careful]

** but what Rovelli does not spell out -- as I have learned from him (see below) -- is that the space K_diff he intends to use is _not_ simply the subspace of S0* consisting of diff-invariant linear functionals, but the even smaller space of _finite-norm_ diff-invariant functionals, that is, the completion P_diff(S0) with respect to the appropriate inner product. Indeed, what is ``clear" to Careful is not even true unless you take into consideration only finite-norm functionals, and even then, only in the extended sense of limits of Cauchy sequences -- certainly not in the sense that ''K_diff is the image of P_diff" as originally claimed in the book. **

But the P_diff( s_i ) still span up the algebraic diff invariant states in S0* (even when you allow S0 to contain countable infinite sums of spin networks), s_i is just one spin network. It is just that you do not allow for infinite norm states (which is what you always do in quantum physics since you want Hilbert spaces H, H can be inbedded in S0* when you allow for these infinite linear combinations while in Rovelli's answer S0* is norm-dense in H). What I said is true (you just did not ask about the Hilbert space). Rovelli's comment concerning your f map is only correct when you do not consider the sum of two s-knot states to be a l-knot state for some l which is what is still done for now I guess, the sum being a purely formal operation.