# Electric charge as ##Q = T_{3} + Y##

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• spaghetti3451
In summary, the conversation discusses the formula ##Q = T_{3} + Y## and how to obtain the electric charges of component fields. The transformation to unitary gauge is also mentioned, with the correct normalisation for ##t_3## being -1, 0, and +1. The concept of unitary gauge is explained as a way to get rid of Goldstone bosons without changing the group generators.
spaghetti3451
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##?

You have gotten the normalisation of ##t_3## wrong. The eigenvalues should be -1, 0, and +1.

Ah! I see!

So, now if you wanted to transform to unitary gauge, how would you do so?

I have difficulty understanding what the unitary gauge transformation explicitly looks like.

spaghetti3451 said:
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##?

Yes! (EDIT: sorry, it seems according to Orodruin's reply to this post that I haven't made my yes clear enough: let's suppose you had the normalisation right, it would indeed be the way you would calculate the charges: the diagonal values give you the ##T_3## charge of each component, to which you just have to add the hypercharge)

spaghetti3451 said:
Ah! I see!

So, now if you wanted to transform to unitary gauge, how would you do so?

Unitary gauge is usually defined when you have Goldstone bosons and want to get rid of them. For instance, for the Higgs field, which you can write ##h=e^{(-i\eta_at^a)}(0, h+v)##, then the change to unitary gauge would be one of parameters the ##\eta##'s to make them disappear. But it does not change anything to the generators of your group.

Last edited:
Q.B. said:
Yes!
No. Why would you bother to give a wrong answer to a question that has already been answered? The normalisation of ##t_3## should not involve the ##1/\sqrt{2}## and the third isospin component of the same multiplet differ by one.

## 1. What is electric charge and how is it measured?

Electric charge is a fundamental property of matter that causes it to experience electrical and magnetic forces. It is measured in units of coulombs (C) and can be positive or negative. The charge of an object is determined by the number of protons and electrons it contains.

## 2. What is the significance of the equation Q = T3 + Y in relation to electric charge?

This equation, also known as the Gell-Mann-Nishijima formula, relates the electric charge (Q) of a particle to its isospin (T3) and hypercharge (Y). It helps to explain the behavior and interactions of subatomic particles, such as quarks and leptons.

## 3. How is electric charge conserved in nature?

Electric charge is conserved in nature, meaning it can neither be created nor destroyed. This is known as the law of conservation of charge. In any interaction or process, the total amount of charge before and after must be equal.

## 4. Can electric charge be transferred from one object to another?

Yes, electric charge can be transferred from one object to another through various means, such as friction, conduction, or induction. This is how static electricity is created, as well as how electrical circuits work.

## 5. How does electric charge affect the behavior of matter?

Electric charge plays a crucial role in determining the behavior of matter. At the atomic level, the arrangement of positively charged protons and negatively charged electrons determines the properties of an element. At the macroscopic level, electric charge is responsible for the attraction and repulsion between objects, as well as the flow of electric current.

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