This will be a first-order differential operator with the property that its
square is the Laplacian
∂/^2 =
∂^2
∂q^21+···+
∂^2
∂qr^2−
∂^2
∂qr^2 +1−···−
∂^2
∂qd^2The Dirac operator∂/acts not on functions but on functions taking values
in the spinor vector spaceSthat the Clifford algebra acts on. Picking a matrix
representation of theγj, the Dirac operator will be a constant coefficient first-
order differential operator acting on wavefunctions with dimScomponents. In
chapter 47 we will study in detail what happens for the case ofr= 3,s= 1 and
see how the Dirac operator there provides an appropriate wave equation with
the symmetries of special relativistic space-time.
34.5 For further reading
The point of view here in terms of representations ofE ̃(3) is not very conven-
tional, but the material here about spin and the Pauli equation can be found
in any quantum mechanics book, see for example chapter 14 of [81]. For more
details about supersymmetric quantum mechanics and the appearance of the
Dirac operator as the generator of a supersymmetry in the quantization of a
pseudo-classical system, see [90] and [1].