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6

This is what happens when physicists try to do group theory but don't bother introducing the proper mathematical notions. There is no isomorphism between $\mathrm{SO}(1,3)$ and $\mathrm{SU}(2)\times\mathrm{SU}(2)$, the former is non-compact, the latter is compact. More around this apparently confusing topic can be found in this answer. Furthermore, using ...

3

When dealing with theories containing both commuting and anti-commuting (even and odd) variables, physicists often use the square bracket notation to denote both commutators and anti-commutators according to the following rule: The brackets are commutators unless both variables are odd, in this case they are anti-commutators, please see footnote no. 3 in ...

1

It seems like you're fixing the representation of $\mathrm{SU}(2)$ to be $T^i=\frac{1}{2}\sigma^{i}$ (i.e., the fundamental representation). This makes sense if you're talking about the Higgs mechanism. Now you want to find the generator $Y$ for the $\mathrm{U}(1)$ part of $\mathrm{SU}(2)\times\mathrm{U}(1)$. Let's call that generator $Y$. Now, any ...

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