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Group Theory and Its Application to Particle Physics

Local Gauge Symmetry

In quantum field theory, if we demand the system to be invariant under the local gauge transformation:

where A is the field for the gauge bosons, gs is the coupling constant, lambda is the generator, and (x) is now a function of space-time. The second term in Eq.(16) is responsible for the various interactions via the exchange of bosons (see more details in "Quantum Field Theory"). The number of gauage bosons depends on the dimension of the generator. Thus, there is only the photon for the U(1) group applicable to electromagnetic interaction; there are three vector bosons - W+, W-, and Zo for the SU(2) group applicable to weak interaction; and there are eight gluons for the SU(3) group applicable to strong interaction (see Table 15-02).

Now let us turn to the physical interpretation of the complex function in Eq.(3). For the U(1) group, fR and fI can be identified with particles carrying different electric charge. The dynamic of these particles is described by the 2-component Klein-Gordon Equation.

Application of the SU(2) group to weak interaction is more complicated because of parity violation. Since there is no right-handed neutrino, the wave functions are split into a left-handed and right-handed parts:

It can be shown that if the U(1) and SU(2) groups are spliced together, i.e., the gauge group becomes U(1)XSU(2), such that

L L' = ei(x)/2 eia(x)t L,
R R' = ei(x) R,

where t is given by the matrices in Eq.(10). Then the electro-weak theory would be invariant under this transformation with corresponding covariant derivative similar to Eq.(16).

The transformation of the quark fields in electro-weak theory is slightly different because the right-handed field involves both the up and down quarks dR, and uR:

L L' = e-i(x)/3 eia(x)t L,
dR dR' = e-4i(x)/3 dR ,
uR uR' = e2i(x)/3 uR .

The SU(3) transformation of the color fields for the quarks in strong interaction theory is similar to the one shown in Eq.(15):

u u' = e u,

where is given by the matrices in Eq.(11). In QCD, each flavor of quark is represented by a triplet of colors. For example the u quark is associated with the triplet

where the subscripts r, g, b label the color charges (red, green, blue, which can be considered as the sources of the strong interaction like the electric charge for the electromagnetic interaction, and has nothing to do with color vision). See more in "Standard Model" for strong interaction.

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