Lecture 5 – Symmetries and Isospin What is a symmetry ? Noether’s theorem and conserved quantities Isospin Obs! Symmetry lectures largely covers sections of the material in chapters 5, 6 and 10 in Martin and Shaw though not always in the order used in the book. FK7003

Symmetry – an intuitive example and a definition FK7003

Types of symmetry Continuous symmetries Depend on some continuous variable, eg linear translation which can be built up out of infinitesimal steps Eg linear spatial translation x’=x+d x, translation in time t’=t+d t Discrete symmetries (subsequent lectures) Two possibilities (all or nothing) Take a process, eg A+B C and measure it! Does this process happen at the same rate if Eg. The mirror image is considered (parity – move from left-handed to right-handed co-ordinate system) Eg. All particles are transformed to their anti-particles: A+B C (charge conjugation) FK7003

Conserved quantities in quantum mechanics FK7003

Conserved quantities – why we need them and how we find them FK7003

Translational invariance x x1 x2 x1+dx x2+dx FK7003

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What have we done ? FK7003

Noether’s theorem Symmetry Conservation Law Translation in space Previous slides showed a quantum mechanical ”version” of Noether’s theorem. If a system of particles shows a symmetry, eg its Hamiltonian is invariant to a continuous transformation then there is a conserved quantity. Symmetry Conservation Law Translation in space Linear momentum Translation in time Energy Rotation in space Angular momentum Gauge transformations Electric, weak and colour charge Discrete symmetries also lead to conserved quantities (to come) FK7003

Charge conservation – (not for lecture/exam) FK7003

Angular Momentum (no spin) y y’ xp P yp ’ x’ xp ’ yp x FK7003

Angular momentum FK7003

Spin Simplest case for studying angular momentum. Algebraically a carbon copy of orbital angular momentum: Useful to use matrices to represent states and operators. A spin ½ particle can have a spin-up or spin-down projection along an arbitary z-axis. General state - linear combination: Operators with 2x2 matrices. FK7003

Transformations Intuitively easy to understand how the components of a vector change under a rotation. Consider how a two component spinor is affected by a rotation of a co-ordinate system. FK7003

Question Find the matrix representing a rotation by p around the y-axis. Show it converts a spin-up to spin-down particle. Can transform between possible states – equivalent to rotating co-ordinate axes. Obs! – in an experiment a rotation would affect all states and there wouldn’t be observable effect in the physics measurement – nature doesn’t care where your axes are. y spin-up spin-down z FK7003

Question Repeat the procedure in the previous question and converts a spin-down to a spin-up particle. y spin-up spin-down z FK7003

Addition of angular momentum Two spin ½ particles can form multiplets A triplet and a singlet can be formed (n=2S+1) If nature chooses to use a multiplet, it must use all ”members”. Eg deuteron (np) – 3 x spin 1 states (SZ=-1,0,1) No stable spin 0 np state triplet (5.41) singlet (5.42) FK7003

SU(2) - isospin Proton and neutron have similar masses: Heisenberg postulated that proton and neutron are two states of the same particle – isospin doublet. Mass differences due to electromagnetic effects. Proton and neutron have different projections in internal ”isospin space”. Strong force invariant under rotations in isospin space. Isospin conserved for strong interactions (Noether’s theorem). FK7003

Testing isospin Best way to understand anything is to look at physical situations Two approaches to testing the hypothesis that isospin is a symmetry of the strong force. Isospin invariance If a strong reaction/decay takes place then the reaction/decay of the isospin- rotated particles must also happen at the same rate. If a particle within an isospin multiplet is found in nature then the other multiplet members must also exist since they correspond to different projections in isospin space and the strong force is invariant to a rotation in isospin space. Isospin conservation Isospin quantum numbers must ”add up” when considering reactions/decays. Both approaches are complementary (Noether’s theorem) FK7003

Isospin multiplets If the electromagnetic field could be turned off the masses of the particles within the isospin multiplets would be the same according to isospin symmetry. p (9383 MeV) n (9396 MeV) p0 (135 MeV) p+ (140 MeV) I3 p- (140 MeV) Multiplicity of states: N=2I+1 (5.43) FK7003

(5.44) (5.45) FK7003

Rotation in isospin space Slightly more interesting than rotations in real space. FK7003

Question FK7003

Gell-Mann-Nishijima Formula Baryon octet (spin ½) Meson nonet (spin 0) -1 -½ ½ +1 -1 -½ ½ I3 +1 I3 FK7003

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Conserved quantities/symmetries Quantity Strong Weak Electromagnetic Energy  Linear momentum Angular momentum Baryon number Lepton number Isospin - Flavour (S,C,B) Charges (em, strong and weak forces) Parity (P) C-parity (C) G-parity (G) CP T CPT FK7003

Summary Symmetry: i.e. under a given transformation certain observable aspects of a system are invariant to the transformation. If the Hamiltonian of a system is invariant to a symmetry operation a conservation law is obtained. Noether’s theorem Isospin is an algebraic copy of spin but covering an internal ”isospin space” Isospin symmetry a powerful way to understand the masses of particles and their reactions. Isospin violation/conservation can be studied by considering a rotation in isospin space or counting a conserved quantum number. FK7003