Foundations of Quantum Field Theory. Klaus D Rothe

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      so that we recover the Schrödinger 1-particle wave function of the non-relativistic theory. Since the argument of the exponential appearing in the Fourier integral is not Galilei-invariant, the corresponding non-relativistic field ϕnr(, t) does not transform as a scalar. In fact, each Fourier component will pick up a phase such as witnessed in Eq. (1.1) of Chapter 1. We therefore see that in this respect the transformation law in the relativistic case is simpler than in the case of Galilei transformations.

3.2Difficulties with the wave equation

      As we now show, the wave equation (3.1) presents a number of problems which will eventually lead us to abandon it.

       Locality

      The wave equation (3.1) has an unwanted property: In order to determine the change in the solution at the point in an infinitesimal time interval (t, t + dt), we must know the function for time t at all points . This property is contained in the Taylor expansion of the Hamilton operator in powers of the Laplacian and is referred to as non-locality. It is also evident from the form of the solution (3.4) by noting that

      which upon using the Taylor expansion in p² becomes

       Probability interpretation

      In order for the wave function to have the interpretation of a probability amplitude, it should have at least the following properties:

      (i) Its normalization with respect to some suitable integration measure should be independent of time, as well as of the choice of inertial frame.

      (ii) The associated probability density should be positive semi-definite.

      The second condition is satisfied by defining the probability density by

      Furthermore, adopting the following definition for the scalar product

      and making use of the equation of motion (3.1), one finds that the normalization of the wave function is preserved in time:

      The probability of finding a particle anywhere in space should, however, also be independent of the choice of inertial frame. Since the wave function ϕ(, t) transforms like a scalar (see Eq. (3.3)) this is not the case due to the Lorentz non-invariance of the measure d³r.

      For the case of a free particle this defect is easily repaired by adopting a new definition for the scalar product:

      Indeed, making use of the Fourier decomposition (3.4) for a free particle wave function, one computes

      Because of the invariance property (2.43) of the integration measure, and the scalar property (3.5) of the Fourier amplitude, we conclude that the normalization of the wave function is a Lorentz invariant with respect to the scalar product (3.6). In fact, one easily shows that it has all the properties expected from a scalar product:

      and

      for solutions of the equation of motion. The scalar product (ϕ, ϕ)_t is also time-independent, as one easily checks using the equation of motion (3.2):

      for ∂V → ∞ and ϕ → 0 at infinity. In fact, the probability density satisfies a continuity equation analogous to that in non-relativistic Quantum Mechanics:

      where

      We may in fact collect and j(, t) into a 4-vector as follows:

      where . Indeed, under a Lorentz transformation we have (compare with (2.5))

      Although we have succeeded in satisfying the requirement (i), this is not the case as far as requirement (ii) is concerned. As a simple example shows, the probability density is not a positive semi-definite quantity. To see this, consider the superposition of two plane waves for two free particles of mass m:

      with k · x = kμxμ, etc.

      Let c and d be real. One then has

      Now, for every 4-vector (k − q)^μ there exists a vector xμ such that (k − q) · x = 0. For this vector

      We may then rewrite the above expression as

      The matrix appearing here is hermitian, and may thus be diagonalized, its eigenvalues being

      Hence, though the total probability is positive, the density in space-time is not. Since λ₋ < 0, there always exist coefficients such that , which proves our claim.

3.3The Klein–Gordon equation

      One