What is time? According to Connes and Rovelli - Part 5
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….
