Lectures on Quantum Field Theory. Ashok Das

Lectures on Quantum Field Theory

we can identify the Hamiltonian for the Dirac equation with (recall the time dependent Schrödinger equation (1.37))

In the particular representation of the γ^µ matrices in (1.91), we note that

We can now determine either from the definition in (1.98) and (1.79) or from the explicit representation in (1.101) that the matrices α, β satisfy the anti-commutation relations

with We can, of course, directly check from this explicit representation that both β and α are Hermitian matrices. But, independently, we also note from the form of the Hamiltonian in (1.100) that, in order for it to be Hermitian, we must have

In terms of the γ^µ matrices, this translates to

Equivalently, we can write

Namely, independent of the representation, the γ^µ 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 γ^µ 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, γ^µ, 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 for any four vector A_µ)

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 γ^µ matrices in (1.91), we can write this explicitly as

This is a set of four linear homogeneous equations (in the four variables u_α(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))

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