﻿ SU(3), the Special Unitary Group of Order 3
San José State University

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Thayer Watkins
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 SU(3), The Special Unitary Group of Order 3

The abstract group SU(3) is represented by a set of eight 3×3 matrices of complex elements which have determinant of unity. These elements of the group can be generated by eight special matrices. These matrices must be Hermitian; i.e., the transpose of their complex conjugates is the same as the matrix. These matrices do not have determinants of unity; instead all have traces (sums of elements on the principal diagonal) of zero. For a proof these proposition see Group Generators.

In the following i represents the imaginary unit √-1. Here are the generators for the matrix representation of SU(3).

#### | 010 | ½ | 100 |     | 000 |     | 0−i0 | ½ | i  00 |     | 0  00 |     | 1  00 | ½ | 0−10 |     | 0  00 |     | 001 | ½ | 000 |     | 100 |     | 00−i | ½ | 00  0 |     | i0  0 |     | 000 | ½ | 001|     | 010 |     | 00  0 | ½ | 00−i |     | 0i  0 |               | 10  0 | 1/(2√3)  | 01  0 |               | 00−2 |

Let these matrices be labeled λ1, λ1, …, λ8. Now consider the computation of the commutations of them.

λ1λ2 is equal to

#### | i 00 | ¼ | 0-i0 |     | 0 00 | which reduces to      | 1 00 | ¼i | 0-10 |      | 0 00 | which is ½iλ3

Likewise λ2λ1 is equal to

#### | -i00 | ¼  | 0i0 |     |  000 | which reduces to       | 1 00 | -¼i | 0-10 |       | 0 00 | which is −½iλ3

Thus the commutation of λ1 and λ2 is equal to

#### [λ1, λ2] = λ1λ2 −λ2λ1 = ½iλ3 − (−½λ3) = iλ3

Similarly the commutations for the other matrices can be found and the results can be represented as

#### [λj, λk] = iεjkmλm

where εjkm is +1 if jkm is an even permutation of values of j,k,m written in the order of their sizes and −1 if jkm is an odd permutation. If there are repeated indices then εjkm is equal to zero.

## The Formulation of Raising and Lowering Operators

(To be continued.)  (To be continued.)