Thanks, Sean. I won't pretend to speak for Carlo Rovelli, though I think I do understand what he means by degrees of freedom in an n-particle state space (I would say n-dimension state space and use the Hilbert space) where things happen discontinuously because the field assumes t = 1 and time coordinates are therefore constant (static).
In a continuous spacetime field, time changes with space -- dynamically, as you say. So I think what Carlo is saying, is that because a continuum theory such as shape dynamics cannot associate a definite dimensionless integer to a point of the space evolving in time, it is just as, if not more, meaningful to speak of the algebraic (Fock space) degrees of freedom as to speak of finite degrees of freedom in your continuously evolving system -- because the two descriptions, absent a simple time parameter of reversible trajectory, are equivalent.
Personally, I don't agree with Carlo, though I understand the Fock space construction. At the end of the day, I think that if shape dynamics is mathematically complete, it allows relative degrees of freedom; i.e., time reversibility that guarantees conservation of information. That's what the 't Hooft and Corda references are all about.
All best,
Tom
