# Time reversal invariance and boundary conditions in electrodynamics

This is really several related questions...

The equations of classical electrodynamics are time-reversal invariant. However, when we solve the equations for a particular system of charges it is common to take the solution with only retarded fields. The reason given for this choice is usually so that the solution agrees with our notion of causality, but this choice of boundary conditions is clearly not time-reversal invariant.

1) According to Feynman (Lectures on Physics, Volume II, chapter 20) taking the theory with time-reversal invariant boundary conditions 'in many circumstances does not lead to physically absurd conclusions'... but this implies than in some circumstances it does -- i.e. it should be possible to test via experiment that the choice of retarded fields only is the physically 'correct' one. What would such an experiment look like?

2) If such an experiment exists isn't it therefore wrong to say that the electromagnetic force is time-reversal invariant?

3) Is the time-reversal invariant theory Feynman mentions the same as Feynman-Wheeler absorber theory?

4) Is the same thing true in Quantum Electodymanics? That is, is there a condition equivalent to fixing the boundary conditions in the classical theory where only retarded solutions are considered in the calculation? And if so, does an alternative with time-reversal invariant boundary conditions exist? And if it does, does the distinction lead to experimentally testable differences?

Even if you can answer only one or a few of these questions it would be very much appreciated!

Update: I worry this question is already too long, but I'll try to be more specific about what I mean by time-reversal invariance. Start with some distribution of (classical) charges with their initial velocities and accelerations specified, and assumed to be isolated from the rest of the universe. The Feynman-Wheeler theory predicts their subsequent motion without any reference to fields. Then, at some time $T$ later measure all the new (positions, velocities, accelerations), reverse all velocities, and take this as a new initial condition for a second isolated system. Time-reversal invariance means that the dynamics of the second system should just be exactly the reverse of the first system (so that if I measure the state of the second system a time $T$ later and reverse all velocities again I get back to the initial condition of the first system).

I'm pretty sure this is true in the Feynman-Wheeler theory, but if I use the usual theory of electrodynamics and solve for both systems considering only retarded fields I guess it will be false. Therefore, in principle at least it is possible to tell the difference between the two theories experimentally; however, I'm not sure this is doable or has been done in practice.

## 1 Answer

1) one way to find out is to measure motion of charged particles in their own fields on the microscale and infer which EM fields give best agreement with the observed motion. This was never done, because there are always EM fields of other sources and the motion is hard to measure accurately enough.

2) it depends on what you mean by "force is time-reversal invariant". Lorentz expression of the EM force is time-reversal invariant in the sense that the time reversal changes both sign of $\mathbf v$ and $\mathbf B$, so $q(\mathbf E + \mathbf v\times\mathbf B)$ remains the same value. However, when you express $\mathbf E, \mathbf B$ as retarded fields in terms of past positions, velocities and accelerations of other particles, the field is eliminated and the resulting expression may not be time-reversal invariant. Only if the fields are chosen as half-advanced half-retarded, is such force time-reversal invariant. Irrespective of the choice of the fields, if the fields are retained in the equations, the theory is time-reversal invariant.

3) most probably yes

• Thanks Jan... 1) When you refer to 'particles in their own fields' are you talking about the self-interaction of fundamental particles? If so, then the quantum theory is relevant here. Do you know of any way to test on the classical level? Or of a reference where the quantum case is discussed? 2) I'm not entirely sure what I mean by 'time-reversal invariant' either :-P... I'll edit the question to try to be a bit more specific. – Mark A Feb 21 '14 at 0:36
• 1) No, I meant interaction of different particles. Perhaps some relevant experiments could be done on particles in a Paul/Penning trap, but I do not know of any such work. – Ján Lalinský Feb 21 '14 at 3:40
• I see... I think I'd better do some more research into it! – Mark A Feb 21 '14 at 6:30