Wheeler–Feynman absorber theory

The Wheeler–Feynman absorber theory (also called the Wheeler–Feynman Time-Symmetric 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. This is referred to in the time travel feature film Deja Vu ( 2006 ).

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 $t_{0}=0$ and point $x_{0}=0$, then the wave of the first solution will arrive at point $x_{1}$ at the instant $t_{1}=x_{1}/c$ after the emission (where $c$ is the speed of light) while the second one will arrive at the same place at the instant $t_{2}=x_{1}/c$ 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 phenomenon 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}}.\$

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,\$

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.

Resolution of causality issue

T.C. Scott and R.A. Moore demonstrated that the apparent acausality caused by the presence of advanced Liénard–Wiechert potentials in their original formulation could be removed by recasting their theory into a fully relativistic many-bodied electrodynamics formulation in terms of retarded potentials only without the complications of the absorber part of the theory. If we consider the Lagrangian acting on particle 1 from the time-symmetric fields generated by particle 2, we have:

$L_{1}=T_{1}-{\frac {1}{2}}\left((V_{R})_{1}^{2}+(V_{A})_{1}^{2}\right)$

where $T_{i}$ is the relativistic kinetic energy functional of particle i, and, $(V_{R})_{j}^{i}$ and $(V_{A})_{j}^{i}$ are respectively the retarded and advanced Liénard–Wiechert potentials acting on particle j from the relativistic electromagnetic fields generated by particle i. Conversely, the corresponding Lagrangian for particle 2 motioned by particle 1 is:

$L_{2}=T_{2}-{\frac {1}{2}}\left((V_{R})_{2}^{1}+(V_{A})_{2}^{1}\right).$

It was originally demonstrated with computer algebra and then proven mathematically that the difference between a retarded potential of particle i acting on particle j and the advanced potential of particle j acting on particle i is simply a total time derivative:

${\frac {dF}{dt}}=(V_{R})_{j}^{i}-(V_{A})_{i}^{j}$

or a divergence as it is called in the Calculus of variations because it contributes nothing to the Euler-Lagrange equations. Thus, by adding the right amount of total time derivatives to these Lagrangians, the advanced potentials can be eliminated. The Lagrangian for the N-body system is therefore:

$L=\sum _{i=1}^{N}T_{i}-{\frac {1}{2}}\sum _{i\neq j}^{N}(V_{R})_{j}^{i}$

in which the advanced potentials make no appearance. For $N=2$ this Lagrangian will generate exactly the same equations of motion of $L_{1}$ and $L_{2}$ and consequently the Physics of the problem is preserved. Therefore, from the point of view of an outside observer viewing the relativistic n-body problem, everything is causal. However, if we isolate the forces acting on a particular body, the advanced potential makes its appearance. This recasting of the problem comes at a price: the N-body Lagrangian depends on all the time derivatives of the curves traced by all particles i.e. the Lagrangian is infinite order. Nonetheless, symmetry under exchange of particles and total Generalized momenta (resulting from the definition of an infinite order Lagrangian) are conserved. The one feature which might seem non-local is that Hamilton's principle is applied to a relativistic many-particle system as a whole but this is as far as one can go with a classical (non quantum-mechanical) theory. However, much progress was made in examining the unresolved issue of quantizing the theory. Numerical solutions for the classical problem were also found.

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 particle that moves in a nonuniform way (for example an oscillating one $x(t)=x_{0}cos(\omega t)$), then 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 $x_{j}$) 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. Moore and Scott showed that the radiation reaction can be alternatively derived using the notion that on average the net dipole moment is zero for a collection of charged particles, thereby avoiding the complications of the absorber theory.

Conclusions

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

$E^{damping}(\mathbf {x} _{j},t)={\frac {e}{6\pi c^{3}}}{\frac {\mathrm {d} ^{3}}{\mathrm {d} t^{3}}}x$

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.

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. This theory is mentioned in the chapter entitled "Monster Minds" in Feynman's autobiographical work Surely You're Joking, Mr. Feynman!. It lead to the formulation of a framework of quantum mechanics using a Lagrangian and action as starting point rather than a Hamiltonian, namely the formulation using Feynman Path integrals.

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," Reviews of Modern Physics, 21, 425–433 (1949).

See also

• Action at a distance (physics)
• Symmetry in physics and T-symmetry
• Wave equation and Maxwell's equations
• Transactional interpretation

References

• Hoyle, F. and Narlikar, J. V., "Cosmology and action-at-a-distance electrodynamics," Reviews of Modern Physics, 67, 113–155 (1995).

External links

• J. A. Wheeler and R. P. Feynman, "Interaction with the Absorber as the Mechanism of Radiation" Caltech Library of Authors