Perturbative QED

Introduction

At this point in the course, the full Hamiltonian structure of quantum electrodynamics is finally on the table.

We have:

• a free photon field,
• a free Dirac field,
• and interaction terms coupling light and charged matter.

That means, in a very real sense, we have arrived at the complete theory of QED.

The lecture says this bluntly and correctly: this is the “End Boss of physics.” Not because QED is the full Standard Model, but because structurally the full Standard Model looks like this — more fields, more interaction terms, but the same kind of operator-valued Hamiltonian field theory. :contentReference[oaicite:1]{index=1}

But reaching the full Hamiltonian is not the same thing as solving it.

In fact, the central practical message of the lecture is that we cannot diagonalize QED exactly. The free Hamiltonian $\hat H_0$ is easy: it is already diagonal in terms of photon, electron, and positron creation and annihilation operators. But the interaction Hamiltonian $\hat H_I$ is far too complicated to solve exactly. It contains terms with products of many field operators, and those terms mix sectors with different particle content.

So why is QED still one of the most successful physical theories ever created?

Because it has a small dimensionless coupling:

$$\alpha = \frac{q^2}{4\pi \varepsilon_0 \hbar c} \approx \frac{1}{137}.$$

That small number lets us organize all calculations as a perturbative expansion. Higher orders contribute smaller and smaller corrections. This is the whole basis of perturbative QED. :contentReference[oaicite:2]{index=2}

But the lecture makes a deeper point too. In ordinary classical particle theory, or even in first-quantized quantum mechanics, the interaction Hamiltonian tells particles how to move. In QFT that is no longer the whole story. In QED, the interaction Hamiltonian is part of the definition of what the particles are, because particles are defined as excitations of the full interacting system. The true vacuum is not the vacuum of $\hat H_0$. The true one-electron state is not just a bare one-electron excitation of $\hat H_0$. Everything gets dressed by the interaction.

That is why perturbation theory in QFT is both ordinary and profound:

• ordinary, because the mathematics is just perturbation theory,
• profound, because what is being perturbed is the very definition of vacuum and particles.

The lecture then works this out explicitly through time-independent perturbation theory:

• first for the vacuum,
• then for the vacuum energy,
• then for the one-electron rest state.

And already at low order, the familiar QFT problems appear:

• the vacuum acquires virtual photons and electron-positron pairs,
• the vacuum energy seems nonzero and even divergent,
• the electron rest energy receives a large correction,
• and one is forced to recognize that the canonically quantized Hamiltonian has missed a $c$-number counterterm that must be included to match nature.

That is the beginning of renormalization. :contentReference[oaicite:3]{index=3}

Learning Objectives

• Recall the full QED Hamiltonian in Coulomb gauge.
• Distinguish the free Hamiltonian $\hat H_0$ from the interaction Hamiltonian $\hat H_I$.
• Identify the fine-structure constant $\alpha$ as the small dimensionless coupling controlling perturbative QED.
• Understand why exact diagonalization of the full QED Hamiltonian is hopeless.
• Explain why the interaction Hamiltonian in QFT affects the definition of particles and vacuum, not just their motion.
• Apply time-independent perturbation theory to the QED vacuum.
• Derive why the first nontrivial correction to the vacuum state comes from $\hat H_{AJ}$, not from $\hat H_{\text{Coulomb}}$.
• Understand why the first-order corrected vacuum contains virtual photon-electron-positron components.
• Compute the second-order vacuum energy shift and explain why it appears large or divergent.
• Explain why the vacuum energy must vanish in a Lorentz-invariant theory and why a $c$-number counterterm must be added.
• Apply the same logic to the one-electron rest state.
• Understand why the interacting one-electron state is a dressed state, not a bare electron.
• Recognize the appearance of ultraviolet sensitivity and the need for renormalization.

Prerequisite Knowledge

• QED Hamiltonian in Coulomb gauge
• Free photon and Dirac fields
• Creation and annihilation operators for photons, electrons, and positrons
• Time-independent perturbation theory
• Normal ordering
• Lorentz invariance at a conceptual level
• Basic dimensional analysis
• Fine-structure constant

1. Review: the QED Hamiltonian in Coulomb gauge

The lecture begins by reviewing the QED Hamiltonian in Coulomb gauge:

$$\hat H_{\text{QED}} = \hat H_0 + \hat H_I.$$

The free part is

$$\hat H_0 = \sum_\pm \int d^3k \left[ \hbar ck\, \hat a_\pm^\dagger(k)\hat a_\pm(k) + \hbar \omega(k) \big( \hat b_\pm^\dagger(k)\hat b_\pm(k) + \hat d_\pm^\dagger(k)\hat d_\pm(k) \big) \right],$$

where

$$\omega(k)=\sqrt{\left(\frac{Mc^2}{\hbar}\right)^2 + c^2k^2}.$$

So $\hat H_0$ is the diagonal Hamiltonian of free photons, electrons, and positrons. :contentReference[oaicite:4]{index=4}

The interaction Hamiltonian is split into two parts:

$$\hat H_I = \hat H_{AJ} + \hat H_{\text{Coulomb}}.$$

The current-field coupling is

$$\hat H_{AJ} = \int d^3r\, \hat A(\vec r)\cdot \hat j(\vec r),$$

and the instantaneous Coulomb term is

$$\hat H_{\text{Coulomb}} = \int d^3r\,d^3r'\, \frac{:\hat\rho(\vec r)\hat\rho(\vec r') :}{4\pi\varepsilon_0 |\vec r-\vec r'|}.$$

The lecture reviews the charge density

$$\hat\rho(\vec r)=q:\hat\psi^\dagger(\vec r)\hat\psi(\vec r):$$

and the Dirac current

$$\hat j(\vec r)=qc:\hat\psi^\dagger(\vec r)\gamma^0\vec\gamma\,\hat\psi(\vec r):.$$

It also reminds us that:

• $\hat H_{\text{Coulomb}}$ contains 16 terms, each quartic in fermionic operators,
• $\hat H_{AJ}$ contains 8 terms, each involving one photon operator and two fermionic operators. :contentReference[oaicite:5]{index=5}

This is the full interacting Hamiltonian of QED in Coulomb gauge.

2. Why exact solution is impossible

The lecture then states the practical truth directly: we cannot diagonalize $\hat H_{\text{QED}}$ exactly.

This may look surprising at first, because the Hamiltonian fits on one slide. But the formal compactness hides enormous complexity. The free Hamiltonian $\hat H_0$ is diagonal and easy. The interaction terms are not. They connect states with different particle content and different momenta, and they do so in infinitely many ways. :contentReference[oaicite:6]{index=6}

This is the key reason perturbation theory is needed.

3. QED has a small dimensionless parameter

The lecture then explains why perturbation theory is possible at all.

The free Hamiltonian $\hat H_0$ contains only one physical parameter, the fermion mass $M$. So we can make everything dimensionless by measuring:

• energies in units of $Mc^2$,
• lengths in units of the Compton wavelength $\hbar/(Mc)$,
• and rescaling the creation and annihilation operators accordingly.

After this rescaling, $\hat H_0$ becomes a parameter-free dimensionless operator. :contentReference[oaicite:7]{index=7}

Then the interaction terms reveal their true dimensionless coupling.

For the Coulomb term, the prefactor becomes

$$\frac{q^2}{4\pi\varepsilon_0 \hbar c} =: \alpha \approx \frac{1}{137}.$$

The lecture explicitly says: This is small! and notes that physicists have simply been lucky that this dimensionless combination of fundamental constants turns out to be so small. :contentReference[oaicite:8]{index=8}

For the $A\cdot j$ interaction, the same underlying coupling appears in the combination $\sqrt{4\pi\alpha}\approx 0.303$, again a small number. So the interaction Hamiltonian is multiplied by small coefficients.

That is the whole basis of perturbative QED.

4. Why perturbation theory dominates almost all QFT results

The lecture then makes the standard but important point: almost all results in QFT are perturbative.

In QED we compute everything as a Taylor expansion in powers of $\alpha$, and higher-order corrections become smaller and smaller. The lecture notes that calculations up to at least eighth order are done in practice, yielding agreement with experiment to parts per billion. :contentReference[oaicite:9]{index=9}

It also contrasts this with strongly coupled QFTs, where the dimensionless coupling is not small — mentioning low-energy QCD as the canonical example. Those theories remain much more mysterious.

So QED is special because it is weakly coupled.

5. Why the Coulomb term is normal ordered

The lecture then explains an important structural choice.

The Coulomb term is quartic in fermion operators. If it were not normal ordered, then commuting the operators into normal-ordered form would generate quadratic terms. But quadratic terms belong to the same class of operators that $\hat H_0$ can diagonalize. Therefore the lecture says we should regard those quadratic contributions as already absorbed into $\hat H_0$. In particular, failing to normal order would effectively add a large correction to the electron/positron rest energy. So by definition, such effects are absorbed into the parameter $Mc^2$ appearing in $\hat H_0$. :contentReference[oaicite:10]{index=10}

This is an early renormalization move already built into the setup.

6. Why perturbation theory in QFT is conceptually deeper

The lecture then shifts from technical to conceptual.

In classical particle mechanics and in ordinary first-quantized quantum mechanics, the particles are defined independently of the interaction Hamiltonian. The interaction merely tells them how to move.

But in QED, particles are defined as excited states of the total system. Saying that we have a certain number of electrons, positrons, and photons with given momenta and spins is really a way of labeling energy eigenstates of the full Hamiltonian. Therefore $\hat H_I$ does not merely modify particle motion — it is part of the definition of what the particles are. :contentReference[oaicite:11]{index=11}

This is one of the deepest points in the lecture.

The same applies to the vacuum: the vacuum is not the ground state of $\hat H_0$. The vacuum is the ground state of $\hat H_{\text{QED}}$. :contentReference[oaicite:12]{index=12}

That is why perturbation theory is computing dressed states, not merely small energy corrections.

7. Perturbation expansion of the QED vacuum

The lecture then introduces the perturbative expansion of the vacuum:

$$|{\rm vac}\rangle = \sum_{n=0}^\infty \alpha^{n/2}\,|{\rm vac}\rangle_n,$$

with energy

$$E_{\rm vac} = \sum_{n=0}^\infty \alpha^{n/2} E_n.$$

For now, it keeps the notation

$$|0\rangle$$

for the ground state of $\hat H_0$, i.e. the state annihilated by all photon, electron, and positron annihilation operators. :contentReference[oaicite:13]{index=13}

So the true QED vacuum is being expanded in the basis of eigenstates of the free Hamiltonian.

8. Zeroth and first half-order for the vacuum

At zeroth order, the time-independent Schrödinger equation gives

$$\hat H_0 |{\rm vac}\rangle_0 = E_{{\rm vac},0}|{\rm vac}\rangle_0,$$

so

$$|{\rm vac}\rangle_0 = |0\rangle, \qquad E_{{\rm vac},0}=0.$$

Then the lecture looks at the next nontrivial order, order $\alpha^{1/2}$. It points out that since $\hat H_{\text{Coulomb}}$ is of order $\alpha$, only $\hat H_{AJ}$ contributes at this order. So the perturbation equation is

$$\hat H_0 |{\rm vac}\rangle_1 + \alpha^{-1/2}\hat H_{AJ}|0\rangle = E_{{\rm vac},1}|0\rangle.$$

Projecting onto $\langle 0|$, the lecture finds

$$E_{{\rm vac},1}=0,$$

because $\langle 0|\hat H_{AJ}|0\rangle=0$ by normal ordering. :contentReference[oaicite:14]{index=14}

So the first nontrivial effect is not an energy shift but a correction to the vacuum state itself.

9. The first correction to the vacuum state

The lecture then computes

$$|{\rm vac}\rangle_1.$$

Acting with $\hat H_{AJ}$ on $|0\rangle$, only one class of terms survives: the ones that create a photon, an electron, and a positron. After the momentum integral over $r$, a delta function imposes momentum conservation, leaving an expression of the form

$$|{\rm vac}\rangle_1 \sim \sum_{\pm_1,\pm_2,\pm_3} \int d^6k\, F(k_1,k_2,\pm_1,\pm_2,\pm_3)\, \hat a^\dagger_{\pm_3}(k_1-k_2)\, \hat b^\dagger_{\pm_2}(k_2)\, \hat d^\dagger_{\pm_1}(-k_1)\, |0\rangle.$$

The coefficient contains the expected spinor and polarization matrix element

$$u^\dagger \gamma^0 \vec\gamma\cdot \vec\epsilon\, v$$

divided by the total intermediate-state energy

$$\hbar c|k_2-k_1| + E(k_1)+E(k_2).$$

The lecture emphasizes that this coefficient is generically not zero. :contentReference[oaicite:15]{index=15}

So the interacting QED vacuum contains amplitudes for states with:

• one photon,
• one electron,
• one positron.

It then asks the obvious question: does this mean there are photons, electrons, and positrons in the vacuum?

The lecture does not answer with a slogan. It lets the calculation itself force the correct interpretation: the true vacuum is a dressed superposition, not the bare free vacuum. The total charge and total momentum still remain exactly zero. :contentReference[oaicite:16]{index=16}

10. Second-order vacuum energy shift

At order $\alpha$, the perturbation equation becomes

$$\hat H_0|{\rm vac}\rangle_2 + \alpha^{-1/2}\hat H_{AJ}|{\rm vac}\rangle_1 + \alpha^{-1}\hat H_{\text{Coulomb}}|0\rangle = E_{{\rm vac},2}|0\rangle.$$

Projecting onto $\langle 0|$, the lecture finds

$$E_{{\rm vac},2} = \alpha^{-1/2}\langle 0|\hat H_{AJ}|{\rm vac}\rangle_1,$$

because

$$\langle 0|\hat H_0|{\rm vac}\rangle_2=0, \qquad \langle 0|\hat H_{\text{Coulomb}}|0\rangle=0.$$

The resulting expression is a negative integral over the virtual photon-electron-positron intermediate states:

$$E_{{\rm vac},2} = - \sum \int d^6k\, \frac{\text{matrix element}^2}{\hbar c|k_2-k_1|+E(k_1)+E(k_2)}.$$

The lecture says explicitly: this is not zero, and in fact it seems to be infinite, or at least very large if one introduces a physically sensible ultraviolet cutoff. :contentReference[oaicite:17]{index=17}

So the vacuum energy is shifted downward by the interaction, and the shift is ultraviolet sensitive.

11. Why the vacuum energy must vanish

The next slide makes a very important conceptual argument.

The vacuum energy must vanish.

Why? Because energy is the 0-component of the energy-momentum 4-vector. If the vacuum energy were nonzero in one frame, then in some other Lorentz frame the vacuum would carry nonzero momentum. But the vacuum must have zero momentum in all frames. Otherwise there would be a preferred frame in which the vacuum is “really at rest,” contradicting Lorentz invariance. :contentReference[oaicite:18]{index=18}

Therefore the theory must contain an additional $c$-number term that the naïve canonical quantization procedure missed. The lecture writes the corrected Hamiltonian schematically as

$$\hat H_{\text{QED}} = \hat H_0 + \hat H_I - \Delta E,$$

where $\Delta E$ is chosen so that the vacuum energy comes out to zero. In particular,

$$\Delta E = \alpha E_{{\rm vac},2} + O(\alpha^2).$$

This is a concrete example of a counterterm: a $c$-number energy shift inserted because the theory must be matched to nature, not merely derived mechanically from a heuristic quantization recipe. :contentReference[oaicite:19]{index=19}

This is the first serious renormalization statement of the lecture.

12. One-electron state at rest

The lecture then asks about the rest energy of an electron.

Define the lowest-energy eigenstate with charge $q=-e$, momentum zero, and spin projection $\pm \hbar/2$ to be $|e_\pm\rangle$. At zeroth order in $\alpha^{1/2}$,

$$|e_\pm\rangle_0 = \hat b^\dagger_\pm(0)|0\rangle, \qquad E_{e,0}=Mc^2.$$

And again one finds

$$E_{e,1}=0.$$

So the bare free electron at rest is the starting point for the interacting one-electron state. :contentReference[oaicite:20]{index=20}

13. The first correction to the one-electron state

The lecture says that, just as for the vacuum, the first correction $|e_\pm\rangle_1$ contains additional electron-positron pairs and photons. In fact, it contains the same cloud of virtual pairs and photons that already dresses the vacuum, now with one additional electron at rest as a spectator. On top of that, there is also amplitude for the electron to recoil and pick up momentum while a photon carries compensating momentum. :contentReference[oaicite:21]{index=21}

So the physical one-electron state is not a bare electron. It is a dressed state:

• a bare electron,
• plus vacuum dressing,
• plus its own interaction cloud.

This is exactly the physical picture one expects in QED.

14. Second-order rest-energy correction

The lecture then states the key result for the second-order electron rest-energy shift:

$$E_{e,2} = E_{{\rm vac},2} + Mc^2 \ln(K\lambda_C),$$

if an ultraviolet cutoff

$$|k'|\le K$$

is imposed on the intermediate-state momentum integration. Here

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

is the Compton wavelength. :contentReference[oaicite:22]{index=22}

So the electron rest energy contains:

• the same vacuum-energy contribution already found for the vacuum sector,
• plus an additional logarithmically divergent self-energy correction.

This is the clearest signal yet that the bare mass parameter $M$ in $\hat H_0$ cannot simply be identified once and for all with the physical electron mass. It must absorb ultraviolet-sensitive interaction effects.

That is the mass-renormalization problem in its earliest appearance.

15. What this lecture really established

The lecture did more than introduce perturbation theory mechanically.

It established three big ideas.

1. QED is weakly coupled

Because $\alpha$ is small, perturbation theory is meaningful and extraordinarily accurate.

2. Interactions redefine the vacuum and the particles

The true vacuum is not the free vacuum. The true one-electron state is not a bare one-electron state. Both are dressed by virtual excitations.

3. Renormalization is unavoidable

The vacuum energy and the electron rest energy receive large or divergent corrections. The theory must therefore be defined with counterterms or parameter redefinitions chosen to match physical requirements such as Lorentz-invariant vacuum and observed particle masses.

That is why this lecture is so important. It shows perturbative QED not as a bag of Feynman-rule tricks, but as a structural redefinition of the theory’s basic states.

Worked Examples

Example 1: Why the first vacuum correction contains a photon, electron, and positron

At order $\alpha^{1/2}$, only $\hat H_{AJ}$ contributes. Since $\hat H_{AJ}$ contains one photon operator and two fermionic operators, the only terms that can act nontrivially on the free vacuum $|0\rangle$ are those with three creation operators:

$$\hat a^\dagger \hat b^\dagger \hat d^\dagger.$$

Therefore

$$|{\rm vac}\rangle_1$$

must be a superposition of photon-electron-positron states. This is the first indication that the interacting vacuum is not empty in the free-particle basis.

Example 2: Why the vacuum energy counterterm must exist

The second-order vacuum energy shift $E_{{\rm vac},2}$ comes out nonzero and large. But the vacuum of a Lorentz-invariant theory must have zero 4-momentum in every frame. If the vacuum energy were nonzero, then in some boosted frame the vacuum momentum would be nonzero, implying a preferred rest frame of the vacuum. That is impossible. Therefore the Hamiltonian must contain a $c$-number counterterm $-\Delta E$ chosen so that the true vacuum energy is exactly zero.

Intuition

This lecture is where QED starts behaving like real quantum field theory instead of an interaction added to already-defined particles.

In a particle mechanics mindset, one starts with particles and then asks how they interact. In QED, that mindset breaks. The interaction Hamiltonian changes what counts as a vacuum and what counts as an electron. The true vacuum contains virtual excitation amplitudes. The true electron carries a cloud of virtual photons and pairs. So “electron” and “vacuum” are not primitive objects. They are energy eigenstates of the full interacting theory.

That is why perturbation theory is more than a calculational approximation. It is how we construct the physical states themselves.

And then the ultraviolet divergences show up immediately, forcing the lesson that the parameters in the bare Hamiltonian are not yet the directly observed ones. Nature gives us renormalized mass and zero vacuum energy, and the Hamiltonian has to be adjusted accordingly.

Common Mistakes

• Thinking perturbation theory in QED is just a small correction to already well-defined particles. In QED the interaction helps define the particles themselves.
• Forgetting that the true vacuum is the ground state of $\hat H_{\text{QED}}$, not of $\hat H_0$.
• Assuming the first vacuum correction should vanish just because the free vacuum is empty. It does not.
• Misreading virtual components in $|{\rm vac}\rangle_1$ as meaning the vacuum has nonzero total charge or momentum. It does not.
• Forgetting why the Coulomb term is normal ordered.
• Thinking the nonzero second-order vacuum energy can simply be accepted physically. The lecture argues it cannot, because of Lorentz invariance.
• Missing that the extra $c$-number term $-\Delta E$ is already a renormalization counterterm.
• Assuming the one-electron state remains a bare one-electron state once interactions are present.
• Missing that the electron rest energy acquires an ultraviolet-sensitive correction.

Short Summary

The lecture begins by reviewing the Coulomb-gauge QED Hamiltonian

$$\hat H_{\text{QED}} = \hat H_0 + \hat H_I, \qquad \hat H_I = \hat H_{AJ} + \hat H_{\text{Coulomb}},$$

where $\hat H_0$ is the diagonal free Hamiltonian of photons, electrons, and positrons, $\hat H_{AJ}$ is the light-matter coupling, and $\hat H_{\text{Coulomb}}$ is the instantaneous Coulomb interaction. It then shows that after expressing all quantities in Compton units, the free Hamiltonian becomes parameter-free, while the interaction terms are multiplied by the small dimensionless fine-structure constant

$$\alpha = \frac{q^2}{4\pi\varepsilon_0\hbar c}\approx \frac{1}{137}.$$

This makes perturbation theory possible. The lecture emphasizes that in QFT the interaction Hamiltonian does not merely govern motion; it also helps define what the vacuum and particle states are. The true QED vacuum is therefore expanded perturbatively in powers of $\alpha^{1/2}$ in the eigenbasis of $\hat H_0$. At first nontrivial order, only $\hat H_{AJ}$ contributes, and the correction to the vacuum state contains superpositions of one photon, one electron, and one positron:

$$|{\rm vac}\rangle_1 \sim \hat a^\dagger \hat b^\dagger \hat d^\dagger |0\rangle.$$

At order $\alpha$, the vacuum energy shift is nonzero and ultraviolet sensitive. The lecture then argues that the vacuum energy must vanish in a Lorentz-invariant theory, so the Hamiltonian must contain an additional $c$-number counterterm

$$\hat H_{\text{QED}} = \hat H_0 + \hat H_I - \Delta E,$$

with $\Delta E$ chosen so that the true vacuum energy is zero. The lecture then applies the same logic to the one-electron rest state. At zeroth order this is just $\hat b^\dagger_\pm(0)|0\rangle$ with energy $Mc^2$, but perturbatively it becomes a dressed state containing additional virtual pairs and photons. Its second-order energy shift contains the same vacuum contribution plus an ultraviolet-sensitive self-energy term,

$$E_{e,2} = E_{{\rm vac},2} + Mc^2 \ln(K\lambda_C),$$

showing already that mass renormalization is unavoidable in QED. Thus the lecture introduces perturbative QED not merely as a computational trick, but as the framework in which the vacuum and particles of the interacting theory are actually constructed. :contentReference[oaicite:23]{index=23}

Practice Problems

1. Why is exact diagonalization of the full QED Hamiltonian impossible even though the Hamiltonian can be written compactly?

2. Why is the fine-structure constant $\alpha$ the natural perturbative expansion parameter in QED?

3. Why is $\hat H_{\text{Coulomb}}$ taken to be normal ordered in the lecture’s Hamiltonian?

4. Explain why the interaction Hamiltonian in QED helps define what particles and vacuum are, not just how they evolve.

5. Why does the first correction to the vacuum state come from $\hat H_{AJ}$ rather than from $\hat H_{\text{Coulomb}}$?

6. Why does $|{\rm vac}\rangle_1$ contain photon-electron-positron states?

7. Why is the first nonzero correction to the vacuum energy second order in the perturbative expansion?

8. Why does the lecture argue that the vacuum energy must vanish exactly?

9. What is the role of the $c$-number counterterm $-\Delta E$ in the Hamiltonian?

10. Why is the interacting one-electron state not just a bare electron at rest?

11. What does the expression

$$E_{e,2}=E_{{\rm vac},2}+Mc^2\ln(K\lambda_C)$$

tell you about mass renormalization?

12. In your own words, explain why perturbation theory in QED is both mathematically ordinary and conceptually deeper than ordinary perturbation theory in first-quantized quantum mechanics.