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In a previous question, I asked what the matrix representation of the electroweak fields is, and I was told they are identical to the Faraday tensors, but come in a set of three ($W_i, i\in \{1,2,3\}$), associated to the $SU(2)$ symmetry and another one ($B$), associated to the $U(1)$ symmetry for a total of four.

I also know that the expressions $||\mathbf{E}||^2-||\mathbf{B}||^2$ and $\mathbf{E}\cdot \mathbf{B}$ are the two fundamental invariants of the electromagnetism. My question is what are the invariants of the $SU(2)$ symmetry and the $U(1)$ symmetry. Based on the similarity, my guess is the $SU(2)$ invariants are 3 orthogonal elements of the same form as the invariants of the electromagnetism:

$$ (||\mathbf{A}_1||^2-||\mathbf{D}_1||^2)\sigma_x\\ (||\mathbf{A}_2||^2-||\mathbf{D}_2||^2)\sigma_y\\ (||\mathbf{A}_3||^2-||\mathbf{D}_3||^2)\sigma_z $$

That can be squared as follows:

$$ (||\mathbf{A}_1||^2-||\mathbf{D}_1||^2)^2+(||\mathbf{A}_2||^2-||\mathbf{D}_2||^2)^2+(||\mathbf{A}_3||^2-||\mathbf{D}_3||^2)^2 $$

and the $U(1)$ symmetry is of the same form as that of electromagnetism:

$$ ||\mathbf{F}||^2-||\mathbf{G}||^2 $$


Would be nice to know $SU(3)$ invariants as wells.

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  • $\begingroup$ You are using maximally opaque and disruptive language for this. In the covariant formulation, you are talking about $F_{\mu\nu} F^{\mu\nu}$ and $\epsilon^{\mu\nu\kappa\lambda}F_{\mu\nu} F_{\kappa\lambda} $. The close analog expression for the YM field strength also holds. What, exactly, is holding you up? $\endgroup$ – Cosmas Zachos Jul 14 at 13:04

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