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To get around the problem of negative energy, Dirac proposed an energy spectrum containing all electrons in the universe (see Figure 03). In addition to the normal positive-energy spectrum, it also contains the negative-energy variety, which spans the spectrum from -m_{0}c^{2} down to negative infinity. All the negative-energy levels are filled, thus the positive-energy particle is inhibited from transition into these lower energy states bypassing the unobserved phenomena of falling into the negative energy levels. Thus, the existence of negative energy has no observable effect in the real world. Only when there is enough energy available, e.g., when E 2m_{o}c^{2}, a real particle and anti-particle pair with positive-energy can be created from this unseen sea of negative-energy particles. The particle is the electron originally resided in the negative energy region, while the anti-particle (positron) can be interpreted as the hole in the vacated energy level acquiring a mass m_{o}. The law of charge conservation demands that this anti-particle carries a positive charge. Many interesting things occur when the electron meets the hole (see Figure 03). Over the past three | |

## Figure 03 Negative Energy [view large image] |
decades, the use of e^{+}e^{-} collisions to probe the vacuum have yielded a great deal of information about the nature of the strong, weak, and electromagnetic interactions, and have played a major role in establishing the Standard Model of |

There are four types of solution for the Dirac equation depending on energy (positive or negative) and spin (up or down) as shown below :

---------- (31) |

Charge conjugation is the operation that turns a field describing a certain particle into a field for the corresponding antiparticle. It can be shown that such operator is in the form :

---------- (32) |

where the exponential part is excluded from the spinors. The operation transforms the types of solution from

According to the theory of electromagnetism a magnetic moment would be generated by any current loop. The Stern-Gerlach experiment

demonstrated that the spinning electrons produce a pattern of two distinct parts corresponding to two opposite spin orientations in the magnetic field. The fine structure of the hydrogen atom (1 energy level splits into 2) further determined that the spin of the electron is 1/2. In the presence of a magnetic field it is the z component which aligns with the field direction having a quantized value of _{}/2 while the entire magnetic moment precesses around the field with a magnitude S = [(s+1)s]^{1/2} (Figure 04). The expression for the magnetic moment derived in quantum theory has the form: M = (e/m)s ---------- (34) | |

## Figure 04 Electron Spin |
where s = (/2)(_{x} + _{y} + _{z}), indicating that the _{i}'s are related to the spin of the electron. |

Eq.(35a) describes a massless particle with direction of spin opposite to the direction of motion (left-handed helicity), while Eq.(35b) describes a massless particle with spin pointing to the same direction of motion (right-handed helicity). This kind of 2-components spinor is often referred to as Weyl spinor. They used to be associated with neutrino (only left-handed neutrino and right-handed anti-neutrino exist in nature) when it was considered as massless. The Weyl spinor has also been applied to the formulation of the Standard Model with u_{L} participated in both the electromagnetic and weak interactions, while u_{R} involves with only the electromagnetic interaction. Both the electron and neutrino are considered to be massless originally, only the electron acquires mass via interaction with the Higgs field in that model. The Weyl spinor also found its way into the theory of supersymmetry, which insists that every boson have a fermion partner (in the form of Weyl spinor). When neutrino has mass, its speed would always be lower than the speed of light, theoretically an observer can move in a speed faster than the left-handed neutrino, overtakes this neutrino and
| |

## Figure 05 Neutrino Mass |
sees a right-handed anti-neutrino (Figure 05,b). That's why the two Weyl spinors mix in case of particle with mass as shown in Eqs.(28a) and (28b). |

The 2-components spinor is related to the isotropic vector R=(r

r

where the r

r

r

r

the inverse of which is :

s

s

For example, let's take R=(1, i, 0), then S=(1, 0) or S=(-1, 0). If R is rotated by an angle such that R => e^{-i}R around r_{3}, then S => e^{-i/2}S around s_{1}. That is, a rotation of R by an angle translates to a rotation of S by an angle /2. It is this peculiar prpperty of spinor that makes it so different from the other kinds of field. Particles associated with spinor are called fermion to distinguish themselves from the other types called boson (Figure 06). | |

## Figure 06 Different Kinds of Spin [view large image] |
This special property of rotation for spinor is also applicable to the internal (invisible) rotation in gauge transformation. |

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