Quantum Mechanics for Mathematicians

however much more convenient, as in the non-relativistic case of chapters 36
and 37, to use the Bargmann-Fock representation, treating the quantum field
theory system as an infinite collection of harmonic oscillators. This requires a
choice of complex structure, which we will discuss in this section.
By analogy with the non-relativistic case, it is tempting to try and think
of solutions of the Klein-Gordon equation as wavefunctions describing a single
relativistic particle. A standard physical argument is that a relativistic single-
particle theory describing localized particles is not possible, since once the po-
sition uncertainty of a particle is small enough, its momentum uncertainty will
be large enough to provide the energy needed to create new particles. If you try
and put a relativistic particle in a smaller and smaller box, at some point you
will no longer have just one particle, and it is this situation that implies that
only a many-particle theory will be consistent. This leads one to expect that
any attempt to find a consistent relativistic single-particle theory will run into
some sort of problem.
One obvious source of trouble are the negative energy solutions. These will
cause an instability if the theory is coupled to other physics, by allowing initial
positive energy single-particle states to evolve to states with arbitrarily negative
energy, transferring positive energy to the other physical system they are coupled
to. This can be dealt with by restricting to the space of solutions of positive
energy, takingH 1 to be the space of complex solutions with positive energy (i.e.,
lettingφtake complex values and settingf(−ωp,−p) = 0 in equation 43.5).
One then has solutions

φ(t,x) =

1

(2π)^3 /^2

∫

R^3

f(ωp,p)e−iωpteip·x

d^3 p

2 ωp

(43.14)

parametrized by complex functionsfofp.
This choice ofH 1 gives a theory much like the non-relativistic free particle
Schr ̈odinger case (and which has that theory as a limit if one takes the speed of
light to∞). The factor ofωprequired by Lorentz invariance however leads to
the following features (which disappear in the non-relativistic limit):

• There are no states describing localized particles, since one can show that
  solutionsφ(t,x) of the form 43.14 cannot have compact support inx. The
  argument is that if they did, the Fourier transforms of bothφand its time
  derivativeφ ̇would be analytic functions ofp. But the Fourier transforms
  of solutions would satisfy

∂

∂t

φ ̃(t,p) =−iωpφ ̃(t,p)

This leads to a contradiction, since the left-hand side must be analytic,

but the right-hand side can’t be (it’s a product of an analytic function,

and a non-analytic function,ωp).

• A calculation of the propagator (see section 43.5) shows that it is non-
  zero for space-like separated points. This implies a potential violation of