Conserved numbers and discrete symmetries

Two different kinds of rule share the word conservation. Charge, energy-momentum and angular momentum are conserved because of Noether's theorem: each follows from a continuous symmetry of the action, and no experiment is expected to break them. CPT invariance is a theorem too, forced by Lorentz invariance, locality and a Hermitian Hamiltonian in any quantum field theory. Everything else on the ledger was invented to explain a missing process. Baryon number exists because the proton is stable — the partial lifetime for p → e⁺π⁰ now exceeds 10³⁴ years, some twenty-four orders of magnitude beyond the age of the universe. Separate electron, muon and tau numbers exist because μ⁻ → e⁻γ has never been seen, down to a branching ratio near 10⁻¹³. Strangeness exists because kaons and hyperons are produced copiously and then decay absurdly slowly. Assigning a quantum number records an absence in arithmetic; it does not explain it.

Given a proposed reaction, work down a fixed list and stop at the first failure. Charge is exact and takes one line. Baryon number is exact at every Standard Model vertex: count quarks minus antiquarks and divide by three. Then run the three lepton-family ledgers separately — this is where μ⁻ → e⁻γ dies, with ΔL_μ = −1 and ΔLₑ = +1 even though the total lepton number is untouched. Then energy: a decay needs a positive Q value, a collision needs invariant mass above threshold. Only after all that do you look at flavour, and flavour asks a different question. A change of strangeness, charm or beauty does not veto a process; it routes it, because the strong and electromagnetic interactions conserve flavour and the charged weak current does not. The first four tests answer whether the process can happen at all. The fifth answers who has to do it, and therefore how slowly.

Strangeness is conserved exactly by the strong and electromagnetic interactions and broken by the weak one, by at most one unit for each W exchanged, so a weak decay has |ΔS| ≤ 1 and |ΔS| = 2 processes, which need two W's, are suppressed far below that. One rule settles the whole strange-particle puzzle of the 1950s. In π⁻ + p → K⁰ + Λ⁰ the strangeness ledger reads 0 → (+1) + (−1) = 0, so the strong interaction is free to make the pair, and it does in about 10⁻²³ s. Neither product can then decay strongly, because that would change S on its own, so the Λ⁰ lives 2.63 × 10⁻¹⁰ s — some 10¹⁴ times its own production time. Isospin is approximate for a different reason: it treats u and d as one doublet, which works only because m_d − mᵤ and α are both small beside the QCD scale. The error is set by those ratios themselves — m_d − mᵤ ≈ 2.5 MeV against a QCD scale near 200 MeV, and α ≈ 1/137, each about one per cent — so isospin predictions are worth about one per cent and no more. The proton–neutron splitting, 1.3 MeV on 939 MeV, is smaller still because the quark-mass and electromagnetic shifts partly cancel.

Parity inverts every spatial coordinate, r → −r. Polar vectors — position, momentum, electric field — change sign under it. Axial vectors, built from a cross product, do not: L = r × p picks up two sign changes, and spin transforms the same way. So a spin dotted into a momentum, ⟨J⟩⋅p̂, changes sign under P where an ordinary scalar does not. It is a pseudoscalar, and if the interaction responsible commuted with P its average would have to vanish exactly. That is what makes the Wu experiment a proof rather than a hint. Cobalt-60 polarised near 0.01 K emits its beta electrons preferentially opposite to the nuclear spin, following 1 − P (v/c) cos θ. With a polarisation near 0.6 and v/c near 0.6 the front-to-back modulation is about 36 per cent, and it reverses when the polarising field reverses, which is what rules out an apparatus artefact.

Write the projector P_L = (1 − γ⁵)/2. The charged weak vertex is ū γ^μ (1 − γ⁵) u, which is 2 ū γ^μ P_L u: the W couples to the left-handed chiral field and has no coupling whatever to the right-handed one. In the massless limit chirality equals helicity, so the neutrino from a weak decay is purely left-handed and its mirror image, a right-handed neutrino, has zero amplitude. A mirror-reflected weak process does not merely run at a different rate — the leading term does not run at all. That is what maximal means, and it is the same statement as the vector and axial couplings being equal in size, g_A/g_V = −1 at the quark level, dressed by QCD to −1.27 for the nucleon. The electromagnetic and strong vertices are pure γ^μ, which commutes with parity, so they cannot generate a cos θ term at all. The neutral weak current is the intermediate case: the Z couples to both chiralities of every charged fermion, with unequal strengths fixed by the fermion's charge, its weak isospin and sin²θ_W ≈ 0.231, so it violates parity by an amount that varies from fermion to fermion — the electron's left–right coupling asymmetry is only about 0.15 — rather than maximally across the board.

Charge conjugation fails as badly as parity: apply C to a left-handed neutrino and you get a left-handed antineutrino, which the weak current does not couple to and which has never been seen. The combination CP maps a left-handed neutrino onto a right-handed antineutrino, which does exist, so CP very nearly survives — and for the seven years after Wu it was taken to be exact. It is not. The long-lived neutral kaon, nominally CP-odd, decays to π⁺π⁻ with a branching ratio of 1.97 × 10⁻³. CPT then forces a T violation of exactly the same size, and the K⁰ ↔ K̄⁰ rate difference has been measured directly. In B mesons the effect is not small at all: sin 2β ≈ 0.70. This matters beyond spectroscopy, because Sakharov showed that a universe ending up with more matter than antimatter needs CP violation. The CKM phase supplies some, and falls short of the observed baryon asymmetry by roughly ten orders of magnitude — one of the clearest places the Standard Model is known to be incomplete.