Particle Ptuesdays (7/16): QED as the Simplest Gauge Theory

In the previous essay, we made a move that at first looked almost too innocent to matter. We took a field with a phase, and instead of allowing that phase to rotate by the same amount everywhere, we allowed it to rotate differently from point to point.

For a complex field, a global phase rotation looks like

where α is constant. But a local phase rotation looks like

where α may depend on spacetime position.

That small change breaks the ordinary derivative. The derivative of the transformed field does not transform in the same clean way as the field itself, because the derivative also acts on α(x). The cure was to replace the ordinary derivative ∂_μ with a covariant derivative D_μ, built so that it transforms properly under local changes of phase.

This essay turns that abstract mechanism into the first complete gauge theory in the series: quantum electrodynamics, or QED.

QED is the quantum field theory of electrons, positrons, photons, and their electromagnetic interactions. It is also the simplest serious example of the principle that will later generate the full Standard Model. The central revelation is this:

Electromagnetism is what local phase symmetry looks like when written as a field theory.

This sentence is worth slowing down for. It does not say that electromagnetism happens to have a symmetry. It says that once we insist that the phase convention of a charged quantum field may be chosen independently at every spacetime point, the mathematical machinery required to compare those phases from point to point has exactly the form of the electromagnetic field.

The photon appears as the gauge field for local U(1) phase symmetry.

The Surface View and the Structural View

At the surface level, QED says that electrons carry electric charge and interact by exchanging photons. This is the familiar particle picture. It is not wrong. It is the view one often meets first, because it gives a useful diagrammatic language: electron lines, photon lines, vertices where an electron emits or absorbs a photon.

But the structural view asks a different question: why does the theory have that form at all?

Why is there a massless spin-1 field A_μ? Why does it couple to a conserved current? Why does the interaction enter by replacing ∂_μ with D_μ? Why does electric charge appear as a coupling constant? Why does the photon not itself carry electric charge?

QED answers these questions in a unified way. It is not a list of separate facts. It is a field theory organized by one compact demand:

The group U(1) is the group of complex phase rotations. Its elements are numbers of the form eᴵᵅ, with α an angle. As a global symmetry, U(1) says that physics does not change if we rotate the phase of a charged field everywhere by the same amount. As a local gauge symmetry, it says that physics does not change if we choose the phase convention independently at every point in spacetime.

The word “gauge” is doing important work here. A local gauge transformation is not an ordinary physical transformation that takes one physical situation into a different physical situation. It is a redundancy in how we describe the same physical situation. The phase of the charged field is partly a convention. The gauge field tells us how to compare that convention from one point to another.

That is the conceptual seed from which QED grows.

The Matter Field: A Relativistic Electron

The matter field in QED is usually written as ψ(x). It represents the electron field, although in relativistic quantum field theory the same field also contains the positron degrees of freedom.

For the purposes of this essay, the most important fact about ψ is that it is charged under U(1). Under a local phase transformation, it changes as

Here q is the charge weight of the field under the U(1) symmetry. Different conventions distribute factors between q, the coupling constant e, and the gauge parameter α(x). The physical idea is the same: a charged field is one whose phase is affected by the U(1) transformation.

Mathematically, this says that ψ transforms in a representation of U(1). For U(1), representations are especially simple. The group acts by multiplication by a phase. The number multiplying α tells us the field’s charge assignment.

Physically, that charge assignment tells us how the field responds to the electromagnetic gauge field. Electric charge is not merely a label pasted onto the electron after the fact. It is representation data: it tells us how the electron field transforms under the gauge symmetry.

Because the electron is a relativistic spin-1/2 particle, ψ is not a single complex number at each spacetime point. It is a spinor field. The fully relativistic kinetic term uses gamma matrices γ^μ, which are the algebraic machinery needed to write a Lorentz-invariant equation for spin-1/2 particles.

We will not make gamma matrices the focus here. For now, it is enough to know that they allow the combination

to serve as the relativistic kinetic term for a free spin-1/2 field. The bar denotes the Dirac adjoint,

chosen so that the expression transforms properly under Lorentz transformations.

Without electromagnetism, the schematic Dirac Lagrangian is

The first term describes propagation. The second term is the mass term. It tells us that the electron field has rest mass m.

But this free theory only respects global phase rotations. If we demand local U(1) invariance, the ordinary derivative must be upgraded.

Why the Ordinary Derivative Fails

The problem with ∂_μ is direct. Start with a local phase transformation,

Now differentiate the transformed field:

The first term is what we would like. It transforms the derivative in the same way the field transforms. The second term is the obstruction. It appears only because α depends on spacetime.

This is the mathematical reason local symmetry is not free. Once the symmetry parameter varies from point to point, the derivative notices. It sees not only how the field changes, but also how our local phase convention changes.

So we introduce a new object: a gauge field A_μ(x). It is arranged to transform in such a way that it cancels the unwanted derivative of α(x).

The covariant derivative is written, in one common convention, as

Here e is the electromagnetic coupling constant, and q is the charge of the field in units of e. For the electron, one often takes q=-1, so its electric charge is -e.

The purpose of D_μ is not mysterious. It is built to satisfy

That is the defining property. The covariant derivative of the field transforms like the field itself.

To make this happen, the gauge field must also transform. Depending on sign conventions, one may write a transformation of the form

Do not get attached to the exact sign in isolation; it depends on how one defines D_μ and the phase transformation. The invariant content is that A_μ shifts by a derivative of the gauge parameter. That shift is what cancels the extra term produced when ∂_μ acts on the local phase.

The electromagnetic potential has appeared as the compensating field that makes local phase symmetry possible.

The Gauge Field Is Not Optional Decoration

There is a common pedagogical trap here. One can make it sound as if we already had electromagnetism, noticed it had a vector potential A_μ, and then found a pleasing symmetry interpretation afterward.

The structural logic is stronger. If we start with a charged matter field and demand local U(1) phase invariance, something with the transformation behavior of A_μ must be introduced. The gauge field is not an optional force field glued onto the theory. It is the connection that allows us to compare phases at neighboring points.

The word “connection” is useful. Imagine trying to compare arrows drawn on different tangent planes of a curved surface. To say whether the arrow has “changed direction,” one needs a rule for transporting it from one place to another. In gauge theory, the issue is not direction in physical space, but phase in an internal space. The gauge field supplies the rule for comparing phase choices at neighboring spacetime points.

Mathematically, A_μ is the U(1) gauge connection.

Physically, it is the electromagnetic potential. Its quantum excitation is the photon.

This is one of the most important identifications in the entire series: the photon is not introduced as a little billiard ball of light. It is the quantum of the gauge field required by local U(1) invariance.

The Field Strength: What Survives the Gauge Redundancy

If A_μ can change under a gauge transformation without changing the physical situation, then A_μ itself cannot be directly identified with a gauge-invariant observable. Different potentials can describe the same electromagnetic field.

The gauge-invariant field strength is

This object packages the electric and magnetic fields into one relativistic tensor. In ordinary three-dimensional language, the electric and magnetic fields can be recovered from components of F_μν. The separation into E and B depends on the observer’s frame, but F_μν is the spacetime object behind both.

Why this particular antisymmetric combination?

Because the derivative shift in A_μ cancels out:

The mixed partial derivatives commute. So F_μν is unchanged by the gauge transformation of A_μ.

This is the first glimpse of a pattern that will become more intricate in Yang–Mills theory. The gauge potential A_μ is not itself gauge invariant, but it builds a field strength that describes physical curvature in the gauge connection. In QED, that curvature is the electromagnetic field.

The field strength also gives the gauge field its own dynamics. A theory with A_μ but no kinetic term for it would not yet describe propagating photons. The simplest Lorentz-invariant, gauge-invariant kinetic term is

This is the field-theoretic form of the electromagnetic field energy and dynamics. In classical language, it contains the familiar energy stored in electric and magnetic fields. In quantum language, it gives the photon field its propagation.

The factor -1/4 is a convention chosen for normalization. The important structure is the contraction of the field strength with itself.

The QED Lagrangian

Putting the pieces together, the QED Lagrangian can be written schematically as

This compact expression is our first small version of the “compressed blueprint” idea. It is not the full Standard Model Lagrangian, but it has the same style. A great deal of physics is compressed into a few terms because the notation is doing structural work.

The first part,

is the kinetic term for the electron field, upgraded from an ordinary derivative to a covariant derivative. It describes how the electron field propagates while respecting local U(1) gauge invariance.

The mass term,

gives the electron its mass. In QED by itself, this mass term is allowed: the left- and right-handed components of the Dirac electron have the same electric charge, so the mass term does not violate the electromagnetic gauge symmetry. Later, in the electroweak theory, this will become much more subtle. Before symmetry breaking, the weak interaction treats left and right differently, and ordinary fermion mass terms are not automatically allowed.

The final term,

is the kinetic term for the electromagnetic field. It gives the photon field its dynamics and encodes the source-free part of Maxwell’s equations.

The entire expression is built to respect Lorentz symmetry and local U(1) gauge invariance. These two demands sharply restrict what the theory can look like.

The Interaction Is Hidden in the Derivative

At first glance, the QED Lagrangian may look like a sum of two separate things: a matter-field term and an electromagnetic-field term. But the interaction between electrons and photons is already present inside the covariant derivative.

Using

the fermion kinetic term becomes

up to the sign convention chosen for D_μ.

The first term is the free propagation of the electron field. The second term is the interaction between the electron field and the electromagnetic potential.

The expression

is the Dirac current. Multiplying it by the charge gives the electromagnetic current. So the interaction has the schematic form

This is a remarkable result. The demand for local phase invariance has not merely told us that some interaction is possible. It has told us the form of the interaction: the electromagnetic gauge field couples to the conserved current associated with the charged matter field.

In the surface view, one says: electrons emit and absorb photons.

In the structural view, one says: the U(1) gauge connection enters the matter kinetic term through the covariant derivative, and expanding that derivative produces the current-gauge-field coupling.

These are not competing descriptions. The first is the particle-language consequence of the second.

Electric Charge as Coupling to the Gauge Field

In elementary physics, electric charge often appears as a property an object carries, like mass or size. In QED, charge becomes more precise.

Electric charge tells us how a field transforms under U(1), and therefore how it couples to the U(1) gauge field.

A neutral field does not pick up a phase under electromagnetic U(1). For such a field, the covariant derivative reduces to the ordinary derivative with respect to electromagnetism. It does not couple directly to the photon.

A charged field transforms nontrivially:

That same q appears in the covariant derivative. The transformation law and the interaction strength are tied together. This is one of the great conceptual economies of gauge theory.

For U(1), the charge assignments are especially transparent because the group is abelian and its irreducible complex representations are one-dimensional. Each charged field carries a phase weight. In more advanced language, electric charge is the representation label for the electromagnetic U(1).

This prepares us for the Standard Model. The familiar charges of particles are not arbitrary decorations. They tell us how the corresponding fields transform under the gauge group. Later, the electron, neutrino, quarks, Higgs field, and gauge bosons will all be understood in terms of representation content under

QED is the training ground where this idea is least cluttered.

Noether’s Theorem Returns

The reader who has followed the previous essays has already met Noether’s theorem: continuous symmetries of the action correspond to conserved currents.

For the free Dirac field, global U(1) phase symmetry gives a conserved current. Schematically,

This is charge conservation. The total electric charge does not change with time.

Gauge theory deepens this story. The global part of the U(1) symmetry is associated with a conserved charge. Making the symmetry local introduces the gauge field and forces the interaction to occur through the current.

So Noether’s theorem provides the bridge between the older conservation-law intuition and the newer gauge-theory structure.

Global U(1) symmetry says: there is a conserved electric charge.

Local U(1) gauge invariance says: the theory contains a gauge connection A_μ, and charged fields interact with it through the covariant derivative.

One should not collapse these two statements into each other. They are related, but they are not identical. A global symmetry maps physical states to physically distinct states with the same energy structure. A gauge symmetry is a redundancy of description. Its local transformations identify different mathematical descriptions of the same physical state.

This distinction matters enormously in modern field theory. Gauge symmetry is not merely a bigger version of ordinary symmetry. It is a principle about how much of our mathematical description is convention and how physical quantities must be built so they do not depend on that convention.

Why the Photon Is Massless in QED

The QED Lagrangian contains a kinetic term for A_μ, but not a mass term of the form

The reason is gauge invariance. Under a gauge transformation, A_μ shifts by a derivative of α. The expression A_μ A^μ is not invariant under that shift. A direct photon mass term would break the local U(1) gauge symmetry.

So in ordinary QED, the photon is massless because gauge invariance forbids a mass term for the gauge field.

This statement should be read carefully. It is not that a massless photon was guessed first and gauge symmetry was added later as an aesthetic preference. Rather, if we build a local U(1) gauge theory in this direct way, the allowed terms do not include a photon mass.

This will make the Higgs mechanism more meaningful when we arrive there. The Standard Model contains massive weak gauge bosons, but gauge invariance does not simply disappear. Instead, the symmetry is realized in a less obvious way after spontaneous symmetry breaking. The mass of the W and Z bosons is not put in by writing forbidden mass terms directly. It emerges through the interaction with the Higgs field.

QED is the clean case: unbroken electromagnetic U(1) gauge symmetry leaves the photon massless.

Why the Photon Does Not Carry Electric Charge

There is another important feature of QED: the photon itself is electrically neutral.

In particle language, photons do not directly emit or absorb other photons through an electric-charge vertex in pure QED. There is no basic three-photon interaction analogous to the electron-photon interaction. Photons can scatter off photons indirectly through quantum loops involving charged particles, but the photon does not carry electric charge in the way the electron does.

Structurally, this comes from the fact that U(1) is abelian. Its group elements commute:

For QED, the field strength is linear in the gauge field:

There is no extra term involving products of A_μ fields. As a result, the pure gauge-field kinetic term does not contain direct photon self-interactions.

This is one of the cleanest ways in which QED differs from the non-abelian gauge theories that come next. In Yang–Mills theory, the gauge fields themselves carry the charge of the gauge symmetry. The field strength includes terms quadratic in the gauge fields, and the gauge bosons interact with one another.

For electromagnetism, the gauge group is simple enough that the gauge boson is neutral under its own force.

For the strong interaction, this will no longer be true. Gluons carry color charge. That one difference changes the entire character of the theory.

QED as a Prototype, Not the Whole Story

QED is extraordinarily successful, but in this series it plays an additional role. It is the simplest working model of the gauge-theoretic logic behind the Standard Model.

It teaches several lessons in their least complicated form.

A charged field is not merely a field with a label. It transforms under a representation of a symmetry group.

Making the phase symmetry local forces the introduction of a gauge connection.

The covariant derivative is the device that lets derivatives respect local gauge redundancy.

The field strength is the gauge-invariant curvature built from the connection.

The interaction between matter and gauge field is encoded inside the matter kinetic term.

The global part of the symmetry connects to charge conservation through Noether’s theorem.

And the abelian nature of U(1) explains why the photon does not carry electric charge and why QED has no direct photon self-coupling at the fundamental Lagrangian level.

This is already a large conceptual shift from the surface view. Electromagnetism is no longer merely a force between charged objects. It is the local geometry of phase in a quantum field theory.

The Compressed Blueprint Begins to Speak

We can now read the QED Lagrangian as more than a formula:

It says:

There is a relativistic spin-1/2 matter field ψ.

It has a mass m.

It carries a U(1) charge.

Because the U(1) symmetry is local, ordinary derivatives are replaced by covariant derivatives.

The covariant derivative contains a gauge field A_μ.

The curvature of that gauge field is F_μν.

That curvature has its own dynamics.

The quantum of the gauge field is the photon.

The interaction between electron and photon is not an extra ornament. It is already present in the demand that the matter kinetic term respect local phase redundancy.

This is the kind of reading skill the series is trying to build. A Lagrangian is not a spell. It is compressed information about fields, symmetries, representations, allowed interactions, and forbidden terms.

QED is the first place where the compression becomes readable.

Toward Yang–Mills Theory

The simplicity of QED comes from the simplicity of U(1). The group is abelian. Its representations are easy to label by charge. Its field strength is linear in the gauge potential. The photon does not carry electric charge. The gauge field does not directly interact with itself.

The Standard Model, however, is not built only from U(1). It also contains SU(2) and SU(3), and these groups are non-abelian. Their elements do not generally commute. Their gauge fields come in multiplets. Their field strengths contain extra terms. Their gauge bosons can carry the very charges they mediate.

This is where gauge symmetry stops looking like electromagnetism with more labels. It becomes Yang–Mills theory.

QED has taught us the grammar: charged fields, covariant derivatives, gauge connections, field strengths, currents, and gauge redundancy.

The next essay changes one word in that grammar: abelian becomes non-abelian.

Almost everything else follows from that change.

Written by ChatGPT-5.5, based on conversations with Arturo Ramírez-Montesinos Krogulski (Control Equis), as part of an ongoing philosophical exploration of AI, consciousness, and the emerging practice of Software Curatorship.

You can read the preceding parts here:

“The Schoolhouse of One”

“Particle Ptuesdays (0/16): A Custom Textbook for the Standard Model”

“Saturday Seminar: The Classroom at the Back of the Book”

“Particle Ptuesdays (1/16): Why the Standard Model Feels Arbitrary Until It Suddenly Doesn’t”

“Saturday Seminar (1/16)”

“Particle Ptuesdays (2/16): What Physicists Mean by Symmetry”

“Saturday Seminar (2/16)”

“Particle Ptuesdays (3/16): Groups, Lie Groups, and Why Continuous Symmetry Needs Algebra”

“Saturday Seminar (3/16)”

“Particle Ptuesdays (4/16): Representations: The Hidden Meaning of a Particle’s Quantum Numbers”

“Saturday Seminar (4/16)”

“Particle Ptuesdays (5/16): Fields, Relativity, and Why Particles Are Not Little Balls”

“Saturday Seminar (5/16)”

“Particle Ptuesdays (6/16): From Global Symmetry to Gauge Symmetry”

“Saturday Seminar (6/16)”