Lectures on Quantum Field Theory. Ashok Das

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 and are, therefore, fermions. To be specific, let us assume that the particles described by the Dirac equation are the spin electrons. Since fermions obey Pauli exclusion principle, any given energy state can accommodate at the most two electrons with opposite spin projections. Taking advantage of this fact, Dirac postulated that the physical ground state (vacuum) in such a theory should be redefined for consistency. Namely, Dirac postulated that the ground state in such a theory is the state where all the negative energy states are filled with electrons. Thus, unlike the conventional picture of the ground state as being the state without any particle (quantum), here the ground state, in fact, contains an infinite number of negative energy particles. Furthermore, Dirac assumed that the electrons in the negative energy states are passive in the sense that they do not produce any observable effect such as charge, electromagnetic field etc. (Momentum and energy of these electrons are also assumed to be unobservable. This simply means that one redefines the values of all these observables with respect to this ground state.)

      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, γ^µ, 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 γ^µ 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 γ^µ matrices, it anti-commutes with any one of them. Namely,

      Given the matrix γ₅, we can define four new matrices as

      Since we know the explicit forms of the matrices γ^µ and γ₅ in our representation, let us write out the forms of γ₅γ^µ also in this representation.

      Finally, we can also define six anti-symmetric matrices, σ^µν, as (µ, ν = 0, 1, 2, 3)

      whose explicit forms in our representation can be worked out to be (i, j, k = 1, 2, 3)

      Here we have used the three dimensional notation ϵ_ijk = ϵ^ijk. We have already seen in (2.71) that the matrices represent the spin operators for the Dirac particle. From (2.99) we conclude,