Wheeler–Feynman absorber theory

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The Wheeler–Feynman absorber theory is an interpretation of electrodynamics that starts from the idea that a solution to the electromagnetic field equations has to be symmetric with respect to time-inversion, as are the field equations themselves. The motivation for such choice is mainly due to the importance that time symmetry has in physics. Indeed, there is no apparent reason for which such symmetry should be broken, and therefore one time direction has no privilege to be more important than the other. Thus, a theory that respects this symmetry appears, at least, more elegant than theories with which one has to arbitrarily choose one time direction over the other as the preferred one.

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[edit] The problem of causality

The first problem one has to face if one wants to construct a time-symmetric theory is the problem of causality. Maxwell equations and the wave equation for electromagnetic waves have, in general, two possible solutions: a retarded solution and an advanced one. This means that if we have an electromagnetic emitter which generates a wave at time t0 = 0 and point x0 = 0, then the wave of the first solution will arrive at point x1 at the instant t1 = x1 / c after the emission (where c is the speed of light) while the second one will arrive at the same place at the instant t2 = x2 / c before the emission. The second wave appears to be clearly unphysical as it means that in a model where it is considered we could see the effect of any phenomena before it happens, and therefore it's usually discarded in the interpretation of electromagnetic waves.

Feynman and Wheeler overcame this difficulty in a very simple way. Consider all the emitters which are present in our universe, then if all of them generate electromagnetic waves in a symmetric way, the resulting field is

E^{tot}(\mathbf{x},t)= \sum_{n}\frac{E_n^{ret}(\mathbf{x},t)+E_n^{adv}(\mathbf{x},t)}{2}\ \quad \text{.}

Then, if you consider that in your universe holds the relation

E^{free}(\mathbf{x},t)=\sum_{n} \frac{E_n^{ret}(\mathbf{x},t)-E_n^{adv}(\mathbf{x},t)}{2}=0 \quad \text{,}

you can freely add this last quantity to the total field solution of Maxwell equations (being this a solution of the homogenous Maxwell equation) and you get

E^{tot}(\mathbf{x},t)= \sum_{n}\frac{E_n^{ret}(\mathbf{x},t)+E_n^{adv}(\mathbf{x},t)}{2}+ \sum_{n}\frac{E_n^{ret}(\mathbf{x},t)-E_n^{adv}(\mathbf{x},t)}{2} =\sum_{n}E_n^{ret}(\mathbf{x},t)

In this way the model sees just the effect of the retarded field, and so causality still holds. The presence of this free field is related to the phenomenon of the absorption from all the particles of the universe of the radiation emitted by each single particle. Still the idea is quite simple as it's the same phenomenon which happens when an electromagnetic wave is absorbed from an object; if you look to the process on a microscopic scale you will see that the absorption is due to the presence of the electromagnetic fields of all the electrons which react to the external perturbation and create fields which cancel it. The main difference here is that the process is allowed to happen with advanced waves.

Finally one could still consider that this formulation is still no more symmetric than the usual one as the retarded time direction still seems to be privileged. However, this is only an illusion as one can always reverse the process simply reversing who is considered as the emitter and who is considered the absorber. Any apparent 'privilege' of a time direction is only due to the arbitrary choice of which is the emitter and which the absorber.

[edit] The problem of self-interaction and damping

The motivation of finding a different interpretation of the electromagnetic phenomena comes even from the need for a satisfying description of the electromagnetic radiation process. The point here is the following: consider a charged puntiform particle that moves in a nonuniform way (for example an oscillating one x(t) = x0cos(ωt)), than is known that this particle radiates, and so loses energy. If you write down the Newton equation of the particle you need a damping term which takes into account this energy loss. The first solution to this problem is mainly due to Lorentz and was later expanded on by Dirac. Lorentz interpreted this loss as due to the retarded self-interaction of such a particle with its own field. Such interpretation though is not completely satisfactory as it leads to divergences in the theory and needs some assumption on the structure of charge distribution of the particle. Dirac generalized the formula given by Lorentz for the damping factor to make it relativistically invariant. While doing so, he also suggested a different interpretation of the damping factor as being due to the free fields generated from the particle at its own position.

E^{damping}(\mathbf{x}_j,t)=\frac{E_j^{ret}(\mathbf{x}_j,t)-E_j^{adv}(\mathbf{x}_j,t)}{2}

The main lack of this formulation is the absence of a physical justification for the presence of such fields.

So absorber theory was formulated as an attempt to correct this point. Using absorber theory, if we assume that each particle does not interact with itself and evaluate the field generated by the particle j at its own position (the point xj) we get:

E^{tot}(\mathbf{x}_j,t)=\sum_{n\neq j} \frac{E_n^{ret}(\mathbf{x}_j,t)+E_n^{adv}(\mathbf{x}_j,t)}{2}\ \text{.}

It's clear that if we now add to this the free fields

E^{free}(\mathbf{x}_j,t)=\sum_{n} \frac{E_n^{ret}(\mathbf{x}_j,t)-E_n^{adv}(\mathbf{x}_j,t)}{2}=0

we obtain

E^{tot}(\mathbf{x}_j,t)=\sum_{n\neq j} \frac{E_n^{ret}(\mathbf{x}_j,t)+E_n^{adv}(\mathbf{x}_j,t)}{2} +\sum_{n} \frac{E_n^{ret}(\mathbf{x}_j,t)-E_n^{adv}(\mathbf{x}_j,t)}{2}

and so

E^{tot}(\mathbf{x}_j,t)=\sum_{n\neq j} E_n^{ret}(\mathbf{x}_j,t)+E^{damping}(\mathbf{x}_j,t)

This interpretation avoids the problem of divergent self-energy for a particle giving a reasonably physical interpretation to the equation of Dirac.

[edit] Conclusions

Still this expression of the damping fields is not free of problems, as written in the non-relativistic limit it gives:

E^{damping}(\mathbf{x}_j,t)=\frac{e}{6\pi c^3}\dot{\ddot{x}}

which is the Lorentz formulation. Since the third derivative with respect to the time appears here, it is clear that to solve the equation is not sufficient to give just the initial position and velocity of the particle, but the initial acceleration of the particle will also be needed, which makes no sense. This problem is solved by observing that the equation of motion for the particle has to be solved together with the Maxwell equations for the field generated by the particle itself. So instead of giving the initial acceleration one can give the initial field and the boundary condition. This restores the coherence of the physical interpretation of the theory. Still some difficulties may arise when one tries to solve the equation and interpret the solution. It is commonly stated that the Maxwell equations are classical and cannot correctly account for microscopic phenomena such as the behavior of a point-like particle where quantum mechanical effects should appear. However with absorber theory, Wheeler and Feynman were able to create a coherent classical approach to the problem.

One last remark should be said on the problem of the self-energy interaction. When they formulated this paper Wheeler and Feynman were trying to avoid this divergent term. Later however, Feynman would come to state that self-interaction is needed as it correctly accounts, within quantum mechanics, for the Lamb shift.

[edit] Key papers

  • J. A. Wheeler and R. P. Feynman, "Interaction with the Absorber as the Mechanism of Radiation," Reviews of Modern Physics, 17, 157–161 (1945).
  • J. A. Wheeler and R. P. Feynman, "Classical Electrodynamics in Terms of Direct Interparticle Action," 21, 425–433 (1949).

[edit] See also