Compton Scattering

Introduction

This lecture marks a real turning point in the course.

Up to now, perturbative QED has been introduced in a very structural way:

• the full Coulomb-gauge QED Hamiltonian was written down,
• the fine-structure constant $\alpha$ was identified as the small expansion parameter,
• and time-independent perturbation theory was used to study the vacuum and one-electron states.

That already revealed something important and uncomfortable: the apparently simplest perturbative questions are not actually the easiest ones.

The vacuum energy shift is large and needs to be cancelled by a $c$-number counterterm. The electron rest energy gets a logarithmically divergent correction that can be absorbed into the physical electron mass. And the photon energy correction is worse: it is quadratically divergent, and there is no photon rest-mass parameter already present in the Hamiltonian that could absorb it. The lecture says this very directly. That is why the “obvious” first perturbative calculations are actually surprisingly problematic.

So instead of beginning with the easiest-looking stationary perturbation problems, the lecture moves to an apparently more advanced process that is actually conceptually cleaner:

Compton scattering, the scattering of a photon by an electron.

This is a first-order-in-$\alpha$ QED process that is finite and physically measurable. Historically, Compton scattering helped confirm Einstein’s photon idea by showing that the change in photon wavelength after scattering depends on the scattering angle exactly as a relativistic particle picture predicts:

$$\Delta \lambda = \lambda_C(1-\cos\theta),$$

with $\lambda_C$ the electron Compton wavelength. But QED should do much more than reproduce the kinematic shift. It should give the amplitude for scattering into any final momentum and polarization state. :contentReference[oaicite:3]{index=3}

The lecture’s real point is not just to derive a cross section. It is to show something much deeper:

• in the Hamiltonian formulation, the individual intermediate-state processes contributing to Compton scattering are not Lorentz invariant by themselves,
• but when the correct set of electron and positron processes is summed coherently, a Lorentz-invariant amplitude appears.

That is the first big glimpse of how relativity and quantum mechanics coexist in interacting QFT. They do not coexist because every intermediate process is individually relativistic in an obvious way. They coexist because the Hamiltonian has a very special structure, enforced by relativity and gauge invariance, and this causes apparently noncovariant pieces to combine into Lorentz-invariant observable results. :contentReference[oaicite:4]{index=4}

The lecture ends by saying this is exactly why a more efficient formalism is needed. The direct Hamiltonian perturbation-theory calculation is already unpleasantly complicated for Compton scattering. That is what motivates the upcoming move to Feynman’s Lagrangian perturbation theory.

Learning Objectives

• Recall the full Coulomb-gauge QED Hamiltonian and identify the free and interaction parts.
• Understand why low-order stationary perturbation theory in QED immediately raises vacuum-energy, electron-mass, and photon-energy problems.
• Explain why the electron mass correction is logarithmically ultraviolet sensitive but still absorbable into the physical mass.
• Explain why the photon energy correction is more problematic than the electron mass correction.
• Understand why gauge-invariant regularization must make the photon-energy correction vanish.
• Explain why Compton scattering is the first clean finite order-$\alpha$ process in perturbative QED.
• Set up Compton scattering as a time-dependent perturbation-theory problem in the interaction picture.
• Distinguish “true scattering” from forward propagation with no momentum exchange.
• Show why $\hat H_{\text{Coulomb}}$ does not contribute to non-forward electron-photon scattering at first order in $\alpha$.
• Explain why second-order TDPT in $\hat H_{\text{int}}$ is required to get the lowest nontrivial Compton amplitude.
• Identify the four intermediate-state processes contributing to Compton scattering at this order.
• Understand the role of Lorentz-invariant $k$-space probability density measures.
• Explain why individual process amplitudes are not Lorentz invariant in the Hamiltonian formalism.
• Show that Lorentz invariance appears only after coherent addition of the appropriate electron and positron channels.
• Understand why this motivates Feynman perturbation theory.

Prerequisite Knowledge

• QED Hamiltonian in Coulomb gauge
• Time-independent and time-dependent perturbation theory
• Interaction picture
• Dirac spinors $u$ and $v$
• Photon polarization vectors
• Lorentz-invariant 4-momentum conservation
• Delta distributions in Fourier integrals
• Basic Compton-scattering kinematics

1. Review: the Coulomb-gauge QED Hamiltonian

The lecture begins by reviewing the full QED Hamiltonian:

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

The free Hamiltonian is

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

with

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

The Coulomb term is

$$\hat H_{\text{Coulomb}} = \alpha \hbar c \int d^3r\,d^3r' \frac{:\hat\psi^\dagger(r)\hat\psi(r)\hat\psi^\dagger(r')\hat\psi(r') :} {|r-r'|},$$

while the light-matter interaction is

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

The lecture also writes out $\hat H_{AJ}$ explicitly in momentum space, where one sees that it contains eight distinct operator processes: photon absorption, photon emission, pair production, and pair annihilation, each with electron/positron variants.

This is the starting point for all perturbative QED calculations.

2. Why the apparently simplest perturbative problems are actually hard

Before turning to Compton scattering, the short B4 deck reviews why the most obvious stationary perturbative problems are already troublesome.

Vacuum energy

Earlier time-independent perturbation theory produced a nonzero second-order vacuum energy correction $E_{{\rm vac},2}$, which must be subtracted because the vacuum energy must vanish by Lorentz invariance. So the Hamiltonian must include the $c$-number subtraction $\Delta E$. :contentReference[oaicite:6]{index=6}

Electron mass

A similar second-order calculation for the one-electron rest state gives an ultraviolet-divergent correction. The lecture states that at large momentum cutoff $K$,

$$M_2 \propto M \ln\!\left(\frac{\hbar K}{Mc}\right).$$

Because this divergence is only logarithmic, one can regard the physical electron mass as the sum of a bare mass parameter and perturbative corrections. In other words, the theory does not predict the electron mass from nothing; it renormalizes a mass parameter already present. :contentReference[oaicite:7]{index=7}

Photon energy

The photon-energy correction is more serious. The lecture says the second-order photon energy shift is quadratically divergent,

$$\sim \alpha K^2,$$

and unlike the electron case there is no bare photon mass parameter in the Hamiltonian that can be adjusted. Worse, the observed photon dispersion is exactly the zeroth-order one, $E=\hbar ck$, so the correction must be made to vanish, not merely to become small. :contentReference[oaicite:8]{index=8}

This is why the lecture says the seemingly simplest perturbative calculations are surprisingly problematic.

3. Gauge-invariant regularization and the photon-mass problem

The short deck then explains the basic resolution strategy.

One must regularize the divergent integrals. Many ad hoc regularizations can make integrals finite, but the lecture says the simple ones typically do not make the bad photon-energy integral vanish, and they also break gauge invariance at very short wavelengths. By contrast, certain more sophisticated gauge-invariant regularizations do force this photon-energy correction to vanish exactly. The lecture is honest that this is not a satisfying dynamical explanation; it is a systematic mathematical modification required to preserve gauge invariance and agree with observations. :contentReference[oaicite:9]{index=9}

This sets the stage for why we leave those stationary problems aside for the moment and study a cleaner observable process.

4. Why start with Compton scattering

The longer lecture then says directly: logically one might have expected to begin with time-independent perturbation theory of the vacuum, one photon, or one electron. But those calculations are surprisingly problematic. So instead we begin with Compton scattering, which is conceptually easier to understand. :contentReference[oaicite:10]{index=10}

Compton scattering is:

• photon-electron scattering,
• finite at order $\alpha$,
• and experimentally important.

The lecture reviews the standard kinematic result obtained from 4-momentum conservation. In the rest frame of the initial electron, if the incoming photon defines the $z$-axis, then the change in photon wavelength is

$$\Delta \lambda = \lambda_C(1-\cos\theta),$$

where $\theta$ is the photon scattering angle and $\lambda_C$ is the electron Compton wavelength. The slide emphasizes that this kinematic conservation law is obviously Lorentz invariant. :contentReference[oaicite:11]{index=11}

But the real question is: can QED compute the probability for scattering into a given angle and final state, and will that probability come out Lorentz invariant?

5. Why time-dependent perturbation theory is the right tool

The lecture next explains why time-dependent perturbation theory (TDPT) is the correct framework.

Electron-photon scattering is a relatively rare event in ordinary conditions. So one imagines the incoming photon and electron as well-separated wave packets in the distant past, which overlap only briefly during the collision and then separate again in the distant future. During that short overlap time, the small interaction Hamiltonian changes the state only perturbatively. Therefore the process is naturally formulated in TDPT, evolving from

$$t_i\to -\infty \qquad\text{to}\qquad t_f\to +\infty.$$

The object of interest is the scattering amplitude from an initial wave-packet electron-plus-photon state to a final one. The lecture then moves to the interaction picture to organize the perturbation expansion. :contentReference[oaicite:12]{index=12}

6. Interaction picture setup

The lecture reviews the interaction picture in the standard way. With

$$\hat H = \hat H_0 + \hat H_I,$$

the interaction-picture time evolution operator is

$$\hat U_I(t_f,t_i) = \mathcal T \exp\left( -\frac{i}{\hbar}\int_{t_i}^{t_f} dt\, \hat H_I^I(t) \right),$$

where $\mathcal T$ denotes time ordering. The interaction-picture creation and annihilation operators acquire the usual free oscillatory phases under $\hat H_0$. The lecture then states that the desired amplitude is a matrix element of $\hat U_I$ between initial and final one-electron/one-photon states. :contentReference[oaicite:13]{index=13}

This is the usual scattering-theory setup.

7. Scattering versus not scattering

The lecture is careful to distinguish true scattering from trivial forward propagation.

Because the full Hamiltonian is translation invariant, total momentum is conserved. But there is always a trivial “no scattering” contribution in which the photon and electron simply pass through without exchanging momentum. In that case each particle’s momentum is separately conserved.

Compton scattering means something different:

• total energy-momentum is conserved,
• but the photon and electron momenta are not separately conserved,
• there is actual momentum exchange.

So from the start the lecture focuses on non-forward scattering, where the incoming and outgoing photon momenta differ. :contentReference[oaicite:14]{index=14}

This matters because some terms in the Hamiltonian can only contribute to self-energy or forward propagation and should not be counted as genuine Compton scattering at this order.

8. Why $\hat H_{\text{Coulomb}}$ does not contribute at lowest order

The lecture then asks: what combinations of the Hamiltonian contribute, at first order in $\alpha$, to Compton scattering?

It answers:

• $\hat H_{\text{Coulomb}}$ is itself order $\alpha$, so first-order TDPT in $\hat H_{\text{Coulomb}}$ would nominally be order $\alpha$,
• but $\hat H_{\text{Coulomb}}$ contains no photon operators,
• so it cannot change the photon momentum,
• therefore it cannot produce non-forward electron-photon scattering.

So $\hat H_{\text{Coulomb}}$ contributes only to self-energy-type effects here, not to the actual Compton process. Meanwhile, $\hat H_{AJ}$ cannot contribute at first order in TDPT because it changes photon number by one, so one insertion cannot take a one-photon initial state to a one-photon final state.

Therefore the first nontrivial Compton amplitude comes from second-order TDPT in $\hat H_{AJ}$, which is still overall order $\alpha$ because $\hat H_{AJ}\sim \sqrt{\alpha}$. :contentReference[oaicite:15]{index=15}

This is a very important structural point.

9. The eight elementary photon-matter processes

The lecture then unpacks $\hat H_{AJ}$. When expanded, it contains eight terms corresponding to:

• photon absorption,
• photon emission,
• pair production,
• pair annihilation,

each with electron/positron channels as appropriate. The lecture uses diagrams on the slide to make this visually obvious. :contentReference[oaicite:16]{index=16}

The question then becomes: which combinations of two such elementary processes can start from an initial electron+photon state and end in a final electron+photon state without reducing to trivial non-scattering?

10. The four second-order processes for Compton scattering

The lecture states that there are four such second-order processes:

1. Absorption, then emission
   The incoming electron absorbs the photon, then emits the outgoing photon.

2. Emission, then absorption
   The electron first emits, then later absorbs.

3. Pair production with absorption, then annihilation with emission
   A pair-production event occurs in the intermediate state.

4. Pair production with emission, then annihilation with absorption
   The most complicated channel, involving a five-particle intermediate state.

The slide explicitly counts the number of particles in the intermediate state:

• 1 for the simplest channel,
• 3 for two of the channels,
• 5 for the most complicated one.

The lecture then emphasizes a deep consequence of particle creation and annihilation: what classically looks like simple two-body scattering becomes, in QFT, a problem involving intermediate sectors with up to five particles even at second order. :contentReference[oaicite:17]{index=17}

This is one of the clearest illustrations of why relativistic QFT is not just relativistic particle mechanics.

11. The actual second-order TDPT amplitude

The lecture then turns from pictures to mathematics.

Expanding the interaction-picture evolution operator to second order gives

$$\hat U_I = 1 -\frac{i}{\hbar}\int dt\, \hat H_I^I(t) -\frac{1}{\hbar^2}\int dt\int_{-\infty}^t dt'\, \hat H_I^I(t)\hat H_I^I(t') + O(\alpha^{3/2}).$$

For non-forward Compton scattering, the first-order term vanishes for the reasons already explained, so one must compute the second-order matrix element

$$\langle f|\hat U_I^{(2)}|i\rangle.$$

The lecture writes this explicitly in terms of the photon and fermion creation/annihilation operators and then notes that the four non-vanishing terms in the expansion correspond exactly to the four intermediate-state processes sketched earlier. :contentReference[oaicite:18]{index=18}

This is the direct Hamiltonian perturbation-theory version of what later becomes a sum over Feynman diagrams.

12. Photon factor and fermion factor

The lecture then evaluates the operator algebra in pieces.

First comes the photon factor, where the only nonzero terms come from pairing creation and annihilation operators in the correct way. This produces the expected delta functions and oscillatory factors. The slide on page 15 explicitly notes that one term vanishes for $q\neq q'$, so it is dropped. :contentReference[oaicite:19]{index=19}

Then comes the fermion factor. For the simplest process, the lecture shows that only one term survives when there are no intermediate positrons. The spinor structure is then simplified using the standard $u$-spinor completeness identity

$$\sum_s u_s(\vec k)\bar u_s(\vec k) = \frac12\left( 1-\frac{\gamma^\mu k_\mu}{Mc} \right)$$

in the lecture’s conventions. The point is that the complicated sums over spin indices collapse into covariant spinor-projector expressions. :contentReference[oaicite:20]{index=20}

This is where the relativistic spinor structure enters.

13. Time integrals and distributions

After doing the momentum delta-function integrals, the lecture tackles the two time integrals $t,t'$. It changes variables to

$$s = t-t',$$

so the integrals separate. The first integral gives the familiar delta function enforcing kinetic-energy conservation in the scattering:

$$2\pi\, \delta(\omega' + c q' - \omega - c q).$$

The second integral is subtler and is treated distributionally, with the standard $i\epsilon$-type convergence prescription justified by the fact that the physical problem really involves wave packets rather than exact plane waves. The lecture carefully explains why the $s$-integral can be treated as a distribution after introducing an exponentially decaying regulator and then removing it. :contentReference[oaicite:21]{index=21}

This produces the usual energy-denominator structure of second-order perturbation theory.

14. Lorentz-covariant $k$-space probability density

The lecture then pauses to discuss Lorentz covariance of momentum-space probabilities.

It shows that the invariant measure in Fourier 3-space is

$$\frac{d^3k}{k_0},$$

and therefore a probability density in momentum space is Lorentz invariant only if the amplitude carries the right factors of $1/\sqrt{k_0}$ and $1/\sqrt{q_0}$. The slide makes this very explicit by computing the Jacobian of a boost in the $x$-direction and showing that $d^3k/k_0$ is invariant. :contentReference[oaicite:22]{index=22}

This is a very important technical point. The Hamiltonian formalism is not manifestly covariant, so one must keep close track of the momentum-space normalization factors if one wants Lorentz-invariant probabilities.

15. The first process amplitude is not Lorentz invariant

For the simplest process, the lecture finally writes the second-order amplitude. It contains:

• 4-momentum-conservation delta functions,
• polarization vectors,
• spinor numerators,
• and a time-ordered energy denominator.

But then the lecture says something crucial:

This amplitude is not Lorentz invariant. :contentReference[oaicite:23]{index=23}

That is not a mistake in the algebra. It is a real feature of the Hamiltonian decomposition. The lecture notes that one can make the polarization vector look more covariant by padding the spatial polarization 3-vector into a 4-vector with zero time component in the Hamiltonian frame, but even then the full expression still fails to be Lorentz invariant by itself.

That is one of the deepest lessons of the lecture.

16. The most complicated process and the second non-invariant partner

The lecture then turns to the most complicated process — the one with five particles in the intermediate state. It works out the corresponding photon factor, fermion factor, and spinor structure. After similar time-integral manipulations, it obtains another amplitude, again not individually Lorentz invariant. :contentReference[oaicite:24]{index=24}

So now we have two separate non-invariant amplitudes.

17. A miracle: the sum is Lorentz invariant

Then the lecture reaches its key point:

When the two non-invariant amplitudes are added together, a miracle occurs — the sum is Lorentz invariant. :contentReference[oaicite:25]{index=25}

The same thing happens for the other pair of second-order processes. Each pair combines into a Lorentz-invariant amplitude with the same overall delta-function prefactor but different internal numerator structure. Finally, the sum of all four processes gives the full first-order Compton-scattering amplitude. :contentReference[oaicite:26]{index=26}

This is the main conceptual result: Lorentz invariance is not visible process by process in the Hamiltonian picture. It appears only after coherent superposition of all relevant channels.

18. Why positron processes are essential

The lecture spells this out in words:

Absorption and emission of photons by an electron alone is not by itself Lorentz invariant. Lorentz invariance only comes through coherent addition of processes in which electrons are annihilated or created together with positrons. Specifically, the two kinds of processes that combine to restore Lorentz invariance are related by replacing an intermediate electron by a positron while reversing the order of photon absorption and emission in time. The lecture explicitly invokes Feynman’s later language: the relativistic process is always a superposition of an electron moving forward in time and a positron moving backward in time. :contentReference[oaicite:27]{index=27}

This is a major conceptual insight into relativistic QFT: relativity and quantum mechanics are reconciled not by making each individual particle process obviously relativistic, but by building a Hamiltonian whose different channels interfere in just the right way.

19. Why this motivates Feynman perturbation theory

The lecture ends by being very frank.

This direct Hamiltonian TDPT calculation is already surprisingly complicated for such a simple process as Compton scattering, and this is only lowest order in $\alpha$. The motivation to find a more efficient formalism is therefore extremely high. The lecture says this directly and points ahead to Feynman’s Lagrangian perturbation theory, which will make these calculations enormously simpler by exploiting Lorentz covariance from the outset. :contentReference[oaicite:28]{index=28}

So the lecture’s final lesson is: interacting relativistic quantum field theory is a very special class of quantum mechanics, and it only gives physically sensible results because of hidden structural conspiracies in the Hamiltonian. Feynman perturbation theory is designed precisely to expose and exploit those conspiracies.

Worked Examples

Example 1: Why $\hat H_{\text{Coulomb}}$ does not contribute to Compton scattering at lowest order

The Coulomb term is already order $\alpha$, so first-order TDPT in $\hat H_{\text{Coulomb}}$ would also be order $\alpha$. But $\hat H_{\text{Coulomb}}$ contains only fermion operators and no photon operators. Therefore it cannot change the momentum of the incoming photon. So it cannot contribute to genuine non-forward electron-photon scattering. At this order it only contributes to self-energy-type corrections.

Example 2: Why second-order TDPT in $\hat H_{AJ}$ is still order $\alpha$

The light-matter interaction $\hat H_{AJ}$ carries one factor of $\sqrt{\alpha}$. A single insertion cannot take a one-photon state to a one-photon state, because it either creates or destroys a photon. Therefore the first nontrivial scattering amplitude comes from two insertions:

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

So Compton scattering appears at order $\alpha$ but through second-order time-dependent perturbation theory.

Intuition

This lecture is really about two things.

First, Compton scattering is the first place where perturbative QED gives a clean observable answer without immediately drowning in renormalization problems. That is why the lecture moves to scattering even though, logically, one might have expected to begin with stationary state corrections.

Second, the lecture gives a deep lesson about relativistic QFT: individual intermediate-state stories are not sacred. They are basis-dependent pieces of the perturbation expansion. What is physically meaningful is the sum of all processes that the Hamiltonian allows. Only that coherent sum is Lorentz invariant.

So QFT is not “classical relativistic particle mechanics plus quantization.” It is a much more constrained quantum theory whose physical consistency relies on interference among processes that look completely different in intermediate steps.

Common Mistakes

• Thinking the simplest perturbative QED problems should be one-photon or one-electron energy shifts. In practice those are already renormalization problems.
• Forgetting why the photon energy correction is more problematic than the electron mass correction.
• Treating Compton scattering as if it should appear already at first order in $\hat H_{AJ}$. It cannot, because one insertion changes photon number by one.
• Forgetting that $\hat H_{\text{Coulomb}}$ has no photon operators and therefore cannot produce non-forward electron-photon scattering at lowest order.
• Thinking the intermediate-state pictures are themselves observable. They are not.
• Assuming a single Hamiltonian process amplitude should be Lorentz invariant by itself. The lecture shows it is not.
• Missing that positron channels are essential even when the initial and final states involve only electrons and photons.
• Confusing gauge-invariant physical amplitudes with the noncovariant appearance of the intermediate Hamiltonian expressions.
• Missing the reason Feynman perturbation theory is introduced: it is not stylistic, it is necessary for practical efficiency and manifest covariance.

Short Summary

The lecture begins by reviewing the Coulomb-gauge QED Hamiltonian and the earlier perturbative QED results showing that the vacuum energy shift must be cancelled, the electron mass correction is logarithmically ultraviolet sensitive, and the photon energy correction is quadratically divergent and can only be made to vanish by a gauge-invariant regularization. This explains why the apparently simplest stationary perturbative calculations are already problematic. The lecture therefore turns to Compton scattering, which is a finite order-$\alpha$ process. In the rest frame of the initial electron, 4-momentum conservation gives the standard Compton wavelength shift

$$\Delta\lambda = \lambda_C(1-\cos\theta),$$

but QED should also give the scattering amplitude itself. Because the photon and electron start and end as widely separated wave packets that overlap only briefly, the process is naturally treated with time-dependent perturbation theory in the interaction picture. The lecture shows that the Coulomb term cannot contribute to non-forward electron-photon scattering at first order in $\alpha$, and that a single insertion of the photon-matter interaction $\hat H_{AJ}$ cannot connect one-photon initial and final states. Therefore the leading Compton amplitude comes from second-order TDPT in $\hat H_{AJ}$, which is still order $\alpha$. Expanding $\hat H_{AJ}$ reveals eight elementary photon-matter processes, and four combinations of two such processes contribute to Compton scattering: absorption then emission, emission then absorption, and two positron-related pair-production/annihilation channels. The lecture computes the corresponding matrix elements directly in the Hamiltonian formalism, carefully handling delta functions, spinor identities, Lorentz-covariant momentum-space measures, and distributional time integrals. It then finds the crucial result: the individual process amplitudes are not Lorentz invariant by themselves, but pairs of them add to Lorentz-invariant expressions, and the full sum of all four processes gives the Lorentz-invariant Compton amplitude. This shows that relativistic quantum field theory is not simply a relativistic version of ordinary quantum mechanics for particles; rather, Lorentz invariance emerges through coherent quantum interference among electron and positron processes allowed by the special structure of the QED Hamiltonian. The lecture ends by motivating Feynman perturbation theory as a more efficient and manifestly covariant formalism for exactly such calculations.

Practice Problems

1. Why does the lecture say that the apparently simplest perturbative QED problems are surprisingly problematic?

2. Why can the electron mass correction be absorbed into a physical mass parameter, while the photon energy correction is more serious?

3. Why must the problematic photon-energy integral be made to vanish rather than merely become finite?

4. Why is Compton scattering a cleaner first perturbative observable than the stationary one-particle energy shifts?

5. Why is time-dependent perturbation theory the natural framework for Compton scattering?

6. Why does $\hat H_{\text{Coulomb}}$ not contribute to non-forward Compton scattering at lowest order?

7. Why does a single insertion of $\hat H_{AJ}$ fail to connect the initial and final one-photon states?

8. What are the four second-order processes that contribute to Compton scattering?

9. Why do intermediate states with up to five particles appear in what classically looks like a two-body scattering problem?

10. Why is the measure $d^3k/k_0$ important for Lorentz-covariant momentum-space probabilities?

11. Why are the individual Hamiltonian process amplitudes not Lorentz invariant?

12. Why do positron-related channels have to be included even though the initial and final states contain only an electron and a photon?