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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 in global gauge transformation can be identified to the three quark fields (u, d, s), which can be thought of as the three directions at 120^{o} to each other. It is called colour isospin or colour spin or colour charge (r, g, b) in local gauge transformation.The SU(3) group has N _{a} = 8, there are eight phase angles and eight 3x3 non-commuting matrices (generators) operating on three complex functions (u, d, s) or (r, g, b) depending on whether the transformation is global or local. There are eight corresponding meditation particles (the gluons) - two of them mediate the strong interaction while the rest go around to swap color charges. (see Figure 02d).
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## Figure 02d SU(3) Gauge Theory |

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

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