Foundations of Quantum Field Theory. Klaus D Rothe
case, we attempted to identify the probability density with the time component of a conserved 4-vector-current. It was found to satisfy all requirements, provided we restricted ourselves to positive energy solutions from the outset. As we now show, this will no longer be possible in the presence of an electromagnetic interaction, which will invariably lead to transitions to states involving negative energy solutions.
Following the general line of approach adopted in the free particle case, we note to begin with that we can define again a conserved current by
This current is gauge-invariant
and thus defines a Lorentz covariant observable.
With the aid of the equation of motion (3.11) one easily checks that this current is conserved:
Since jμ transforms like a 4-vector density, we take its zero component to define the probability density:
Unfortunately this definition of the probability density already violates positivity for “positive energy” solutions; indeed, consider the stationary wave function
Substitution into (3.19) yields
Hence, even if E > 0, this density is not positive semi-definite, since the sign of the Coulomb potential can be either positive or negative.
The above considerations lead us to abandon at this stage our search for a relativistic scalar wave equation conforming to the principles of non-relativistic quantum mechanics. We shall, however, return to the field equation (3.11) after having learned in Chapter 7 to interpret ϕ(x) as an operator-valued field acting on a Hilbert space of Fock states.
Chapter 4
The Dirac Equation
We begin this chapter by obtaining the relativistic “Schroödinger” equation for a free spin-1/2 field by following first the historical approach, and then presenting a derivation based on Lorentz covariance and space-time parity alone. This leads us to a four-component wave equation which is first-order in space and time coordinates. We present the solution of this equation for three choices of basis: The Dirac, Weyl (or chiral), and Majorana representations. The latter representation is shown to be particularly useful for the case of Majorana fermions, i.e. fermions which are their own anti-particles. We show that the Dirac equation allows for the notion of a probability density after suitable interpretation of the negative energy states.
4.1Dirac spinors in the Dirac and Weyl representations
In this section we present the derivation of the Dirac equation by following the historical path, as well as a purely group-theoretical approach relying on Lorentz transformation properties alone. We then obtain the general solution of these equations in terms of the four independent Dirac spinors.
Dirac equation: historical derivation
Since in a manifestly Lorentz-covariant wave equation, space and time variables should appear on equal footing, Dirac demanded that the hamiltonian in the equation
should depend linearly on the momentum
The triple
Here the bracket {A, B} denotes the anticommutator of two objects:
In order to get the desired energy momentum relation, this equation has to reduce to the Klein–Gordon equation, which is the case if
From here we deduce the following properties of the matrices:
Tracelessness
Since
or
Dimensionality
Since
Minimal dimension
The Pauli matrices
together with the identity matrix 1 represent a complete basis for 2 × 2 hermitian matrices. Of these, the Pauli matrices satisfy the first of the conditions (4.3); however, the identity matrix cannot be identified with β, since trβ = 0. Since the dimension of the matrices must be even, we conclude that the dimension of these matrices must be at least four.
The following 4 × 4 matrices satisfy all the requirements (4.3):
The same applies of course to matrices obtained from the above ones via a unitary transformation (unitary, in order to preserve the hermiticity of the matrices). For the choice of basis (4.5), the equation reads