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Wave Equations


Interpretation of the Dirac Equation

Negative Energy States 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 -m0c2 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 2moc2, 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 mo. 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
elementary particles (see more in "Quantum Vacuum"). Positron was first detected in 1932 by C. Anderson. The development of quantum field theory in the 1930's made it possible to treat the positron as a "real" particle rather than the absence of a particle, and makes the vacuum the state in which no particles exist instead of an infinite sea of particles. It recaptures all the valid predictions of the Dirac sea, such as electron-positron annihilation. But problem with infinite vacuum energy is still around and has to be removed as unobservable.

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)
where V is the volume containing the free particle, the p and x are now 3 dimensional c-numbers, and the expressions have been reverted from the natural units. The probability density is the usual * except that the wave function is now the 4-components Dirac spinor. Since the probability density is positively defined, it is not necessary to have it re-interpreted as charge density (as in the case of the scalar field, see Eq.(22)). The negative energy states is propagating in the oposite direction (to the positive energy ones) as shown in Eq.(31). It is often interpreted as a positive energy anti-particle moving "backward" in time.

In non-relativistic limit, p << mc, E = |E| = [m2c4 + p2c2]1/2 ~ mc2 + p2/2m, the spinor is reduced to the simple form :
N.B. The spinor with superscript denotes type such as + energy and spin up, while the subscripts signify the various components within the type of spinor, for example :


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)
The antiparticle field is obtained by:
C = eiC * ---------- (33)
where the exponential part is excluded from the spinors. The operation transforms the types of solution from (4) ... (1) to (1) ... (4) with the phase factor having the values of 3/2 for 4 to 1, 2 to 3, and 0 for 3 to 2, 1 to 4. It can be shown that the eigenvalue for this charge conjugation operator C is c = 1.

According to the theory of electromagnetism a magnetic moment would be generated by any current loop. The Stern-Gerlach experiment
Electron Spin 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
[view large image]


where s = (/2)(x + y + z), indicating that the i's are related to the spin of the electron.



In case the mass of the particle equals to zero, i.e, m=0, then Eqs.(28a) and (28b) are de-coupled. The 2-components spinors are often re-labelled as u = uL and v = uR, the equations can be re-written as :
Neutrino 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 uL participated in both the electromagnetic and weak interactions, while uR 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
[view large image]

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=(r1, r2, r3) with zero length, i.e.,
r12 + r22 + r32 = 0 ---------- (36)
where the ri's can be a complex number making a total of 6 dimensions. The R vector can be compressed into a 2-components spinor S=(s0, s1) by defining :
r1 = s02 - s12 ---------- (37a)
r2 = i(s02 + s12) ---------- (37b)
r3 = -2s0s1 ---------- (37c)
the inverse of which is :
s0 = [(r1 - ir2)/2]1/2 ---------- (38a)
s1 = [(-r1 - ir2)/2]1/2 ---------- (38b)
Different Kinds of Spin 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-iR around r3, then S => e-i/2S around s1. 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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