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PHYS3036 · Condensed Matter and Particle Physics

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Chapter 12 of 13 · PHYS3036

Symmetries, Conservation Laws & the Quark Model

This chapter of University of Sydney PHYS3036 makes symmetry the organising principle. Noether's theorem ties continuous symmetries to conservation laws; the discrete symmetries C, P and T and the gauge/flavour quantum numbers (charge, baryon and lepton number, strangeness) decide which processes are allowed. The quark model organises hadrons into strong-isospin multiplets on (Iz, Y) axes, with the Gell-Mann–Nishijima relation linking charge to isospin and hypercharge, while QCD adds colour, gluons and confinement. Exam questions apply conservation laws and multiplet reasoning.

In this chapter

What this chapter covers

01Noether's theorem: every continuous symmetry gives a conservation law (time → energy, space → momentum, rotation → angular momentum)
02Discrete symmetries C (charge conjugation), P (parity), T (time reversal); intrinsic parity; the fall of parity in weak decays
03Conserved quantum numbers: electric charge, baryon number, lepton number; strangeness (conserved by strong/EM, violated by weak)
04Gell-Mann–Nishijima relation Q/e = Iz + ½(Y + C + B + T); strong hypercharge Y = B + S
05Strong isospin (I, Iz) and multiplet diagrams on (Iz, Y) axes; isospin partners have similar masses
06Clebsch–Gordan coefficients for combining strong isospin (from the exam data sheet)
07QCD: colour charge (three colours) resolving the Δ⁺⁺ problem, gluons, confinement and asymptotic freedom

Worked example · free

Gell-Mann–Nishijima: charges of the proton and neutron from quark content

Q [4 marks]. The Gell-Mann–Nishijima relation for light hadrons is Q/e = Iz + ½Y with strong hypercharge Y = B + S (B = baryon number, S = strangeness). The up quark has Iz = +½ and the down quark Iz = −½. Using the quark contents p = uud and n = udd, compute Iz, Y and the electric charge Q of the proton and the neutron, and check against their known charges. (4 marks)

+1Proton (uud) third component of isospin: Iz = (+½) + (+½) + (−½) = +½ (two u at +½, one d at −½). [+1]
+1Proton hypercharge: it is a baryon so B = +1, and it contains no strange quarks so S = 0 ⇒ Y = B + S = 1. Then Q/e = Iz + ½Y = ½ + ½(1) = +1. ✓ (the proton has charge +1). [+1]
+1Neutron (udd) third component of isospin: Iz = (+½) + (−½) + (−½) = −½ (one u, two d). Hypercharge: B = +1, S = 0 ⇒ Y = 1. [+1]
+1Neutron charge: Q/e = Iz + ½Y = (−½) + ½(1) = 0. ✓ (the neutron is neutral). Both results match, confirming the relation and the quark assignments. [+1]

Proton: Iz = +½, Y = 1, Q = +½ + ½ = +1. Neutron: Iz = −½, Y = 1, Q = −½ + ½ = 0. The Gell-Mann–Nishijima relation reproduces the known charges from the quark content, with the proton and neutron sitting as an isospin doublet (same Y, Iz = ±½).

Sia tip — Add the quark Iz values, get Y from B + S, then Q = Iz + ½Y. The proton and neutron form an isospin doublet (same Y, opposite Iz), which is why their masses are so close. All the quantum numbers you need are on the data sheet. Ask Sia to run the same check on the Σ or Ξ baryons to practise non-zero strangeness.

Glossary

Key terms

Noether's theorem: The result that every continuous symmetry of a system implies a conserved quantity — time-translation gives energy, space-translation momentum, rotation angular momentum.
C, P, T: The discrete symmetries charge conjugation (particle ↔ antiparticle), parity (spatial inversion) and time reversal; the weak interaction violates P (and CP).
Baryon / lepton number: Additively conserved quantum numbers counting baryons and leptons (minus their antiparticles); their conservation forbids many otherwise-allowed processes.
Gell-Mann–Nishijima relation: Q/e = Iz + ½(Y + C + B + T): the electric charge of a hadron in terms of its isospin projection and flavour hypercharge; for light hadrons Q/e = Iz + ½Y with Y = B + S.
Strong isospin (I, Iz): An approximate symmetry treating u and d as two states of one particle; hadrons fall into multiplets of definite I, plotted on (Iz, Y) axes, whose members have similar masses.
Colour (QCD): The strong-force charge carried by quarks and gluons in three varieties; it resolves the Δ⁺⁺ spin-statistics puzzle and underlies confinement and asymptotic freedom.

FAQ

Symmetries, Conservation Laws & the Quark Model FAQ

How do I decide whether a reaction is allowed?

Check the conserved quantities at once: electric charge, baryon number, lepton number and energy–momentum must balance for any interaction; strangeness (and other quark flavours) must balance for strong and electromagnetic processes but may change by one unit in weak decays. If any strictly conserved quantity does not balance, the process is forbidden; if only a flavour changes, it must go by the weak interaction. This checklist is a staple of the exam.

What does Noether's theorem give me in practice?

It explains why the conservation laws exist: energy conservation from time-translation symmetry, momentum from space-translation, angular momentum from rotational symmetry. Recognising a symmetry therefore tells you a conserved quantity, and vice versa — a powerful shortcut when reasoning about which processes can occur and which quantum numbers to track.

Why do isospin partners like the proton and neutron have nearly equal masses?

Because strong isospin is an approximate symmetry under which the up and down quarks are almost interchangeable. Hadrons that differ only by swapping u ↔ d sit in the same isospin multiplet, and the strong force treats them almost identically, so their masses are close; the small splitting comes from the up–down mass difference and electromagnetism. The quark model plots these multiplets on (Iz, Y) axes.

What's examined from this chapter?

Applying conservation laws to allow/forbid processes, using Gell-Mann–Nishijima and quark content to get charges and quantum numbers, reading and predicting multiplet diagrams, and the basics of QCD (colour, the Δ⁺⁺ argument, confinement). The data sheet supplies quantum numbers and Clebsch–Gordan coefficients, so drill the reasoning. Confirm the paper's structure on Canvas and the unit outline.

Study strategy

Exam move

Make conservation-law bookkeeping automatic: for any process, line up charge, baryon number, lepton number and (for strong/EM) strangeness, and know that the weak interaction is the one that can change flavour and violate parity. Practise Gell-Mann–Nishijima (Q = Iz + ½Y, Y = B + S) both ways — from quark content to charge and back — and rehearse placing hadrons on (Iz, Y) multiplet diagrams, using the isospin-partner rule to predict similar masses. Add the QCD essentials: colour resolving the Δ⁺⁺ puzzle, gluons, confinement and asymptotic freedom. Because the data sheet carries quantum numbers and Clebsch–Gordan tables, focus your revision on the reasoning rather than recall, and keep it warm for quizzes and the final. When a multiplet or an allowed/forbidden call is unclear, ask Sia to walk the conserved quantities one by one.

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