In this paper:

it says the following in p.2,

It says for $3+1$ dimensional spacetime, the Weyl spinor with $SU(2)$ isospin $1/2$ , "Lorentz signature always carry a real structure; if a fermion field appears in the Lagrangian, so does its hermitian adjoint"

What does it mean to be **real** ? If :

- Weyl spinor is
**complex** in $\text {Spin}(3,1)$, and
- its $SU(2)$ isospin $1/2$ is
**pseudoreal** in $SU(2) = \text {Spin} (3)$,

why do we get a Lorentz signature always carry a real structure (instead of just complex or pseudoreal)? Does it mean the whole Weyl spinor is in a real representation (4 component) of $\text {Spin} (3,1)$ and $SU(2) $ together?

What is the emphasis to say "In the Euclidean signature, nothing like that is true in general; what happens depends on the spacetime dimension"? Isnt that the Lorentz signature : real, pseudoreal, or complex also depends on the spacetime dimension ?

This post imported from StackExchange Physics at 2020-12-03 13:06 (UTC), posted by SE-user annie marie heart