What is time? According to Connes and Rovelli - Part 6
Let’s continue our adventures with Connes and Rovelli. We are on page 3 now.
In a general covariant theory, in which no preferred dynamics and no preferred hamiltonian are given, a flow alpha_t^rho, which we will call the thermal time of rho, is determined by any thermal state rho. In this general case, one can postulate (see: C. Rovelli, Class. and Quant. Grav. 10, 1549 (1993)) that the thermal time alpha_t^rho defines the physical time.
Most interesting! First we have a theory in which no preferred dynamics is given! Preferred? Preferred by whom? By Connes? By Rovelli? By the Pope? Or by "majority of physicists"? Or, perhaps, "preferred by Nature herself? What is the definition of a hamiltonian in a generally covariant theory? What class of theories are we talking about? Anyway, what the authors state is the triviality: given any normal faithful state on a von Neumann algebra, the Tomita Takesaki theorem associates with this state a one-parameter group of automorphisms. And, so called thermal states are usually considered to be faithful normal states. That’s all. As for postulating, sure, we can postulate that the Moon is made of sugar, but that will not make the Moon sweet! I believe that Alain Connes is here completely innocent, and that he would never write such a nonsense, but Alain Connes is a mathematician, and he probably feels some respect for the physicist, he probably thinks that Rovelli certainly knows what he is talking about. While Rovelli is happy that he can have Alain Connes as his support!
By the way, if we really believe that the one-parameter group of algebra automorphisms, induced via a faithful normal state of the algebra in its standard form, is the"physical time flow," then, once we have said "A", we should also say "B". And "B" is that the spectrum of the generator of this flow, which must be associated with the "energy" is necessarily symmetric. The antiunitary involution J maps perfectly positive energy states into negative energy states. Thus we have both matter AND antimatter in a perfect symmetry. My guess is that Dan Brown, 
before writing his "Angels and Demons" must have read Connes and Rovelli paper! I am at present reading the book, and I am on page 137. "Angels and Demons" was written before the bestselling Da Vinci Code. The story, as much as I have read it till now, is about an irresponsible genius physicist, a Catholic priest, Leonardo Vetra, who was working at CERN and invented a cheap and easy method of producing antimatter. (He was then murdered and quarter of a gram of antimatter was stolen.) My guess is that Leonardo Vetra (or perhaps his daughter, also a physicist working at CERN) produced antimatter by applying the antiunitary involution J to matter. Indeed, J reverses the direction of the time flow! If only Connes and Rovelli have paid attention to this fact - their paper could have become a bestseller long before Angels and Demons!
