1. 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.
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2. Symmetry – an intuitive example and a definition
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3. 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)
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4. Conserved quantities in quantum mechanics
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5. Conserved quantities – why we need them and how we find them
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6. Translational invariancex x1 x2
x1+dx
x2+dx
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8. What have we done ?
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9. 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)
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10. Charge conservation – (not for lecture/exam)
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11. Angular Momentum (no spin)y y’ xp P yp ’ x’ xp ’ yp x
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12. Angular momentum
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13. 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.
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14. 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.
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15. 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
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16. Question
Repeat the procedure in the previous question and converts a spin-down to a spin-up particle. y
spin-up
spin-down z
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17. 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)
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18. 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).
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19. 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)
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20. 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)
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21. (5.44)
(5.45)
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22. Rotation in isospin space
Slightly more interesting than rotations in real space.
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23. Question
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24. Gell-Mann-Nishijima Formula
Baryon octet (spin ½)
Meson nonet (spin 0) -1 -½ +1 -1 -½ I3 +1 I3
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25. Question
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26. 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
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27. 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.
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