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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?

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