all elements and inverts the basis vectors. The multivector bilinear form $\overline{\Psi}\Psi$ has the advantage of being Lorentz invariant (as the Dirac bar conjugate operation inverts the bivector generators of Lorentz transformations). On the other hand, the scalar part of this form is not in general positive definite, in contrast to the form $\Psi^{\dagger}\Psi$. Both forms will in general have non-scalar portions.

Functional Solutions of the Massive Dirac Eqn.
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The adjective “massive” should be superflourious, because historically Dirac was describing the electron, a particle with mass. It is therefore a source of confusion for the physicist to encounter the use of the term “Dirac equation” in Clifford analysis sometimes applied to the [*monogenic*]{} equation $\Box\Psi=0$, rather than eq. (3.2a) below. When $\Psi$ is restricted to be a bivector, the monogenic equation would be called the (sourceless) [Maxwell equation]{}, describing the spin-one (i.e. electromagnetic waves). If $\Psi$ is projected onto a minimal ideal (i.e. a [*column spinor*]{}), it would describe a spin-half (again massless) [*neutrino*]{}, often called the [Majorana equation]{}. These are special cases which have many interesting properties, unfortunately which vanish once mass is included, or a source added (which makes the equation non-homogeneous). In this paper, we will 