Special Unitary Groups In Physics

Date: 11/09/98 at 21:20:19
From: David Winsemius
Subject: Group theory in physics

As a student, I left physics before I could see the applications of 
abstract algebra. Before I left physics, I did pick up a bit of group 
theory, but not enough to know what SU(2) and SU(3) groups are. These 
abstract objects are central to modern descriptions of quantum 
chromodynamics, so I was wondering if you could perhaps describe these 
entities, and what areas of study I should pursue?

Thank you,
David Winsemius, MD

Date: 11/09/98 at 22:32:03
From: Doctor Tom
Subject: Re: Group theory in physics

Hi David,

SU(2) and SU(3) are the special unitary groups of dimensions 2 and 3.

SU(2) is the set of all 2x2 matrices that are unitary and have 
determinant 1. SU(3) is the same thing, but with 3x3 matrices.

A unitary matrix is a matrix whose conjugate transpose is its own
inverse.

The transpose of a matrix is gotten by flipping everything across the 
main diagonal. That is, you make the rows into the columns and the 
columns into the rows. So for example, the transpose of:

       [ a b c ]           T   [ a d g ]    
   A = [ d e f ]    is    A  = [ b e h ]    
       [ g h i ]               [ c f i ]

The conjugate is gotten by replacing every element by its complex 
conjugate.

This set of matrices forms a group under normal matrix multiplication.

For example, the matrix:

  [ 0  i ]
  [ i  0 ]

is unitary, since its transpose is itself, and the complex conjugate of 
the transpose is:

  [  0  -i ]
  [ -i   0 ]

and when you multiply that matrix by the one above, you get the
identity.

Unitary matrices are like (and include) the standard rotation matrices 
of real numbers. For example, the following real matrix is also 
unitary:

  [ cos(x)  -sin(x) ]
  [ sin(x)   cos(x) ]

The transpose is:

  [  cos(x)  sin(x) ]
  [ -sin(x)  cos(x) ]

and the conjugate leaves it unchanged, since all the entries are real. 
It's easy to check that the product of these two matrices is also the 
identity.

In 3 dimensions, of course, it's much more complicated.

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/