(1.79) or from the explicit representation in (1.101) that the matrices α, β satisfy the anti-commutation relations
with
In terms of the γµ matrices, this translates to
Equivalently, we can write
Namely, independent of the representation, the γµ matrices must satisfy the Hermiticity properties in (1.105). (With a little bit of more analysis, it can be seen that, in general, the Hermiticity properties of the γµ matrices are related to the choice of the metric tensor and this particular choice is associated with the Bjorken-Drell metric.) In the next chapter, we would study the plane wave solutions of the first order Dirac equation.
1.5References
The material presented in this chapter is covered in many standard textbooks and we list below only a few of them.
1.J. D. Bjorken and S. Drell, Relativistic Quantum Mechanics, McGraw-Hill, New York (1964).
2.A. Das, Lectures on Quantum Mechanics, (second edition) Hindustan Publishing, India and World Scientific, Singapore (2011).
3.C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, New York (1980).
4.L. I. Schiff, Quantum Mechanics, McGraw-Hill, New York (1968).
CHAPTER 2
Solutions of the Dirac equation
2.1Plane wave solutions
The Dirac equation in the momentum representation (see (1.80))
or in the coordinate representation
defines a set of matrix equations. Since the Dirac matrices, γµ, are 4 × 4 matrices, the wave function ψ, in this case, is a four component column matrix (column vector). From the study of angular momentum, we know that multicomponent wave functions suggest a nontrivial spin angular momentum for the particle. (Other nontrivial internal symmetries can also lead to a multicomponent wavefunction, but here we are considering a simple system without any nontrivial internal symmetry.) Therefore, we expect the solutions of the Dirac equation to describe particles with spin. To understand what kind of particles are described by the Dirac equation, let us look at the plane wave solutions of the equation (which are supposed to describe free particles). Let us denote the four component wave function as (x stands for both space and time)
with
Substituting this back into the Dirac equation, we obtain (we define
where the four component function, u(p), has the form
Let us simplify the analysis by restricting to motion along the z-axis. In other words, let us set
In this case, equation (2.5) takes the form
Taking the particular representation of the γµ matrices in (1.91), we can write this explicitly as
This is a set of four linear homogeneous equations (in the four variables uα(p), α = 1, 2, 3, 4) and a nontrivial solution exists only if the determinant of the coefficient matrix vanishes. Thus, requiring,
we obtain,
Thus, we see that a nontrivial plane wave solution of the Dirac equation exists only for the energy values
Furthermore, we see from (2.11) that each of these energy values is doubly degenerate. Of course, we would expect the positive and the negative energy roots in (2.12) from Einstein’s relation. However, the double degeneracy seems to be a reflection of the nontrivial spin structure of the wave function as we will see shortly.
The energy eigenvalues (and the degeneracy) can also be obtained in a simpler fashion by noting that (in the gamma matrix representation of (1.91))
This is identical to (2.11) and the energy eigenvalues would then correspond to the roots of
