From Global to Local: Deriving the QED Interaction from Dirac Theory

Welcome back. In our previous lesson, we constructed the essential tools of gauge theory: the covariant derivative $D_{μ}$ and the gauge field $A_{μ}$ . We established that by replacing the ordinary derivative with a covariant one, we could create a derivative $D_{μ} ψ$ that transforms "nicely" under a local U(1) phase transformation, picking up the same phase as the field $ψ$ itself.

This lesson puts those tools to work. Our objective is to show that the simple, physically motivated requirement of local gauge invariance is a powerful constructive principle. By imposing this symmetry on the free Dirac Lagrangian, we will see that the theory is forced to include an interaction. This procedure will not only restore the broken symmetry but will also automatically generate the precise interaction term that describes how electrons couple to photons—the heart of Quantum Electrodynamics (QED).

The Minimal Coupling Prescription

We begin with the Lagrangian for a free Dirac field, which, as we know, is invariant under a global U(1) transformation:

L_{Dirac} = ψ ˉ (i γ^{μ} \partial_{μ} - m) ψ

(We will work in natural units where $ℏ = c = 1$ , a convention you are familiar with from your QFT course).

As we saw, this Lagrangian is not invariant if the phase rotation $ψ \to e^{i q α (x)} ψ$ is local (i.e., if $α$ depends on spacetime coordinates). The problem lies entirely in the derivative term, $ψ ˉ i γ^{μ} \partial_{μ} ψ$ .

The solution is a beautifully simple prescription known as minimal coupling: to make a Lagrangian with a global symmetry invariant under the corresponding local symmetry, replace all ordinary derivatives with covariant derivatives.

For our U(1) case, this means making the substitution:

\partial_{μ} ⟶ D_{μ} = \partial_{μ} - i q A_{μ}

Applying this to the Dirac Lagrangian gives us a new, candidate Lagrangian:

L = ψ ˉ (i γ^{μ} D_{μ} - m) ψ = ψ ˉ (i γ^{μ} (\partial_{μ} - i q A_{μ}) - m) ψ

The name "minimal coupling" reflects that this is the simplest modification we can make to restore the symmetry, introducing the new gauge field $A_{μ}$ in a way that is linear and involves no higher derivatives.

The Magic of Cancellation

Let's now explicitly verify that this new Lagrangian is indeed invariant under the local gauge transformations:

$ψ (x) \to ψ^{'} (x) = e^{i q α (x)} ψ (x)$
$A_{μ} (x) \to A_{μ}^{'} (x) = A_{μ} (x) + \partial_{μ} α (x)$

While we could use the transformation property of $D_{μ} ψ$ that we derived in the last lesson, it is instructive to see the cancellation happen term by term. The following reading provides an exceptionally clear algebraic demonstration of how invariance is achieved.

The Gauge Principle and Foundations of the Standard Model - arXiv

This section from the pedagogical review "The Gauge Principle and Foundations of the Standard Model" provides a crystal-clear, step-by-step derivation. It shows precisely how adding a new term involving a gauge field Aμ to the Dirac Lagrangian cancels the problematic term that arises from a local phase transformation. (Note: the reading uses a phase convention of e^{i\theta}, a charge q, and a transformation A_\mu' = A_\mu - \frac{\hbar c}{q}\partial_\mu \theta. The logic is identical to our convention, just with different notation).

Please read the subsection titled "How to enforce local phase invariance." Your focus should be on: The initial setup: The modified Lagrangian in equation (30) that includes the new field A_\mu. The explicit calculation showing how the unwanted term from the derivative piece, -\hbar c\,\bar{\psi}\,\gamma^{\mu}(\partial_{\mu}\theta)\,\psi, is perfectly cancelled by a new term that arises from transforming A_\mu. The subsequent argument for why a kinetic term for the A_\mu field, -\frac{1}{4\mu_0}F_{\mu\nu}F^{\mu\nu}, is necessary and why this specific form is chosen (gauge and Lorentz invariance).

As the reading demonstrates, the minimal coupling prescription works perfectly. By expanding our new Lagrangian, we can see exactly what we have created:

L = ψ ˉ (i γ^{μ} \partial_{μ} - m) ψ + q ψ ˉ γ^{μ} A_{μ} ψ

This expression is profound. It contains two distinct parts:

$L_{free} = ψ ˉ (i γ^{μ} \partial_{μ} - m) ψ$ , the original Lagrangian for a free-propagating fermion.
$L_{int} = q ψ ˉ γ^{μ} A_{μ} ψ$ , a completely new interaction term.

The gauge principle has forced an interaction into our theory! Furthermore, we recognize the term $J^{μ} = ψ ˉ γ^{μ} ψ$ as the conserved Noether current associated with the original global U(1) symmetry. The interaction term is therefore precisely of the form $q J^{μ} A_{μ}$ , which you know from classical electromagnetism describes the interaction of a four-current with the electromagnetic four-potential.

Assembling the Full QED Lagrangian

Our Lagrangian is still incomplete. It describes a fermion field $ψ$ interacting with an external, non-dynamical field $A_{μ}$ . To make $A_{μ}$ a physical, propagating field (i.e., the photon), we must add a kinetic term for it. As the reading pointed out, this kinetic term must itself be gauge-invariant and Lorentz-invariant.

The simplest object that satisfies these criteria is built from the field strength tensor:

F_{μν} = \partial_{μ} A_{ν} - \partial_{ν} A_{μ}

This object is gauge-invariant because the extra derivative terms from the transformation of $A_{μ}$ cancel out due to the commutativity of partial derivatives: $\partial_{μ} \partial_{ν} α - \partial_{ν} \partial_{μ} α = 0$ . The only Lorentz scalar we can form is $F_{μν} F^{μν}$ . With the conventional normalization factor, the kinetic term for the gauge field is the Maxwell Lagrangian:

L_{Maxwell} = - \frac{1}{4} F_{μν} F^{μν}

Putting all the pieces together, we arrive at the full Lagrangian for Quantum Electrodynamics:

L_{QED} = ψ ˉ (i γ^{μ} \partial_{μ} - m) ψ - \frac{1}{4} F_{μν} F^{μν} + q ψ ˉ γ^{μ} A_{μ} ψ

This can be written more elegantly using the covariant derivative we started with:

L_{QED} = ψ ˉ (i γ^{μ} D_{μ} - m) ψ - \frac{1}{4} F_{μν} F^{μν}

The following video provides an excellent summary of this entire chain of logic, from identifying the problem to assembling the final Lagrangian.

Deriving The QED Lagrangian | Quantum Electrodynamics

This video, "Deriving The QED Lagrangian", provides a concise walkthrough of the entire procedure we've just discussed, from the local transformation of the Dirac Lagrangian to the final QED Lagrangian.

Please watch the core mathematical derivation from 4:29 to 9:05. As you watch, map the steps in the video to the steps we've just read about and discussed: Identifying the non-invariant term when the phase parameter depends on spacetime. Introducing the gauge field and covariant derivative to restore invariance. Identifying the gauge field with the electromagnetic potential and adding the Maxwell kinetic term to build the full Lagrangian. The presenter uses slightly different notation (e.g., λ for the phase parameter), but the logic is identical.

The general procedure of promoting derivatives to covariant derivatives to enforce local symmetry is a cornerstone of modern physics, referred to as the principle of minimal coupling. As David Tong notes in his QFT lecture notes (a valuable resource for your continued study), this trick works for any theory. For instance, applying it to the Lagrangian for a complex scalar field yields the theory of "scalar QED," which is relevant to the Higgs sector of the Standard Model. Our focus on the Dirac field has led us directly to the foundational theory of electrons and light.

Conclusion

In this lesson, we have seen the gauge principle in action as a powerful engine for discovery. What began as a seemingly aesthetic demand—that the physics should not depend on a local choice of phase—has led to a concrete, predictive physical theory.

Key Takeaways:

Imposing local U(1) gauge invariance on the free Dirac Lagrangian is achieved via the minimal coupling prescription: replace the ordinary derivative $\partial_{μ}$ with the covariant derivative $D_{μ} = \partial_{μ} - i q A_{μ}$ .
This procedure automatically generates an interaction term, $L_{int} = q ψ ˉ γ^{μ} ψ A_{μ} = q J^{μ} A_{μ}$ , which describes the coupling of the conserved fermion current to the gauge field.
To describe the propagation of the gauge field itself, we must add a gauge-invariant kinetic term, $L_{Maxwell} = - \frac{1}{4} F_{μν} F^{μν}$ , where $F_{μν}$ is the gauge-invariant field strength tensor.
The complete Lagrangian for Quantum Electrodynamics (QED) is the sum of these pieces: $L_{QED} = ψ ˉ (i γ^{μ} D_{μ} - m) ψ - \frac{1}{4} F_{μν} F^{μν}$ . The interaction is not added by hand; it is an unavoidable consequence of the symmetry principle.

We have successfully derived the interaction from the symmetry. In our next and final lesson for this module, we will explore one of the most important consequences of this symmetry in the quantum theory: the Ward-Takahashi identity. This identity, a direct result of the conservation of the current $J^{μ}$ , places powerful constraints on the structure of Feynman diagrams in QED and provides a quantum statement of charge conservation.

Recommended resources

The Minimal Coupling Prescription

The Magic of Cancellation

Assembling the Full QED Lagrangian

Conclusion