p+p p+p p-n p-n p-p p-p Consider the scattering reactions: The strong force does not discriminate between nucleon or pion charge. What can we expect for the cross section of these three reactions?

But p+p p+p p-n p-n p- + p  p- + p <p-p|L|p-p> = M + M If we enforce conservation of isospin we can only connect initial and final states of the same total I, I3 p+p p+p p-n p-n | 3/2, 3/2 > | 3/2, -3/2 > | I,I3 > 1 ½ 1 ½ -1 -½ -1 -½ But p- + p  p- + p -1 ½ -1 ½ means combining: | 1 -1 > | 1/2, 1/2 > = |3/2, -1/2 > - |1/2, -1/2 > ) this interaction involves two matrix elements <p-p|L|p-p> = M + M

p+ + p p+ + p a. b. p- + p p- + p c. p- + p p0 + n 1,1 ½ ,½ elastic scattering p- + p p- + p 1,-1 ½ ,½ but only one of the above can also participate in a p- + p p0 + n ? charge exchange process ? 1,-1 ½ ,½ 1, 0 ½ ,-½ This IS observed! So all strong interactions not SIMPLY charge independent. I3 ISOSPIN independence is more general.

p+ + p p+ + p a. b. c. p- + p p- + p p- + p p0 + n p+ p p- p p0 n 1,1 ½ ,½ elastic scattering p- + p p- + p 1,-1 ½ ,½ p- + p p0 + n charge exchange process 1,-1 ½ ,½ 1, 0 ½ ,-½ These three interactions involve the ISOSPIN spaces: p+ p p- p 1 3 2 3 - p0 n 2 3 1 3 + 2 2 M½ Recall: = Mfi M3/2 Let’s denote: same by I3-indep.

a. b. c. p+ + p p+ + p p- + p p- + p p- + p p0 + n p+ p p- p p0 n elastic scattering p- + p p- + p p- + p p0 + n charge exchange process p+ p M½ p- p 1 3 2 3 - M3/2 p0 n 2 3 1 3 + a. b. c. a  M3/2 2 b | M3/2 + M1/2| 2 1 3 2 3 c | M3/2 - M1/2| 2 2 3 2 3

a : b : c = : : p+ + p p+ + p a. a  M3/2 b. c. b | M3/2 + M1/2| 2 p- + p p- + p 1 3 2 3 p- + p p0 + n c | M3/2 - M1/2| 2 2 3 2 3 a : b : c = : : 2 M3/2 1 9 |M3/2 +2M1/2|2 2 9 |M3/2 - M1/2|2 for the combined cross section of both processes Now if M3/2=M1/2 then +p = -p total but also -p0n=

 Measured the depletion of pion beam 2H target S1 S2 S3 S4 Measured the depletion of pion beam repeated with the tank full, empty repeated with + and  - beam for KE  195 MeV (the resonance of the 3/2-spin ) 200 300 Total Cross Section sT (10-27 cm2) p- + p p+ + p gp po p p0p 180 250 160 200 140 120 150 100 100 80 50 60 40 40 60 100 200 400 Photon Beam Energy (MeV) 20 40 100 200 400 Lab Energy of Pion Beam (MeV)

a : b : c = : : a : b + c = : a : b : c = 9 : 1 : 2 M3/2 |M3/2 +2M1/2|2 2 9 |M3/2 - M1/2|2 a : b + c = : 2 M3/2 a : b : c = 9 : 1 : 2

 (U) (U) U† U U†H U= H [H ,U]= 0 [H ,U]= 0 [H ,G]= 0 Symmetry implies any transformation  still satisfies the same Schrödinger equation, same Hamiltonian:  (U) (U) U† U U†H U= H means we must demand: [H ,U]= 0 Which means that the operator U must be associated with a CONSERVED quantity! Though U are UNITARY, not necessarily HERMITIAN, but remember: where the G is Hermitian! since you’ve already shown [H ,U]= 0 [H ,G]= 0 The GENERATOR of any SYMMETRY OPERATION is an OPERATOR of a CONSERVED OBSERVABLE (quantum number!)

Mesons Baryons isospin mass charge pion 139.569 + +1 134.964 0 0 Particle I3 MeV/c2 states Q pion 139.569 + +1 134.964 0 0 139.569 - -1 +1 -1 eta 548.8 0 0 rho 770. + +1 0 0 - -1 +1 -1 omega 783.0 0 0 Baryons Spin-1/2 nucleon 938.280 p +1 939.573 n 0 +1/2 -1/2 Spin-3/2 delta 1232. ++ +2 + +1 0 0 - -1 +3/2 +1/2 -1/2 -3/2

“hypercharge” or BARYON NUMBER Q = I3 + ½Y “hypercharge” or BARYON NUMBER because =1 for baryons 0 for mesons

cloud chamber cosmic ray event 1947 Rochester and Butler cloud chamber cosmic ray event of a neutral object decaying into two pions K0  + +

cloud chamber cosmic ray event 1947 Rochester and Butler cloud chamber cosmic ray event of a neutral object decaying into two pions K0  + + m = 497.72 MeV 1949 C. F. Powell photographic emulsion event K+  + + m = 493.67 MeV

  p +  -  - p 1950 Carl Anderson (Cal Tech) m=1115.6 MeV mp=938.27 MeV

1952 Brookhaven Cosmotron 1st modern accelerator artificially creating these particles for study 1954 6.2-GeV p synchrotron Lawrence,Berkeley 1960 28-GeV p synchrotron CERN, Geneva 33-GeV p synchrotron Brookhaven Lab 1962 6-GeV e synchrotron Cambridge 1963 12.5-GeV p synchrotron Argonne Lab 1964 6.5-GeV p synchrotron DESY,Germany 1966 21-GeV e Linac SLAC (Standford)

isospin mass charge Spin-0 Pseudoscalar Mesons Particle I3 MeV/c2 states Q pion 139.569 + +1 134.964 0 0 139.569 - -1 +1 -1 kaon 493.67 K+ +1 497.72 K0 0 +1/2 -1/2 kaon 497.72 K0 0 493.67 K- -1 +1/2 -1/2 eta 548.8 0 0 rho 770. + +1 0 0 - -1 +1 -1 omega 783.0 0 0 Spin-1/2 Baryons nucleon 938.280 p +1 939.573 n 0 +1/2 -1/2 lambda 1115.6  0 Sigma 1385. + +1 0 0 - -1 +1 -1 Cascade 1533. + +1 - -1 +1/2 -1/2

Spin-3/2 Baryons isospin mass charge Delta 1232. ++ +2 + +1 0 0 Particle I3 MeV/c2 states Q Delta 1232. ++ +2 + +1 0 0 - -1 +3/2 +1/2 -1/2 -3/2 Sigma-star 1385. + +1 0 0 - -1 +1 -1 Cascade-star 1533. *+ +1 *- -1 +1/2 -1/2