Chapter 43

The Klein-Gordon Equation

and Scalar Quantum Fields

In the non-relativistic case we found that it was possible to build a quantum
theory describing arbitrary numbers of particles by taking as dual phase space
Mthe single-particle spaceH 1 of solutions to the free particle Schr ̈odinger
equation. To get the same sort of construction for relativistic systems, one
possibility is to take as dual phase space the space of solutions of a relativistic
wave equation known as the Klein-Gordon equation.
A major difference with the non-relativistic case is that the equation of mo-
tion is second-order in time, so to parametrize solutions inH 1 one needs not
just the wavefunction at a fixed time, but also its time derivative. In addition,
consistency with conditions of causality and positive energy of states requires
making a very different choice of complex structureJ, one that is not the com-
plex structure coming from the complex-valued nature of the wavefunction (a
choice which may in any case be unavailable, since in the simplest theory Klein-
Gordon wavefunctions will be real-valued). In the relativistic case, an appropri-
ate complex structureJris defined by complexifying the space of solutions, and
then takingJrto have value +ion positive energy solutions and−ion negative
energy solutions. This implies a different physical interpretation than in the
non-relativistic case, with a non-negative energy assignment to states achieved
by interpreting negative energy solutions as corresponding to positive energy
antiparticle states moving backwards in time.

43.1 The Klein-Gordon equation and its solu-

tions

To get a single-particle theory describing an elementary particle with a unitary
action of the Poincar ́e group, one can try to take as single-particle state space
any of the irreducible representations classified in chapter 42. There we found