# Comparison between spinor representations in $\operatorname{SL}(2,\mathbb C)=\operatorname{Spin}(1,3)$ and $\operatorname{Spin}(4)$

$$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SU{SU}$$We know that $$\Spin(1,3)=\SL(2,\mathbb C)$$ and $$\Spin(4)=\SU(2) \times \SU(2).$$

The $$\Spin(1,3)$$ is the Lorentz version of Spin group, while $$\Spin(4)$$ is the Euclidean version of Spin group.

The spinor representations of $$\Spin(4)$$ is essentially the spinor representation of $$\Spin(4)=\SU(2) \times \SU(2)$$ thus it is labeled by two components of spinors, each for the spinor representation of $$\SU(2) \times$$.

• How do we understand the spinor representations of $$\Spin(1,3)$$? Note that $$\Spin(1,3)$$ is non-compact. How is this $$\Spin(1,3)$$ spinor related to the spinor of the compact group $$\Spin(4)$$?

• How do we understand the 2 dimensional spinor as complex Weyl and 4 dimensional spinor as complex Dirac but NO 4 dimensional real Majorana in $$\Spin(4)$$?

• It is also said $$\Spin(1,3)$$ is the complexification of $$\SU(2)$$. How to see this from spinor representation living in the vector spaces?

• How do we understand the 2 dimensional spinor as complex Weyl and 4 dimensional spinor as complex Dirac or real Majorana in $$\Spin(3,1)$$?

• Of your three questions I understand only the first one. I understand the two "spinor" representations of the group $G={\rm SL}(2,{\Bbb C})$ (regarded as a *real* algebraic group) as the two 2-dimensional complex representations $$\rho,\bar\rho\,\colon G\to {\rm GL}(2,{\Bbb C})$$ given by the formulas $\,\rho(g)=g\,$ and $\,\bar\rho(g)=\bar g$, where the bar over $g$ denotes the complex conjugation. – Mikhail Borovoi Jul 27 at 17:52