Foundations of Quantum Field Theory. Klaus D Rothe
way out to save Lorentz covariance in the presence of interaction would be to treat space and time from the outset on equal footing by working with a differential equation of second order in space as well as time. This equation should nevertheless contain the solution discussed previously. Noting that
one is thus led in the absence of interaction, to the Klein–Gordon equation
The general solution of this equation is now given by
where
The solution (3.9) thus represents in general two wave packets moving away from each other with time. If we choose the Fourier amplitudes a(+)(k) and a(−)(k) to be concentrated around
which could be interpreted as a two-particle state. Since
we may regard the solution ϕ(−) as solutions of the Schrödinger equation for negative energy. These negative energy solutions do not fit into our probabilistic interpretation, since with the scalar product (3.6),
The negative energy solutions of the KG equation thus carry negative norm with respect to the scalar product (3.6). In the free case we may nevertheless ignore their existence, since they satisfy the orthogonality property
and as a result we have no mixing of positive and negative energy solutions:
Thus we may restrict ourselves to the positive energy sector of the theory. This will, however, no longer be true if we allow for interactions with an external potential, which will induce transitions between positive and negative energy states.
3.4KG equation in the presence of an electromagnetic field
The interaction of a charge q with an external electromagnetic field is introduced in the Klein–Gordon equation by the usual minimal substitution
where Aμ = (Φ,
This leads us to consider the equation of motion
with the covariant derivative
It is important to realize that unlike the free particle case, this equation can no longer be factorized in the form (3.7):
where H is the Hamiltonian for a relativistic particle moving in an external electromagnetic field:
Eq. (3.11) is covariant under the following gauge transformation
where Λ(x) denotes an arbitrary function of x. Indeed, under this transformation
or equivalently
In particular
Hence defining the gauge-transformed wave function ϕ′(x) by
Eq. (3.14) implies
The transformation law (3.15) can be restated in the following way: The wave function ϕ(x) is a functional of the vector potential Aμ(x):
The transformation law (3.15) for the covariant derivative then implies that under the gauge transformation (3.13) the functional ϕ(x; Aμ) transforms as follows:
The gauge covariance of the equation of motion (3.11) allows us to choose in particular the covariant Lorentz gauge ∂·A = 0. In this gauge the 4-tuplet Aμ = (A0,
Negative energy solutions and antiparticles
Consider the case where the vector potential is independent of time. In that case there exist stationary solutions
suggesting again a two-particle interpretation for a general wave packet. Labelling these solutions by the charge q appearing in the covariant derivative (3.12), substitution into Eq. (3.11) leads to the equations
where we have set Φ = A0. From here we see that
This suggests the identification of the “negative energy” solution ϕ(−) with the respective antiparticle. The transformation (3.18) for scalar fields is referred to as charge conjugation.
Probability interpretation
In