Connection between the SU(2) group for the spin 1/2

[captain]

I wanted to ask if there is any connection between the SU(2) group for the spin 1/2 and the gauge group of weak interactions. I feel there isn't much of a connection other than the fact that they share the same group properties, but I am not sure. Thanks to anyone in advance that can clarify this for me.

 

[malawi_glenn]

A group is a mathematical "thing", SU(2) is the set of all unitary 2x2 matrices with determinant 1. So there is the same group describing both spin 1/2 and the gauge properties of the weak interactions. But there is no physical connection.

Another example is how we use SU(3), we use it as gauge group for the strong interactions (SU(3)_colour) and as an (approximative) symmetry of the strong interactions of the three lightest quarks (SU(3)_isospin)

In the strong interaction, a quark has three colours so the spinor is (q_red, q_blue, q_green), thus this spinor transform in colour space and from the requirement that the Lagrangian should be invariant under this transformation, we can derive the gluons and their interactions with themselves and the quarks by their colour charge. So we can say that the strong force is independent of colour, we can transform the quarks to any colour we want and still have the same end result.

In the flavour symmetry, we say that the force between quarks are flavour independent, i.e it does not matter what kinds of quarks that participate in the interaction. up-down is same as strange-up etc. Thus the spinor is (u,d,s), and by imposing this SU(3) symmetry, we end up with the light hadron spectrum.

As you can see, SU(3) is used for two different things. i) The transformations of what colour quarks have, and ii) the transformation on what flavour quarks have.

Now this second symmetry is first of all approximative, due to the quark mass differences. And more, there are more than 3 quarks. So this is just an approximative, but illuminating, symmetry.

So now the SU(2) case, the SU(2)_spin is rotation of a 2-spinor in spin-space, and SU(2)_weak is rotations in weak-isospin space. The spaces are different, but works in a similar manner.

I can describe the number of CD-records I have, and the number of books I have with the same mathematical tool (arithmetic), but that does not say that there is a connection between books and cd's. The mathematical tool can be used for anything that I expect it to be useful for. Same with groups.

 

[Entropia]

There is indeed a connection between the SU(2) group for spin 1/2 and the gauge group of weak interactions. In fact, the SU(2) group is a fundamental part of the Standard Model of particle physics, which describes the interactions between particles and the forces that govern them.

The SU(2) group is one of the three gauge groups in the Standard Model, along with the U(1) group for electromagnetism and the SU(3) group for strong interactions. Each of these groups is associated with a specific force, and the SU(2) group is responsible for the weak force.

The weak force is responsible for processes such as beta decay, where a neutron decays into a proton, electron, and anti-neutrino. This force is mediated by particles called W and Z bosons, which are described by the SU(2) group. The SU(2) group also plays a role in the Higgs mechanism, which gives particles their mass.

Additionally, the SU(2) group is related to spin in quantum mechanics. Spin is a fundamental property of particles and is described by the SU(2) group. This is why the SU(2) group is often referred to as the "spin group."

In summary, the SU(2) group for spin 1/2 and the gauge group of weak interactions are closely connected through their role in the Standard Model of particle physics. They share properties and play important roles in describing the fundamental forces and properties of particles.