Home Page |
Overview |
Site Map |
Index |
Appendix |
Illustration |
About |
Contact |
Update |
FAQ |

where

In SU(2) the phase angle

---------- (10) |

For non-Abelian unitary groups, the number of phase angles (parameters) is determined by the formula N

It can be shown that the absolute square for the determinant of

- SU(2) also has other representations for different kinds of spin in 3-dimensional space (see Figure 01e):
- For the trivial (identity) representation all the generators
**t**_{i}0. It can be identified to particle with spin 0 such as the Higgs boson. In this representation, the particle looks the same from every direction. - The fundamental representation of SU(2) is for leptons with spin 1/2. This is the one that has to rotate 2 x 360
^{o}to return to its original form. - There is the representation of SU(2) by the usual 3-dimensional rotations (the SO(3) group) acting on three dimensional vectors. The generators can be produced from the definition in Eq.(9), for example:

This is the spin 1 representation and can be used to describe gauge particles such as photon, vector boson, and gluon.**t**_{z}= -i (dR_{z}/d)|_{=0}=

a SU(2) transformation of

It can be shown that the magnitude of the complex fields:

In SU(3), the three complex fields can be identified to the three quark fields, which can be thought of as the three directions at 120

---------- (11) |

The SU(2) and SU(3) transformations are more like the non-commuting 3-dimensional rotations of real space because of the non-commuting generators, e.g., e

If we define

It can be shown that the phase transformations are characterized by the eigenvalue of

where t can be determined by the formula n = 2t +1 (n is the dimension of the generator) and for a given t, the value of m can be -t, -t+1, ..., t-1, t; i.e., there are 2t+1 degenerate states for a given t. For example, in the case of SU(2) n = 2, t = 1/2, and m = -1/2 or +1/2 (an isodoublet). The eigenvalue t is often referred to as isospin to indicate that it is similar to spin but in an abstract space. The eigenvalue for

A

or to Top of Page to Select

or to Main Menu