# Relationship between group theory and particle physics?

1. Apr 18, 2010

### Ulagatin

I'm just a year 12 student with an interest in mathematics and physics, but I have a question (or rather a few) for particle physicists/mathematicians out there.

I have read a little about abstract algebra - to about the extent of knowing the definition of a group, its relationship to symmetry and such - and I came across the group SU(n), which I understand to have a relationship with the standard model of particle physics with particular values of n. I have read a little on SU(n), enough to find that it is a group of n*n unitary matrices where the determinant is 1. In this sense, first of all, what is a unitary matrix? I understand the concept of a determinant. I also have read that SU(n) has dimension $$n^2 - 1$$. I do not follow this argument however. If the group is defined by n*n matrices, how can the group have this number of dimensions?

What is the relationship, if any, between SU(n) and SO(n) - the latter which I understand (if I am not mistaken) to be the group of rotations of n-space about axes through the origin of $$\Re^n$$ (a fairly abstract but understandable concept)? And how do these concepts explain phenomena in modern particle physics? I believe it is down to symmetry. but, symmetry of what? I understand (but may be far off) that it is "symmetry operations" that create a group, and so, I am asking what these operations are, and what the symmetry itself is.

Also, what do these SU groups relate to within physics in particular? Is it spin/spin angular momentum? Or perhaps colour changes (what are colours in terms of particles)?

Is there any helpful way to visualise these particle physics concepts - with relation to the symmetries - in any simple way, and can the concepts be described well without getting bogged down in difficult mathematics? If so, I'd ask you to, please, have a go.

Sorry for the many questions, but I became very curious recently about this link between what I would have considered before now an obscure (?) branch of mathematics and an exciting field of physics. Thanks in advance for any help in this. :tongue2:

2. Apr 18, 2010

### ansgar

a unitary matrix

http://en.wikipedia.org/wiki/Unitary_matrix

you can show that a specual unitary nxn matrix has n^2 - 1 independet entries from the very definition of special unitary matrix.

some of the differences between SU(n) and SO(n) is that U refers to untariy matrices, and O to orthogonal matrices, i,e U are comples and O are real (google Orthogonal matrix= http://en.wikipedia.org/wiki/Orthogonal_matrix)

one can posulate a lot of symmetries, let me take SU(3)c symmetry, it relates quarks with 3 charges (the color charges) to each other (there is no particular direction in color space if this is a symmetry in the real world). one then derives experimental and theoretical implications of such symmetry and compre with experiment.

Thus you can think of these symmetries as rotations in an abstract space (i.e. not space-time).

these two references should help you understand symmetry and particle physics quite good I hope:

http://arxiv.org/abs/0810.3328

http://www.physics.ucsb.edu/~mark/qft.html