What are the SU groups? I.e SU(2), SU(3)?

[RyanTG]

I'm revising for my introductory particle physics exam and I've noticed SU(3) pop up a few times, it was never really explained and I don't really understand what it means.

For example when talking about the conservation law of Colour, it says the symmetry is phase invariance under SU(3) group of QCD. What does that mean?

I don't think I'll be rigorously examined on this but I'd like to be able to have a basic understanding of what it means and the ramifications of it.

So if anyone could explain it and why that means for me at the moment in particle physics, that would be tremendously helpful!

 

[stevendaryl]

I'm revising for my introductory particle physics exam and I've noticed SU(3) pop up a few times, it was never really explained and I don't really understand what it means.

For example when talking about the conservation law of Colour, it says the symmetry is phase invariance under SU(3) group of QCD. What does that mean?

I don't think I'll be rigorously examined on this but I'd like to be able to have a basic understanding of what it means and the ramifications of it.

So if anyone could explain it and why that means for me at the moment in particle physics, that would be tremendously helpful!

SU stands for "Special Unitary".

If you have a matrix $A$, it is "unitary" if $A^\dagger A = I$, where $I$ is the identity matrix. A "special unitary" matrix has the additional constraint that $det(A) = 1$

 

[hilbert2]

A somewhat similar group is the SO(3), one representation of which are the 3d rotation matrices and which can be parametrized with Euler angles. Here the "O" means that the matrices in the representation are orthogonal and the "S" that their determinant has value 1 (the operations involve only rotation, not reflections).

 

[Matterwave]

Appending on the above:

SU(3) specifically means the group of 3x3 complex matrices who's Hermitian adjoint is its own inverse ($A^\dagger=A^{-1}$). These matrices conserve the inner product of 3-component complex vectors. Given two 3 component complex column vectors $x$ and $y$, their inner product $x_1^*y_1+x_2^*y_2+x_3^*y_3$ is constant under the transformation $x'=Ax$ and $y'=Ay$ for $A\in SU(3)$.

This property of preserving the inner product is why we like unitary groups in quantum mechanics. It is also the property that makes SU(3) similar to SO(3) since SO(3) does the same for 3-component real vectors. But I should warn you that looking beyond this simple analogy, the two groups SU(3) and SO(3) are quite different both algebraically and topologically. SU(3) has 8 dimensions (spanned for example by the 8 Gell-Mann matrices) while SO(3) is of dimension 3.

 

[PhilDSP]

We should remember that SU(2) is a double cover of SO(3). Simply speaking there are 2 different instances of an SU(2) value that give you an equivalent of an SO(3) value. This is because of the characteristics of rotating an object modeled in SU(2) differs from rotating an object in SO(3).

Pauli originally developed the handling of "spin" using SU(2) matrices. Standard QM requires only SU(2) modeling while QCD requires SU(3) as you mention RyanTG.

This should help in explaining the mathematics of how the "physics of a particle in a field is invariant under certain local transformations of the phase" in SU(1) and SU(2):

https://www.physicsforums.com/library.php?do=view_item&itemid=136