we will be using in this book. We will introduce new notations as they arise in the context of our discussions.
1.3Klein-Gordon equation
With all these basics, we are now ready to write down the simplest of the relativistic equations. We recall that in the case of a non-relativistic particle, we start with the non-relativistic energy-momentum relation
and promote the dynamical variables (observables) to Hermitian operators to obtain the time-dependent Schrödinger equation (see (1.1))
Let us consider the simplest of relativistic systems, namely, a relativistic free particle of mass m. In this case, we have seen that the energy-momentum relation is none other than the Einstein relation (1.30), namely,
Thus, as before, promoting these to operators, we obtain the simplest relativistic quantum mechanical equation to be (see (1.33))
Setting ħ = 1 from now on for simplicity, the equation above takes the form
Since the operator in the parenthesis is a Lorentz scalar and since we assume the quantum mechanical wave function, ϕ(x, t), to be a scalar function, this equation is invariant under Lorentz transformations.
This equation, (1.40), is known as the Klein-Gordon equation and, for m = 0, or when the rest mass of the particle vanishes, it reduces to the wave equation (recall Maxwell’s equations). Like the wave equation, the Klein-Gordon equation also has plane wave solutions which are characteristic of free particle solutions. In fact, the functions
with kµ = (k0, k) are eigenfunctions of the energy-momentum operator, namely, using (1.33) (remember that ħ = 1) we obtain
so that ±kµ are the eigenvalues of the energy-momentum operator. (In fact, the eigenvalues should be ±ħkµ, but we have set ħ = 1.) This shows that the plane waves define a solution of the Klein-Gordon equation (1.39) or (1.40) provided
Thus, we see the first peculiarity of the Klein-Gordon equation (which is a relativistic equation), namely, that it allows for both positive and negative energy solutions. This basically arises from the fact that, for a relativistic particle (even a free one), the energy-momentum relation is given by the Einstein relation which is a quadratic relation in E, as opposed to the case of a non-relativistic particle, where the energy-momentum relation is linear in E. If we accept the Klein-Gordon equation as describing a free, relativistic, quantum mechanical particle of mass m, then, we will see shortly that the presence of the negative energy solutions would render the theory inconsistent.
To proceed further, let us note that the Klein-Gordon equation and its complex conjugate (remember that a quantum mechanical wave function is, in general, complex), namely,
would imply
Defining the probability current density four vector as
where
we note that equation (1.45) can be written as a continuity equation for the probability current, namely,
The probability current density,
of 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
