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Notes on Time

2020-2-3
2020-1-6

The following comes from Satosi Watanabes article on Time. [1] I'll be studying this article in the next few days.

Reversibility and Retrodictability

It is well-known that all the basic microscopic laws of classical physics are symmetrical with respect to the two directions of the time variable. This does not mean that only small-scale physical phenomena are temporarily symmetrical. It is not a question of the scale of the system but it is one of the mode of description. Even large-scale phenomena will show the symmetry if they are described minutely in every detail. This situation raises immediately a pair of twin questions. How is it possible that something essentially symmetric appear asymmetrical, without being distorted in some sense? Which one of the two equivalent directions receives the privileged meaning of "past-to-future"?

What precisely do we mean by saying that the microscopic laws are symmetrical with respect to the two directions of time? We mean by this what is usually called "reversibility of physical laws. [2] In anticipation of what follows, I may mention that in classical physics reversibility implies retrodictability [3] but retrodictability does not necessarily imply reversibility. That is to say, reversibility is a much more restricted notion.

To explain reversibility, it is probably better to start with the notions of reversed phenomenon and reversed state. Suppose we take a motion picture of a physical phenomenon, and run the film backwards through the projector. The phenomenon one sees on the screen is the reversed phenomenon, and the state of the system at an instant in the reversed phenomenon is the reversal of the state of the origin phenomenon at the corresponding instant. Thus, in a reversed state, an object will have the same position but it will be moving in the opposite direction, that is, the velocity changes its sign. In general, all physical quantities can be classified in two classes, those which keep the same sign and those which change the sign in the reverse state. To determine the class of a physical quantity, we have to agree that the spatial positions of all objects keep the same sign, and that the basic attributes of objects, such as mass, electric charge, etc., also keep the same sign.

Now we are prepared to define "reversibility". If a phenomenon and its reversed phenomenon are both allowed by a theory under consideration, then the phenomenon is reversible. All the basic theories (mechanical and electromagnetic) of classical physics from which all other laws are supposed to be derivable are reversible. Laws of friction in mechanics, Ohm's Law, Newton's Law of Heat Conduction, which are secondary macroscopic laws supposedly derivable in principle from the more basic laws, are not reversible.

All these irreversible laws are in agreement with a general law called the Second Law of Thermodynamics. This law says that if the entropy of an (adiabatically) isolated system changes, it can only increase.

This conflict between the basic reversibility and the macroscopic irreversibility is the classical way of formulating the essential problem of the direction of time. In this connection, I have to make two important remarks. First, we have to note, in agreement with many physicists in the past, that the alleged conflict originates from the applications of statistical consideration in the macroscopic description. Second, we want to point out, in a departure from the accepted view, that the crucial conflict is not between reversibility and irreversibility, but between retrodictability and irretrodictability since the direct consequence of statistical consideration is not irreversibility but irretrodictability.

[2] Aside from considerations of entropy increase, the suggestion has been made that the expansion of the universe does itself give a physical phenomenon that defines the direction of time. This definition would indeed associate time with what appears to be a universal feature of our universe.

[3] In quite a different way, Karl R. Popper has argued on the basis of our lack of definite knowledge of cosmological structure that we should not give any cosmic significance to entropy decrease, or to connect that decrease with the arrow of time.

Somehow I got the notion that time is reversible in a quantum particle if you consider it as a single system. I don't know if I read it somewhere or I just made it up. Mathematically, you can predict the propagation of a pure (simple) wave forward or backward in time. Interacting waves can be superimposed. When this happens information about the waves before the interaction are lost if the wave interaction changes the energy states of the interacting waves. That is, if energy leaves the interacting wave system. So you cannot make predictions about waves going backwards in time if their interaction has a net energy change.

Generally, entropy in a system changes whenever energy enters or leaves the system. It follows then, that the informational content in a system changes with energy changes in the system. So you can make predictions about a quantum particle considered as a single system if it never interacts with another system. Hence this is an idealized situation.

In the quantum world, we observe quantum particles, but not the manifestion of the medium in which quanta arises from. So we can make predictions about the exact trajectories of quanta in a magnetic field for example. But we can't make exact predictions about where a quantum particle will arise out of a field because we don't have enough specific information about the "state" of wave fields that will produce the creation of a quantum. We cannot take enough measurements about the medium or field from which the quantum arises from to know enough about the system to make exact predictions. As observers of the quantum world, we are limited to making statistical predictions.

What's important to remember is that everything depends on the states of the interacting systems. The word "state" is associated with "information" in the description of interacting systems. The description of a system as a function of time thus depends on the changes in its state parameters.

There is a lot of stuff here to figure out. I'm not sure how correct I am on anything here. But I know this is really fundamental and important to think about. It's taken me years of thinking about this to come to this point - surely because of my own limitations.

[1] Time and the Probabilistic View of the World, Satosi Watanabe, p. 527, in The Voices of Time, ed. J. T. Fraser., 1966.