- #1

spaghetti3451

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- 34

Let us say that there is some complex scalar field that transforms as a triplet of ##SU_{L}(2)##; i.e.

##\psi = \begin{pmatrix} \psi_{1}\\ \psi_{2} \\ \psi_{3} \end{pmatrix}##

and

##\delta_{2}\psi = i\omega_{2}^{a}t_{a}\psi##

with

##t_{1} = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}, \qquad

t_{2} = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & -i & 0 \\ i & 0 & -i \\ 0 & i & 0 \end{pmatrix}, \qquad

t_{3} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix}##

Let us suppose also that the hypercharge, ##Y##, of the field ##\psi## is zero.

How do we now obtain the electric charges of the component fields ##\psi_{1}##, ##\psi_{2}##, and ##\psi_{3}##?

Is it ##1/\sqrt{2}##, ##0## and ##-1/\sqrt{2}##, because these are the eigenvalues of the eigenstates ##\psi_{1}##, ##\psi_{2}## and ##\psi_{3}## of ##\psi##?