Dirac spinor definition

Question

is it right to say that the Dirac spinor is a mathematical representation of a wave-function that satisfy the Dirac equation? or are there more requirements to it?

Answer 1

Honestly, I somehow dislike this point of view for the simple reason that in order to state the differential equation you must first have a definition of the object which should solve it. Let me state this differently. A differential equation is an equation of the form ${\scr D}\Psi=0$, where $\scr D$ is one differential operator. But to properly define and make sense of $\scr D$ you must know its domain first. Without defining a spinor you cannot write the Dirac eqution.

That said, a Dirac spinor is one field $\Psi:\mathbb{R}^{1,3}\to \mathbb{C}^4$ on Minkowski spacetime which takes values in one vector space which carries one specific representation of the universal cover of the Lorentz group, ${\rm Spin}(1,3)\simeq {\rm SL}(2,\mathbb{C})$.

The spin group ${\rm Spin}(1,3)$ has irreducible representations labelled by $(A,B)$ where $A$ and $B$ are integers or half-integers greater or equal to zero. In particular, the objects living in the representations $\left(\frac{1}{2},0\right)$ and $\left(0,\frac{1}{2}\right)$ are respectively called left-handed Weyl spinors and right-handed Weyl spinors. Finally, the objects living in the direct sum $\left(\frac{1}{2},0\right)\oplus \left(0,\frac{1}{2}\right)$ are called Dirac spinors.

The representation space of the both left and right-handed Weyl spinors is $\mathbb{C}^2$, so the representation space of Dirac spinors is $\mathbb{C}^4$ as anticipated.

After you have a proper definition of a spinor field you make sense of the operator appearing in the Dirac equation ${\scr D} = \gamma^\mu \partial_\mu + m$.

To fully appreciate this story I like sections 5.4 and 5.6 of Weinberg's The Quantum Theory of Fields.