Subtleties

You compute the loop integral in $d = 1, 2, 3, 4, \dots$ dimensions, and then continue that sequence to non-integer $d$.

A function is not determined by its values on the integers, so there would be nothing unique about the result. The uniqueness theorem for analytic continuation asks for agreement on a nondiscrete set — an open region, or at minimum a line or a ray — and the integers are discrete. $\sin(\pi d)$ vanishes at every integer and is emphatically not the zero function; adding it to any candidate answer would give another one. What is actually continued is the formula the angular and radial integrations produce, which is a function of a complex variable $d$ already defined on a region. Four is simply where that function happens to have a pole.

The phrase "$d$ dimensions" invites you to picture a spacetime and then to imagine sliding the number of directions in it, which makes the integers the obvious starting point — they are the only values you can picture. The continuation being unique is also usually stated as a reassurance rather than a theorem with hypotheses, so there is nothing in the telling to check the picture against. And the mistake is almost harmless in practice: the answer you get is right, because the formula was the thing being continued all along. It matters when you try to say why the answer is allowed to be unique, which is the question dimensional regularization stands on.

The $+i\varepsilon$ prescription is what makes the propagator causal.

It makes it time-ordered. The Feynman propagator is small but genuinely nonzero outside the light cone — it falls off like $e^{-m|x-y|}$ rather than vanishing. What does vanish at spacelike separation is the commutator $[\phi(x),\phi(y)]$, and that is the object carrying causality.

Both statements are about the same theory, and both are things you want to be true, so the wrong one slides past unchallenged — especially since “nothing outruns light” is a belief you arrived with. The resolution is that a nonzero amplitude outside the light cone is not the same as an influence outside it: the amplitude for a particle to go from $x$ to $y$ is exactly cancelled by the amplitude for its antiparticle to go from $y$ to $x$, which is what makes the commutator vanish while the propagator does not. Some sources do call a propagator “the causal propagator” — but they mean the Pauli–Jordan commutator function, not this one, which is precisely the collision that keeps the confusion alive.

For an unpolarized cross section, sum over all the spins and divide by the number of spin states — here $\tfrac14 \sum_{\text{spins}} |\mathcal{M}|^2$.

The two halves are doing opposite jobs. Over the initial spins you average, because a real unpolarized beam delivers each combination a quarter of the time and you want the expected rate. Over the final spins you sum, because a detector that cannot tell them apart counts every one of them as an event. The $\tfrac14$ therefore belongs to the initial state alone — it is $2 \times 2$ incoming combinations, not the four outgoing ones.

The formula is almost always written as one compressed expression, $\tfrac14\sum_{\text{spins}}$, in which a single sum sign covers both jobs and the $\tfrac14$ floats in front of everything — so it reads as one operation with a normalization rather than two operations with opposite meanings. It also happens that this process has four initial and four final spin states, which makes the numbers look symmetric exactly where the logic is not. The error is quiet when it happens: dividing by the final states too costs a clean factor of four, which is easy to mistake for a slip in the algebra rather than a misread of what the average is for.

Summing over photon polarizations gives $\sum_\lambda \varepsilon^\mu \varepsilon^{*\nu} = -g^{\mu\nu}$ in Feynman gauge.

The genuine completeness relation over the two physical polarizations is transverse — schematically $\sum_r \varepsilon^i_r \varepsilon^j_r = \delta^{ij} - p^i p^j/|\vec p|^2$, which is not $-g^{\mu\nu}$ and cannot be made into it by any choice of gauge. Replacing it with $-g^{\mu\nu}$ is legitimate only inside an amplitude contracted with a conserved current, where the leftover $p^\mu p^\nu$ pieces vanish because $p_\mu \mathcal{M}^\mu = 0$.

The replacement is used so constantly, and stated so briefly, that it reads as a definition of the sum rather than a step with a precondition — and it is almost always written with an equals sign. What makes it durable is that its two halves live in different places: a book states the transverse relation when it quantizes the photon, then performs the replacement two chapters later when it squares an amplitude, and rarely on the same page. It is also the same mechanism that lets the gauge parameter $\xi$ drop out of every observable, so a reader who has already accepted that has met the argument once without being told they would need it again.

Those are two different diagrams — the photon lines cross in one and not the other.

They are the same diagram. A Feynman diagram is a topological object — which lines join which vertices, what particle each line is, which way the arrows run. Position on the page is not part of it, and neither is whether two lines happen to cross.

The converse is the half that catches people: two drawings can differ almost invisibly and still be different diagrams. The s- and u-channel here share their external legs, their vertices and their internal electron, differing only in which photon attaches where — and that difference is real, with a different propagator denominator behind it.

Because diagrams are introduced as pictures and drawn by hand, so everything visible about them feels meaningful: where a vertex sits, whether lines cross, which way the page is oriented. Most of that is decoration. But some visual features do encode topology — which leg meets which vertex, the direction of an arrow — and nothing on the page marks which kind you are looking at. Worse, the two failure modes point opposite ways: a reader can see a difference that is not there, or miss one that is. The skill is reading a drawing for its graph and discarding the rest, and it is not learned by being told once. It is learned by meeting cases where appearance and topology disagree — which is why this lesson draws its second diagram the unconventional way on purpose.

The Klein–Nishina formula was the first calculation in quantum electrodynamics.

It was the first application of the Dirac equation to the scattering of light, which is not the same claim. Klein and Nishina treated the electron quantum-mechanically and the electromagnetic field as a classical external wave — a semi-classical calculation. Quantum electrodynamics as a theory of a quantized photon field did not exist in 1929; the covariant formulation is a further two decades away. The result nonetheless agrees exactly with the modern field-theoretic one, and the historical record shows why: their method used a quadratic form of the equation that never required the negative-energy states to appear explicitly.

Because the formula is correct, and because it is taught today from the field-theoretic derivation — squaring an amplitude with two external photons is a standard exercise in every QED course, this one included. Working backwards from how it is now derived to how it was originally obtained is a natural inference, and a wrong one. The name reinforces it: "the first QED calculation" is a tidy story, and the tidier version has been repeated often enough to sound settled. What makes the truth more interesting than the myth is what it says about the formula — that it does not actually depend on the photon field being quantized, which is exactly why the same expression falls out of a soft-photon classical limit at one end and a full field theory at the other.

For $\Delta < 0$ the logarithm is $\ln\Delta = \ln|\Delta| + i\pi$, by the principal branch.

A true statement about the principal branch, and a false one about the amplitude. Three things are collapsed there and they have different statuses: the branch point at $\Delta = 0$ is intrinsic to the logarithm; the cut drawn from it is a free choice, and moving it changes no physics; the side you approach from is fixed by the Feynman prescription, and reversing it flips the sign of $\mathrm{Im}\,\Pi$ — a negative cross section, which is the time-reverse rather than another convention. Strictly, $\ln\Delta$ for $\Delta < 0$ is not defined until you say how you got there; what the calculation contains is $\lim_{\delta\to0^{+}}\ln(\Delta - i\delta) = \ln|\Delta| - i\pi$. What survives every choice is the discontinuity across the cut, $2\pi i$, whichever cut you drew.

The principal branch is so standard that "the branch cut" reads as a feature of the logarithm rather than a decision someone made, so where the cut is and which side you are on merge into one question — and only one of them is physics. The notation is complicit: $\ln\Delta$ records neither, so a genuinely ambiguous expression looks fully determined and gets resolved by whichever rule the reader happens to recall. It is also easy to be right for the wrong reason. Invoking the principal branch and landing on $-i\pi$ is the same error as invoking it and landing on $+i\pi$; the first one just agrees with the answer.

The on-shell condition is $q^2 = m^2$ and the scalar propagator is $i/(q^2-m^2)$.

True in the mostly-minus signature $(+,-,-,-)$, which this course uses along with Peskin and Tong. In the mostly-plus signature $(-,+,+,+)$ — used by Srednicki among others — the same physics is written $q^2 = -m^2$ with the propagator $-i/(q^2+m^2)$.

Neither choice is more correct, and no calculation depends on it: every observable comes out the same. But the two conventions produce different-looking formulas for the objects you are most likely to compare between books, and the difference appears exactly where a beginner is least able to tell a convention from an error — inside a propagator denominator, next to a mass. The tell is the on-shell condition: find how a source writes $q^2$ for a real particle, and you know which world you are in before you compare anything else.

The photon propagator carries a gauge parameter $\xi$, with $\xi = 1$ Feynman gauge and $\xi = 0$ Landau gauge.

That is this course’s convention, and a common one. But Tong writes the same parameter as $\alpha$ (with the identical meaning: $\alpha = 1$ Feynman, $\alpha = 0$ Landau), and other sources use a $\lambda$ that is the reciprocal, so that Landau gauge sits at $\lambda \to \infty$ rather than at zero.

A named parameter feels like part of the physics, so it is reasonable to assume the name travels with it. This one does not, and because it is unphysical — it cancels from every observable — nobody is ever forced to reconcile the conventions, which is exactly why several survive. Check which gauge the source calls “Feynman” before trusting any formula containing the parameter; that single value pins down the convention, since $\xi = 1$ and $\lambda = 1$ agree there and diverge everywhere else.

The gauge transformation is $\psi \to e^{i\alpha}\psi$ with $A_\mu \to A_\mu - \tfrac{1}{e}\partial_\mu\alpha$.

That is this course’s form. Tong writes the same transformation as $\psi \to e^{-ie\lambda}\psi$ with $A_\mu \to A_\mu + \partial_\mu\lambda$. These are not two transformations — they are one, with $\lambda = -\alpha/e$.

The choice is about whether the coupling $e$ lives in the phase or in the shift of the potential, and each convention makes a different downstream formula look tidy, so both are well motivated and both are common. To a reader comparing two books it presents as a disagreement about a sign and a factor of $e$ in the covariant derivative — the one place where a genuine sign error would also show up. Absorbing the coupling into the redefinition of an arbitrary function is always available, which is why nothing forces the field to settle on one.

A one-loop ultraviolet divergence shows up as a $1/\varepsilon$ pole, so two sources quoting a pole coefficient should agree on it.

Only if they define $\varepsilon$ the same way. This course, along with Peskin, writes $d = 4 - 2\varepsilon$; Srednicki writes $d = 4 - \varepsilon$. The physics is identical and every observable agrees, but a residue that reads $2/\varepsilon$ here reads $1/\varepsilon$ there, and the scale factor that keeps the coupling dimensionless is $\mu^{\varepsilon}$ in one convention and $\tilde\mu^{\varepsilon/2}$ in the other.

$\varepsilon$ is not a physical quantity — it is a bookkeeping variable that goes to zero at the end — so nothing ever forces two authors to agree on it, and both choices have a reason. The factor of two is chosen so that $d/2 = 2 - \varepsilon$ comes out clean; the factor of one is chosen so that $\varepsilon$ is literally how far from four dimensions you are. The trap is that both produce formulas of exactly the same shape, so there is no visual tell the way an inverted propagator denominator gives away a signature difference. The check takes one line: find how the source writes $d$ before comparing any coefficient.

Every free Lagrangian is quadratic in the same way: $\mathcal{L}_0 = \tfrac{1}{2}\phi K \phi$, with $K$ the operator from the equation of motion.

That $\tfrac{1}{2}$ is there for a real field, where the quadratic form is symmetric and would otherwise be counted twice. A complex field has no such double count: $\psi$ and $\bar\psi$ are independent, and the Dirac Lagrangian is $\bar\psi K \psi$ with no $\tfrac{1}{2}$.

The recipe genuinely is the same in every case that matters — invert the kinetic operator — so the natural move is to take the whole template literally, including the factor. It survives the photon, since $A_\mu$ is real too, which is quietly reassuring right up until the electron. Nothing downstream changes: the propagator is $i K^{-1}$ either way, and no factor of two appears in it. The mismatch is only in how the Lagrangian is written, which is why it is easy to notice and hard to interpret.

Every process has a total cross section $\sigma$: integrate $\mathrm{d}\sigma/\mathrm{d}\Omega$ over all angles.

Not when the mediator is massless. For $e^-\mu^- \to e^-\mu^-$ the propagator contributes $1/t^2$, and $t \to 0$ in the exact forward direction — a pole sitting inside the physical region. The angular integral diverges, so the total cross section does not exist as a number. What exists, and what an experiment reports, is $\sigma(\theta > \theta_{\min})$: the rate above some angular cut.

A total cross section is the first thing you learn to compute, and for annihilation it always works — there $q^2 = s$ is bounded below by a threshold, so nothing ever blows up. The divergence appears only once the exchanged momentum can reach zero, which is a feature of the channel rather than of the process, and nothing about the diagram announces the change. The instinct to treat the infinity as a mistake is also natural and wrong: it is the statement that a massless photon mediates a long-range force, so arbitrarily distant particles are deflected a little, and there are arbitrarily many of them. The infinity is physics. The cut is what turns it into a measurement.

There is no two-photon vertex because that would need a term quadratic in $A$ with no derivatives, and such a term is not gauge invariant.

The conclusion is right for QED and the reason is too broad. A bare $A_\mu A^\mu$ is indeed forbidden — it is a photon mass term. But "quadratic in $A$, no derivatives" is not in general non-invariant: scalar QED contains exactly such a term, the seagull $e^2 A_\mu A^\mu \phi^\dagger\phi$, and it is perfectly gauge invariant. The real reason is narrower: the covariant derivative acting on a Dirac field produces a coupling linear in $A$, so no two-photon vertex can arise.

The photon-mass argument is one of the most memorable results in the subject — it is the punchline of the gauge principle — so reaching for it whenever a term quadratic in $A$ appears is a well-trained instinct rather than a careless one. It happens to give the right answer here, which removes the pressure to check it. The counterexample is not obscure either: it sits one subsection after the QED Feynman rules in the same lecture notes this course cites, where the scalar seagull vertex is written down explicitly.

The fine-structure constant is $\alpha \approx 1/137$.

That is $\alpha$ at low energy, near the Thomson limit. It grows with the scale you probe at, reaching about $1/128$ at the mass of the $Z$ — and that quoted number is additionally scheme-dependent: the PDG value $\hat\alpha^{(5)}(M_Z)^{-1} = 127.930$ is an $\overline{\text{MS}}$ number, and quoting it without saying so leaves out part of its definition.

It is called a constant, it is measured to twelve significant figures, and it is the first number in the subject — three good reasons to treat it as a fixed property of nature rather than a function of the energy you are asking about. The scheme dependence is subtler still, because it does not come from the physics but from how the divergences were organised on the way to the number, so two correct calculations can legitimately report different values. This is not exotic: LEP measured the running directly, in Bhabha scattering, the process this course builds in lesson 2.3.

If $D \ge 0$ the diagram diverges, and if $D < 0$ it converges.

Neither half is safe. A diagram with $D \ge 0$ can be finite — light-by-light scattering counts as $D = 0$ and is convergent, because gauge invariance forces four powers of external momentum out of the answer that the counting never knew about. And a diagram with $D < 0$ can still diverge, if some sub-piece of it has its own $D \ge 0$: the overall count sees only the whole integral, not the loop hiding inside it.

The count is honest about what it does — it compares powers of the loop momentum upstairs and downstairs — and within that scope it is exact. What it cannot see is a cancellation, because a cancellation happens between terms of the same degree, and it cannot see a subdiagram, because it integrates every loop momentum to infinity at once. So the word doing the work is “superficial”, and it is easy to read as a hedge rather than as the precise technical qualifier it is. The practical rule survives regardless: the count tells you which finite list of amplitudes to worry about, and something else — a symmetry, usually — tells you which members of that list were never a problem.

The condition $\partial_\mu A^\mu = 0$ is the Lorentz gauge.

It is the Lorenz gauge, after the Danish physicist Ludvig Valentin Lorenz, who published it in 1867. Hendrik Antoon Lorentz is a different person — Dutch, later, and separately famous.

Both men worked on electromagnetism, both have their names on things a student meets in the same month, and they share credit for the Lorenz–Lorentz relation, so the confusion has a real basis rather than being a simple misspelling. It is also self-sustaining: the condition is manifestly Lorentz-invariant, which makes “Lorentz gauge” sound like it is describing the physics rather than crediting a person. Even careful authors slip — Tong titles the section “Lorentz Gauge” and then footnotes that it is named after Lorenz, “who had the misfortune to be one letter away from greatness”.

Compton scattering is the Compton effect — a photon bounces off an electron and comes back longer, by the Compton wavelength.

Three distinct objects share the name, and they are not even the same kind of object.

• The Compton effect is the 1923 result $\lambda' - \lambda = \lambda_C(1-\cos\theta)$. It is pure energy–momentum conservation — no field theory, no cross section. It says where the light goes.
• Compton scattering, in this course, is the QED process $e^-\gamma \to e^-\gamma$, whose observable is a cross section: how much light goes where. That is the Klein–Nishina formula, and it needs the Dirac equation.
• The Compton wavelength $\lambda_C = h/m_ec \approx 2.43$ pm is a fixed length set by the electron mass. It is the coefficient in the shift formula, not the shift — the actual shift runs from zero (forward) to $2\lambda_C$ (backward).

All three came out of the same experiment and are introduced on the same page of most treatments, so nothing signals that one is a kinematic identity, one a dynamical prediction, and one a constant. The shift formula makes it worse by putting the constant and the effect in a single equation, which invites reading $\lambda_C$ as "the amount it shifts". There is also a concrete trap for anyone searching a text rather than reading it: Tong's notes contain the string "Compton" only in "Compton wavelength" and never compute the process at all, so a reader looking for the scattering calculation finds the length scale and reasonably concludes the topic is covered.

QED was constructed first, and the gauge principle was recognised afterwards as the deep reason it works.

The order is the other way round. Fock wrote down the local-phase/potential link in 1926, and Weyl elevated it to a stated principle in 1929 — both roughly two decades before covariant QED was completed by Tomonaga, Schwinger, Feynman and Dyson between 1946 and 1950.

The tidy story — physics first, principle later — is the shape most discoveries actually have, and it is the shape this material is usually taught in, since a course derives QED and then remarks on the symmetry. There is also a true statement nearby that is easy to slide into the false one: the classical current–potential coupling did come first, and the gauge principle did come after that. Getting the order right matters for more than credit, because it means the principle was a genuine prediction about how to build a theory rather than a description of one already built — which is why it worked again for the weak and strong forces.