Action at A Distance (physics) - Electricity and Magnetism

Electricity and Magnetism

Efforts to account for action at a distance in the theory of electromagnetism led to the development of the concept of a field which mediated interactions between currents and charges across empty space. According to field theory we account for the Coulomb (electrostatic) interaction between charged particles through the fact that charges produce around themselves an electric field, which can be felt by other charges as a force. The concept of the field was elevated to fundamental importance in Maxwell's equations, which used the field to elegantly account for all electromagnetic interactions, as well as light (which, until then, had been a completely unrelated phenomenon). In Maxwell's theory, the field is its own physical entity, carrying momenta and energy across space, and action at a distance is only the apparent effect of local interactions of charges with their surrounding field.

Electrodynamics can be described without fields (in Minkowski space) as the direct interaction of particles with light-like separation vectors. This results in the Fokker-Tetrode-Schwarzchild action integral. This kind of electrodynamic theory is often called "direct interaction" to distinguish it from field theories where action at a distance is mediated by a localized field (localized in the sense that its dynamics are determined by the nearby field parameters). This description of electrodynamics, in contrast with Maxwell's theory, explains apparent action at a distance not by postulating a mediating entity (the field) but by appealing to the natural geometry of special relativity.

Various proofs, beginning with that of Dirac have shown that direct interaction theories (under reasonable assumptions) do not admit Lagrangian or Hamiltonian formulations (these are the so-called No Interaction Theorems). Consequently, the Fokker-Tetrode action is mostly a historic novelty. Still, attempts to recapture action at a distance without a field, which is often difficult to quantize, lead directly to the development of the quantum electrodynamics of Feynman and Schwinger.