U(1)U(1) Define a property characterized by a Define a property characterized by a

single value.single value. E.G. Objects colorE.G. Objects color Change it uniformally (globally) without Change it uniformally (globally) without

changing physicschanging physics Change is QM change of phaseChange is QM change of phase

ie

SU(2)SU(2) Mixture of two thingsMixture of two things Same as misture of spin ½ z-componentsSame 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.

Perform a global SU(2) transformation. You change the direction of the internal symmetry arrow. This could mean that you mix an amount of neutron into a proton or add some up quark to a down quark or you add some neutrino components to the electron.

LocalLocal Transformation depends on positionTransformation depends on position

Space time symmetries

Momentum P Translation; Conserved ; labels states of a particle

Energy H Time evolution: Conserved; labels states of a particle

Angular momentum

L Rotations: Sometimes, labels composite state rotation

Spin S Rotations: Magnitude S for a particle identifies particles; z-projection possible

Total J=L+S Rotations: Conserved

Time reversal Sym. Change time direction things remain same

Isotopic space Sym. 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

Charge Q Electric charge ,Conserved; labels particles

Phase Always introduce an overall phase

Particle same as an anti particle

Charge Conjugation

C Good Q# when particle=antiparticle Almost always conservved; labels particles sometimes.

Particle-anti Sym. Change particle to anti without impact (partially true)

Flavor approximate

Flavor u,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

strangeness S Minus # strange quarks; labels particles see above comments

charm C Number of charmed labels particles see above comments

beauty B~ Minus number of bottom labels particles see above comments

topness T~ Number of top labels particles see above comments

Baryon No B 1/3[N(q)-N(qbar)] labels particles see above comments

Charge Q Defined as above but=2/3[Nu+Nc+Nt] - 1/3[Nd+Ns+Nb]

Complete flavor

S,C,T~,B~,Q,B establish flavor uniquelyOne choice of Q#s that id a quark flavor.

Isospin I Combination 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

CPT Any 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,νe,νμ,ντ, Lepton particles; labels particles

Class Invariance Conserved quantityProper orthochronousLorentz symmetry

translation in time  (homogeneity) energy

translation in space  (homogeneity) linear momentum

rotation in space  (isotropy) angular momentum

Discrete symmetry P, coordinate inversion spatial parityC, charge conjugation charge parityT, time reversal time parityCPT product

Internal symmetry U(1) gauge transformation electric charge

U(1) gauge transformationlepton generation number (accidental symmetry)electron, muon Tau number

U(1) gauge transformation Baryon number (accidental symmetry)

U(1) gauge transformation hypercharge B+S to label quark combos i.e. decuplet octet members

U(1)Y gauge transformation weak hyperchargeU(2) [U(1)xSU(2)] electroweak forceSU(2) gauge transformation isospinSU(2)L gauge transformation weak isospinPxSU(2) G-paritySU(3) "winding number" baryon numberSU(3) gauge transformation quark colorSU(3) (approximate) quark flavorS((U2)xU(3))[ U(1)xSU(2)xSU(3)] Standard Model

Image Description Particle Represented

straight line, arrow to the right electron

straight line, arrow to the left positron

wavy line photon

An electron emits a photon

An electron absorbs a photon

A positron emits a photon

A positron absorbs a photon

A photon produces an electron and a positron (an electron-positron pair)

An electron and a positron meet and annihilate (disappear), producing a photon

Rotating the arms through Rotating the arms through 9090oo

A+BA+BC+DC+DAAC+D+BbarC+D+Bbar

A

B

D

A

BD

Different time ordering same Different time ordering same topologytopology

A

B

D

A

B D

CA

DB

A+B=C+D

DB

D

BC

AA+B=C+D

time

A C

A+B=C+Dt-channel u-channel

s-channel

WeakWeak

QCDQCD pion exchange pion exchange

exampleexampleeePN

Weak interaction

ddu

udu

neutron proton

dquark(-13)

uquark(+23)

e

(e)bar

W

d=u+e-+ebar

W

e

(e)bar

fundamental vertex spectator quarks added

ExampleExampleoP

Weak

sdu

ddu

Sigma proton

squark(-13)

dquark(-13)

u ord

ubar or dbar

Zo

=Po

Z

u ord

ubar or dbar

fundamental vertex spectator quarks added