Lectures on Quantum Field Theory. Ashok Das

course, has the same form as in non-relativistic quantum mechanics. However, we note that the form of the probability density (which results from the requirement of covariance)

      is quite different from that in non-relativistic quantum mechanics (namely, ρ = ϕ^∗ϕ) and it is here that the problem of the negative energy states shows up. For example, even for the simplest of solutions, namely, plane waves of the form

      we obtain

      Since energy can take both positive and negative values, it follows that ρ cannot truly represent the probability density which, by definition, has to be positive semi-definite. It is worth noting here that this problem really arises because the Klein-Gordon equation, unlike the time dependent Schrödinger equation, is second order in time derivatives. This has the consequence that the probability density involves a first order time derivative and that is how the problem of the negative energy states enters. (Note that if the equation is second order in the space derivatives, then covariance would require that it be second order in time derivative as well. This would, in turn, lead to the difficulty with the probability density being positive semi-definite.) One can, of course, ask whether we can restrict ourselves only to positive energy solutions in order to avoid the difficulty with the interpretation of ρ. Classically, we can do this. However, quantum mechanically, we cannot arbitrarily impose this for a variety of reasons. The simplest way to see this is to note that the positive energy solutions alone do not define a complete set of (basis) states in the Hilbert space and, consequently, even if we restrict the states to be of positive energy to begin with, negative energy states may be generated through quantum mechanical corrections. It is for these reasons that the Klein-Gordon equation was abandoned as a quantum mechanical equation for a relativistic single particle. However, as we will see later, this equation is quite meaningful as a relativistic field equation.

1.3.1 Klein paradox. Let us consider a charged scalar particle described by the Klein-Gordon equation (1.40) in an external electromagnetic field. We recall that the coupling of a charged particle to an electromagnetic field is given by the minimal coupling

      where we have used the coordinate representation for the momentum as in (1.33) and A_µ denotes the vector potential associated with the electromagnetic field. In this case, therefore, the scalar particle will satisfy the minimally coupled Klein-Gordon equation (e > 0, namely, the particles are chosen to carry positive charge)

      As a result, the probability current density in (1.46) can be determined to have the form

      where we have defined

      

      With this general description, let us consider the scattering of a charged scalar (Klein-Gordon) particle with positive energy from a constant electrostatic potential. In this case, therefore, we have

      For simplicity, let us assume the constant electrostatic potential to be of the form

      and we assume that the particle is incident on the potential along the z-axis as shown in Fig. 1.2.

Figure 1.2: Klein-Gordon particle scattering from a constant electrostatic potential.

      The dynamical equations will now be different in the two regions, z < 0 (region I) and z > 0 (region II), and have the forms (see (1.54))

      In region I, there will be an incident as well as a reflected (plane) wave so that we can write (remember that the incident particle has positive energy)

      while in region II, we only expect a transmitted wave (traveling to the right) of the form

      where A, B are related respectively to reflection and transmission coefficients. We note here that the continuity of the wave function at the boundary z = 0 requires that the energy be the same in the two regions.

      For the wave functions in (1.60) and (1.61) to satisfy the respective equations in (1.59), we must have

      Here we have used the fact that the energy of the incident particle is positive and, therefore, the square root in the first equation in (1.62) is with a positive sign. However, the sign of the square root in the second relation remains to be fixed.

      Let us note from the second relation in (1.62) that p′ is real for both E − eΦ₀ > m (weak potential) and for E − eΦ₀ < −m (strong potential). However, for a potential of intermediate strength satisfying −m < E − eΦ₀ < m, we note that p′ is purely imaginary. Thus, the behavior of the transmitted wave depends on the strength of the potential. As a result, in this second case, we must have

      in order that the wave function is damped in region II. To determine the sign of the square root in the cases when p′ is real, let us note from the second relation in (1.62) that the group velocity of the transmitted wave is given by

      Since we expect the transmitted wave to be travelling to the right, we determine from (1.64) that