# Action at a distance

In physics, action at a distance is the interaction of two objects which are separated in space with no known mediator of the interaction.

This term was used most often in the context of early theories of gravity and electromagnetism to describe how an object responds to the influence of distant massive or charged bodies. More generally “*Action at a distance*” describes the break between human intuition, where objects have to touch to interact, and physical theory. The exploration and resolution of this problematic phenomenon led to significant developments in physics, from the concept of a field, to descriptions of quantum entanglement and the mediator particles of the standard model.

### Electricity

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-Schwartzchild 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 in which two events in spacetime can be physically distinct and still have “*zero*” separation. Perceived action at a distance is a result of human bias for spatial separation, charged particles can be separated in space, and yet geometrically connected.

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.

### Gravity

**Newton**

Newton’s theory of gravity offered no prospect of identifying any mediator of gravitational interaction. His theory assumed that gravitation acts instantaneously, regardless of distance. Kepler’s observations gave strong evidence that in planetary motion angular momentum is conserved. (The mathematical proof is only valid in the case of a Euclidean geometry.) Gravity is also known as a force of attraction between two objects because of their mass.

A related question, raised by Ernst Mach, was how rotating bodies know how much to bulge at the equator. This, it seems, requires an action-at-a-distance from distant matter, informing the rotating object about the state of the universe. Einstein coined the term Mach’s principle for this question.

**Einstein**

According to Albert Einstein’s theory of special relativity, instantaneous action-at-a-distance was seen to violate the relativistic upper limit on speed of propagation of information. If one of the interacting objects were to suddenly be displaced from its position, the other object would feel its influence instantaneously, meaning information had been transmitted faster than the speed of light.

One of the conditions that a relativistic theory of gravitation must meet is to be mediated with a speed that does not exceed c, the speed of light in a vacuum. It could be seen from the previous success of electrodynamics that the relativistic theory of gravitation would have to use the concept of a field or something similar.

This problem has been resolved by Einstein’s theory of general relativity in which gravitational interaction is mediated by deformation of space-time geometry. Matter warps the geometry of space-time and these effects are, as with electric and magnetic fields, propagated at the speed of light. Thus, in the presence of matter, space-time becomes non-Euclidean, resolving the apparent conflict between Newton’s proof of the conservation of angular momentum and Einstein’s theory of special relativity. Mach’s question regarding the bulging of rotating bodies is resolved because local space-time geometry is informing a rotating body about the rest of the universe. In Newton’s theory of motion, space acts on objects, but is not acted upon. In Einstein’s theory of motion, matter acts upon space-time geometry, deforming it, and space-time geometry acts upon matter.

### Quantum mechanics

Since the early 20th century, quantum mechanics has posed new challenges for the view that physical processes should obey locality. The collapse of the wave function of an electron being measured, for instance, is presumed to be instantaneous. Whether this counts as action-at-a-distance hinges on the nature of the wave function and its collapse, issues over which there is still considerable debate amongst scientists and philosophers. One important line of debate originated with Einstein, who challenged the idea that the wave function offers a complete description of the physical reality of a particle by showing that such a view leads to a paradox. Einstein, along with Boris Podolsky and Nathan Rosen, proposed a thought experiment to demonstrate how two physical quantities with non-commuting operators (e.g. position and momentum) can have simultaneous reality. Since the wave function does not ascribe simultaneous reality to both quantities and yet they can be shown to exist simultaneously, Einstein, Podolsky and Rosen (EPR) argued that the quantum mechanical description of reality must not be complete.

This thought experiment, which came to be known as the EPR paradox, hinges on the principle of locality. A common presentation of the paradox is as such: two particles interact briefly and then are sent off in opposite directions. One could imagine an atomic transition that releases two photons A and B (spin-1 particles) with no overall change in momentum. The photons end up so far away from each other that one can no longer influence the other (this is the principle of locality). As long as the photons act only locally, the perfect anticorrelation of their momenta will hold. That is, if photon A has a momentum of 1 (in appropriate units) then by the conservation of momentum photon B must have a momentum of -1. Therefore, EPR’s argument goes, we could measure the position of photon A, and also simultaneously know photon A’s momentum by measuring photon B (since A’s momentum must be the opposite of B’s).

Because EPR’s proposal involved properties that were not captured in the wave equation and which were local and real, it became known as a local ‘hidden variables’ theory. After the EPR paper, several scientists such as de Broglie took up interest in local hidden variables theories. In the 1960s John Bell derived an inequality that showed a testable difference between the predictions of quantum mechanics and local hidden variables theories. Experiments testing Bell-type inequalities in situations analogous to EPR’s thought experiments have been consistent with the predictions of quantum mechanics, suggesting that local hidden variables theories can be ruled out. Whether or not this is interpreted as evidence for nonlocality depends on one’s interpretation of quantum mechanics. In the standard interpretation the wave function is still considered a complete description so the nonlocality is generally accepted, but there is still debate over what this means physically.

One important question raised by this ambiguity is whether Einstein’s theory of relativity is compatible with the experimental results demonstrating nonlocality. Relativistic quantum field theory requires interactions to propagate at speeds less than or equal to the speed of light, so “quantum entanglement” cannot be used for faster-than-light-speed propagation of matter, energy, or information. Measurements of one particle will be correlated with measurements on the other particle, but this is only known after the experiment is performed and notes are compared, therefore there is no way to actually send information faster than the speed of light. On the other hand, relativity predicts causal ambiguities will result from the nonlocal interaction. In terms of the EPR experiment, in some reference frames measurement of photon A will cause the wave function to collapse, but in other reference frames the measurement of photon B will cause the collapse.

Non-standard interpretations of quantum mechanics also vary in their response to the EPR-type experiments. Bohm interpretation gives an explanation based on nonlocal hidden variables for the correlations seen in entanglement. Many advocates of the many-worlds interpretation argue that it can explain these correlations in a way that does not require a violation of locality, by allowing measurements to have non-unique outcomes.