time derivative), the probability density is independent of time derivative much like in the Schrödinger equation. Consequently, the probability density, as we have seen explicitly in (2.38) and (2.40), can be defined to be positive definite even in the presence of negative energy solutions. This is rather different from the case of the Klein-Gordon equation that we have studied in chapter 1.
2.5Dirac’s hole theory
We have seen that the Dirac equation leads to both positive and negative energy solutions. In the free particle case, for example, the energy eigenvalues are given by
Thus, even for this simple case of a free particle the energy spectrum has the form shown in Fig. 2.1. We note from Fig. 2.1 (as well as from the equation above, (2.88)) that the positive and the negative energy solutions are separated by a gap of magnitude 2m (remember that we are using c = 1).
Even when the probability density is consistently defined, the presence of negative energy solutions leads to many conceptual difficulties. First of all, in such a case, we note that the energy spectrum is unbounded from below. Since physical systems have a tendency to go to the lowest energy state available, this implies that any such physical system (of Dirac particles) would make a transition to these unphysical energy states thereby leading to a collapse of all stable systems such as the Hydrogen atom. Classically, of course, we can restrict ourselves to the subspace of positive energy solutions. But as we have argued earlier within the context of the Klein-Gordon equation, quantum mechanically this is not acceptable. Namely, even if we start out with a positive energy solution, any perturbation would cause the energy to lower, destabilizing the physical system and leading to an ultimate collapse.
Figure 2.1: Energy spectrum for a free Dirac particle.
In the case of Dirac particles, however, there is a way out of this difficulty. Let us recall that the Dirac particles carry spin
This redefinition of the vacuum automatically prevents the instability associated with matter. For example, a positive energy electron can no longer drop down to a negative energy state without violating the Pauli exclusion principle since the negative energy states are already filled. (Note that this would not work for a bosonic system such as particles described by the Klein-Gordon equation. It is only because fermions obey Pauli exclusion principle that this works for the Dirac equation.) On the other hand, such a redefinition of the ground state does predict some new physical phenomena which are experimentally observed. For example, if enough energy is provided to such a ground state, a negative energy electron can make a transition to a positive energy state and can appear as a positive energy electron. Furthermore, the absence of a negative energy electron can be thought of as a “hole” which would have exactly the same mass as the particle but otherwise opposite internal quantum numbers. This “hole” state is what we have come to recognize as the antiparticle – in this case, a positron – and the process under discussion is commonly referred to as pair creation (production). Thus, the Dirac theory predicts an anti-particle of equal mass for every Dirac particle. (The absence of a negative energy electron in the ground state can be thought of as the ground state plus a positive energy “hole” state with exactly opposite quantum numbers to neutralize its effects. The amount of energy necessary to excite a negative energy electron to a positive energy state is E ≥ 2m.)
This is Dirac’s theory of electrons and works quite well. However, we must recognize that it is inherently a many particle theory in the sense that the vacuum (ground state) of the theory is defined to contain infinitely many negative energy particles. (This unconventional definition of the vacuum state can be avoided in a second quantized field theory which we will study later.) In spite of this, the Dirac equation passes as a one particle equation primarily because of the Pauli exclusion principle. On the other hand, this is a general feature that combining quantum mechanics with relativity necessarily leads to a many particle theory.
2.6Properties of the Dirac matrices
The Dirac matrices, γµ, were crucial in taking a matrix square root of the Einstein relation and, thereby, in defining a first order equation. In this section, we will study some of the useful properties of these matrices. As we have seen, the four Dirac matrices satisfy (in addition to the Clifford algebra)
Since these are 4 × 4 matrices, a complete set of Dirac matrices must consist of 16 such matrices. Of course, the identity matrix will correspond to one of them.
To obtain the other basis matrices, let us define the following sets of matrices. Let
where
represents the four-dimensional generalization of the Levi-Civita tensor. Note that in our particular representation for the γµ matrices given in (1.91), we obtain
where we have used the property of the Pauli matrices
We recognize from (2.92) that we can identify this with the matrix ρ defined earlier in (2.60). Note that, by definition,
and that, since it is the product of all the four γµ matrices, it anti-commutes with any one of them.
