Cliff(3,1) is isomorphic to the algebraM(4,R) of 4 by 4 real matrices. Several
conventional identifications of the generatorsγjwith 4 by 4 complex matrices
satisfying the relations of the algebra were described in chapter 41. Each of
these gives an identification of Cliff(3,1) with a specific subset of the complex
matricesM(4,C) and of the complexified Clifford algebra Cliff(3,1)⊗Cwith
M(4,C) itself. The Dirac operator in Minkowski space is thus
∂/=γ 0∂
∂x 0
+γ 1∂
∂x 1
+γ 2∂
∂x 2
+γ 3∂
∂x 3
=γ 0∂
∂x 0
+γ·∇and it will act on four-component functionsψ(t,x) =ψ(x) on Minkowski space.
These functions take values in the four dimensional vector space that the Clifford
algebra elements act on (which can beR^4 if using real matrices,C^4 in the
complex case).
We have seen in chapter 42 that−P 02 +P 12 +P 22 +P 32 is a Casimir operator for
the Poincar ́e group. Acting on four-component wavefunctionsψ(x), the Dirac
operator provides a square root of (minus) this Casimir operator since
∂/^2 =−
∂^2
∂x^20+
∂^2
∂x^21+
∂^2
∂x^22+
∂^2
∂x^23=−(−P 02 +P 12 +P 22 +P 32 )
For irreducible representations of the Poincar ́e group the Casimir operator acts
as a scalar (0 for massless particles,−m^2 for particles of massm). Using the
Dirac operator we can rewrite this condition as
(−
∂^2
∂x^20+ ∆−m^2 )ψ= (∂/+m)(∂/−m)ψ= 0 (47.1)This motivates the following definition of a new wave equation:
Definition(Dirac equation). The Dirac equation is the differential equation
(∂/−m)ψ(x) = 0 (47.2)for four-component functions on Minkowski space.
Using equation 40.3, for the Minkowski space Fourier transform, the Dirac equa-
tion in energy-momentum space is
(i/p−m)ψ ̃(p) = (i(−γ 0 p 0 +γ 1 p 1 +γ 2 p 2 +γ 3 p 3 )−m)ψ ̃(p) = 0 (47.3)Note that solutions to this Dirac equation are also solutions to equation 47.1,
but in a sense only half of them. The Dirac equation is first-order in time, so
solutions are determined by the initial value data
ψ(x) =ψ(0,x)ofψat a fixed time, while equation 47.1 is second-order, with solutions deter-
mined by specifying bothψand its time derivative.