The Nature of Time and Semigroups
By Tom Brown
We attempt to explain to a layman in a very basic way from a mathematics' viewpoint the concepts of a semigroup and later on a dynamical system, the nature of time and some related ideas.
The terminology is probably not familiar to the reader however should appeal to intuition and the reasoning may be followed well enough. Don't give up too easily. You may perhaps think of the unknown words as names simply, like Tom, Dick or Harry and at this stage not try to understand everything. One may also find it worthwhile to consult a dictionary or a good general encyclopedia.
The sketches should be of help to understand the essential ideas. Someone with training in advanced mathematics shouldn't have a problem.
Our first contact with semigroups is usually in abstract algebra which is often the first, a basic subject of post-graduate studies for an honours degree. The theory illustrates the incredible power of abstraction in modern mathematics, it is usually at this point where you fall in love.
One of the axioms for a group is relaxed, wherein the definition is weakened. This axiom to do with inverses is just mentioned in the algebra course at the start and in itself is not very interesting.
The concept is found again in operator theory in functional analysis but there in a much more structured and complicated situation.
Defining properties, cause and effect
The concepts here are explained in a classical setting and it is a family of linear operators on a vector space. The phase space would give all possible configurations at a given time, t , of a model system that being a complete description at that instant: time, position, velocity, and acceleration of each particle and giving all variables and of other relevant information.
Time is of a theological scale as I see it, and an “external” divine being is the only thing that could stop it or interfere should it want to.
We consider mathematical descriptions of evolution paths in a phase space. An evolution of operators P(t) determines a trajectory in phase space from a given initial position. Altering the present would have to alter the future. What is Time? For now we ignore relativity and think of physics in a classical framework.
Our causality is a deterministic cause and effect and concepts arise as of semigroups or of dynamical systems. In a semigroup you have in effect two essential properties, the nature of time is encapsulated in,
E( 0) = I and E ( t + s) = E( t) E( s) = E( s) E( t)
These equations are really just ordinary common sense.
I is the identity operator it maps a point in the phase space into itself. The first equation thus asserts that after zero time or zero seconds one ends up at the same place you've started. At time zero the trajectory is itself and proceeds from there.
The second equation can in principle also easily be understood, as can seen in an example: If I am fifty years old and a decade passes I will then be sixty, but if I am ten years old and fifty years pass I also am sixty. It would result in the same age (the destination).
An evolution operator E determines a curve in phase space from a given initial position. It too may be very simply understood from a graph of a typical trajectory. A trajectory evolves in the manner explained, which sounds quite reasonable. The properties now should seem obvious.
Illustrations can be handy. In one dimension it would be the exponential function of our graphs and of course t is time. Our example is easily adapted to the euclidean plane, for a two dimensional cartesian phase space.
Considering only semigroup evolutions we don't make the invertability assumption of dynamical systems. That is really not terribly interesting in our current time-travel scenarios or in our current discussion.
In conclusion, Relativity
All this was in a Newtonian setting. I believe the ideas can be adapted to General Relativity by adding time itself as another variable.
In effect there will be one coordinate more, one more dimension in the phase space and in this process the idea of simultaneity has to be abandoned as it is in relativity theory.
This would mean the evolution is in an extra dimension namely time, and therefore the equations would be defined as implicit. In principle there shouldn't be a problem. The same idea, the evolution is now just a trajectory in phase space but just with one more variable.