What is time? Part 2
In the Introduction the authors, Alain Connes and Carlo Rovelli, write:
Our approach is based on a key structural property of von Neumann algebras. The links between some of the problems mentioned and central aspects of von Neumann algebras theory have already been noticed. A prime example is the relation between the KMS theory and the Tomita-Takesaki theorem [7]}. Rudolf Haag describes this connection as “a beautiful example of `prestabilized harmony’ between physics and mathematics" ([7], pg. 216). Here, we push this relation between a deep mathematical theory and one of the most profound and unexplored areas of fundamental physics much further.
It is clear from the above that the authors will be seeking the solution in the formal mathematical concepts, not in expanding our conceptual framework. That should be not a surprise as Alain Connes is a mathematician and Carlo Rovelli did not show a deep understanding of philosophical and conceptual problems either. Consider for instance the following sentence from page 2:
The problem we consider is the following. The physical description of systems that are not generally covariant is based on three elementary physical notions: observables, states, and time flow.
Let us analyze this: systems that are not generally covariant. This expression is meaningless. To say that a "system is generally covariant" is to convey no information at all and is meaningless. For instance, our solar system is a physical system. But is it generally covariant? Simply the concept of general covariance does not apply here. Our solar system system is neither generally covariant nor "not generally covariant". The concept of "general covariance" applies only to certain descriptions of certain mathematical models. But even then the concept is rather tricky. Every model, even one that at first may look as being not generally covariant, can be, if we wish so and if we are smart enough, to be reinterpreted as "generally covariant." I am sure that Connes and Rovelli know about it, as they must have read discussions of general covariance published in the literature and talked about at length at conferences, and yet here they take an easy path and forget about the necessity of being clear an precise. They go on to say:
Observables and states determine the kinematics of the system, and the time flow (or the 1-parameter subgroups of the Poincare’ group) describes its dynamics.
Here we have another wrong statement. For a physical system described through a classical mechanical model it is not so much observables and states that describe kinematics, but the symplectic manifold and its symplectic structure. We may have the same set of observables and states - and yet completely different kinematics, if we choose a different symplectic structure. Moreover, for a time dependent systems the time flow is not described by a 1-parameter subgroups of the Poincare’ group. What more one has to distinguish between passive and active "time flow". For instance for a dissipative system we can have a 1-parameter group of "passive time translations" and "1-parameter semi-group" of active time translations. These subtleties are important and it is not a good start when the authors make errors and oversimplifications already at the very start of their paper.
Next laputan statement:
In a general covariant theory there is no preferred time flow, and the dynamics of the theory cannot be formulated in terms of an evolution in a single external time parameter.
A general covariant theory may have a preferred time flow as one of its variables. It may have, for instance, a vector field - as a kinematical or a dynamical variable. This vector field will define a "preferred time flow" and yet the theory will be "general covariant". Moreover, as mentioned above, the question of whether a given theory is generally covariant or not is a tricky question. If I declare that my vector field is "fixed" - I will be told that my theory is not generally covariant. But if, for exactly the same theory, I will declare that my vector field is "arbitrary" and that my "configuration space" includes "all possible vector fields" - then my theory (the same as before) may be even admitted into the "generally covariant zoo." Some will still argue and ask me whether my vector filed is "dynamical" or "only kinematical", but even so they will not be able to define precisely what they mean by "dynamical" as opposite to "purely kinematical". They will try all kind of little tricks - but they will not succeed. Connes and Rovelli, I am sure, know about it, yet they repeatedly fall into a trap of following the popular slang that really deep thinkers have to "unlearn" if they want to really understand what is this theoretical physics all about.
