Digital Philosophy

Chapter 30: DM and CPT

We have shown how to construct DM models that handle CPT symmetry perfectly.  To understand this we have to look at the microscopic aspects of discrete time and state.

In DP we state: “If time is stopped, then properly started up in the opposite direction, then as a consequence of that single action, all charges will be complemented, parity will be complemented and the laws of physics will remain the same.”

How is it that changing the direction of time causes charge to be conjugated?   When an electron is moving through time in the positive direction (towards the future) that motion through time causes the particle to have a negative charge.  If the motion of time were reversed, then we would expect that same electron to be a positron with a positive charge.  In this case, we see a great difference between the dimensions of space and of time.  If an electron is going west, it will remain an electron if it stops and proceeds to the east.  If time is reversed, an electron that was going west must become a positron going east.  Further, we know that the reversal of the direction of time must have the effect of changing the handedness of everything.

Let us define what is meant by “…one instant in time.”  It is the smallest interval of time where the dynamic properties of particles are properly represented.  Therefore, at one instant in time, we expect particles to have properties such as charge, momentum and spin.  This is related to Zeno’s paradox about the arrow in flight.  “If, at one instant of time an arrow is perfectly still, why doesn’t it just fall straight down to the Earth?”  The answer must be that at that one instant of time, the dynamical information of a particle is somehow represented.  Further, there must be a process that looks at the dynamical information and uses that information to move the particle.  Yet the state of a system, at that point in time must be different in the two cases: 1. Where the system proceeds forward in time from that point; and 2.  Where the system proceeds in reverse from that point.  This is a subtle yet important point.  If we assume that, hypothetically, we can look at the state of a system at one instant in time and determine the charge of a particle, then reversing time must be more complex than just going in the other temporal direction because whenever time is stopped while going in the reversed direction we must find that the charges have been complemented.  This is tricky, because we insist that reversing time must not introduce anything new into the laws of physics.

Assume we are in control of a small closed DM system that operates in ways consistent with the laws of physics.  We can stop the process and then we can start it up in either the forward or the reverse direction.  If the system is running forwards and we stop the system, we can look at the bits that define the momentum of each particle and we can look at other static representations of the dynamical state information.  We can re-start in the forward direction and all is well.  If the system is running in the backwards direction, we can look at the state of each particle and note that each particle is the conjugate of the forward running particle.  Again, when we stop the backwards-running system, we can see that the static representation of the dynamical information is consistent with CPT symmetry.  If we restart it continuing in the reverse direction, all is well.  Between going forwards and going backwards there must be some kind of transformation that can change the static representation of the dynamical state information of a stopped forward going system into the proper static representation for a stopped, backwards going system.  Only after that transformation can the system proceed in reverse according to the laws of physics.  That transformation itself ought to be nothing more than an application of the laws of physics, as opposed to some special ad hoc process.  To summarize, the way that time must be reversed in a system that obeys the laws of physics seems to be that the system must be stopped, the information that is the static representation of the dynamical state must be changed to be consistent with CPT reversal, then time can proceed in the reverse direction.  While we find it difficult to imagine how this might happen in a model with continuous space-time, it is easy to imagine in a sub-class of RUCAs (with discrete space-time).

All that is necessary in order to accomplish the task of reversing time perfectly is that the reversal be accomplished in the following manner (at time α_t we apply R_t):

1.      Time flows as follows:   …α2t-3, α2t-2, α2t-1, α2t,

2.      Time is stopped after the completion of α2t and before α2t+1.  

3.      Then time again flows as follows: α2t, α2t-1, α2t-2, α2t-3…

4.      The last rule applied in the forwards direction was R2t(S2t, S2t-1) → (S2t, S2t+1).

At that point in time, after R_2t and before continuing with the next step, if we examine the state of the RUCA we can determine that the system was stopped while time was progressing in the forward direction.  All of the static representation of dynamic information indicates time is moving forwards.

We now put time into reverse and the first rule applied is again R_2t.

R_2t(S_2t, S_2t+1) → (S_2t, S_2t-1)

At this point in time, after the second application of R_2t and before R_2t-1 if we examine the state of the RUCA we can determine that the system is stopped while time is progressing in the reverse direction.  The second application of R_2t reversed the static representation of all of the dynamic information!  All particles have become their conjugates.  All spin and momentum have been reversed.  What is beautiful is that the R’s are the laws of physics.  The simple sequencing of the laws of physics:

1.      Drives the system forwards in normal time.

2.      Can change the static representation of all the dynamic information from forward to reverse or from reverse to forwards.

3.      Drives the system backwards in reversed time.

4.      All we need for perfect CPT symmetry is …α_2t-2, α_2t-1, α_2t, α_2t, α_2t-1, α_2t-2…

5.      Of course it also works perfectly for ...α_2t-1, α_2t, α_2t+1, α_2t+1, α_2t, α_2t-1…

There is a wonderful consequence of what we have just described.  This DM model has T symmetry.  However the T symmetry in this DM model is exactly equivalent to CPT symmetry in ordinary physics.  If a model like this were to reflect the physics of the real world, then T symmetry would be restored to physics as consistent with all the laws of physics and all experimental evidence!  Wonderful discoveries like this that pop up out of the DP approach to physics, encourage us to keep exploring DM models of physics.  If the sequence of time steps is ... α_2t-1, α_2t, α_2t+1, α_2t, α_2t-1, …α_2t-k, then what happens to the system is that a one step, non-physical discontinuity is introduced.  From then on the system continues to obey the laws of physics, but it’s on a new trajectory.  Reversibility is not compromised so a proper reverse into forwards, α_2t-k, α_2t-k+1...  and then ...α_2t-1, α_2t, α_2t+1, α_2t+1, α_2t… will return the haywire system to the point of the discontinuity and send it into reverse properly, with the damage undone.