I am reading this article discussing quantum field theory. The author mentioned that 'a particle has a certain spin because its underlying field has a respective spin'. I think intuitively this makes sense because particles are excitations of the underlying quantum field. However, I was thinking if I can come up with a specific example to illustrate this. Assume we have the Dirac lagrangian in chiral representation: $$ \mathcal L = \bar\Psi(i\not\partial-m)\Psi $$
Since we know that the electron is described by a Dirac field, why this field has a half-integer spin? How can we say the mathematical structures of $\Psi$ make electrons spin 1/2?
