Renormalization

Introduction

This lecture is where perturbative QED stops being just a clever method and starts confronting its main conceptual difficulty.

The previous lecture showed that QED scattering amplitudes can be computed from a Lorentz-invariant path integral by expanding in powers of the coupling and translating the surviving terms into Feynman diagrams. That was the big simplification. But it left one obvious problem unresolved:

what happens when the integrals represented by those diagrams are infinite?

That problem is not a side issue. It is built into relativistic quantum field theory.

Every internal loop in a Feynman diagram introduces an unconstrained integration over a four-momentum $k^\mu$. Those loop integrals may fail to converge at large $k^\mu$. So the real question is not whether divergences occur. The real question is whether they are so uncontrolled that the theory becomes meaningless, or whether they can be absorbed into a controlled redefinition of a small number of physical parameters.

The lecture’s answer is the second one, and that is exactly what renormalization means here. :contentReference[oaicite:2]{index=2}

The lecture starts by reviewing the invariant amplitude $M$ and the Feynman rules for QED. Then it explains precisely how to decode a Feynman diagram into algebra, paying careful attention to fermion-line orientation, spinor ordering, gamma matrices, traces around fermion loops, and photon-line contractions. Only after that bookkeeping is firmly in place does it turn to the real issue: loops.

The lecture then makes several strong structural claims:

• loops are the only hard parts of Feynman diagrams, because each loop introduces a free four-momentum integral,
• loops are also what make QFT genuinely quantum,
• and for QED there are remarkably few divergent loop structures.

In fact, after using power counting plus some nontrivial cancellations such as Furry’s theorem, the lecture concludes that there are exactly three divergent loop integrals in QED, all only logarithmically divergent:

1. electron self-energy,
2. vacuum polarization,
3. vertex correction. :contentReference[oaicite:3]{index=3}

That is the decisive simplification.

The lecture then shows why this implies renormalizability. Whenever one of these divergent subloops appears inside a larger diagram, it can be replaced by a lower-order basic structure of the same type:

• a fermion propagator correction,
• a photon propagator correction,
• or a vertex correction.

So the effect of all ultraviolet divergences can be absorbed into rescaling the basic propagators and vertices, together with a shift of the electron mass parameter. The lecture explicitly introduces the renormalization constants $Z_0,Z_1,Z_2,Z_3$ and explains that by allowing them to depend on the ultraviolet cutoff $\Gamma$, one can make all final physical results independent of $\Gamma$. :contentReference[oaicite:4]{index=4}

That is the whole content of renormalization in this lecture: not magic removal of infinities, but systematic absorption of divergent pieces into the small set of quantities that were never directly observable in the first place.

Learning Objectives

• Recall the Lorentz-invariant amplitude $M$ and its relation to scattering probabilities.
• Review the QED Feynman rules for external lines, propagators, and vertices.
• Decode a fermion path and a fermion loop into algebraic expressions.
• Understand the extra sign rules for fermion loops and fermion-line crossings.
• Explain why loops are the only hard part of perturbative QED diagrams.
• Understand why each loop introduces an unconstrained four-momentum integration.
• Explain why loop order is the genuinely quantum part of QFT.
• Derive the topological relation

$$V = (E-2)+2L$$

for connected QED diagrams.

• Show why higher loop number corresponds to higher order in $\alpha$.
• Use power counting to estimate which loop topologies may diverge.
• Understand the role of Furry’s theorem in eliminating odd-fermion-loop diagrams.
• Identify the three divergent loop structures in QED.
• Explain why the existence of only three divergent structures implies renormalizability.
• Understand mass renormalization, charge renormalization, and wave-function renormalization.
• Explain the role of the renormalization constants $Z_0,Z_1,Z_2,Z_3$.
• Understand why the continuum limit is singular and why that is not a fatal problem.

Prerequisite Knowledge

• QED Feynman diagrams and Feynman rules
• Lorentz-invariant amplitudes
• Dirac spinors and gamma matrices
• Photon and fermion propagators
• Loop momentum integrals
• Basic ultraviolet divergence idea
• Charge conservation in QED
• Path-integral perturbation theory

1. Review: invariant amplitude $M$

The lecture begins by reviewing the covariant perturbation-theory setup.

A transition amplitude between initial and final momentum states is the square root of a probability density in 3D momentum space. But a probability density in 3D $k$-space is not Lorentz invariant. Therefore one factors out the appropriate $1/\sqrt{k_0}$-type normalization to define the Lorentz-invariant amplitude $M$. The slide states that $M$ has dimensions of one power of length for each initial or final particle. Perturbative QED is then used precisely to compute this invariant amplitude $M$. :contentReference[oaicite:5]{index=5}

This is an important reminder because renormalization is about the quantities that enter actual scattering amplitudes, not just arbitrary intermediate expressions.

2. Review: Feynman rules for QED

The next slide reviews the diagrammatic ingredients:

• initial/final electrons and positrons with momentum $k$ and spin $\pm$,
• initial/final photons with momentum $k$ and helicity $\pm$,
• the fermion propagator,
• the photon propagator,
• and the QED vertex. :contentReference[oaicite:6]{index=6}

The lecture is about divergences, but it starts by reminding you exactly what the objects are that can diverge.

3. How to decode diagrams: fermion paths and loops

Pages 4–6 are a very practical rulebook for translating diagrams into formulas.

The lecture says every fermion line must belong to either:

• a continuous path connecting initial/final electrons or positrons,
• or a closed internal loop. :contentReference[oaicite:7]{index=7}

For a fermion loop:

• start at any vertex,
• write the corresponding gamma-matrix factor,
• move around the loop in either direction,
• write propagators and vertices in order,
• then take the trace of the resulting gamma-matrix product. :contentReference[oaicite:8]{index=8}

For a fermion path connecting external states:

• one end is always a row spinor,
• the other end is a column spinor,
• start at the row-spinor end,
• write the row spinor,
• then write the propagators and vertices in sequence,
• and finish with the column spinor. :contentReference[oaicite:9]{index=9}

The lecture also stresses an important charge-conservation rule: a fermion path whose arrow directions do not line up consistently is not a valid QED Feynman diagram. The page 5 figure explicitly marks such a diagram as invalid. :contentReference[oaicite:10]{index=10}

Then page 6 explains photon lines:

• an external photon line contributes a polarization vector attached to the exposed Lorentz index of its vertex gamma matrix,
• an internal photon line contributes the photon propagator with its two Lorentz indices contracted into the two attached vertices. :contentReference[oaicite:11]{index=11}

This decoding review matters because the divergences depend on exactly how many propagators and vertices a loop contains.

4. Extra sign rules

Page 7 adds the familiar but important extra rules:

• a factor of $-1$ for every fermion loop,
• a factor of $-1$ for every crossing of fermion lines,
• and some permutation symmetry factors. :contentReference[oaicite:12]{index=12}

These are part of the exact diagrammatic bookkeeping.

5. Loops are the interesting things

Page 8 makes one of the lecture’s main claims: loops are the interesting things.

Why?

Because without loops, the delta functions at the vertices fix all internal momenta in terms of the external momenta, leaving no further integrations. But with a loop, there is always one free loop momentum $k^\mu$ that remains unconstrained by the vertex delta functions, so one must perform a full 4D integral over that momentum. :contentReference[oaicite:13]{index=13}

The slide also makes a deeper point:

• no loops corresponds to classical physics,
• loops represent higher-order quantum corrections.

So in this sense, the difficult loop integrals are the genuinely quantum part of perturbative QFT. :contentReference[oaicite:14]{index=14}

That is a strong and useful way to think about the loop expansion.

6. Loop order and powers of $\alpha$

Page 9 makes the relation precise.

Each vertex contributes a factor of $\alpha^{1/2}$, so the order of a diagram in powers of $\alpha$ is one half the number of vertices:

$$\text{order} = \frac{V}{2}.$$

For connected diagrams, the lecture then gives the topological relation

$$V = (E-2)+2L,$$

where:

• $V$ is the number of vertices,
• $E$ is the number of external legs,
• $L$ is the number of loops. :contentReference[oaicite:15]{index=15}

The lecture proves this by induction:

• for tree diagrams, attaching a new vertex increases both $V$ and $E$ by one,
• creating a loop removes two external legs while increasing $L$ by one, keeping $E+2L$ fixed. :contentReference[oaicite:16]{index=16}

So for fixed external process:

• the minimum power of $\alpha$ comes from the minimum number of vertices,
• each additional loop adds two more vertices,
• therefore each additional loop adds one more power of $\alpha$.

This is why the loop expansion is simultaneously:

• a quantum expansion,
• and an $\alpha$-expansion.

7. Connected diagrams and $W=\ln Z$

Pages 10–11 then make a structural point about connectedness.

The sum over all diagrams includes many disconnected diagrams. But the contribution of a disconnected diagram is just the product of the contributions of the connected pieces it contains. Therefore, once one knows all connected diagrams, the disconnected ones are trivial to reconstruct. :contentReference[oaicite:17]{index=17}

The lecture then defines $W[J^\mu,\eta,\bar\eta]$ to be the functional corresponding to the sum over all connected Feynman diagrams and states the theorem: the generator of all diagrams is related to connected diagrams by

$$Z = e^{iW}.$$

The slide says the factor of $i$ is just a convention, and that $W$ matters because it lets us prove theorems about connected diagrams using the path integral. :contentReference[oaicite:18]{index=18}

This is not the main renormalization result, but it is an important structural statement for diagrammatics.

8. Loop expansion as semiclassical approximation

Pages 12–13 explain another deep interpretation of loops.

Rescale all fields and sources in the path integral by $\alpha^{-1/2}$. Then the entire action acquires an overall factor of $\alpha^{-1}$. For small $\alpha$, the path integral can be analyzed by steepest descents. The lecture then identifies:

• the zeroth-order saddlepoint term with the classical action evaluated on the classical field history,
• the no-loop diagrams with quantum transition amplitudes built only from classical field dynamics,
• and the loops as the nontrivial quantum corrections. :contentReference[oaicite:19]{index=19}

Page 13 then gives the clean slogan: “Quantum = Loops.” :contentReference[oaicite:20]{index=20}

That is a powerful way to interpret the whole perturbative structure.

9. Why loop integrals may diverge

Page 14 states the next obvious issue: loops are problematic because their 4D integrals may not converge.

Sometimes symmetries force the large-$k^\mu$ pieces to cancel. The lecture gives one universal example: Furry’s theorem — any fermion loop with an odd number of photon insertions vanishes exactly, because the diagram cancels against the one with the opposite orientation around the loop, which gives the opposite trace. :contentReference[oaicite:21]{index=21}

But these cancellations are not enough to save every loop. Some loop integrals really are divergent.

The lecture then asks: which ones?

10. Power counting for loop integrals

Page 15 sets up the power counting.

For large loop momentum $k^\mu$:

• each fermion propagator contributes roughly $1/k$,
• each photon propagator contributes roughly $1/k^2$,
• and the 4D momentum-space measure contributes $d^4k \sim k^3\,dk$. :contentReference[oaicite:22]{index=22}

So if a loop contains:

• $n_F$ fermion lines,
• $n_\gamma$ photon lines,

then the integral behaves schematically like

$$\int^\infty dk\, k^m, \qquad m = 3 - n_F - 2n_\gamma.$$

This is only the superficial degree of divergence; further cancellations may improve it. But it is enough to identify the dangerous cases. :contentReference[oaicite:23]{index=23}

11. Listing all possible loop types

Pages 16–17 systematically list the possible loop topologies by $(n_F,n_\gamma)$.

The lecture uses the basic QED fact that each vertex joins two fermion ends and one photon end, so one can have $n_F \ge n_\gamma$, but never $n_\gamma > n_F$. It then goes case by case:

• some loops are impossible,
• some vanish by Furry’s theorem,
• some are superficially worse but are softened by cancellations,
• and many are convergent. :contentReference[oaicite:24]{index=24}

The summary table on page 17 is the key result: only three loop topologies remain divergent, and all of them only logarithmically. :contentReference[oaicite:25]{index=25}

The slide explicitly names them:

1. electron self-energy
2. vacuum polarization
3. vertex correction :contentReference[oaicite:26]{index=26}

That is the decisive classification.

12. Exactly three divergent loop integrals

Page 18 states the conclusion in a boxed sentence:

There are exactly three divergent loop integrals in QED, all logarithmic. :contentReference[oaicite:27]{index=27}

The slide shows the three corresponding one-loop structures:

• the fermion propagator correction,
• the photon propagator correction,
• the three-point vertex correction.

Everything else converges, vanishes, or reduces to these cases when nested inside larger diagrams.

This is the key reason QED is renormalizable rather than hopeless.

13. Why this implies renormalizability

Page 19 makes the renormalizability logic explicit.

Whenever one of these divergent loops appears inside a larger diagram, one can remove the loop and replace it by the corresponding lower-order basic structure:

• electron self-energy loop $\to$ corrected fermion propagator,
• vacuum-polarization loop $\to$ corrected photon propagator,
• vertex loop $\to$ corrected vertex. :contentReference[oaicite:28]{index=28}

So the divergent subdiagram always has the same form as something already present in the original QED rules.

That is exactly what renormalizability means in practice: the divergences do not force you to invent an infinite number of new interaction structures. They can be absorbed into the same basic ingredients already in the theory.

14. Ultraviolet cutoff and logarithmic divergences

Page 20 then introduces the practical regularization picture.

Interpret the divergent loop integrals as finite integrals with an ultraviolet cutoff $\Gamma$, to be taken to infinity at the end:

$$\int d^4k \;\to\; \int^{\Gamma} dk\, k^3 \int d\Omega_k.$$

Then each of the three divergent loop classes has the form

$$(\text{finite, cutoff-independent part}) + \alpha \ln(\Gamma)\, A_n \times (\text{basic structure}),$$

for some cutoff-scheme-dependent constants $A_1,A_2,A_3$. :contentReference[oaicite:29]{index=29}

The lecture notes that the exact coefficients depend on how the cutoff is implemented, and that a literal 4-sphere cutoff is not Lorentz invariant, so there are several regularization options.

That is an important honesty point: the cutoff is a temporary regulator, not the final physics.

15. Changing the Feynman rules instead of keeping divergent loops

Page 21 explains the core renormalization move.

Including all the logarithmically divergent loops is equivalent to:

• discarding the divergent $\ln(\Gamma)$ parts from those loop integrals,
• and instead modifying the Feynman rules for the three basic ingredients:
  • fermion propagator,
  • photon propagator,
  • vertex. :contentReference[oaicite:30]{index=30}

The slide illustrates this schematically: the same physical effect as the divergent loop contributions can be reproduced by changing the lower-order building blocks.

This is renormalization in diagrammatic form.

16. The big idea: say we already did that

Page 22 then states the renormalization philosophy directly, in big text:

LET’S SAY THAT WE ALREADY DID THAT. :contentReference[oaicite:31]{index=31}

The lecture explains the meaning:

• the electron mass is an empirically tuned constant anyway, so there is no reason the bare mass parameter in the Lagrangian should equal the observed electron mass before higher-order corrections are included,
• the electric charge in the vertex is also an empirical parameter fixed by experiment,
• therefore both mass and charge should be tuned so that the final, higher-order-corrected answers match experiment and remain finite. :contentReference[oaicite:32]{index=32}

This is a very practical statement: the bare parameters were never observables. So it is not a contradiction if they depend on the regulator in just the way needed to make the physical quantities finite.

17. Wave-function renormalization

Pages 23–24 then address a subtler point: what does it mean to rescale the fermion and photon propagators?

The lecture answers: we do not really want transition amplitudes between noninteracting particle states. We want amplitudes between real photons and fermions. The variational construction of external states used the noninteracting vacuum and noninteracting creation operators. But for the real interacting theory, the correctly normalized one-particle excitations are not created with unit amplitude by those bare source variations.

So to compute amplitudes between real particle states, the external lines must be multiplied by:

• $\sqrt{Z_1(\alpha,\Gamma)}$ for fermions,
• $\sqrt{Z_3(\alpha,\Gamma)}$ for photons. :contentReference[oaicite:33]{index=33}

The lecture explicitly notes that the fermion factor must be the same for electrons and positrons because of charge-conjugation symmetry. :contentReference[oaicite:34]{index=34}

This is the meaning of wave-function renormalization.

18. The traditional $Z$-constants

Pages 24–25 then explain the traditional notation.

The lecture identifies:

• $Z_0$ with mass renormalization,
• $Z_1$ with fermion field-strength renormalization,
• $Z_2$ with vertex/charge renormalization,
• $Z_3$ with photon field-strength renormalization,

up to the lecture’s notation conventions. It then explains the standard rewriting where the effective vertex renormalization factor is decomposed so that:

• each fermion line attached to a vertex contributes a $\sqrt{Z_1}$,
• each photon line attached to a vertex contributes a $\sqrt{Z_3}$,
• and the remaining pure vertex factor is $Z_2$. :contentReference[oaicite:35]{index=35}

The lecture then says that renormalizing every external line with $\sqrt{Z_{1,3}}$ and every vertex appropriately is equivalent to:

• leaving external lines unrenormalized,
• renormalizing every fermion propagator by $Z_1$,
• renormalizing every photon propagator by $Z_3$,
• renormalizing every vertex by $Z_2$,
• and renormalizing the mass parameter separately via $Z_0$. :contentReference[oaicite:36]{index=36}

So effectively there are four tuning parameters $Z_0,Z_1,Z_2,Z_3$, all allowed to depend on $\Gamma$.

19. Finite physical results from singular bare parameters

The lecture then reaches its main conclusion.

By choosing the $Z$-factors as cutoff-dependent functions, one can cancel the divergent $\ln\Gamma$ pieces from loop integrals so that the final physical results do not depend on $\Gamma$. Therefore the $\Gamma\to\infty$ limit is finite even though the bare parameters themselves become singular. :contentReference[oaicite:37]{index=37}

Page 26 then gives the final perspective: the continuum limit is typically singular. Even if $\Gamma$ were finite — i.e. even if spacetime were effectively discrete at very short scales — one would still need renormalization because the parameters in the theory have to be tuned nontrivially to fit experiment. The only disturbing extra fact is that the required renormalization factors become logarithmically divergent as $\Gamma\to\infty$. But the lecture reminds us that this is exactly what already happened for classical fields in the continuum limit: the continuum limit is singular there too. Quantum fields just need a few more singular redefinitions. :contentReference[oaicite:38]{index=38}

The closing rhetorical question is: Is that really so bad?

That is the tone the lecture wants: renormalization is not a scandal. It is how continuum field theories are defined.

Worked Examples

Example 1: Why a loop always introduces an integral

Take any closed loop in a Feynman diagram. The delta functions at the vertices constrain momentum conservation locally, but one can still add the same four-momentum $k^\mu$ to every internal line around the loop and keep all those vertex constraints satisfied. Therefore one four-momentum remains free, and each loop contributes one integral

$$\int d^4k.$$

That is why loops are the only genuinely hard parts of perturbative QED.

Example 2: Why the three divergent loop structures are enough

Suppose a divergent electron self-energy loop appears somewhere inside a larger diagram. Remove the loop. What remains is just a fermion propagator line. Likewise:

• remove a vacuum-polarization loop and you get a photon propagator,
• remove a vertex-correction loop and you get a vertex.

So the divergent pieces never force new structures beyond:

• fermion propagator,
• photon propagator,
• vertex, plus the fermion mass parameter already present in the propagator. This is exactly why QED is renormalizable.

Intuition

The cleanest intuition is this:

QED has infinitely many possible Feynman diagrams, but only a tiny number of ways they can go ultraviolet-bad.

That is the miracle.

If every higher-order loop produced a genuinely new divergent structure, the theory would need infinitely many new parameters and would lose predictive power. But in QED that does not happen. All the bad ultraviolet behavior folds back into the same few basic ingredients that were already there:

• the electron propagator,
• the photon propagator,
• the interaction vertex,
• and the electron mass.

So renormalization is really just retuning the basic knobs of the theory so that the infinite ultraviolet contributions never show up in the final measurable quantities.

That is why renormalizability matters so much. It is what keeps QED predictive.

Common Mistakes

• Thinking every loop diagram is divergent. Many are convergent, and some vanish exactly.
• Forgetting Furry’s theorem, which kills fermion loops with an odd number of photon insertions.
• Confusing superficial power counting with the actual divergence after cancellations.
• Thinking renormalization means adding infinitely many arbitrary fixes. In QED the lecture shows only a small finite set of structures need tuning.
• Forgetting that bare mass and bare charge were never directly observable quantities.
• Thinking wave-function renormalization is optional. It is needed because the external states in the perturbative construction are not already normalized as real interacting one-particle states.
• Confusing the transition amplitude between bare states with the physical amplitude between real particle states.
• Treating the singularity of the continuum limit as unique to quantum fields; the lecture explicitly says continuum classical fields already had singular limits too.

Short Summary

The lecture begins by reviewing Lorentz-invariant perturbative QED and the invariant amplitude $M$, together with the QED Feynman rules. It then explains in detail how to decode Feynman diagrams into formulas by following fermion paths, taking traces around fermion loops, contracting photon lines with the exposed Lorentz indices at vertices, and including the extra $-1$ factors for fermion loops and fermion-line crossings. The central observation is that loops are the only hard part of perturbative QED, because every loop introduces one unconstrained integration over a four-momentum $k^\mu$. The lecture further shows that loop order is both the order in quantum corrections and the order in the $\alpha$-expansion: for connected diagrams

$$V=(E-2)+2L,$$

so each extra loop adds two vertices and therefore one extra power of $\alpha$. After interpreting the loop expansion as a semiclassical steepest-descents expansion, the lecture analyzes which QED loop integrals can diverge. Using power counting plus exact cancellations such as Furry’s theorem, it concludes that there are exactly three divergent loop structures in QED, all logarithmically divergent:

• electron self-energy,
• vacuum polarization,
• vertex correction. :contentReference[oaicite:39]{index=39}

Whenever one of these divergent subloops appears inside a larger diagram, it can be replaced by the corresponding lower-order basic structure: a fermion propagator, a photon propagator, or a vertex. This means all ultraviolet divergences can be absorbed into rescalings of the basic ingredients of the theory together with the electron mass parameter. Introducing an ultraviolet cutoff $\Gamma$, the lecture shows that the divergent parts are proportional to $\alpha\ln\Gamma$ times those basic structures. Therefore, by allowing the renormalization constants $Z_0,Z_1,Z_2,Z_3$ to depend on $\Gamma$, one can absorb the divergent pieces into:

• electron mass renormalization,
• charge/vertex renormalization,
• fermion wave-function renormalization,
• photon wave-function renormalization. The lecture emphasizes that bare mass and charge are empirical tuning parameters anyway, and that the external lines in amplitudes must also be renormalized because the bare noninteracting one-particle states are not the same as the real interacting particle states. The final result is that all physical predictions can be made finite and cutoff-independent, even though the bare parameters become singular in the continuum limit. The lecture closes by arguing that this is not a disaster but the normal way continuum field theories must be defined. :contentReference[oaicite:40]{index=40}

Practice Problems

1. Why does each loop in a Feynman diagram introduce one unconstrained four-momentum integral?

2. Explain why loop order is also the order of quantum corrections in QED.

3. Derive or explain the topological relation

$$V=(E-2)+2L$$

for connected diagrams.

4. Why are no-loop diagrams interpreted as the classical part of the field dynamics?

5. What does Furry’s theorem say, and why is it useful for divergence counting?

6. Why is superficial power counting not always the same as the actual divergence of a loop integral?

7. Why are there exactly three divergent loop structures in QED?

8. Why does the existence of only those three divergent loop structures imply renormalizability?

9. What is the difference between mass renormalization, charge renormalization, and wave-function renormalization?

10. Why must external fermion and photon lines be multiplied by $\sqrt{Z_1}$ and $\sqrt{Z_3}$ when computing amplitudes for real particles?

11. Why is it acceptable that the bare parameters depend on the ultraviolet cutoff $\Gamma$?

12. In what sense does the lecture claim that the continuum limit being singular is not actually shocking?