Feynman Diagrams

Introduction

The previous lectures built the machinery. This lecture finally turns that machinery into a practical language.

The problem is familiar by now. In Hamiltonian perturbation theory, QED works, but it works in an ugly way. Even something as basic as Compton scattering required a long operator calculation, with several time-ordered intermediate processes that were not Lorentz invariant one by one. Only after everything was added together did the invariant amplitude emerge. That is not a good way to think about a relativistic quantum field theory. :contentReference[oaicite:1]{index=1}

The path-integral reformulation fixed the conceptual issue. By rewriting QED in Lagrangian language, the theory became manifestly Lorentz invariant at every step. But the path-integral formulas by themselves are still not convenient enough for everyday perturbative calculations. One still needs a systematic way to know:

• which source variations matter,
• which terms survive,
• how to read off propagators and vertices,
• and how to turn the generator into a concrete scattering amplitude.

That is exactly what Feynman diagrams do.

The lecture begins by reviewing the Gaussian generating functionals for bosonic and fermionic degrees of freedom. Then it recalls how those become the Dirac and electromagnetic generating functionals for free fields. After that, it states the key structural fact: once the Coulomb term has been absorbed and the theory is written in a covariant path-integral form, perturbative QED is just variational calculus on a Lorentz-invariant Gaussian generator. :contentReference[oaicite:2]{index=2}

At that point, an enormous combinatorial simplification appears. The vast collection of source derivatives that could in principle arise does not survive arbitrarily. The surviving terms follow a simple pattern, and that pattern can be represented graphically. Those graphs are the Feynman diagrams.

The lecture then defines the diagrammatic building blocks:

• external photon, electron, and positron lines,
• internal photon and fermion propagators,
• and the QED vertex connecting one photon line with two fermion lines.

Then it states the practical algorithm: to compute a transition amplitude to order $\alpha^n$,

1. draw all distinct diagrams with $2n$ vertices and the right external lines,
2. translate each diagram into a formula using the Feynman rules,
3. add them all.

This is the real purpose of Feynman diagrams: they are a compressed bookkeeping language for Lorentz-invariant perturbative QED. :contentReference[oaicite:3]{index=3}

Learning Objectives

• Explain why Hamiltonian perturbation theory motivates a covariant diagrammatic method.
• Recall the Gaussian generating functionals for bosonic and fermionic path integrals.
• Understand how transition amplitudes in the noninteracting theory are generated by source derivatives.
• Recognize the free Dirac path integral as a Gaussian functional.
• Recognize the free electromagnetic path integral as a Gaussian functional.
• Understand the role of the fully Lorentz-invariant noninteracting QED generating functional $Z_0$.
• Explain how interactions are added perturbatively by source variation.
• Understand why the physically relevant object is the Lorentz-invariant amplitude $M$, not the raw noncovariant transition matrix element by itself.
• Identify the graphical building blocks of QED Feynman diagrams.
• Distinguish external lines, internal propagators, and vertices.
• Understand the topology rules for QED diagrams.
• Apply the order-counting rule that order $\alpha^n$ corresponds to diagrams with $2n$ vertices.
• Explain why diagrammatic perturbation theory is so much more efficient than direct Hamiltonian TDPT.

Prerequisite Knowledge

• Path integrals for bosons and fermions
• Coherent-state and Grassmann generating functionals
• Free Dirac field and free electromagnetic field
• Lorentz-invariant source generating functionals
• Perturbative expansion in $\alpha$
• Compton scattering and the idea of invariant amplitudes
• Basic propagator language

1. Review: why path integrals help

The lecture opens by reviewing the whole logic of the previous lectures.

Path integrals provide an equivalent representation of quantum time evolution. For relativistic quantum field theory, they have one major advantage over the Hamiltonian formalism: they use the Lagrangian instead of the Hamiltonian, so every step of the calculation is Lorentz invariant. :contentReference[oaicite:4]{index=4}

The lecture repeats the three structural reasons this is useful for weakly interacting quantum fields:

1. without interactions, the path integrals can be computed explicitly as Gaussian functions of initial and final states,
2. with arbitrary linear classical sources added to every field, the transition amplitudes become Gaussian functionals of the sources,
3. interactions can then be added perturbatively, in a manifestly covariant way, by taking source variations of the noninteracting generating functional and then setting the sources to zero. :contentReference[oaicite:5]{index=5}

That review is not filler. It is the precise foundation of the Feynman-diagram construction.

2. Review: Gaussian functionals for bosons and fermions

The lecture then recalls the two elementary building blocks from B5.

For a driven harmonic oscillator, the bosonic path integral is Gaussian and can be evaluated by using the Green’s function and completing the square. The result is the bosonic Gaussian generating functional. The slide on page 3 summarizes this without redoing the whole derivation. :contentReference[oaicite:6]{index=6}

Then, on page 5, the lecture recalls the analogous result for a driven two-state system, using a Grassmann path integral. Again the result is a Gaussian functional, now in Grassmann sources. :contentReference[oaicite:7]{index=7}

The next review slide emphasizes the practical point: all noninteracting transition amplitudes can be obtained from the vacuum-to-vacuum amplitude of the driven system by taking source variations and then setting the sources to zero. :contentReference[oaicite:8]{index=8}

That is the source-language that later becomes the diagrammatic one.

3. Adding interactions perturbatively in the Lagrangian language

The lecture then writes the generic perturbative expansion

$$L = L_0 + \epsilon L_1,$$

and shows that the interacting generating functional can be expanded as

$$Z_\epsilon[J] = \sum_{n=0}^{\infty} \frac{1}{n!} \left(\frac{i\epsilon}{\hbar}\right)^n \left[ \int dt\,L_1\!\left(-i\hbar\frac{\delta}{\delta J(t)}\right) \right]^n Z_0[J].$$

The review slide says this explicitly: this is time-dependent perturbation theory, using the Lagrangian instead of the Hamiltonian. :contentReference[oaicite:9]{index=9}

That sentence is the right way to think about Feynman perturbation theory. It is not a different physical approximation scheme; it is the same perturbative idea reorganized in a covariant way.

4. The Feynman propagator already appears in the Gaussian

The lecture then points out that the central Gaussian in $Z_0[J]$ has a very useful representation.

For the bosonic oscillator, the time-ordered Green’s function that appears in the Gaussian is identified on page 8 as the Feynman propagator. The lecture shows it first in time-domain form with step functions and then in frequency space. :contentReference[oaicite:10]{index=10}

This is a key conceptual point: propagators are not added later by hand. They are already encoded in the Gaussian generating functional of the free theory.

5. From single degrees of freedom to quantum fields

The lecture then moves from one bosonic or fermionic mode to whole quantum fields.

On page 9 it states that the Dirac path integral is just multiplying many fermionic two-state systems together: one for each momentum $\vec k$, each spin, and each particle type. The resulting fermionic functional integral becomes

$$\int \mathcal D\Psi\,\mathcal D\bar\Psi\; e^{\frac{i}{\hbar}\int d^4x\, \bar\Psi(i\hbar \gamma^\mu\partial_\mu - Mc)\Psi},$$

or equivalently the Fourier-space version with

$$\bar\Psi(k_\mu \gamma^\mu - Mc/\hbar)\Psi.$$

The slide explicitly labels this as the Dirac path integral. :contentReference[oaicite:11]{index=11}

So the free fermion sector of QED is already a Gaussian functional integral.

6. The Dirac generating functional

The next slide says this plainly: by evaluating the Gaussian path integral by completing the square, one obtains the Dirac generating functional. The visual formula on page 10 makes clear that the result has the standard source-quadratic structure with the Dirac propagator kernel in the exponent. The slide even marks the vacuum normalization factor as $=1$. :contentReference[oaicite:12]{index=12}

This is the free fermionic half of $Z_0$.

7. The electromagnetic path integral

The lecture then turns to the photon sector.

On pages 11–13 it rewrites the electromagnetic coherent-state path integral using new variables and completes the square to perform the momentum integration. The point made on page 12 is that one still initially has only the two transverse components of $A_\mu$, since this construction begins from Coulomb gauge. :contentReference[oaicite:13]{index=13}

Then page 13 recalls the Coulomb-term trick: the Coulomb factor can be written as a path integral over an additional $A_0$ component, which neatly completes the Lorentz-invariant interaction term in the Maxwell–Dirac Lagrangian by adding the zeroth-component pieces. The slide explicitly says this yields the Lorentz- and gauge-invariant Maxwell–Dirac action, though an infinite product of delta functions still remains enforcing Coulomb gauge. :contentReference[oaicite:14]{index=14}

So at this stage the full QED path integral is almost covariant, but not yet in the final unconstrained $d^4A_\mu$ form.

8. Restoring full $A_\mu$ integration

Pages 14–17 handle this remaining issue.

The lecture explains that one can perform a gauge transformation on the integration dummy variables $A_\mu$ and $\Psi$, with an arbitrary real function $\phi(\vec r,t)$, chosen so that the transformed fields satisfy the desired relation that removes the Coulomb-gauge delta-function constraint. Since the Maxwell–Dirac action is gauge invariant, the path integral is unchanged. :contentReference[oaicite:15]{index=15}

Then, in “Step 2: Using the gauge freedom,” the lecture notes that the transformed expression appears to depend on the arbitrary function $\phi$, but actually does not — different $\phi$ values are only different choices of integration dummy variables. One can then multiply by a suitable factor and integrate over the gauge function, which produces only an overall constant. :contentReference[oaicite:16]{index=16}

Finally, in “Step 3,” the lecture says that this constant does not matter, because the physical object we want is the generator

$$Z_\alpha[f_\mu] = \frac{\langle 0|\hat U^\alpha_{f_\mu}(-\infty,\infty)|0\rangle} {\langle 0|\hat U^\alpha_{0}(-\infty,\infty)|0\rangle},$$

and any source-independent constant cancels between numerator and denominator. Therefore one is effectively free to use a fully Lorentz-invariant path integral for the generator. :contentReference[oaicite:17]{index=17}

This is the crucial move that frees the path integral from the gauge-fixed Hamiltonian origin.

9. The crucial Gaussian generating functional for free QED

Page 18 states the central object of the lecture explicitly: the crucial Gaussian generating functional for noninteracting QED.

After setting the coupling $\alpha\to 0$, the remaining path integral is Gaussian. The displayed formula contains:

• the Dirac propagator in the fermionic source sector,
• and the photon propagator in the electromagnetic source sector.

The slide also notes that one arbitrary number appears in the photon propagator structure. The lecture says the final results never depend on the chosen value, and notes that Feynman liked one particular choice while another is also possible. :contentReference[oaicite:18]{index=18}

This is the free QED generator $Z_0[J,\bar J,f_\mu]$. Everything that follows in perturbative QED is generated from it.

10. Perturbative QED is variational calculus on $Z_0$

The next slide gives the real slogan of the lecture: Perturbative quantum electrodynamics as variational calculus on a Lorentz-invariant Gaussian generator. :contentReference[oaicite:19]{index=19}

The interacting Maxwell–Dirac action is split into:

• the noninteracting part,
• plus the cubic interaction term proportional to $\sqrt{4\pi\alpha}\int d^4x\, A_\mu \bar\Psi \gamma^\mu \Psi$.

Then the interacting generator is written as the exponential of source-derivative operators acting on the Gaussian free generator $Z_0$. This is the full QED version of the earlier perturbative identity from single oscillators. :contentReference[oaicite:20]{index=20}

So conceptually, QED perturbation theory is no longer “sum over awkward time-ordered Hamiltonian processes.” It is:

• differentiate a Gaussian,
• collect the surviving terms,
• and interpret them.

11. Transition amplitudes and the invariant amplitude $M$

The lecture then makes an important distinction.

The actual transition amplitude between specific initial and final states is not Lorentz invariant, because the initial and final states are usually eigenstates of the noninteracting Hamiltonian, which itself is not Lorentz invariant. That is fine. What is Lorentz invariant is the amplitude $M$ that factors out of the kinematic normalization.

The slide on page 20 writes the transition matrix element schematically as a kinematic prefactor times the invariant amplitude $M$. It explicitly points out that $M$ is obtained by Lorentz-invariant combinations of source variations acting on $Z_0$. :contentReference[oaicite:21]{index=21}

This is a very important conceptual cleanup:

• raw state-normalized transition elements depend on conventions and frame-dependent normalizations,
• the invariant amplitude $M$ is the physically central covariant quantity.

12. Which source-variation terms survive?

Now comes the decisive simplification.

The lecture asks: after taking the necessary source derivatives and then setting all sources to zero, what kinds of terms actually survive?

It says there is a fairly simple pattern, and that pattern can be expressed graphically in terms of certain diagrams. :contentReference[oaicite:22]{index=22}

That is the birth of the Feynman-diagram language in the lecture.

13. Feynman diagram components

Page 21 lists the graphical building blocks very explicitly.

The diagram components are:

• an initial-state electron with momentum $k$,
• an initial-state positron with momentum $-k$,
• a final-state electron with momentum $k$,
• a final-state positron with momentum $-k$,
• an electron-positron propagator (internal fermion line),
• a photon propagator or an external photon line,
• and a vertex connecting one photon line to two fermion lines. :contentReference[oaicite:23]{index=23}

The lecture then states several topology rules:

• lines connect only to vertices, not directly to other lines,
• propagators can be drawn curved,
• vertices are topological: their orientation or exact shape is unimportant,
• time runs from left to right only in the weak bookkeeping sense that incoming external lines are drawn on the left and outgoing ones on the right. :contentReference[oaicite:24]{index=24}

The slide also gives visual examples of:

• electron-positron annihilation,
• Compton scattering,
• light-by-light scattering,
• photon self-energy. :contentReference[oaicite:25]{index=25}

This page is the practical dictionary of the whole method.

14. Feynman rules for QED

Page 22 then refines the diagram components into QED Feynman-rule labels:

• initial and final electrons/positrons carry momentum $k$ and spin $\pm$,
• initial and final photons carry momentum $k$ and helicity $\pm$,
• internal lines correspond to propagators,
• vertices are the basic $A_\mu\bar\Psi\gamma^\mu\Psi$ interaction points. :contentReference[oaicite:26]{index=26}

The slide is mostly visual, but its role is clear: the diagrams are not merely pictures. They are code for exact algebraic factors.

15. Order counting in $\alpha$

The lecture then states the main perturbative counting rule.

For a contribution of order $\alpha^n$, one must write down all possible diagrams with

$$2n$$

vertices and the appropriate incoming and outgoing external lines. Then each diagram is decoded into a formula using the Feynman rules, and the sum of all distinct diagrams gives the order-$\alpha^n$ contribution to the invariant amplitude $M$. :contentReference[oaicite:27]{index=27}

This matches the structure of the QED interaction: each vertex brings one factor of $\sqrt{\alpha}$, so $2n$ vertices correspond to $\alpha^n$.

16. Compton scattering revisited

The last two pages apply this logic to Compton scattering.

The lecture says that, to the relevant order, one simply writes down all distinct diagrams with the correct external electron and photon lines and the right number of vertices. Then one decodes them with the Feynman rules and adds them.

The final slide states the practical payoff clearly: using the new path-integral and generator method, the correct Lorentz-invariant result for Compton scattering is obtained much faster than in the old Hamiltonian time-dependent perturbation theory. It also says that the simplicity of the diagrammatic representation allows us to understand general features of QED instead of getting hopelessly lost in enormous calculations for every possible term. :contentReference[oaicite:28]{index=28}

That is exactly the point of Feynman diagrams.

17. What this lecture really established

This lecture established three things at once.

1. Free QED is controlled by one Gaussian generator

Once the free Dirac and photon sectors are assembled covariantly, the whole noninteracting theory is encoded in one Lorentz-invariant Gaussian generating functional $Z_0$.

2. Interacting QED is generated by source differentiation

The cubic interaction $A_\mu\bar\Psi\gamma^\mu\Psi$ acts as a differential operator on $Z_0$. Perturbation theory is therefore a controlled expansion in source derivatives.

3. The surviving terms have a graphical pattern

That pattern is captured by Feynman diagrams, and the diagrams package the entire combinatorics of perturbative QED into a small set of graphical rules.

So the diagrams are not decoration. They are a compressed form of the covariant perturbation theory derived from the path integral.

Worked Examples

Example 1: Why every QED vertex has one photon line and two fermion lines

The interaction term in the Maxwell–Dirac action is

$$\sqrt{4\pi\alpha}\int d^4x\, A_\mu \bar\Psi \gamma^\mu \Psi.$$

This contains:

• one factor of the photon field $A_\mu$,
• one factor of $\bar\Psi$,
• one factor of $\Psi$.

So every interaction insertion must connect exactly:

• one photon line,
• and two fermion lines.

That is why the basic QED vertex always has that topology.

Example 2: Why order $\alpha^n$ corresponds to $2n$ vertices

Each QED interaction vertex contributes one factor of $\sqrt{\alpha}$, because the coupling is proportional to $\sqrt{4\pi\alpha}$. Therefore a diagram with $V$ vertices contributes at order

$$(\sqrt{\alpha})^V = \alpha^{V/2}.$$

So if we want order $\alpha^n$, we must include diagrams with

$$V=2n$$

vertices.

Intuition

The intuition here is simple once you stop thinking of the diagrams as pictures of particles literally moving around.

The real object is the Lorentz-invariant generating functional of the free theory. Interactions are added by differentiating it with respect to source fields. If you did all that differentiation blindly, you would get a huge number of terms. But almost all of them vanish once the sources are set to zero. The survivors follow a very rigid connectivity pattern.

Feynman diagrams are just the bookkeeping device for those surviving connectivity patterns.

So a line is not “a little particle trajectory” in the naive sense. It is the graphical stand-in for either:

• an external state insertion,
• or a propagator coming from the Gaussian source functional.

And a vertex is the graphical stand-in for one insertion of the interaction term in the action.

That is why the method is so powerful: it translates an enormous symbolic differentiation problem into topology.

Common Mistakes

• Thinking Feynman diagrams are merely heuristic cartoons. In the lecture they arise from exact variational structure of the path integral.
• Forgetting that the real starting point is the Gaussian generating functional $Z_0$, not the diagrams themselves.
• Thinking the raw transition matrix element must itself be Lorentz invariant. The lecture says the invariant object is $M$, after factoring out the kinematic normalization.
• Treating external lines and internal propagators as the same thing. They play different roles in the rules.
• Forgetting that in QED the vertex always joins one photon line with two fermion lines.
• Miscounting perturbative order: each vertex contributes $\sqrt{\alpha}$, not $\alpha$.
• Assuming the arrows and shapes in the diagrams have rigid geometric meaning. The lecture explicitly says vertices are topological; orientation and exact shape do not matter.
• Forgetting that the point of the whole formalism is to replace long Hamiltonian calculations by manifestly covariant ones.

Short Summary

The lecture begins by reviewing the path-integral logic developed in the previous lectures: path integrals give an equivalent formulation of quantum time evolution, and in relativistic quantum field theory their great advantage is that they use the Lagrangian rather than the Hamiltonian, making every step Lorentz invariant. It recalls the Gaussian generating functionals for driven bosonic and fermionic degrees of freedom and the fact that all noninteracting transition amplitudes can be generated by source derivatives of the vacuum-to-vacuum amplitude. It then extends this logic from single modes to full quantum fields: the Dirac path integral becomes a Gaussian functional integral over $\Psi$ and $\bar\Psi$, while the electromagnetic field yields the corresponding Gaussian functional integral for the photon sector. After the Coulomb-term trick and the restoration of full $A_\mu$ integration from the previous lecture, the noninteracting QED theory is encoded in one fully Lorentz-invariant Gaussian generating functional

$$Z_0[J,\bar J,f_\mu].$$

Interacting QED is then obtained by treating the Maxwell–Dirac interaction

$$\sqrt{4\pi\alpha}\int d^4x\, A_\mu \bar\Psi \gamma^\mu \Psi$$

as a differential operator acting on $Z_0$. The lecture emphasizes that the transition amplitude between specific external states is not itself Lorentz invariant because the noninteracting external states are not defined covariantly, but it is proportional to a Lorentz-invariant amplitude $M$, which is obtained from Lorentz-invariant combinations of source variations. The huge set of possible source-differentiation terms turns out to collapse into a simple connectivity pattern, and this pattern is represented graphically by Feynman diagrams. The lecture defines the QED diagrammatic components:

• external electron, positron, and photon lines,
• internal fermion and photon propagators,
• and the QED vertex joining one photon line to two fermion lines. It then states the practical rule: to compute a contribution of order $\alpha^n$, draw all diagrams with $2n$ vertices and the required incoming and outgoing lines, translate each diagram into a formula using the Feynman rules, and add the results. Applied to Compton scattering, this method reproduces the correct Lorentz-invariant answer far more efficiently than direct Hamiltonian time-dependent perturbation theory, and the lecture ends by stressing that the real power of the diagrammatic method is that it lets one see general features of QED instead of getting lost in massive algebraic expansions. :contentReference[oaicite:29]{index=29}

Practice Problems

1. Why does the lecture say that path-integral perturbation theory is manifestly Lorentz invariant while Hamiltonian perturbation theory is not?

2. What role does the Gaussian generating functional $Z_0$ play in perturbative QED?

3. Why are propagators already present before any interaction is added?

4. Why does the QED interaction term imply that every vertex has one photon line and two fermion lines?

5. Why is the physically central quantity the invariant amplitude $M$ rather than the raw transition matrix element?

6. Why does a diagram with $V$ vertices contribute at order $\alpha^{V/2}$?

7. What is the difference between an external line and an internal propagator in a Feynman diagram?

8. Why does the lecture say that lines connect only to vertices and not directly to other lines?

9. What does it mean that QED vertices are topological?

10. Why do source variations produce a huge number of terms, but only a simple subset survive when the sources are set to zero?

11. Why does the diagrammatic method simplify Compton scattering so much compared with Hamiltonian TDPT?

12. In your own words, explain why Feynman diagrams are not just pictures but a compressed bookkeeping language for Lorentz-invariant perturbation theory.