What advantages does action-at-a-distance description have over the field view of forces?

Question

It is written in Jackson (page 3) :

In fact, though there are recurring attempts to eliminate explicit reference to the fields in favor of action-at-a-distance descriptions of the interaction of charged particles, the concept of the electromagnetic field is one of the most fruitful ideas of physics, both classically and quantum mechanically.

What does he mean?

What advantages can a action-at-a-distance description have over the field view of forces, that makes (at least some) people try toward that description?

Answer 1

What advantages can a action-at-a-distance description have over the field view of forces, that makes (at least some) people try toward that description?

In classical electromagnetic theory one has both charged particles and the EM field. While state of a particle - its position and momentum - can be chosen freely, the state of the EM field is constrained a lot by the Maxwell equations, namely by the equations $$ \nabla \cdot \mathbf E = \rho, $$ $$ \nabla \cdot \mathbf B = 0. $$ Thus the field is not an entirely independent object to describe; it is partially dependent on the particles and thus its numerical representation is partially superfluous in describing the state of the system.

On the other hand, even with these restrictions, the fields can be chosen in an infinite number of ways and it is not immediately clear which initial and boundary conditions should be supplemented to the equations to solve for, say, the motion of the electron around the proton.

If the particles are points, which is the simplest case and electrons may be such according to current knowledge, this formulation with one field has also very serious problem that it is not capable to determine force acting on the charged particle (the field is singular at the position of all point particles). This lead to great confusion about the equation of motion of a charged particle - see the most famous attempt, so-called Lorentz-Abraham-Dirac equation, which has serious problems though.

Action-at-a-distance theory developed in works by Tetrode, Fokker, Frenkel and by Feynman and Wheeler and others removes both the superfluousness and ambiguity of the field by replacing it by a new principle that determines how the particles move. For example, the above works concentrated on a theory in which the force acting on a particle is a sum of the Lorentz forces determined by individual half-advanced, half-retarded EM fields due to all other particles. More intuitively, one can set up such theory with purely retarded forces. As a result, one way or another, one obtains consistent equations of motion of charged point-like mutually-interacting particles. The mathematical function field is still useful as an instrument to find the force and derive the consequences of the theory, but it has no degree of freedom of its own, because it is entirely determined by the past motion (or future or both) of the charged particles. There is no "free field" in such theory; particles just interact with each other, which can be aesthetically pleasing to some.

Answer 2

In the first place, let me stress that the term "action-at-a-distance" is a misnomer that has generated lots of confusion. A better term is direct-particle-action. The advantages of this model over the contact-action model of the field-picture are:

1. Conceptual simplicity. There are only real particles interacting directly. In the field-picture there are real particles, bare particles, clouds of virtual particles, and fields.

2. Economy of description. The only degrees of freedom are those of particles. In the field-picture, there are both particle and field degrees of freedom, and fields have an infinite number of them.

3. Physical model. There are no difficulties associated to infinite self-energies, violations of causality and inertia, infinite vacuum energies, etc.

4. Computational accuracy. The simplicity and rigor of the direct-particle-action model allows the development of accurate computational models. This is is specially true on the study of strong interactions, where the particle-based models can make many useful quantitative statements about nuclear or hadronic dynamics of realistic systems, even beyond the scope of lattice-based field-theoretic computational models.