Open System for Geniuses - by Chronostalker

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!

Let us continue our adventure with Connes and Rovelli. On the road again!

Let us illustrate here the core of this idea — a full account is given in sec. 3 below. Consider classical statistical mechanics. Let rho be a thermal state, namely a smooth positive (normalized) function on the phase space, which defines a statistical distribution in the sense of Gibbs (see Gibbs, Elementary Principles in Statistical Mechanics, Yale University Press, 1902).

Why do they think that a state should be a smooth function? Why do they think that is should be a function an the phase space? In time-dependent systems we have to replace the phase space by a contact manifold. Don’t they know it? And what if the phase space itself changes with time? And what is a momentum when there is no time? What is the meaning of phase space?

In a conventional non-generally covariant theory, a hamiltonian H is given and the equilibrium thermal states are Gibbs states rho = exp[ - beta H].

You can have a conventional non-generally covariant theory and no Hamiltonian at all. Or you may have a Hamiltonian that depends explicitly on time. What is an equilibrium state in this case? And what about special relativity? Special relativity is non-generally covariant. Which phase space will Connes and Rovelli take? And what is the definition of "thermal equilibrium"? Thermal equilibrium of what with what? And why should it be described by a Gibbs state and no by something else? There are many theories of "thermal equilibria" - why only one temperature? What if have a system with more than one temperatures? Don’t the authors know the book Information Dynamics and Open Systems: Classical and Quantum Approach.

Notice that the information on the time flow is coded into the Gibbs states as well as in the hamiltonian. Thus, the time flow alpha_t can be recovered from the Gibbs state rho (up to a constant factor beta, which we disregard for the moment).

What a discovery! It would be really useful for the authors to consider systems that are explicitly time-dependent and/or disssipative - as most system are. It would be really useful to take a special relativistic system. That would save us from trivial statements like the one above is, and it could, perhaps, teach us something.

This fact suggests that in a thermal context it may be possible to ascribe the dynamical properties of the system to the thermal state, rather than to the hamiltonian: The Gibbs state determines a flow, and this flow is precisely the time flow.

Now, notice this words "in a thermal context"! What is the meaning of this qualification? When our system is in a thermal context and when it is not? Is our universe in a thermal context? Or was it? And who puts it in such a context? Moreover, the Gibbs state determines time flow only when you know the temperature and when you know the value of the Boltzmann constant. And how do you know the temperature when you do not know time? Mystery after mystery….

Connes and Rovelli continue:

In this paper, we consider a radical solution to this problem. This is based on the idea that one can extend the notion of time flow to general covariant theories, but this flow depends on the thermal state of the system.

Now, just think about it: what a radical solution! I propose even more radical and simpler solution: to extend the notion of time flow to general covariant theories, but this flow depends on the time flow!

More in detail, we will argue that the notion of time flow extends naturally to general covariant theories, provided that: i. We interpret the time flow as a 1- parameter group of automorphisms of the observable algebra (generalised Heisenberg picture); ii. We ascribe the temporal properties of the flow to thermodynamical causes, and therefore we tie the definition of time to thermodynamics; iii. We take seriously the idea that in a general covariant context the notion of time is not state-independent, as in non-relativistic physics, but rather depends on the state in which the system is.

"We interpret the time flow as a 1- parameter group of automorphisms of the observable algebra (generalised Heisenberg picture)" This is a radical solution indeed! Except that a) it is nothing new, and b) it applies only to closed system - while our universe seems to be an open system - otherwise nothing will ever ever ever happen - except of Cointon’s time evolution. No bit of information will ever be transfered or recorded. It is a radically dead world! We ascribe the temporal properties of the flow to thermodynamical causes, and therefore we tie the definition of time to thermodynamics. And what are thermodynamic causes ? Something related to heat? And what is heat? Is it related to chaotic motion? And what is motion? Is it, perhaps, change of a state with time? Wait, But what is time? We take seriously the idea that in a general covariant context the notion of time is not state-independent, as in non-relativistic physics, but rather depends on the state in which the system is. Now, question, is state a subjective or an objective property? There is a whole lot of arguments in the foundations of quantum theory that support the idea that there is no "objective state," and so there is no "reduction of the wave packet" - because what is being reduced, what jumps, is knowledge - which is subjective (or partly subjective). If so, then the flow of time is also subjective? Connes and Rovelli seem not to notice a serious problem here. Moreover, if state is objective, and if time is what they say time is, then, because our universe is evidently NOT in a thermodynamical equilibrium, therefore time does NOT flow in our universe. A radical concept indeed!