Main issue: What are the legal and possible values of the quantum field can take?
Clarify by examples:
(1) For example, for the spin-0 Klein Gordon field $\phi$, we may choose it to be:
- real $\mathbb{R}$.
- complex $\mathbb{C}$.
(2) For the spin-1/2 fermion field $\psi$, we may choose it to be a spinor which needs to be
- Grassman variable
but can also be
- complex $\mathbb{C}$. (Dirac or Weyl spinor/fermion)
- We can ask: Can it be in real $\mathbb{R}$? (Majorana or Majorana-Weyl spinor/fermion)
(3) For the spin-1/ boson field $A_\mu$, we may choose it to be a vector which needs to be
real $\mathbb{R}$ usually for photon field.
but can it be complex $\mathbb{C}$?
(3) How about the spin-3/2 fermion field $\psi_\mu$?
- can it be in real $\mathbb{R}$, complex $\mathbb{C}$, quaternion $\mathbb{H}$, ..?
