Foundations of Quantum Field Theory. Klaus D Rothe

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      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 tentative way out to save Lorentz covariance in the presence of interaction would be to treat space and time from the outset on equal footing by working with a differential equation of second order in space as well as time. This equation should nevertheless contain the solution discussed previously. Noting that

      one is thus led in the absence of interaction, to the Klein–Gordon equation

      The general solution of this equation is now given by

      where

      The solution (3.9) thus represents in general two wave packets moving away from each other with time. If we choose the Fourier amplitudes a⁽⁺⁾(k) and a⁽⁻⁾(k) to be concentrated around , then these two wave packets will separate from each other with twice the group velocity

      which could be interpreted as a two-particle state. Since

      we may regard the solution ϕ⁽⁻⁾ as solutions of the Schrödinger equation for negative energy. These negative energy solutions do not fit into our probabilistic interpretation, since with the scalar product (3.6),

      The negative energy solutions of the KG equation thus carry negative norm with respect to the scalar product (3.6). In the free case we may nevertheless ignore their existence, since they satisfy the orthogonality property

      and as a result we have no mixing of positive and negative energy solutions:

      Thus we may restrict ourselves to the positive energy sector of the theory. This will, however, no longer be true if we allow for interactions with an external potential, which will induce transitions between positive and negative energy states.

3.4KG equation in the presence of an electromagnetic field

      The interaction of a charge q with an external electromagnetic field is introduced in the Klein–Gordon equation by the usual minimal substitution

      where Aμ = (Φ,

) is related to the electric field E in the usual way:

      This leads us to consider the equation of motion

      with the covariant derivative

      It is important to realize that unlike the free particle case, this equation can no longer be factorized in the form (3.7):

      where H is the Hamiltonian for a relativistic particle moving in an external electromagnetic field:

      Eq. (3.11) is covariant under the following gauge transformation

      where Λ(x) denotes an arbitrary function of x. Indeed, under this transformation

      or equivalently

      In particular

      Hence defining the gauge-transformed wave function ϕ′(x) by

      Eq. (3.14) implies

      The transformation law (3.15) can be restated in the following way: The wave function ϕ(x) is a functional of the vector potential Aμ(x):

      The transformation law (3.15) for the covariant derivative then implies that under the gauge transformation (3.13) the functional ϕ(x; Aμ) transforms as follows:

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