References: Mark Srednicki, Quantum Field Theory, (Cambridge University Press, 2007) – Chapter 1.
One possibility for converting the non-relativistic Schrödinger equation for a free particle to a relativistic equation is to replace the classical energy by the relativistic energy
Using the quantum mechanical momentum operator , the new equation becomes
This equation as it stands involves an operator on the RHS that produces a differential equation that is very difficult to solve, and in fact gives rise to a number of other problems we won’t go into here.
However, this equation in its simplest form says that the operator on the LHS is formally equivalent to the operator on the RHS. So we can, in effect, multiply this equation through by the same operator which has the effect of squaring the operators on both sides of the equation, giving
This is the Klein-Gordon equation. It is consistent with special relativity because it is Lorentz invariant. To see this, we need to show that it is invariant under a Lorentz transformation. The most general Lorentz transformation is
where is the usual Lorentz transformation matrix which depends on the relative velocity of the two inertial frames and the vector is a constant translation of coordinates. The notation denotes an event in 4-d spacetime, so that
One of the principles of relativity is that physics should look the same in all inertial frames. This means that the wave function must have the same value for a given event (a specific set of time and space coordinate values) in both inertial frames. In other words, if is the wave function in the barred system at an event with coordinates in the barred system, we must have
where consists of the spacetime coordinates for that event in the unbarred system.
Since we’re dealing only with special relativity, the metric tensor is the flat space metric
which we’ve seen is invariant under Lorentz transformations. That is
This condition is derived from the requirement that the interval is invariant.
where the in stands for the four components and
If we transform 10 into the barred frame, we have (since , and are all invariants)
Using 6 we have
so in order for this equation to be invariant, we need to show that . Suppose that a single derivative transforms in the same way as the spacetime vector, that is
Therefore, the relation 15 gives the correct value for . This means
The problem is that because of the second-order time derivative on the LHS, the equation doesn’t have the same form as the Schrödinger equation, where the time derivative is first order. One consequence of this is that the normalization condition of the wave function isn’t conserved. If you review the notion of probability current you’ll see that the rate of change of the probability of a particle being in the interval is
In any physical situation, the wave function goes to zero at infinity, so as we extend and , we get which says simply that the probability of the particle being somewhere is constant (that is, 1). If you review the derivation of this result, it came about because we could replace the first order derivative with respect to time by the second order derivative with respect to by using the Schrödinger equation. With the Klein-Gordon equation and its second order time derivative, this derivation doesn’t work any more, with the result that we can’t state categorically that . This is a fundamental violation of the statistical interpretation of the wave function.