Motivation for SU(2) x U(1) and charge operator

[jdstokes]

I've read that the choice of Gauge group $\mathrm{SU}(2)_\mathrm{L}\times \mathrm{U}(1)_\mathrm{Y}$ can be justified by the fact that the electromagnetic charge and the flavour-changing weak charge do not form a closed SU(2) current algebra. The solution is to tack on an additional U(1) group and to interpret the photon as a superposition of the U(1) and the neutral SU(2) gauge fields.

I saw this explained somewhere by the fact that since the charge operator is

$Q = \left( \begin{matrix} 0 & 0 \\ 0 & -1 \end{matrix} \right)$,

it does not commute with any of the Pauli matrices. Can anyone explain where this definition of the charge operator comes from? I'm failing to understand even what the charge operator is.

 

[QuantumDevil]

Maybe it comes from Gell-Mann–Nishijima phenomenological formula which relates SU(2) weak isospin $$I_z$$, weak hypercharge $$Y$$ and electric charge $$Q$$?

$$\qquad Q = I_z + {1 \over 2} Y$$

 

[sir-pinski]

The motivation for using the gauge group SU(2) x U(1) in particle physics is rooted in the need to explain the observed behavior of particles under the weak and electromagnetic interactions. The weak interaction is responsible for the decay of particles and the electromagnetic interaction is responsible for the interaction between charged particles.

The choice of SU(2) x U(1) gauge group can be justified by looking at the charges associated with the weak and electromagnetic interactions. The electromagnetic charge, also known as electric charge, is associated with the U(1) group. On the other hand, the weak charge, which is responsible for flavor-changing interactions, does not form a closed SU(2) current algebra. This means that the weak charge does not behave like a traditional SU(2) charge, and therefore an additional U(1) group is needed to explain its behavior.

The charge operator is a mathematical representation of the electric charge. In the case of SU(2) x U(1) gauge group, the charge operator is given by the matrix Q, as shown in the provided equation. This operator is used to describe the behavior of particles under the electromagnetic interaction. The fact that this operator does not commute with any of the Pauli matrices is a consequence of the non-commutative nature of the SU(2) group.

To understand the definition of the charge operator, it is important to first understand what the charge of a particle is. In particle physics, the charge of a particle is a fundamental property that determines how it interacts with other particles. The charge operator is simply a mathematical representation of this property. It is defined in such a way that it describes how the particle's charge changes under a particular gauge transformation.

In summary, the motivation for using the SU(2) x U(1) gauge group and the charge operator is rooted in the need to explain the behavior of particles under the weak and electromagnetic interactions. The charge operator is a mathematical representation of the electric charge and is defined in a way that describes its behavior under gauge transformations.