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Dear Prof. Rovelli,
Your invocation of Tomita flow requires that the algebra of operators be, as you say, a von Neumann algebra, as well as requiring a state over the algebra. A von Neumann algebra has a Norm, by definition, which, I claim, must have a *timeless* meaning for your argument to go through.
The Norm of an algebra of observables decides what measurements are close to each other -- that is, the topology -- which allows us, by continuity, to decide whether we expect, in a given state, that the results of one measurement will be close to the results of another experiment. We can only verify that two given experimental procedures are close to each other by applying them to many different states, thereby determining that we get almost the same results (according to some Norm on the space of results) in every state. Given the statistical nature of a state -- as much in classical statistical mechanics as in quantum theory -- this requires us to construct many ensembles. Now, how are we to construct these multiple ensembles timelessly?
As an ideal world, of course, you are quite entitled to posit any mathematical structure you like, but I would like to see a Physical interpretation include at least a schematic for an operational correspondence with the world for every significant mathematical structure. In any case, given its significance to your account of time, I would like to see a relatively full account of how to understand the Norm of the von Neumann algebra in a timeless way, whether operational or not.
I apologize if this question is well understood. I don't follow the literature on quantum gravity closely at present. Please feel free just to cite a reference.
I'm perhaps simply at cross-purposes with you, since in the last sentence of section VI you assert that the world is "in" a given state rho, which suggests that you understand QFT in terms of a non-ensemble interpretation of probability. However, I would personally take a non-operational definition of probability also to be problematic for your program.