# How is the complexification of the Lorentz Lie algebra related to the need for Dirac's 4-component spinor in QFT?

There have been several questions with good answers in physics.stackexchange about the motivation of the complexification of the Lorentz Lie algebra, basically as a mathematically nice way to deal with the situation generated by the non-existence of finite dimensional unitary representations of the Lorentz group.

But I'm interested in a clear outline about how to link this with the need of the Dirac spinor in QFT and how the complexification prompts one to go from a 2-spinor to a 4-spinor.

I think this is related to the need to go from the spin structure (Spin group) to the spin-c structure (Spin-c group with complex reresentation) in 4-dimensional Minkowski space with charged spin 1/2 particles carrying a unitary representation but I'm not sure exactly how.

Furthermore, there is no need, a priori, for Dirac spinors in QFT. If one demands invariance under the orthochronous Lorentz group, the Dirac spinor is reducible, and we may instead consider Weyl fermions. Dirac Spinors are irreducible if one enlarges the $$Spin$$-group to a $$Pin$$-group containing spatial reflections or time-reflections.
For charged particles, the spinors fields should take value in $$\mathbb{C} \otimes V$$, where $$V$$ is a representation space of the $$Spin$$ group and the $$\mathbb{C}$$-factor describes charge. This is an example of a $$Spin^c$$-structure, but note that there is no need to do this unless the particle is charged, and even if it is charged, we don't have to consider a general $$Spin^c$$ structure.
• I didn't write that the Dirac spinor representation was unitary, the infinite-dimensional linear representations of the Lorentz($SO^+(1,3)$) and Poincare groups are. I hope you can see that complexification is related to the role of classifying more easily these infinite-dimensional representations. My question was referring to charged fermions in its respect to Dirac spinors, yes, as they are quite important in QFT and SM. I didn't understand what you meant about not considering a general $Spin^c$ structure in your more helpful in relation to my specific question third paragraph. – bonif Oct 19 '18 at 16:39