The Kleingordon Equation

In the last chapter we brought the subject to the point where the electromagnetic field was quantized, and the production and annihilation of the field quanta (photons in that case) could be treated. The treatment of the EM field was fully relativistic, even if it was not manifestly covariant (because of the Coulomb gauge). However, the treatment of the particles (electrons and protons) remained nonrelativistic, and we had no way to describe the production and annihilation of these particles (which must always occur in particle-antiparticle pairs because both electron and baryon numbers are conserved). With this chapter we begin to develop the tools necessary to describe "classical" particles, such as electrons, covariantly. The development will eventually lead to the construction of field theories for electrons and other classical particles (Chapter 7), with the capability tn describe the production of particle-antiparticle pairs. Only then will the distinction between classical particles and fields disappear, with the recognition that the quantum field is the single entity suitable for the description of all matter and energy.

However, before we can introduce these new quantum fields we must first understand how to describe single particles in a covariant fashion. This is the subject of the next three chapters. In this chapter we begin with the simplest relativistic equation, the Klein-Gordon {KG) equation. Then we discuss the Dirac equation (Chapter 5) and applications of the Dirac equation (Chapter 6).

4.1 THE EQUATION

We begin our systematic study of relativistic equations by briefly considering the following equation:

where, as in the previous chapters, x represents both the time and space dependence of the wave function, so that ip{x) = rp(r,t) is understood. This equation follows the traditional "rules" of quantum mechanics in that it can be obtained

from the relativistic energy relation E = i/m2 + p2 by the substitution

There is no problem in principle with the operator J5y = \/m2 — V2; this operator is defined by either (i) expanding any function in terms of the eigenfunctions of V (the momentum eigenfunctions), on which the operation £y is easily carried out, or (ii) defining J5y by its power series expansion

While this series may not always converge, we may consider its analytic continuation to be the definition of the operation of Ey on any function.

The disadvantage of Eq. (4.1) is that it is not manifestly covariant. To be manifestly covariant, we must know how to transform the equation not only in time and space, described by the infinitesimal generators H and P, but also under the homogeneous Lorentz group, which includes rotations, generated by the angular momentum operators J, and the boosts, generated by the operators K (the Lorentz group and its generators will be discussed in Sec. 5.8). While these transformations can sometimes be worked out for equations of the type (4.1), many problems are encountered, and therefore this route was not the one taken in the original developments which led to the quantum field theory of elementary /articles [except that the Dirac equation can be regarded as arising from the linearization of the square root in (4.1); see Chapter 5]. Now we know that many particles originally supposed to be "elementary" (such as the proton) are in fact complicated composite structures of valence quarks and a sea of quark-antiquark pairs and hence cannot be described by a single local quantum field. In the search for approximate methods of describing such particles, interest in equations of the type (4.1) has been rekindled and is an active area of current research. Such an approach sometimes is identified as relativistic Hamiltonian dynamics and will not be discussed further here.*

The alternative route is to introduce equations which are manifestly covariant, and this method is sometimes referred to as manifestly covariant dynamics. It is the route which is traditionally taken to relativistic field theory. A simple, manifestly covariant wave equation is obtained if the substitutions (4.2) are made into the mass energy relation E2 = m2 +p2. This gives the Klein-Gordon (KG) equation for a free particle

In these equations, the d/dxM operator operates all the way to the right, so that in the first term it operates on both A (or V) and ip. While all the terms in (4.8) are scalars, the first two terms consist of a vector potential which is contracted with the operator to make an overall scalar, while the last term is a scalar by itself. Note that the symmetrized form of the vector term is required in order to maintain the hermiticity of the interaction. In the most general case, the scalar, S, and vector, VM, parts of the potential could be independent interactions, but for electromagnetism they are related by

In some of the following discussion, the vector and scalar parts will be treated as independent interactions, and at other times we will specialize the discussion to electromagnetism, Eq. (4.9).

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