Gauge Theory I

Introduction

This lecture is where the course starts to become recognizably electrodynamics in the modern field-theory sense.

Up to now, the Dirac field has been built as a relativistic spin-$\tfrac12$ field with a global phase symmetry, a conserved charge, and a Hamiltonian that only becomes physically sensible after fermionic quantization. But none of that yet explains electromagnetic interaction. The next question is obvious:

How do we couple the charged Dirac field to electromagnetism?

The old-fashioned answer is “replace derivatives by covariant derivatives and add the Maxwell field.” But this lecture wants the more important answer:

local $U(1)$ gauge symmetry determines the structure of the theory.

That is the conceptual core.

The lecture begins by reviewing the Dirac Lagrangian density, the dimensions of the field, and the previously derived momentum and charge operators. It then recalls the familiar gauge invariance of the electromagnetic potentials

$$\mathbf A \to \mathbf A + \nabla f, \qquad \Phi \to \Phi - \partial_t f,$$

or in 4-vector notation

$$A_\mu \to A_\mu + \partial_\mu f.$$

From there it examines the Maxwell Lagrangian and notes an old classical subtlety: gauge invariance of the source coupling requires charge continuity,

$$\partial_\mu j^\mu = 0.$$

This becomes the key bridge into Maxwell–Dirac theory. For the Dirac field, the current

$$j^\mu = qc\,\bar\psi\gamma^\mu\psi$$

is automatically conserved when the Dirac equation holds. So the lecture introduces the coupled Dirac Lagrangian

$$\mathcal L_{D+A} = i\hbar c\,\bar\psi\gamma^\mu \left(\partial_\mu - i\frac{q}{\hbar}A_\mu\right)\psi - Mc^2 \bar\psi\psi,$$

showing that it preserves continuity, preserves global phase symmetry, and in fact has a stronger local gauge symmetry:

$$\psi \to e^{iqf/\hbar}\psi, \qquad A_\mu \to A_\mu + \partial_\mu f.$$

The lecture then steps back and says the modern point of view is the reverse of the historical one. We do not start with Maxwell–Dirac theory and discover local symmetry as an accident. We start from local $U(1)$ symmetry, and that symmetry fixes the form of the interaction.

Then the lecture changes direction slightly. It says that for the rest of the first half of the course, $A_\mu$ will be treated as classical, while the Dirac field remains quantum. That is, one studies the quantum Dirac field in a classical electromagnetic background.

Finally, the last pages foreshadow a deeper quantum issue: how should one understand the charge density operator of the Dirac field, its eigenstates, and the ground state? The lecture introduces point-charge creation operators, relates them to the momentum-space $a$- and $c$-operators, and states a striking result: the Dirac kinetic-energy ground state is full of correlated pairs of positive and negative point charges, with structure set by the Compton wavelength. This is where normal ordering and vacuum structure start to matter. :contentReference[oaicite:1]{index=1}

So this lecture is really two beginnings:

1. the beginning of gauge theory,
2. and the beginning of the nontrivial structure of the Dirac vacuum.

Learning Objectives

• Review the Dirac Lagrangian density and the basic dimensions and operator structure of the Dirac field.
• Recall how momentum and charge arise from translation and global phase symmetry.
• State electromagnetic gauge transformations in 3-vector and 4-vector language.
• Understand why gauge invariance of the classical Maxwell source coupling requires continuity of the current.
• Define the Dirac current $j^\mu = qc\,\bar\psi\gamma^\mu\psi$ and explain why it obeys continuity when the Dirac equation holds.
• Construct the Maxwell–Dirac Lagrangian using minimal coupling.
• Derive the gauge-covariant Dirac equation in an electromagnetic background.
• Explain why global phase symmetry is preserved and why local $U(1)$ symmetry is stronger.
• Understand the modern viewpoint that local gauge symmetry determines the form of electrodynamics.
• Explain why the course now treats $A_\mu$ as classical while keeping the Dirac field quantum.
• Understand the role of normal ordering and why the charge density operator of the Dirac field has a nontrivial vacuum structure.
• Interpret the lecture’s claim that the kinetic-energy ground state of the Dirac field contains correlated positive and negative point-charge pairs.

Prerequisite Knowledge

• Dirac Lagrangian and Dirac Hamiltonian
• 4-spinor notation and gamma matrices
• Noether charges
• Fermionic quantization of the Dirac field
• Basic Maxwell theory
• Electromagnetic gauge transformations
• Continuity equation and charge conservation
• Normal ordering at a conceptual level

1. Review: the free Dirac field

The lecture begins by recalling the free Dirac Lagrangian density

$$\mathcal L_D = i\hbar c\, \bar\psi \gamma^\mu \partial_\mu \psi - Mc^2 \bar\psi \psi.$$

It also reminds the reader of the conventions

$$\partial_0 = \frac{1}{c}\frac{\partial}{\partial t}, \qquad \bar\psi = \psi^\dagger \gamma^0,$$

and the explicit $4$-component form of the spinor field $\psi$ together with the chosen gamma matrices. The slide asks a useful dimensional question: what are the units of $\psi$? It also recalls the equal-time canonical anti-commutation relation

$$\{\hat\psi_m(r),\hat\psi_n^\dagger(r')\} = \delta_{mn}\delta(r-r').$$

So the lecture starts by grounding the gauge-theory discussion firmly in the free Dirac field already developed. :contentReference[oaicite:2]{index=2}

This matters because the interaction theory is not being built from scratch. It is a deformation of an already established relativistic spinor field theory.

2. Review: momentum and charge of the Dirac field

The next slide reviews two continuous symmetries of the free Dirac field:

• translation,
• global phase rotation.

Under translation,

$$\psi(r,t)\to \psi(r-a,t),$$

and under a global phase rotation,

$$\psi(r,t)\to e^{-i\theta}\psi(r,t).$$

The corresponding generators are

$$G = -\nabla\psi, \qquad Q = -i\psi.$$

Using the canonical momentum field, the lecture recalls the Noether charges:

$$P = \int d^3r\, \Pi G, \qquad Q = \int d^3r\, \Pi(-i\psi).$$

In momentum space, after quantization, these become

$$\hat P = \sum_\pm \int d^3k\, \hbar k \left( \hat a_\pm^\dagger(k)\hat a_\pm(k) + \hat c_\pm^\dagger(k)\hat c_\pm(k) \right),$$

and

$$\hat Q = \hbar\int d^3k \left( \hat a_\pm^\dagger(k)\hat a_\pm(k) - \hat c_\pm^\dagger(k)\hat c_\pm(k) \right).$$

So the $a$-quanta and $c$-quanta have the same momentum structure but opposite charge. :contentReference[oaicite:3]{index=3}

This review is exactly what gauge theory needs: a charged field with a global $U(1)$ symmetry.

3. Electromagnetic gauge invariance in classical form

The lecture then switches to electromagnetism. It recalls the usual definitions

$$\mathbf B = \nabla \times \mathbf A, \qquad \mathbf E = -\dot{\mathbf A} - \nabla \Phi.$$

These remain invariant if

$$\mathbf A \to \mathbf A + \nabla f, \qquad \Phi \to \Phi - \dot f.$$

In 4-vector language,

$$A_\mu = (A_0,\mathbf A) = (-\Phi/c,\mathbf A),$$

and the gauge transformation is

$$A_\mu \to A_\mu + \partial_\mu f.$$

The lecture also writes the Maxwell Lagrangian density with source coupling:

$$\mathcal L_M = -\frac{1}{4\mu_0} g^{\lambda\nu}g^{\mu\rho} (\partial_\mu A_\nu - \partial_\nu A_\mu) (\partial_\lambda A_\rho - \partial_\rho A_\lambda) + A_\mu j^\mu.$$

It asks about the units of $A_\mu$ and $f$, and notes in particular that

$$[f] = \frac{\hbar}{C},$$

which is exactly the right dimension for appearing in a phase factor once multiplied by charge. :contentReference[oaicite:4]{index=4}

That dimensional observation is not incidental. It foreshadows the local phase transformation of the matter field.

4. Why classical gauge invariance requires continuity

The lecture then points out a classical subtlety that is often hidden in quick derivations.

For the source-coupled Maxwell theory to be gauge invariant, the current must obey continuity:

$$\partial_\mu j^\mu = 0.$$

The lecture emphasizes that the Maxwell equations also require this. In classical point-charge electrodynamics, this is enforced by charge conservation and by defining the charge and current densities from moving point particles:

$$\rho(r,t) = \sum_n Q_n \delta(r-r_n(t)),$$ $$j(r,t) = \sum_n Q_n \dot r_n(t)\delta(r-r_n(t)),$$

with $\dot Q_n=0$, which implies

$$\dot \rho = -\nabla\cdot j.$$

The lecture then makes an important conceptual statement: in this classical point-charge theory, charge conservation does not follow from gauge invariance. Rather, charge conservation is needed to ensure gauge invariance of the source coupling. :contentReference[oaicite:5]{index=5}

This sets up the key question: how does this work in quantum electrodynamics?

5. The Dirac current and continuity

The lecture answers that question by recalling the classical Dirac current:

$$\rho = q\psi^\dagger \psi, \qquad j^\mu = qc\,\bar\psi \gamma^\mu \psi.$$

It then asks whether this current obeys continuity. The answer is yes, though not identically: it does so if the Dirac equation holds. The lecture sketches the calculation:

$$\partial_\mu(\psi^\dagger \gamma^0 \gamma^\mu \psi) = \bar\psi \gamma^\mu \partial_\mu \psi + (\gamma^0\gamma^\mu \partial_\mu \psi)^\dagger \psi,$$

using the property

$$(\gamma^0 \gamma^\mu)^\dagger = \gamma^0 \gamma^\mu.$$

So the conserved Dirac current arises dynamically from the Dirac equation. :contentReference[oaicite:6]{index=6}

This is the field-theoretic replacement for classical point-particle continuity.

6. Minimal coupling: Maxwell–Dirac theory

Now the lecture proposes the coupled theory. Replace the free Dirac Lagrangian by

$$\mathcal L_D \to \mathcal L_{D+A} = i\hbar c\, \bar\psi \gamma^\mu \left( \partial_\mu - i\frac{q}{\hbar}A_\mu \right)\psi - Mc^2 \bar\psi \psi.$$

This is the standard minimal-coupling substitution

$$\partial_\mu \to D_\mu := \partial_\mu - i\frac{q}{\hbar}A_\mu.$$

Then the full Lagrangian is

$$\mathcal L = \mathcal L_M + \mathcal L_{D+A},$$

with the source current identified as

$$j^\mu \to q\,\bar\psi \gamma^\mu \psi.$$

The lecture then writes the resulting Dirac equation in the electromagnetic background:

$$\gamma^\mu(i\hbar \partial_\mu + qA_\mu)\psi = Mc\,\psi.$$

This is the gauge-covariant Dirac equation. :contentReference[oaicite:7]{index=7}

So the electromagnetic interaction appears simply by replacing the ordinary derivative with a gauge-covariant derivative.

7. Continuity, global phase symmetry, and gauge invariance

The lecture then asks three checks in sequence:

1. Do we still have continuity?
   Yes.

2. Do we still have the global phase symmetry

$$\psi(r,t)\to e^{-i\theta}\psi(r,t)?$$

Yes.

3. Do we still have gauge invariance, up to a total 4-divergence?
   Yes.

But then the lecture says something more important: actually now we have a stronger symmetry. :contentReference[oaicite:8]{index=8}

This is where the modern gauge-theory viewpoint enters.

8. Local $U(1)$ symmetry as the first principle

The stronger symmetry is the local transformation

$$\psi \to e^{iq f(r,t)/\hbar}\psi, \qquad A_\mu \to A_\mu + \partial_\mu f.$$

The lecture states that this is now an exact symmetry, not merely one that holds up to a total 4-divergence. :contentReference[oaicite:9]{index=9}

Then it makes the conceptual point explicit:

From the modern viewpoint, this is not a coincidence. It is not that we start with Maxwell–Dirac theory and then discover that it happens to have a local symmetry. Rather, local $U(1)$ symmetry is the first principle, and it is what determines the form of the Lagrangian and therefore the structure of electrodynamics itself. :contentReference[oaicite:10]{index=10}

That is the central conceptual lesson of the lecture.

Electromagnetism is not merely “a force field added to charged matter.” It is the gauge field required by local phase symmetry of the matter field.

9. Global symmetry remains inside local symmetry

The lecture then points out that the old global phase rotation remains as the special case in which $f$ is constant. So the global $U(1)$ charge symmetry is not lost; it is contained inside the larger local gauge symmetry. :contentReference[oaicite:11]{index=11}

That is exactly how gauge theory should work:

• global symmetry gives conserved charge,
• local symmetry demands a gauge field and fixes the coupling structure.

10. Classical electromagnetic background, quantum Dirac field

The lecture then states a practical restriction for the rest of the first half of the course: for now, $A_\mu$ will be treated as classical. The quantum field under study is the Dirac field in a classical electromagnetic background. Full quantization of the electromagnetic field comes later. :contentReference[oaicite:12]{index=12}

This is an important simplification. It allows one to study charged quantum matter interacting with external electromagnetic fields before introducing photons as dynamical quantum excitations.

11. Normal ordering and the charge density operator

The later slides shift attention from classical gauge symmetry to the quantum structure of the Dirac field itself.

The lecture revisits the Dirac spinors $u_\pm(k)$ and $v_\pm(k)$ and asks about the eigenvalues and eigenstates of the charge density operator $\hat\rho$. The exact algebra on the slides is not fully parsed in plain text, but the visual content makes the point clear: the lecture is now reorganizing the Dirac field in terms of operators that create point-charge eigenstates rather than momentum eigenstates. :contentReference[oaicite:13]{index=13}

The image on page 11 states that in the continuum limit, the charge density operator can be written in terms of creation operators for positive and negative point charges — labeled informally there as “$q$-electrons” and “$q$-positrons.” It also states that just as the Dirac field operator destroys momentum-space electrons and creates momentum-space positrons, it also destroys the point-charge eigenstates of the charge density operator. The slide then gives explicit Fourier-transform relations between the momentum-space operators and the point-charge operators. :contentReference[oaicite:14]{index=14}

This is an important conceptual warning. The momentum-energy eigenstates of the Dirac field are not the same thing as the eigenstates of local charge density. Different operator bases reveal different physical structures.

12. The Dirac kinetic-energy ground state

The last slide contains the lecture’s most striking statement:

The Dirac kinetic energy ground state is full of pairs of positive and negative point charges. :contentReference[oaicite:15]{index=15}

The slide shows that the kinetic-energy vacuum $|0\rangle_K$ is not the naive “state with no charges” $|0\rangle_Q$. Instead it is obtained by acting on the no-charge state with an exponential of pair-creation operators that create correlated positive- and negative-charge point excitations. The slide explicitly writes the vacuum as an exponential pair condensate, first in momentum space and then in position space, with kernels $P_{ss'}(k)$ and $F_{ss'}(x)$. It also highlights that the characteristic length scale controlling the spatial structure is the Compton wavelength

$$\lambda_C = \frac{\hbar}{Mc}.$$

The large-distance behavior shown on the slide decays exponentially on that scale. :contentReference[oaicite:16]{index=16}

This is a big conceptual result. The free Dirac vacuum is not empty in the naive point-charge basis. It contains correlated positive/negative charge-pair structure. This is one of the reasons normal ordering becomes important: the vacuum of the physically relevant Hamiltonian is not the trivial no-excitation state in every basis.

The lecture does not fully develop the interpretation here, but it clearly wants to prepare the reader for the idea that the fermionic vacuum is structurally nontrivial.

13. What this lecture really established

So the lecture has two layers.

First layer: gauge theory

It establishes that coupling a charged Dirac field to electromagnetism is governed by local $U(1)$ symmetry. The gauge principle determines the covariant derivative, the current coupling, and the form of the Maxwell–Dirac Lagrangian.

Second layer: quantum Dirac vacuum structure

It begins showing that once the Dirac field is quantized, local charge density and kinetic-energy eigenstates do not line up trivially. The vacuum in one basis contains structured charge-pair content in another basis, with correlations controlled by the Compton scale.

That second point is not yet the main theme of gauge theory, but it is exactly the kind of quantum-operator subtlety that will matter once interacting QED is developed.

Worked Examples

Example 1: Why local phase symmetry forces minimal coupling

Start with the free Dirac kinetic term

$$i\hbar c\,\bar\psi\gamma^\mu \partial_\mu \psi.$$

Under a local phase transformation

$$\psi \to e^{iqf/\hbar}\psi,$$

the derivative hits both $\psi$ and the position-dependent phase, producing an extra term proportional to $\partial_\mu f$. To cancel that term, one must introduce a field $A_\mu$ that transforms as

$$A_\mu \to A_\mu + \partial_\mu f,$$

and replace $\partial_\mu$ by

$$D_\mu = \partial_\mu - i\frac{q}{\hbar}A_\mu.$$

That is minimal coupling. So the gauge field is not an arbitrary addition; it is what local phase symmetry demands.

Example 2: Why the Dirac current is the source of electromagnetism

For the Dirac field, the natural conserved current is

$$j^\mu = qc\,\bar\psi\gamma^\mu\psi.$$

Using the Dirac equation and its adjoint, one finds

$$\partial_\mu j^\mu = 0.$$

Therefore this current can consistently appear as the source in the Maxwell Lagrangian. This is the quantum-field-theory replacement of the classical point-particle current.

Intuition

This lecture says something very simple and very deep:

a charged quantum field has a global phase symmetry, and once you demand that this phase symmetry be local, electromagnetism appears.

That is the gauge principle in one sentence.

The vector potential $A_\mu$ is not introduced because we already know classical electromagnetism and want to bolt it onto the Dirac field. In the modern view, it is introduced because local phase changes of the charged field must be physically allowed, and $A_\mu$ is the compensating field that makes this possible.

Then the lecture adds a second lesson: once the Dirac field is quantized, even the vacuum is more complicated than it first looks. In the local charge basis, the kinetic-energy ground state is a correlated sea of positive and negative point-charge pairs. So gauge theory is being built not on a trivial empty background, but on a vacuum with real fermionic structure.

Common Mistakes

• Thinking gauge invariance is just a trick for rewriting Maxwell’s equations instead of a symmetry principle that determines the interaction.
• Forgetting that in the classical source-coupled Maxwell theory, current continuity is needed for gauge invariance.
• Assuming the free Dirac global $U(1)$ symmetry disappears after coupling to electromagnetism; it survives as the constant-$f$ subset of the local symmetry.
• Treating minimal coupling as an arbitrary substitution rather than the result of demanding local phase invariance.
• Confusing the Dirac field operator with a one-particle wavefunction.
• Assuming that the momentum eigenstate basis is the same thing as the local charge-density eigenbasis.
• Forgetting that the lecture only treats $A_\mu$ classically at this stage.
• Thinking the Dirac vacuum is empty in every physically relevant basis.
• Missing the importance of normal ordering once vacuum charge-pair structure appears.

Short Summary

The lecture begins by reviewing the free Dirac field, its Lagrangian density

$$\mathcal L_D = i\hbar c\,\bar\psi\gamma^\mu\partial_\mu\psi - Mc^2\bar\psi\psi,$$

and the momentum and charge operators that arise from translation and global phase symmetry. It then recalls electromagnetic gauge invariance in terms of the potentials,

$$A_\mu \to A_\mu + \partial_\mu f,$$

and notes that gauge invariance of the classical Maxwell source coupling requires current continuity. For the Dirac field, the natural current

$$j^\mu = qc\,\bar\psi\gamma^\mu\psi$$

obeys continuity when the Dirac equation holds. This makes it possible to couple the Dirac field consistently to electromagnetism by replacing

$$\partial_\mu \to \partial_\mu - i\frac{q}{\hbar}A_\mu$$

in the Dirac Lagrangian, giving the Maxwell–Dirac theory

$$\mathcal L_{D+A} = i\hbar c\,\bar\psi\gamma^\mu \left(\partial_\mu - i\frac{q}{\hbar}A_\mu\right)\psi - Mc^2\bar\psi\psi.$$

The resulting Dirac equation in the background field is

$$\gamma^\mu(i\hbar\partial_\mu + qA_\mu)\psi = Mc\,\psi.$$

The lecture then emphasizes the modern viewpoint: local $U(1)$ symmetry is the first principle, and it determines the structure of electrodynamics. For the remainder of the first half of the course, $A_\mu$ is treated as classical while the Dirac field is quantum. Finally, the lecture turns to normal ordering and the charge density operator, showing that the kinetic-energy ground state of the Dirac field is not the naive no-charge state but contains correlated pairs of positive and negative point charges, with spatial structure controlled by the Compton wavelength. :contentReference[oaicite:17]{index=17}

Practice Problems

1. Why does the free Dirac field have a global $U(1)$ phase symmetry?

2. Why does gauge invariance of the classical Maxwell source coupling require the continuity equation?

3. Show conceptually why the Dirac current

$$j^\mu = qc\,\bar\psi\gamma^\mu\psi$$

is the natural source current for Maxwell–Dirac theory.

4. Why does replacing

$$\partial_\mu \to \partial_\mu - i\frac{q}{\hbar}A_\mu$$

restore local phase invariance?

5. Explain why the global phase symmetry remains as a special case inside the local gauge symmetry.

6. Why does the lecture treat $A_\mu$ as classical at this stage instead of quantizing it immediately?

7. What is the conceptual difference between momentum eigenstates of the Dirac field and eigenstates of the local charge density operator?

8. Why is normal ordering relevant once one starts discussing the charge density operator and the Dirac vacuum?

9. What does the lecture mean by saying the kinetic-energy ground state is full of pairs of positive and negative point charges?

10. Why does the Compton wavelength appear as the natural length scale in the vacuum pair structure?