Quantum Mechanics for Mathematicians

• The spin polarization vectors and equation 34.9 can be used to parametrize
  solutions of fixed energyEin terms of two functionsαE,+(p),αE,+(p)
  on the sphere|p|^2 = 2mE. This gives an explicit decomposition into
  irreducible representations ofE ̃(3), with the representation space a space
  of complex functions on the sphere, with invariant inner product for each
  helicity choice given by

〈αE,±(p),α′E,±(p)〉=

1

4 π

∫

S^2

αE,±(p)†α′E,±(p) sin(φ)dφdθ

In each case the space of solutions is a complex vector space and one can
imagine trying to take it as a phase space (with the imaginary part of the
Hermitian inner product providing the symplectic structure) and quantizing.
This will be an example of a quantum field theory, one discussed in more detail
in section 38.3.3.

34.4 The Dirac operator

The above construction can be generalized to the case of any dimensiondas
follows. Recall from chapter 29 that associated toRdwith a standard inner
product, but of a general signature (r,s) (wherer+s=d,ris the number
of + signs,sthe number of−signs) we have a Clifford algebra Cliff(r,s) with
generatorsγjsatisfying
γjγk=−γkγj, j 6 =k
γ^2 j= +1 forj= 1,···,r γ^2 j=− 1 ,forj=r+ 1,···,d

To any vectorv ∈Rd with componentsvj, recall that we can associate a
corresponding element/vin the Clifford algebra by

v∈Rd→v/=

∑d

j=1

γjvj∈Cliff(r,s)

Multiplying this Clifford algebra element by itself and using the relations above,
we get a scalar, the length-squared of the vector

v/^2 =v^21 +v^22 ···+v^2 r−v^2 r+1−···−v^2 d=|v|^2

This shows that by introducing a Clifford algebra, we can find an interesting
new sort of square root for expressions like|v|^2. We can define:

Definition(Dirac operator).The Dirac operator is the operator

∂/=

∑d

j=1

γj

∂

∂qj