Electromagnetism is a relativistic theory. Indeed, it had been relativistic, or Lorentz invariant, before Einstein and Minkowski understood that this somewhat peculiar symmetry of Maxwell’s equations was not accidental but expressive of a radically new structure of time and space. Minkowski spacetime, in contrast to Newtonian spacetime, doesn’t come with a preferred space-like foliation, its geometric structure is not one of ordered slices representing “objective” hyperplanes of absolute simultaneity. But Minkowski spacetime does have an objective (geometric) structure of light-cones, with one double-light-cone originating in every point. The most natural way to define a particle interaction in Minkowski spacetime is to have the particles interact directly, not along equal-time hyperplanes but along light-cones
In other words, if zi(τi) and zj(τj) denote the trajectories of two charged particles, it wouldn’t make sense to say that the particles interact at “equal times” as it is in Newtonian theory. It would however make perfectly sense to say that the particles interact whenever
(zμi zμj)(zμi zμj) = (zi – zj)2 = 0 —– (1)
For an observer finding himself in a universe guided by such laws it might then seem like the effects of particle interactions were propagating through space with the speed of light. And this observer may thus insist that there must be something in addition to the particles, something moving or evolving in spacetime and mediating interactions between charged particles. And all this would be a completely legitimate way of speaking, only that it would reflect more about how things appear from a local perspective in a particular frame of reference than about what is truly and objectively going on in the physical world. From “Gods perspective” there are no fields (or photons, or anything of that kind) – only particles in spacetime interacting with each other. This might sound hypothetical, but, it actually is not entirely fictitious. for such a formulation of electrodynamics actually exists and is known as Wheeler-Feynman electrodynamics, or Wheeler-Feynman Absorber Theory. There is a formal property of field equations called “gauge invariance” which makes it possible to look at things in several different, but equivalent, ways. Because of gauge invariance, this theory says that when you push on something, it creates a disturbance in the gravitational field that propagates outward into the future. Out there in the distant future the disturbance interacts with chiefly the distant matter in the universe. It wiggles. When it wiggles it sends a gravitational disturbance backward in time (a so-called “advanced” wave). The effect of all of these “advanced” disturbances propagating backward in time is to create the inertial reaction force you experience at the instant you start to push (and cancel the advanced wave that would otherwise be created by you pushing on the object). So, in this view fields do not have a real existence independent of the sources that emit and absorb them. It is defined by the principle of least action.
Wheeler–Feynman electrodynamics and Maxwell–Lorentz electrodynamics are for all practical purposes empirically equivalent, and it may seem that the choice between the two candidate theories is merely one of convenience and philosophical preference. But this is not really the case since the sad truth is that the field theory, despite its phenomenal success in practical applications and the crucial role it played in the development of modern physics, is inconsistent. The reason is quite simple. The Maxwell–Lorentz theory for a system of N charged particles is defined, as it should be, by a set of mathematical equations. The equation of motion for the particles is given by the Lorentz force law, which is
- The electromagnetic force F on a test charge at a given point and time is a certain function of its charge q and velocity v, which can be parameterized by exactly two vectors E and B, in the functional form:
describing the acceleration of a charged particle in an electromagnetic field. The electromagnetic field, represented by the field-tensor Fμν, is described by Maxwell’s equations. The homogenous Maxwell equations tell us that the antisymmetric tensor Fμν (a 2-form) can be written as the exterior derivative of a potential (a 1-form) Aμ(x), i.e. as
Fμν = ∂μ Aν – ∂ν Aμ —– (2)
The inhomogeneous Maxwell equations couple the field degrees of freedom to matter, that is, they tell us how the charges determine the configuration of the electromagnetic field. Fixing the gauge-freedom contained in (2) by demanding ∂μAμ(x) = 0 (Lorentz gauge), the remaining Maxwell equations take the particularly simple form:
□ Aμ = – 4π jμ —– (3)
□ = ∂μ∂μ
is the d’Alembert operator and jμ the 4-current density.
The light-cone structure of relativistic spacetime is reflected in the Lorentz-invariant equation (3). The Liénard–Wiechert field at spacetime point x depends on the trajectories of the particles at the points of intersection with the (past and future) light-cones originating in x. The Liénard–Wiechert field (the solution of (3)) is singular precisely at the points where it is needed, namely on the world-lines of the particles. This is the notorious problem of the electron self-interaction: a charged particle generates a field, the field acts back on the particle, the field-strength becomes infinite at the point of the particle and the interaction terms blow up. Hence, the simple truth is that the field concept for managing interactions between point-particles doesn’t work, unless one relies on formal manipulations like renormalization or modifies Maxwell’s laws on small scales. However, we don’t need the fields and by taking the idea of a relativistic interaction theory seriously, we can “cut the middle man” and let the particles interact directly. The status of the Maxwell equation’s (3) in Wheeler–Feynman theory is now somewhat analogous to the status of Laplace’s equation in Newtonian gravity. We can get to the Gallilean invariant theory by writing the force as the gradient of a potential and having that potential satisfy the simplest nontrivial Galilean invariant equation, which is the Laplace equation:
∆V(x, t) = ∑iδ(x – xi(t)) —– (4)
Similarly, we can get the (arguably) simplest Lorentz invariant theory by writing the force as the exterior derivative of a potential and having that potential satisfy the simplest nontrivial Lorentz invariant equation, which is (3). And as concerns the equation of motion for the particles, the trajectories, if, are parametrized by proper time, then the Minkowski norm of the 4-velocity is a constant of motion. In Newtonian gravity, we can make sense of the gravitational potential at any point in space by conceiving its effect on a hypothetical test particle, feeling the gravitational force without gravitating itself. However, nothing in the theory suggests that we should take the potential seriously in that way and conceive of it as a physical field. Indeed, the gravitational potential is really a function on configuration space rather than a function on physical space, and it is really a useful mathematical tool rather than corresponding to physical degrees of freedom. From the point of view of a direct interaction theory, an analogous reasoning would apply in the relativistic context. It may seem (and historically it has certainly been the usual understanding) that (3), in contrast to (4), is a dynamical equation, describing the temporal evolution of something. However, from a relativistic perspective, this conclusion seems unjustified.