# Legal values of spin-1 field can take: $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, ..?

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. The field strength $$F= dA$$ is also real. Same for the nonabelian case $$F= dA+A^2$$ is also real.

• but can $$A$$ be complex $$\mathbb{C}$$?

• but can $$A$$ be quaternion $$\mathbb{H}$$?

What are the legal values of spin-1 field can take? real $$\mathbb{R}$$, complex $$\mathbb{C}$$, quaternion $$\mathbb{H}$$ , ..? And what may be the QFT like in these cases?