21 Relativistic Field Equations
In the preceding chapters matter, such as electrons and nuclei, was treated as non-relativistic
and distinguishable, while electro-magnetic radiation, in the form of emissions and absorptions,
is relativistic and was treated field theoretically. Indeed, in field theory, the number of particles,
such as photons, does not have to be conserved, while the number of non-relativistic electrons and
nuclei was conserved. Relativity, especially through its equivalence of matter and energy, makes it
impossible for the number of particles to be conserved, whether they be photons, electrons, protons,
neutrons or nuclei. For example, electrons and positrons can collide and annihilate one another
by producing photons, and leaving no charged particles behind. The same can happen to protons
and anti-protons, and even the collision of two protons at high energy can produce new particles,
including protons, electrons and positrons. Thus, the dynamics of electrons (and later on of all
particles) will have to be formulated field theoretically, so that the number of electrons need not
be conserved throughout physical processes.
In this chapter, we shall begin by reviewing those aspects ofspecial relativity that will be
needed here, and then proceed to constructing relativisticinvariant field theory equations in a
systematic manner, including Maxwell’s equations. The field equations for electrons, namely the
Dirac equation, will be constructed in the subsequent chapter.
21.1 A brief review of special relativity
Special relativity is based on two basic postulates
- The laws of Nature and the results of all experiments in anytwo inertial frames with relative
velocityvare the same. - The speed of light is independent of the relative speedv.
To make the consequences of these postulates explicit, we spell out the properties of inertial frames
and the relations between two such frames.
An inertial frame in special relativity is a coordinate systemR(t,x) in which Maxwell’s equa-
tionsin the absence of matterhold true. The coordinates of two inertial framesR(t,x) andR′(t′,x′)
are related to one another by affine transformations, which include translations (the affine part),
rotations and boosts the linear part).
To make this more precise, we define the Minkowski distances^2 between two events (t 1 ,x 1 ) and
(t 2 ,x 2 ),
s^2 =−c^2 (t 1 −t 2 )^2 + (x 1 −x 2 )^2 (21.1)The physical interpretation ofds^2 depends upon its sign;
- s^2 = 0, the events are causally related by the propagation of light;
- s^2 >0, the events are causally unrelated;