Chapter 34

The Pauli Equation and the

Dirac Operator

In chapter 33 we considered supersymmetric quantum mechanical systems where
both the bosonic and fermionic variables that get quantized take values in an
even dimensional spaceR^2 d=Cd. There are then two different operatorsQ 1
andQ 2 that are square roots of the Hamiltonian operator. It turns out that there
are much more interesting quantum mechanics systems that can be defined by
quantizing bosonic variables in phase spaceR^2 d, and fermionic variables inRd.
The operators appearing in such a theory will be given by the tensor product of
the Weyl algebra in 2dvariables and the Clifford algebra indvariables, and there
will be a distinguished operator that provides a square root of the Hamiltonian.
This is equivalent to the fact that introduction of fermionic variables and the
Clifford algebra provides the Casimir operator−|P|^2 for the Euclidean group
E(3) with a square root: the Dirac operator∂/. This leads to a new way to
construct irreducible representations of the group of spatial symmetries, using
a new sort of quantum free particle, one carrying an internal “spin” degree of
freedom due to the use of the Clifford algebra. Remarkably, fundamental mat-
ter particles are well-described in exactly this way, both in the non-relativistic
theory we study in this chapter as well as in the relativistic theory to be studied
later.

34.1 The Pauli-Schr ̈odinger equation and free

spin^12 particles ind= 3

We have so far seen two quite different quantum systems based on three dimen-
sional space:

• The free particle of chapter 19. This had classical phase spaceR^6 with
  coordinatesq 1 ,q 2 ,q 3 ,p 1 ,p 2 ,p 3 and Hamiltonian 21 m|p|^2. Quantization us-
  ing the Schr ̈odinger representation gave operatorsQ 1 ,Q 2 ,Q 3 ,P 1 ,P 2 ,P 3