- #1
spaghetti3451
- 1,344
- 34
My question is about the formula ##Q = T_{3} + Y##.
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##?
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##?