Chapter 40

Minkowski Space and the

Lorentz Group

For the case of non-relativistic quantum mechanics, we saw that systems with an
arbitrary number of particles, bosons or fermions, could be described by taking
as dual phase space the state spaceH 1 of the single-particle quantum theory.
This space is infinite dimensional, but it is linear and it can be quantized using
the same techniques that work for the finite dimensional harmonic oscillator.
This is an example of a quantum field theory since it is a space of functions that
is being quantized.
We would like to find some similar way to proceed for the case of rela-
tivistic systems, finding relativistic quantum field theories capable of describ-
ing arbitrary numbers of particles, with the energy-momentum relationship
E^2 =|p|^2 c^2 +m^2 c^4 characteristic of special relativity, not the non-relativistic

limit|p|mcwhereE=|p|

2
2 m. In general, a phase space can be thought of as
the space of initial conditions for an equation of motion, or equivalently, as the
space of solutions of the equation of motion. In the non-relativistic field theory,
the equation of motion is the first-order in time Schr ̈odinger equation, and the
phase space is the space of fields (wavefunctions) at a specified initial time, say
t= 0. This space carries a representation of the time-translation groupRand
the Euclidean groupE(3) =R^3 oSO(3). To construct a relativistic quantum
field theory, we want to find an analog of this space of wavefunctions. It will be
some sort of linear space of functions satisfying an equation of motion, and we
will then quantize by applying harmonic oscillator methods.
Just as in the non-relativistic case, the space of solutions to the equation
of motion provides a representation of the group of space-time symmetries of
the theory. This group will now be the Poincar ́e group, a ten dimensional
group which includes a four dimensional subgroup of translations in space-time,
and a six dimensional subgroup (the Lorentz group), which combines spatial
rotations and “boosts” (transformations mixing spatial and time coordinates).
The representation of the Poincar ́e group on the solutions to the relativistic