Chapter 7 Quantum electrodynamics (QED)

This chapter studies the coupling between the electromagnetic potential and the Dirac field, leading to quantum electrodynamics (QED). Some simple examples of scattering amplitudes in QED are discussed.

7.1 Minimal gauge coupling

We would like to build a QFT whose field content is given by a Dirac field \(\Psi (x)\) and the electromagnetic potential \(A_\mu (x)\), and whose dynamics includes a suitable interaction between these fields. The physics we wish to capture by this QFT is that of the interactions between electrons/positrons, which arise as the quantum particles/antiparticles of \(\Psi \), and photons, which arise as the quantum particles of \(A_\mu \). The essential ingredient that is needed for defining such a QFT is an action functional \(S_{\mathrm {QED}}^{}[\Psi ,\overline {\Psi },A]\) on the whole collection of fields \(\Psi \), \(\overline {\Psi }\) and \(A\). Taking for the quadratic parts (i.e. the free actions) the Dirac and Maxwell actions from (5.39) and (6.11), the problem reduces to finding a suitable interaction term \(S_{\mathrm {int}}\) for

\begin{flalign} \nn S_{\mathrm {QED}}^{}[\Psi ,\overline {\Psi },A] \,&=\, S_{\mathrm {Dirac}}^{}[\Psi ,\overline {\Psi }] + S_{\mathrm {MW}}^{}[A] + S_{\mathrm {int}}^{}[\Psi ,\overline {\Psi },A]\\[6pt] \, &=\,\int _{\bbR ^d} -\bigg (\overline {\Psi }\,\big (\slashed {\partial } + m \big )\,\Psi +\frac {1}{4}\, F^{\mu \nu }\,F_{\mu \nu } \bigg )\,\dd x~+~ S_{\mathrm {int}}^{}[\Psi ,\overline {\Psi },A]\quad .\label {eqn:QEDactionpre} \end{flalign} Due to the presence of gauge symmetries (6.13) for the electromagnetic potential, which as we have seen are crucial for removing the redundant/unphysical degrees of freedom of \(A_\mu \), any consistent choice of interaction term must be gauge invariant. (Otherwise, the interactions would re-introduce unphysical degrees of freedom.) While there are still infinitely many possibilities to define a gauge invariant \(S_{\mathrm {int}}\), there is one particularly nice choice that is not only mathematically very pleasing, as it is obtained from a symmetry principle, but also physically realized in nature. This choice is obtained via a construction that is called gauging and minimal coupling, which we will now describe in some detail.

To motivate the gauging construction, let us recall that the Dirac action \(S_{\mathrm {Dirac}}^{}\) is invariant under global phase rotations

\begin{flalign} \label {eqn:Diracglobaltransformation} (T_{\alpha }\Psi )(x) \,=\, e^{-\ii \,q\,\alpha } \,\Psi (x)~~,\quad (T_{\alpha }\overline {\Psi })(x) \,=\, e^{\ii \,q\,\alpha } \,\overline {\Psi }(x)\quad , \end{flalign} for \(\alpha \in \bbR \) a constant. We have introduced here also a parameter \(q\in \bbR \) that has the physical interpretation of the electric charge of the quantum particles associated with \(\Psi \). Indeed, in the presence of \(q\), the equation for charge eigenstates (5.90) reads as \(\noor {Q}\ket {k,s,\pm } = \pm q\, \ket {k,s,\pm }\), i.e. the particle/antiparticle has charge \(\pm q\). (For electrons/positrons, one would choose \(q =-e\), where \(e\) is the elementary charge.) On the other hand, the Maxwell action \(S_{\mathrm {MW}}^{}\) is invariant under the local gauge transformations

\begin{flalign} \label {eqn:Maxwelllocaltransformation} (T_{\alpha }A)_\mu (x) = A_\mu (x) + \partial _\mu \alpha (x)\quad , \end{flalign} for \(\alpha (x)\) a function on the Minkowski spacetime \(\bbR ^d\). While the type of parameter (\(\alpha \) is a scalar) is the same in these transformation laws, there is the huge difference that (7.2) is a global (i.e. \(x\)-independent) transformation and (7.3) is a much more general local (i.e. \(x\)-dependent) transformation. So why don’t we try to extend (7.2) to a local transformation

\begin{flalign} (T_{\alpha }\Psi )(x) \,=\, e^{-\ii \,q\,\alpha (x)} \,\Psi (x)~~,\quad (T_{\alpha }\overline {\Psi })(x) \,=\, e^{\ii \,q\,\alpha (x)} \,\overline {\Psi }(x)\quad , \end{flalign} which in the literature is often described as gauging the global symmetry. Let us recall that we haven’t done this previously because the kinetic term in the Dirac action \(S_{\mathrm {Dirac}}^{}\) isn’t invariant under such local transformations, which is a consequence of

\begin{flalign} \partial _\mu (T_\alpha \Psi )(x) = \partial _\mu \Big (e^{-\ii \,q\,\alpha (x)} \,\Psi (x)\Big ) = e^{-\ii \,q\,\alpha (x)}\, \Big (\partial _\mu \Psi (x) -\ii \,q\,\partial _\mu \alpha (x) \,\Psi (x)\Big )\quad . \end{flalign} But wait: Since we now have learned about the electromagnetic potential, we recognize that the “bad” second term looks suspiciously similar to the transformation law (7.3) of \(A_\mu \) under gauge transformations. It can be compensated by introducing the (gauge) covariant derivative

that combines the partial derivative with the electromagnetic potential. Indeed, under a combined local gauge transformation

\begin{flalign} (T_{\alpha }\Psi )(x) \,&=\, e^{-\ii \,q\,\alpha (x)} \,\Psi (x)\quad ,\\ (T_{\alpha }\overline {\Psi })(x) \,&=\, e^{\ii \,q\,\alpha (x)} \,\overline {\Psi }(x)\quad ,\\ (T_{\alpha }A)_\mu (x) \,&=\, A_\mu (x) + \partial _\mu \alpha (x)\quad \end{flalign} on all the fields of the theory, the covariant derivative transforms very nicely as

\begin{flalign} T_{\alpha }\,:\, D_\mu \Psi (x) ~\longmapsto ~ e^{-\ii \,q\,\alpha (x)}\,D_{\mu }\Psi (x)\quad . \end{flalign} As a side-remark, note that the gauge covariant derivative (7.6) has a similar structure as the covariant derivative of tensor fields in general relativity. From a geometric perspective, both of these covariant derivatives can be understood as connections on certain types of fiber bundles.

Using the concept of gauge covariant derivatives, there is an extremely simple method, called minimal coupling, to promote the free Dirac action \(S_{\mathrm {Dirac}}^{}\) to an action that is invariant under the combined local gauge transformations (7.7): We simply replace the partial derivative \(\partial _\mu \) by the covariant derivative \(D_\mu \). Together with the free Maxwell action \(S_{\mathrm {MW}}^{}\), which we recall is already gauge invariant and hence doesn’t need to be altered, we obtain the action

that is clearly invariant under the gauge transformations (7.7). Writing \(\slashed {D} = \slashed {\partial } + \ii \,q\,\slashed {A}\) and comparing with (7.1), we observe that minimal coupling leads to the interaction term

\begin{flalign} S_{\mathrm {int}}^{}[\Psi ,\overline {\Psi },A] \,=\, - \ii \,q\,\int _{\bbR ^d} \overline {\Psi }\slashed {A}\Psi \,\dd x \,=\, - \ii \,q\,\int _{\bbR ^d} A_{\mu }\,\overline {\Psi }\gamma ^\mu \Psi \,\dd x\quad . \end{flalign} Comparing this further with (6.11), we note that \(A_\mu \) couples to the current \(j^\mu = -\ii \,q\,\overline {\Psi }\gamma ^\mu \Psi \) that is build out of the Dirac field and its adjoint.

It is somewhat surprising that this simple gauging and minimal coupling construction, leading to the action functional (7.9), describes the physics of quantum electrodynamics extremely well. What is even more surprising is that all interactions in the standard model of particle physics (electromagnetic, weak and strong) are modeled by gauging and minimal coupling a certain global symmetry group, namely the \(\mathsf {U}(1)\times \mathsf {SU}(2) \times \mathsf {SU}(3)\) group. We will discuss this in more detail in Chapter 9.