Building the QED Lagrangian

1. From Global to Local Symmetry

Let's start with the free Dirac Lagrangian, which describes a non-interacting electron:
${\mathcal{L}}_{\text{Dirac}}=\bar{\psi }(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi$
As we've seen, this Lagrangian is invariant under a global phase transformation:
$\psi (x)\to {\psi }^{\prime }(x)=e^{-i\alpha }\psi (x)$
where $\alpha$ is a constant. This is a U(1) symmetry. The phase change is the same at every point in spacetime. By Noether's theorem, this global symmetry leads to a conserved quantity: electric charge, carried by the current $j^{\mu }=\bar{\psi }{\gamma }^{\mu }\psi$.

But this seems oddly restrictive. The choice of phase is a matter of convention; why should we be forced to choose the same convention everywhere in the universe at all times? What if we demand a more powerful, local symmetry, where we can choose the phase independently at each spacetime point?
$\psi (x)\to {\psi }^{\prime }(x)=e^{-ie\lambda (x)}\psi (x)$
Here, $\lambda (x)$ is now an arbitrary function of $x$, and $e$ is a constant that will eventually be identified as the coupling strength (the electric charge).

Does the Dirac Lagrangian respect this local symmetry? As you will see, it does not. The derivative term ${\partial }_{\mu }$ spoils the invariance because it acts on the now-local phase factor $\lambda (x)$.

The following video provides an excellent walkthrough of this entire procedure, which is the foundation of all modern gauge theories.

2. Restoring Invariance: The Covariant Derivative

The failure of the ordinary derivative ${\partial }_{\mu }$ to respect local gauge invariance signals that it is not the correct way to compare the field $\psi$ at infinitesimally separated points. The problem is that a local phase transformation changes the "internal direction" of the field differently at each point.

To fix this, we introduce a new type of derivative, the covariant derivative $D_{\mu }$, which is designed to transform "covariantly" with the field. Its job is to compensate for the change in phase convention from point to point. We define it as:
$D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$
Here, $A_{\mu }$ is a new vector field, called the gauge field (or connection field), which must transform in a very specific way to cancel the unwanted term generated by ${\partial }_{\mu }\lambda (x)$. The required transformation is:
$A_{\mu }(x)\to A_{\mu }^{\prime }(x)=A_{\mu }(x)-{\partial }_{\mu }\lambda (x)$
(Note: The sign convention for $e$ and $A_{\mu }$ can vary. We use the convention common in particle physics textbooks, which might differ slightly from the video, but the physics is identical).

With this machinery, the combination $\bar{\psi }i{\gamma }^{\mu }{D}_{\mu }\psi$ becomes gauge invariant. This entire procedure is known as minimal coupling.

A Deeper Look: Analogy with General Relativity

Given your background in General Relativity, this concept of a covariant derivative should be very familiar. In GR, you cannot simply take the partial derivative of a vector field because the basis vectors themselves change from point to point in curved spacetime.

• In GR, you introduce the Christoffel symbols ${\Gamma }_{\mu \nu }^{\lambda }$ to define a covariant derivative ${\nabla }_{\mu }$ that accounts for the changing basis vectors.
• In QED, you introduce the gauge field $A_{\mu }$ to define a covariant derivative $D_{\mu }$ that accounts for the changing internal phase convention.

The gauge field $A_{\mu }$ acts as a connection on the U(1) internal space, much like the Christoffel symbols act as a connection on the spacetime manifold. This is a deep and beautiful geometric parallel. The following resource explores this geometric interpretation of gauge invariance as a form of parallel transport in an internal "charge space".

Now, let's see this in action. The next segment of the video shows precisely how the transformation of $A_{\mu }$ is chosen to make the new Lagrangian invariant.

3. The Complete QED Lagrangian

By demanding local gauge invariance, we have been forced to do two things:

1. Replace ${\partial }_{\mu }$ with $D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$ in the Dirac Lagrangian.
2. Introduce a new vector field, $A_{\mu }$, which we will identify with the photon field.

Let's expand the gauge-invariant Dirac Lagrangian:
${\mathcal{L}}_{\text{matter}}=\bar{\psi }(i{\gamma }^{\mu }{D}_{\mu }-m)\psi =\bar{\psi }(i{\gamma }^{\mu }({\partial }_{\mu }+ie{A}_{\mu })-m)\psi$
${\mathcal{L}}_{\text{matter}}=\bar{\psi }(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi +ie\bar{\psi }{\gamma }^{\mu }{A}_{\mu }\psi =\underset{{\mathcal{L}}_{\text{Dirac}}}{\underbrace{\bar{\psi }(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi }}\underset{{\mathcal{L}}_{\text{Interaction}}}{\underbrace{-e\bar{\psi }{\gamma }^{\mu }{A}_{\mu }\psi }}$
The requirement of local symmetry has automatically generated an interaction term, ${\mathcal{L}}_{\text{int}}=-e(\bar{\psi }{\gamma }^{\mu }\psi )A_{\mu }=-e{j}^{\mu }{A}_{\mu }$. This term couples the conserved current of the Dirac field to the newly introduced gauge field. This is the fundamental interaction vertex of QED.

To complete the theory, the gauge field $A_{\mu }$ must be allowed to propagate through spacetime—it needs its own kinetic term. This kinetic term must itself be gauge invariant. As you know from classical E&M, the appropriate term is built from the field strength tensor, $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$.

The following resource summarizes this final step.

Putting it all together, the full, classical, gauge-invariant Lagrangian for Quantum Electrodynamics is:
${\mathcal{L}}_{\text{QED}}=\bar{\psi }(i{\gamma }^{\mu }{D}_{\mu }-m)\psi -\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }$
When expanded, this reveals its three essential parts:
${\mathcal{L}}_{\text{QED}}=\underset{\text{Free Electron/Positron}}{\underbrace{\bar{\psi }(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi }}-\underset{\text{Free Photon}}{\underbrace{\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }}}\underset{\text{Interaction}}{\underbrace{-e\bar{\psi }{\gamma }^{\mu }\psi A_{\mu }}}$

For quantization, we would also add the gauge-fixing term $-\frac{1}{2\xi }({\partial }_{\mu }{A}^{\mu }{)}^2$ that we studied in the last lesson. However, this term only affects the free photon part of the theory; the crucial interaction term remains unchanged.

4. The Interaction Vertex

The interaction Lagrangian density, ${\mathcal{L}}_{\text{int}}=-e\bar{\psi }{\gamma }^{\mu }\psi A_{\mu }$, is the heart of QED. It tells us how the fundamental particles interact. It is a three-field interaction involving one fermion field ($\psi$), one anti-fermion field ($\bar{\psi }$), and one photon field ($A_{\mu }$).

In the context of perturbation theory and Feynman diagrams, this term corresponds to the QED vertex.

The interaction term contains a rich physical picture. Each field operator ($\bar{\psi }$, $\psi$, $A_{\mu }$) contains both creation and annihilation operators. Their product describes all possible fundamental processes, such as:

• An electron emitting a photon ($e^{-}\to e^{-}+\gamma$).
• An electron absorbing a photon ($e^{-}+\gamma \to e^{-}$).
• An electron and positron annihilating into a photon ($e^{-}+e^{+}\to \gamma$).
• A photon creating an electron-positron pair ($\gamma \to e^{-}+e^{+}$).

The following video decomposes the interaction term to illustrate these fundamental processes.

Conclusion

In this lesson, we have derived the complete Lagrangian for Quantum Electrodynamics from the fundamental principle of local U(1) gauge invariance. This powerful idea not only forced the existence of the electromagnetic field but also dictated the exact form of its interaction with charged matter.

Key Takeaways:

• Demanding that the Dirac Lagrangian be invariant under local phase transformations $\psi \to e^{-ie\lambda (x)}\psi$ is the guiding principle.
• This requires replacing the ordinary derivative ${\partial }_{\mu }$ with the covariant derivative $D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$.
• The new field $A_{\mu }$ is the gauge field (photon), which must transform as $A_{\mu }\to A_{\mu }-{\partial }_{\mu }\lambda (x)$ to maintain invariance. This procedure is called minimal coupling.
• The complete QED Lagrangian is ${\mathcal{L}}_{\text{QED}}=\bar{\psi }(i{\gamma }^{\mu }{D}_{\mu }-m)\psi -\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }$.
• Expanding the covariant derivative reveals the interaction term: ${\mathcal{L}}_{\text{int}}=-e\bar{\psi }{\gamma }^{\mu }\psi A_{\mu }$, which corresponds to the fundamental QED vertex. This vertex couples the fermion current $j^{\mu }=\bar{\psi }{\gamma }^{\mu }\psi$ to the photon field $A_{\mu }$.

Preview of the Next Lesson:
We have now derived all the pieces of our theory: the free Lagrangians, the interaction term, and the propagators for both the fermion and the photon. In the next lesson, we will formalize this knowledge by stating the complete set of Feynman rules for QED in momentum space. These rules are the dictionary that translates our Lagrangian into a practical toolkit for calculating physical quantities, like scattering cross-sections.