U(1) Define a property characterized by a single value. Define a property characterized by a single value. E.G. Objects color E.G. Objects color Change it uniformally (globally) without changing physics Change it uniformally (globally) without changing physics Change is QM change of phase Change is QM change of phase

SU(2) Mixture of two things Mixture of two things Same as misture of spin ½ z-components Same as misture of spin ½ z-components Equivalent to a rotation in a internal 3-4 space. Equivalent to a rotation in a internal 3-4 space. Changes the direction of the axis of quantization. Changes the direction of the axis of quantization.

Local Transformation depends on position Transformation depends on position

Space time symmetries MomentumPTranslation; Conserved ; labels states of a particle EnergyHTime evolution: Conserved; labels states of a particle Angular momentum LRotations: Sometimes, labels composite state rotation SpinSRotations: Magnitude S for a particle identifies particles; z-projection possible TotalJ=L+SRotations: Conserved Time reversalSym.Change time direction things remain same Isotopic spaceSym.Rotation and translation symmetry Lorentz Inv.Sym.Relativity and frames of reference equivalence Time evolution Sym.Physics is same at all times U(1) Symmetry ChargeQElectric charge,Conserved; labels particles PhaseAlways introduce an overall phase Particle same as an anti particle Charge Conjugation CGood Q# when particle=antiparticle Almost always conservved; labels particles sometimes. Particle-antiSym.Change particle to anti without impact (partially true)

Flavor approximate Flavoru,d,s,c,t,b; ; labels particles; good for E&M and QCD, NOT weak since (u,d) and due to CKM mass matrix. Flavor mixing, Partially for the QCD strangenessSMinus # strange quarks; labels particles see above comments charmCNumber of charmed labels particles see above comments beautyB~Minus number of bottom labels particles see above comments topnessT~Number of top labels particles see above comments Baryon NoB1/3[N(q)-N(qbar)] labels particles see above comments ChargeQDefined as above but =2/3[Nu+Nc+Nt] - 1/3[Nd+Ns+Nb] Complete flavor S,C,T~,B~,Q,B establish flavor uniquely One choice of Q#s that id a quark flavor. IsospinICombination of u-ness and d-ness Lepton number conservation Electron no.Conserved as far we know; labels particles Muon no.Conserved as far we know; labels particles Tau no.Conserved as far we know; labels particles The neutrinos are now believed to have mass. Once we allow neutrino mass, neutrino oscillations destroy strict lepton no Q# conservation except for the total. There is no decay CPTAny field theory will require that the product of CPT is conserved Chirality helicity Right handed left handed, for m=0 this character is maintained e,μ,τ,ν e,ν μ,ν τ,Lepton particles; labels particles

ClassInvarianceConserved quantity Proper orthochronous Lorentz symmetry Lorentz symmetry translation in time (homogeneity)homogeneity energy translation in space translation in space (homogeneity)homogeneity linear momentum rotation in space rotation in space (isotropy)isotropy angular momentum Discrete symmetryP, coordinate inversionspatial parity C, charge conjugationcharge conjugationcharge parity T, time reversaltime parity CPTproduct Internal symmetryU(1)U(1) gauge transformationgauge transformationelectric charge U(1)U(1) gauge transformationgauge transformation lepton generation numberlepton generation number (accidental symmetry) electron, muon Tau number U(1)U(1) gauge transformationgauge transformationBaryonBaryon number (accidental symmetry) U(1)U(1) gauge transformation hyperchargehypercharge B+S to label quark combos i.e. decuplet octet members U(1) U(1) Y gauge transformationgauge transformationweak hypercharge U(2) [U(1)xSU(2)]SU(2)electroweak force SU(2) gauge transformationisospin SU(2) SU(2) L gauge transformationweak isospin PxSU(2)G-parity SU(3) "winding number"baryon number SU(3) gauge transformationquark color SU(3)SU(3) (approximate)quark flavor S((U2)xU(3))[ U(1)xSU(2)xSU(3)]U(1)xSU(2)xSU(3)Standard Model

Rotating the arms through 90 o

A+B  C+D A  C+D+Bbar

Different time ordering same topology

Weak

QCD  pion exchange

example

Example