Analyticity and the Antiparticles · Elementary Particle Physics

One function describes an electron going one way and a positron going the other, so the two are not separate objects — they are the same object evaluated in different places.

🎯 Why this matters

Antiparticles are therefore not an extra assumption anyone made. Relativity allows a boost to reverse the order of two space-like events, and a theory without antiparticles cannot survive that.

Three pages, and the best idea in the chapter. It starts as a remark about complex analysis and ends by proving that antiparticles have to exist.

One function, several processes

Take any two-body scattering $a + b \to c + d$ and form the two invariants of §1.4 — they are what name the s and t channels :

\htmlClass{t-s}{s} = \left(E_a + E_b\right)^2 - \left(\mathbf{p}_a + \mathbf{p}_b\right)^2, \qquad \htmlClass{t-t}{t} = \left(E_b - E_a\right)^2 - \left(\mathbf{p}_b - \mathbf{p}_a\right)^2

(5.40–5.41)

Bettini p. 201. Two numbers, and s ≥ 0 while t ≤ 0.

Every symbol, one at a time

Hover or tap a symbol above — it lights up in the equation and its meaning, units and type appear here.

The amplitude is an analytic function of these two variables. That is not a technical nicety: analytic continuation means one function, evaluated in different regions, describes different physical processes.

s channel — e⁺e⁻ → μ⁺μ⁻

Click a vertex or an internal line.

The virtual photon has a positive mass², so it can hit a real one.

t channel — e⁻μ⁻ → e⁻μ⁻

Click a vertex or an internal line.

Same function, different region. Here mass² is negative and nothing can resonate.

💡 What this really says — “virtual” is a matter of degree

The two diagrams above are drawn differently and describe different experiments, but they are the same analytic function evaluated at different arguments. Move continuously from one region to the other and you move continuously between the processes.

Which forces a conclusion the book states in one sentence and is easy to read past: the difference between a virtual particle and a real one is quantitative, not qualitative. A virtual photon with mass² far from zero is deeply virtual and lives no time at all. One with mass² close to zero is barely virtual. And when $s$ lands exactly on a real mass, the “virtual” particle is the real one — the propagator diverges, the cross-section peaks, and you have a resonance.

That is why §4.1’s Breit–Wigner and this section’s propagator are the same object: a pole in an analytic function, seen from two sides.

The two channels, and what each can and cannot do

channel	$\text{mass}^2\ \text{of the exchanged particle}$	can it resonate?	where you meet it
s	s ≥ 0	yes	Annihilation. Every resonance in Chapter 4 was found this way — scan √s and watch the cross-section jump when it crosses a real mass. The J/ψ, the ϒ, the Z.
t	t ≤ 0	no	Exchange. Rutherford scattering, Bhabha scattering, electron–muon scattering — the cross-section falls smoothly with \|t\| and has no structure at all.

<strong>When a = c and b = d the two channels contribute to the SAME process</strong> — e⁺e⁻ → e⁺e⁻ is Bhabha scattering, and Fig. 5.20 shows it drawn both ways. The cross-section is then the squared modulus of the <em>sum</em>, so the two interfere. §5.8 uses exactly that: at LEP the Bhabha t channel measures α at spacelike momenta, and staying away from the Z peak is what keeps the s channel from swamping it.

The argument that produces antiparticles

Now the fermion propagator, and Compton scattering as the example. Look at the t-channel diagram and ask when each vertex happened.

two events, one boost · when does "which came first" have an answer?

Δt = 1.00Δx = 2.20boost β = 0.000

interval Δt² − Δx²	-3.840 · space-like
in the boosted frame	Δt′ = 1.000, Δx′ = 2.200
interval, boosted	-3.840 — unchanged
which happened first?	A, then B

The interval is the same in both frames — it is the checksum of§1.4 applied to a separation rather than a momentum. What is notthe same is the sign of Δt, and only when the separation is space-like.

The separation is space-like (Δx² > Δt²), so B is outside the light cone of A. Nothing travelling at or below c can connect them — and yet the amplitude for the virtual particle to do so is not zero. Boost past β = Δt/Δx and the order reverses.

and this is where antiparticles come from

In this frame an electron is emitted at A and absorbed at B. Boost past β = 0.455 and the same two events happen in the opposite order — so a second observer sees a photon materialise at B into two electrons, one going forward and one going backward in time, the latter annihilating at A.

Neither description is privileged, so the theory must contain both — the virtual particle of one observer is the virtual antiparticle of the other. The sum over all configurations, which is what the diagram stands for, is Lorentz invariant. Every particle therefore has an antiparticle, fermion or boson, and that is a consequence of relativity and quantum mechanics together rather than an extra assumption.

⚙️ Engineer’s bridge — a sign that is not invariant, and one that is

The widget is one statement about a linear map. A boost is a hyperbolic rotation mixing $t$ and $x$ , and it preserves the quadratic form $\Delta t^2 - \Delta x^2$ exactly — that is the invariant, the checksum of §1.4, applied to a separation instead of a momentum.

What it does not preserve is the sign of $\Delta t$ on its own, and the condition is sharp: the sign survives every boost when $|\Delta t| > |\Delta x|$ , and can be reversed by a boost with $\beta > \Delta t/\Delta x$ when it is not. Causal order is frame-independent inside the light cone and frame-dependent outside it.

For an engineer, that is the difference between a quantity your protocol may rely on and one it may not. Two events at a space-like separation have no agreed ordering — there is no global clock, and any code that assumed one is wrong in some frame. Distributed systems make the same distinction and call it happens-before: events not causally connected simply have no order, and Lamport’s argument is the same argument.

Where it goes further than the analogy: in a distributed system the missing order is a limitation of what you can know. Here it is a statement about what exists — because the theory must give the same answer to both observers, the particle description and the antiparticle description have to be the same description, and antiparticles are forced into the theory.

💡 What this really says — the conclusion, spelled out

Set the widget to the space-like preset and push the boost past the critical value. Then read the same picture twice.

Observer 1 sees an electron created at A, travelling to B — outside the light cone, faster than light, which the amplitude permits because a virtual particle is not on the mass shell — and absorbed at B.

Observer 2, moving relative to the first, sees B happen first: a photon materialises at B into two particles. One goes forward in time. The other has the opposite charge and goes forward in time too — and it annihilates at A with the original electron.

Both descriptions are of the same events. Neither observer is privileged. So the theory has to contain both, which means it has to contain a particle with the electron’s mass and the opposite charge.

Antiparticles are not an extra postulate. They are what you get when you insist that quantum mechanics — which lets amplitudes leak outside the light cone — and special relativity — which denies a universal time ordering out there — hold at once. Every particle has one, fermion or boson. The photon is its own, which is why the wavy line in a diagram carries no arrow.

Helicity is conserved at a vertex

The last page of the section proves a rule used constantly from here on. At energies far above the masses, an electromagnetic vertex cannot flip a fermion’s helicity.

The chirality structure of the vector current, in 4×4 matrices

import numpy as np

# Dirac representation
I2 = np.eye(2); Z2 = np.zeros((2,2))
sx = np.array([[0,1],[1,0]], complex)
sy = np.array([[0,-1j],[1j,0]])
sz = np.array([[1,0],[0,-1]], complex)
g0 = np.block([[I2, Z2], [Z2, -I2]]).astype(complex)
gs = [np.block([[Z2, s], [-s, Z2]]).astype(complex) for s in (sx, sy, sz)]
gam = [g0] + gs
g5 = 1j * g0 @ gs[0] @ gs[1] @ gs[2]

PR = (np.eye(4) + g5) / 2          # right-chiral projector
PL = (np.eye(4) - g5) / 2

print("  projectors: PL + PR = 1?", np.allclose(PL + PR, np.eye(4)),
      " PL PR = 0?", np.allclose(PL @ PR, 0))
print("  gamma5 anticommutes with every gamma^mu?",
      all(np.allclose(g5 @ g + g @ g5, 0) for g in gam))

print("\n  the four chirality combinations of the vector current psibar gamma^mu psi:")
print("  (psibar_a means psibar P_b with b the OPPOSITE projector, because P gamma0 = gamma0 P')")
for la, Pa, name_a in [("L", PR, "psibar_L"), ("R", PL, "psibar_R")]:
    for lb, Pb, name_b in [("L", PL, "psi_L"), ("R", PR, "psi_R")]:
        big = max(np.abs(Pa @ g @ Pb).max() for g in gam)
        verdict = "VANISHES" if big < 1e-12 else "survives"
        print(f"    {name_a} gamma^mu {name_b:7s}: max|P gamma P| = {big:.1e}   {verdict}")

print("\n  so the current is DIAGONAL in chirality: a vertex cannot flip it.")
print("  At energies far above the mass, chirality = helicity, and that is why")
print("  Mott scattering vanishes at 180 degrees -- backscattering would need the")
print("  spin to keep its direction while the momentum reverses, i.e. a flipped helicity.")

prints

  projectors: PL + PR = 1? True  PL PR = 0? True
gamma5 anticommutes with every gamma^mu? True

the four chirality combinations of the vector current psibar gamma^mu psi:
(psibar_a means psibar P_b with b the OPPOSITE projector, because P gamma0 = gamma0 P')
  psibar_L gamma^mu psi_L  : max|P gamma P| = 5.0e-01   survives
  psibar_L gamma^mu psi_R  : max|P gamma P| = 0.0e+00   VANISHES
  psibar_R gamma^mu psi_L  : max|P gamma P| = 0.0e+00   VANISHES
  psibar_R gamma^mu psi_R  : max|P gamma P| = 5.0e-01   survives

so the current is DIAGONAL in chirality: a vertex cannot flip it.
At energies far above the mass, chirality = helicity, and that is why
Mott scattering vanishes at 180 degrees -- backscattering would need the
spin to keep its direction while the momentum reverses, i.e. a flipped helicity.

Two zeros in a 4 × 4 calculation, and they carry the whole rule. The reason is one line of algebra: $\gamma^\mu$ anticommutes with $\gamma^5$ , so $P_R\gamma^\mu = \gamma^\mu P_L$ , and $P_L P_R = 0$ .

📏 Chirality and helicity are not the same thing — they only coincide at high energy

The calculation above is about chirality, the eigenvalue of $\gamma^5$ , which is a Lorentz-invariant label. What experiments measure is helicity, the projection of spin on momentum, which is not invariant: overtake a massive particle and its momentum reverses while its spin does not, so its helicity flips.

The two agree only when the particle moves so fast that no observer can overtake it — that is, when $E \gg m$ . §2.9 made this distinction; this section is where it starts doing work, and the “with a very good approximation at ultra-relativistic energies” in Bettini’s statement is exactly this caveat.

For a strictly massless particle the two are identical and the rule is exact, which is why it is stated so confidently for neutrinos in Chapter 7.

Two consequences drawn immediately, and both are checkable

consequence	why it follows
Mott scattering vanishes at 180°	Backscattering reverses the momentum but leaves the spin alone, so the outgoing helicity is opposite to the incoming one. A vertex cannot do that. The zero in Eq. (1.95) that looked like an accident of the algebra is this rule.
in annihilation, the pair must have OPPOSITE helicities	The same statement read in the other kinematic region: a fermion and an antifermion couple to a photon only if their helicities are opposite. That is what fixes the (1 + cos²θ) angular shape of §5.7 — and the shape is measured, so the rule is measured.

Both come from one matrix identity, and both are experimental facts. That is the pattern of this whole chapter: a structural statement about the Lagrangian turning into a number a detector can read.

Erratum — the figure reference after the chirality proof

Having proved the annihilation case from Fig. 5.23(b), the book writes: “In a similar way one proves that the initial and final fermions in Fig. 5.26 are coupled only if their helicities are equal.”

It should be Fig. 5.23(a). “Initial and final fermions” describes the scattering vertex, which is 5.23(a) — the companion panel of the very figure just used. Fig. 5.26 is the four polarization configurations of $e^+e^-\to\mu^+\mu^-$ , two pages later, and is not a vertex diagram at all.

The sentence immediately following confirms it: “we can now understand why the probability of the elastic scattering of an electron by a massive target becomes zero at 180°” — elastic scattering, i.e. 5.23(a) again. The two panels of Fig. 5.23 are precisely the two cases being contrasted, same helicity for scattering and opposite for annihilation, so pointing at neither of them breaks the parallel the paragraph is drawing.

Aside — which mediators are their own antiparticles

The section closes by applying the antiparticle argument to the gauge bosons themselves.

The photon is completely neutral, so photon and antiphoton are the same particle. That is why the wavy line carries no arrow: emitted at the upper vertex and travelling backwards, or emitted at the lower and travelling forwards, are the same statement.
The Z⁰ is likewise its own antiparticle.
W⁺ and W⁻ are each other’s. A W⁺ running backwards in time is a W⁻ running forwards, so strictly the weak lines should carry arrows — the book notes this and then, reasonably, does not bother.
Of the eight gluons, two are completely neutral and the other six form three particle–antiparticle pairs. Chapter 6.

🔑 If you remember only three things

Nothing quantum enters until the last line. The argument runs entirely on what a boost does to two events, and the physics arrives only when the picture is reinterpreted.
Crossing is a labour-saving device as much as a principle. Compute once, move to another region of the same function, and a scattering result answers a question about annihilation.
Every antiparticle in the catalogue is now accounted for. What Chapters 2 to 4 recorded as a list of coincidences becomes a requirement of the framework.

Where this goes next

§5.7 computes e⁺e⁻ → μ⁺μ⁻ with everything this section established: an s-channel photon, helicity conservation fixing the angular distribution, and a cross-section falling as 1/s.
§5.8 uses the other channel — Bhabha scattering measures α at space-like momentum transfer, where t < 0 and there is no resonance to contend with.
§4.1 is worth rereading now: a Breit–Wigner is a pole in this section’s analytic function, approached from the s-channel side.
Chapter 7 adds the axial current to the vector one, and the helicity rule proved here becomes the reason the weak interaction violates parity.

✅ Check yourself — analyticity, antiparticles and helicity

0/5 answered · 0 correct

1.Why can an s-channel process resonate while a t-channel one never can?
2.Set the light-cone widget to the space-like preset and boost past the critical β. What changes and what does not?
3.How does that reversal force antiparticles to exist?
4.The snippet finds that two of the four chirality combinations of $\bar\psi\gamma^\mu\psi$ vanish identically. What is the one-line reason, and what does it buy?
5.Why does the section say helicity conservation holds only 'to a very good approximation at ultra-relativistic energies'?