# Poincare Recurrence and the Klein-Gordon Equation

1. Aug 5, 2012

### JPBenowitz

There exists Green's Functions such that the solutions appear to be retro-causal. The Klein-Gordon equation allows for antiparticles to propagate backwards in time. Does this mean the future can influence the past and present?

Then again The Poincare Recurrence Theorem states that over a sufficiently long enough time a dynamic system will return very close to its initial conditions. Is it possible that Poincare Recurrence Time can be reconciled with Green's Functions with non-zero values at negative t such that the function is describing a recurrence in the system? In other words could antiparticles be artifacts of a Poincare Recurring Universe pre-dating the big bang?

2. Aug 5, 2012

### Bill_K

JP, Not just the Klein-Gordon equation, all the fundamental equations of physics are time-symmetric. Given any solution, you can produce another equally valid one by replacing t → -t. However that does not mean that every solution is physically acceptable. In addition to satisfying the equations, there are boundary conditions which must also be satisfied. Time flows in one direction and one direction only -- this is an observed fact, and to satisfy it we impose "retarded" boundary conditions on our solutions.
This is not correct. Antiparticles propagate forward in time just like normal particles. (What about particles that are their own antiparticle, like photons. Which way would they propagate?) The Klein-Gordon equation has both positive and negative frequency solutions. An early interpretation of it assumed that negative frequency meant negative energy, and a negative energy solution would indeed represent a particle traveling backwards in time. But within a few years it was recognized that the interpretation was wrong, and such solutions were replaced by antiparticle solutions traveling forward.
The Poincare Recurrence Theorem is a statement of ergodicity, and applies to closed systems with constant energy. It does not, for example, apply to the expanding universe.