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The modern theory of elementary particles depends heavily on the concept of gauge invariance, which was used to describe some changes that do not have any effect on observation. For example, if the electric voltage throughout the circuit is raised uniformly by the same amount, there would not be any observable effect. Such instantaneous global change seems to be somewhat un-natural as it takes a definite time to signal the change. Theory with local gauge invariance is more | |

## Figure 15-06a Global and Local Transformations [view large image] |
realistic, but much more stringent. The difference between a global and local transformation is illustrated pictorially in Figure 15-06a (just for visualization purpose - it's not the gauge variety). Since local gauge symmetry can change in a |

"internal space". The photon (the gauge boson) is the response to restore the appearance (see Figure 15-06b), which signifies mathematically the invariance of the "Action" under this internal rotation. Theoretical physicists are fond of putting similar objects together called a group. For the case of electromagnetic interaction, there is only one kind of objects -- the two dimensional internal rotation. The different rotational displacements form a group, this particular group is called U(1). The symbol U indicates that the transformation (the internal rotation) is unitary, which preserves the normalization (probability). This U(1) group has the property that the internal rotation operations are commutative -- mathematicians call such kind of group an Abelian group. Similar gauge invariance exists for the strong and weak interactions, the "internal rotation" depends on more than one parameter in these cases. Group of objects can be formed from these generalized "rotational displacements". However, these elements are no longer commutative. Such groups are called non-Abelian. The gauge theory for the U(1) is called Quantum Electrodynamics (QED). | |

## Figure 15-06b Local Gauge Field Invariance [view large image] |

In superconductivity, it is known that the Meissner effect, where the magnetic field is expelled from the superconducting material, can be interpreted as the electromagnetic field (the photon) acquiring mass while trying to disrupt the ordered structure of the Copper pairs (see Figure 15-06c). But the gauge bosons in weak interaction have mass even in empty space, and no form of mass generating substance has been identified so far for such boson mass. We are therefore led to postualate that there is a new form of matter doing the job. Accordingly, what we perceive as empty space is in | |

## Figure 15-06c Higgs Condensate |
fact filled with an exotic, suitably aligned substrate: the Higgs condensate. |

In the mathematical formalism, these Higgs fields are added to the "Action". At the transition temperature (early in the Big Bang),

these Higgs fields^{5} move to more stable states in lower energy level. However, unlike the electromagnetic field, which has its minimum energy at zero field strength; the Higgs field, in contrast, has its minimum energy at a nonzero field strength (see Figure 15-06d, in this diagram the "ball" represents the preferred state of the universe). Thus, the universe, in its natural lowest energy state, is permeated by that nonzero value of the Higgs field. Once this happens, all the particles (both bosons and fermions) would acquire mass by interacting with the Higgs field. There is now a convincing consensus of experimental results supporting this electroweak theory.
| |

## Figure 15-06d Higgs Field [view large image] |

When the internal rotation is generalized to SU(3), The gauge theory can be applied to the case of strong interaction. There are eight parameters for this group corresponding to eight gauge bosons called gluons. The participating particles are the quarks with 3 different colour charges -- red (r), green (g), and blue (b). Three quarks with different colour charges combine to form a baryon. Each quark can carry different colour charges at different time, provided the colour combination is "white". Unlike the case of the U(1) group where the gauge boson (the photon) does not carry charge, the gluons do themselves carry the colour charges. Such difference produces phenomenon such as asymptotic freedom and quark confinement. The gauge theory for the SU(3) group is called Quantum Chromodynamics (QCD). The formulism for QCD and electroweak interaction together is known as the Standard Model, which describes all the phenomena associated with leptons and quarks.

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