1.1.2 Physics Implied in Dirac Equation

The conclusions of the Dirac equation were highly controversial when they were first described in 1928, but in a curious way, those criticisms appeared to simply bounce off Dirac, which is perhaps a consequence of his deep previsional personality.

Now you are suggested to perceive the profound physical implications hidden in the Dirac equation by making various discussions.

(1)The implication of can be seen by means of a simple calculation

Likewise, you have r=cα. Now you have learnt that the operator is the velocity operator in unit of the velocity of light.

(2)The momentum is conserved, which is an immediate conclusion due to

(3)The angular momentum is not conserved. As you know =r ×, it is not difficult for you to calculate

Thus =, H]=0, which means that the angular momentum is not conserved. For a free particle, its angular momentum should be con-served as the corresponding Hamiltonian possesses rotational symmetry. Therefore, there must be something new or something wrong here. What is it? You will soon find that it is quite subtle.

(4)The existence of intrinsic angular momentum. If you define

you can easily know that

Then you perceive that the“total”angular momentum is actually conserved. This brings in a profound concept that a particle characterized by the Dirac equation (we call a Dirac particle) must carry intrinsic an-gular momentum . Thus, electrons are Dirac particles, which naturally resolves the originality of electron spin (in Pauli equation, the spin must be introduced phenomenologically). Strictly speaking, if here are inter-preted as the operators for intrinsic angular momentum, they must follow the standard commutation relations that angular momenta obey. This is left as an exercise for readers.

(5) Plane wave solution and the energy-momentum relation for free Dirac particles. The states with definite value of momentum are plane waves

where u and v are two-row matrices. Then the stationary Dirac equation (1.19)gives rise to the following algebraic equations

The existence of nonzero solution for the above algebraic equation requires that

With the help of the formula

you obtain from the secular equation(1.26)the energy eigenvalues as fol-lows

which is plotted in Fig. 1.1. Substituting these two eigenvalues into the eigenequation (1.25), you can get solutions of u and v, then you will ob-tain four eigenfuctions (I omit here, interested readers are suggested to carry out the concrete calculations by themselves). Eventually, here you encounter with solutions of negative energy for free particles! Since there exists a continuum of negative-energy states from-mc2to-∞, electrons would be unstable and fall into a negative-energy state by emitting pho-ton spontaneously. Furthermore, once reaching a negative-energy state, it would keep on lowering its energy indefinitely by emitting photons again and again. This is of course a subtle and challenging issue ought to be comprehended.

(6) The Dirac sea and the prediction of positron. Dirac proposed, in 1930, a very clever picture which we call“Dirac sea”. In his proposal, all the negative-energy states are completely occupied that prevent the afore-mentioned catastrophic transitions due to the Pauli exclusive principle. Thus, what we usually call the vacuum can be regarded as an infinite sea of negative-energy electrons.

You can imagine that, once one of the negative-energy electrons in the Dirac sea absorbs a quanta of energy more than 2mc2, it becomes an E＞0 positive-energy state. Consequently, a hole is created in the Dirac sea. Then you may wonder whether a hole in the imaginary Dirac sea is of any realistic physical significance. Actually, the existence of a hole in the Dirac sea for electrons is equivalent to the existence of a particle with positive charge above the sea. To understand this point, you assume the total charge and the total energy of the vacuum (i.e., the Dirac sea) are Qvacuumand Evacuumrespectively, then the total charge Q and total energy E of the Dirac sea with a hole are given by

Q=Qvacuum-(-|e|)=Qvacuum+|e|

E=Evacuum-(-|ε|)=Evacuum+|ε|

This is depicted in Fig.1.2.

At first, Dirac identified the hole to be proton. But he was bewildering for a long time at the big mass difference between electrons and protons. However, the mathematician Wigner never worried about the mass differ-ence while regarded the masses of the hole and electron as the same from the aesthetics point of view in mathematics. Finally, Dirac gave up the at-tempts to solve the puzzle of the mass difference and audaciously predicted that there must exist positively charged particles of mass identical to elec-trons. He named such particles as positrons. The idea of negative energy states and the consequent hole theory were finally confirmed by Carl David Anderson's discovery of the positron in 1932. The unique characteristics of electrons make such devices as transistors, solid-state lasers, and iPhones possible for us.

(7) Particles and antiparticles. For Dirac particles of charge q in an electromagnetic field, the stationary Dirac equation reads

Making a complex conjugation to the above equation(1.29)and multiply-ing α2from the left, you obtain that

where = -α2, α2β=-βα2and α2αi= -αiα2(i=1,3) have been employed. Again, multiplying β from the left, you get the following equa-tion

Then you perceive that you can define an operation by introducing

so that Ψcis also a solution of the stationary Dirac equation of energy-E and charge-q if Ψ is a solution of energy E and charge q. Thus the operation(1.32)is called charge conjugation that maps a wavefunction to another wavefunction describing a particle with energy and charge of the opposite sign. Now you are in the position to reach a new point of view:the positron can be regarded as the antiparticle of electron. Actually, one can associate each particle, in nature, with an antiparticle of equal mass and opposite charge. When the particle and antiparticle meet, they an-nihilate each other producing light in the process. If a particle of neutral charge is its own antiparticle, i.e., Ψc=Ψ, it is called Majorana particle. The concept of Majorana particle was earlier introduced in the study of quantum field theory, while it is realized in various systems of condensed matter recently.