PHY3004:NuclearandParticlePhysicsMarcelMerk,Jacco deVries

StandardModel:particlesandforcesClassificationofparticles

• Lepton: fundamentalparticle• Hadron: consistofquarks• Meson: 1quark+1antiquark(𝜋",𝐵$%,…)• Baryon:3quarks(𝑝 ,𝑛 , Λ,…)• Anti-baryon:3anti-quarks

• Fermion:particlewithhalf-integerspin.• Antisymmetricwavefunction:obeysPauli-exclusionprincipleandPauli-Diracstatistics

• Allfundamentalquarksandleptonsarespin-½• Baryons(S=) *⁄ ,, *⁄ )

• Boson:particlewithintegerspin• Symmetricwavefunction:Bose-Einsteinstatistics• Mesons:(S=0,1),Higgs (S=0)• Forcecarriers: 𝛾,𝑊,𝑍,𝑔 (S=1);graviton(S=2)

WaveEquationsContents:1. Waveequationsa) Waveequationsforspin-0fields• Schrödinger(nonrelativistic),Klein-Gordon(relativistic)

b) Waveequationforspin-½fields• Diracequation(relativistic)• Fundamentalfermions

c) Waveequationsforspin-1fields• Gaugebosonfields;eg.electromagneticfield

2. Gaugefieldtheorya) Variational CalculusandLagrangiansb) LocalGaugeinvariancei. QEDii. Yang-MillsTheory(Weak,Strong)

Griffithschapter7andPP1chapter1

• RequiredQuantumMechanicsknowledge:• Angularmomentumandspin:studyGriffithssections4.2,4.3,InparticularPauliMatrices

Griffithschapter10andPP1chapter1

Part1WaveEquationsandProbability

1a)Spin-0

Griffiths§7.1– §7.3

SchrödingerEquationandprobability• Quantizationofclassicalnon-relativistictheory:• Take𝐸 = 7

9

*:andsubstituteenergyandmomentumbyoperatorsthatoperateon𝜓:

• ResultisSchrödinger’sequation:𝑖ℏ >>?𝜓 = − ℏ9

*:𝛻*𝜓

• Planewavesolutions:𝜓 = 𝑁𝑒D 7EFG? /ℏ withthekinematicrelation𝐸 = 𝑝*/2𝑚

• MultiplybothsidesSchrödingerby 𝜓∗ andadditscomplexconjugate

𝜓∗ >>?𝜓 = 𝜓∗ Dℏ

*:𝛻*𝜓

𝜓 >>?𝜓∗ = 𝜓 FDℏ

*:𝛻*𝜓∗

>>?

𝜓∗𝜓K

= −𝛻 ⋅ Dℏ*:

𝜓𝛻𝜓∗ − 𝜓𝛻𝜓M

+

𝜕𝜌𝜕𝑡

𝛻 ⋅ 𝑗Recognize“continuity”equation:

𝜕𝜌𝜕𝑡+ 𝛻 ⋅ 𝚥 = 0

Lawofconservedcurrents,with:𝜌 ≡ 𝜓∗𝜓 = 𝑁 *

𝚥 ≡ Dℏ*:

𝜓𝛻𝜓∗ − 𝜓∗𝛻𝜓 = U 9

:��

• Interpret:probabilitywaves!

𝐸 → 𝐸W = 𝑖ℏ >>?

;𝑝 → �� = −𝑖ℏ𝛻

*Use:𝛻 ⋅ 𝜓∗𝛻𝜓 − 𝜓𝛻𝜓∗ = 𝜓∗𝛻*𝜓 − 𝜓𝛻*𝜓∗

Relativistic:Klein-Gordonequation• Quantizationofrelativistictheory• Startwith𝐸* = 𝑝*𝑐* + 𝑚*𝑐Z andsubstituteagain𝐸 → 𝑖ℏ >

>?and𝑝 → −𝑖ℏ𝛻

• ResultisKlein-Gordonequation:− )[9

>9

>?9𝜙 = −𝛻*𝜙 + :9[9

ℏ9𝜙

• Planewavesolutions:𝜓 = 𝑁𝑒D 7EFG? /ℏ withrelativisticrelation𝐸* = ��* + 𝑚*

• Usethecovariantnotation:

𝑝]𝑝] = 𝑚*

• Klein-Gordoninfour-vectornotation:𝜕]𝜕]𝜙 +𝑚*𝜙 = 0• Planewavesolutions:𝜓 = 𝑁𝑒FD 7^E^

• Timeandspacecoordinatesarenowtreatedfullysymmetric• Thisisneededinarelativistictheorywheretimeandspacefordifferentobservesarelinearcombinationsofeachother

Usenow:ℏ = 𝑐 = 1

𝜕] = >>?, −𝛻 ;𝜕] =

>>?, 𝛻

𝜕]𝜕] ≡)[9

>9

>?9− 𝛻* (usuallytake𝑐 = ℏ = 1)

𝑝% = 𝐸 and𝑥% = 𝑡

Klein-Gordonconservedcurrents• SimilartotheSchrödingercasemultiplybothsidesby−𝑖𝜙∗ fromleftandaddtheexpressiontoitscomplexconjugate

−𝑖𝜙∗ − >9b>?9

= −𝑖𝜙∗ −𝛻*𝜙 +𝑚*𝜙

𝑖𝜙∗ − >9b∗

>?9= 𝑖𝜙 −𝛻*𝜙∗ + 𝑚*𝜙∗

>>?𝑖 𝜙∗ >b

>?− 𝜙 >b∗

>?K

= 𝛻 ⋅ 𝑖 𝜙∗𝛻𝜙 − 𝜙𝛻𝜙∗

FM

• Thequadraticequationleadstodoublesolutions:𝐸* = ⋯ ⇒ 𝐸 = ±⋯• Positiveandnegativeenergysolutions• Negativesolutionsimplynegativeprobabilitydensity𝜌• ThisbotheredDiracandthereforehelookedforanequationlinearin𝐸 and 𝑝 …

Againrecognize“continuity”equation,thelawofconservedcurrents:

𝜕]𝑗] = 0Withnow:j] = 𝜌, 𝚥 = 𝑖 𝜙∗ 𝜕]𝜙 − 𝜙 𝜕]𝜙∗

Itgivesforplanewaves:𝜌 = 2 𝑁 *𝐸𝚥 = 2 𝑁 *��

Orin4-vector: 𝑗] = 2 𝑁 *𝑝]

+

Antiparticles

• Feynman-Stückelberg interpretation• Chargecurrentofanelectronwithmomentum�� andenergy𝐸𝑗] −𝑒 = −2𝑒 𝑁 *𝑝] = −2𝑒 𝑁 * 𝐸, ��

• Chargecurrentofapositron𝑗] +𝑒 = +2𝑒 𝑁 *𝑝] = −2𝑒 𝑁 * −𝐸,−��

Thepositroncurrentwithenergy−𝐸 andmomentum−�� isthesameastheelectroncurrentwith𝐸 and��

• Thenegativeenergyparticle solutionsgoingbackwardintimedescribethepositive-energyantiparticle solutions.• Thewavefunction𝜙 = 𝑁𝑒FDE^7^ staysinvariantfornegativeenergyandgoingbackwardsintime

• Considereg.𝑒FD FG F? = 𝑒FDG?

• Apositronis anelectrontravellingbackwardsintime

≡

𝑒" 𝑒F

𝐸 < 0 𝐸 > 0

Part1WaveEquationsandProbability

1b)Spin-½

Griffiths§7.1– §7.3

DiracEquation• Diracdidnotlikenegativeprobabilitiesandlookedforawaveequationoftheform𝐸 = 𝑖 >

>?𝜓 = 𝐻𝜓 = ? ,butrelativistically correct.

• Try:𝐻 = �� ⋅ �� + 𝛽𝑚 where�� ⋅ �� = 𝛼)𝑝E + 𝛼*𝑝m + 𝛼,𝑝n ;��? 𝛽?• Weknowthat:𝐻*𝜓 = 𝐸*𝜓 = ��* + 𝑚* 𝜓• Writeitout:

• Thisworksoutif:• 𝛼)* = 𝛼** = 𝛼,* = 𝛽* = 1• 𝛼D, 𝛼*, 𝛼,, 𝛽 anti-commute:ie.:𝛼)𝛼* = −𝛼*𝛼) etc

• Anti-commutator: 𝛼D, 𝛼o = 2𝛿Do ; 𝛼D, 𝛽 = 0 ;𝛽* = 1• Usingdefinition: 𝐴, 𝐵 = 𝐴𝐵 + 𝐵𝐴:

𝐻* = ∑ 𝛼D𝑝D + 𝛽𝑚�D ∑ 𝛼o𝑝o + 𝛽𝑚�

o= ∑ 𝛼D𝛼o𝑝D𝑝o + ∑ 𝛼D𝛽𝑝D𝑚 + ∑ 𝛽𝛼D𝑝D𝑚�

D +�D 𝛽*𝑚*�

D,o

= t𝛼D*𝑝D*�

D

+t 𝛼D𝛼o + 𝛼o𝛼D 𝑝D𝑝o

�

Duo

+t 𝛼D𝛽 + 𝛽𝛼D 𝑝D𝑚�

D

+ 𝛽*𝑚*

= 0

Dirac’sidea

• Clearly𝛼D and𝛽 cannotbenumbers.Letthembematrices!• Inthatcasetheyoperateonawavefunctionthatisacolumnvector• Thesimplestcasethatallowstherequirementsare4x4matrices.• Dirac’sequationbecomes:

• ItispossiblemakinguseofthePaulispinmatrices• 𝛼D =

0 𝜎D𝜎D 0 and𝛽 = 𝟙 0

0 −𝟙 with 𝜎) = 𝜎E =0 11 0 ;𝜎* = 𝜎m =

0 −𝑖𝑖 0 ;𝜎, = 𝜎n =

1 00 −1

• Notethat𝛼 and𝛽 arehermitian:𝛼Dx = 𝛼D and 𝛽x = 𝛽 (SinceHamiltonianhasreal𝐸 eigenvalues.)

• Thisisaverycomplicatedequation!• Whatdoesitmeanthatthewavefunction𝜓 isnowa1-by-4columnvector?• 𝜓 isnot a4-vector,sincetheindicesdonotrepresentkinematicvariables,butmatricesindices!

𝑖𝜕𝜕𝑡

𝜓)𝜓*𝜓,𝜓Z

= −𝑖

. . . .

. . . .

. . . .

. . . .z{

⋅ 𝛻D +

. . . .

. . . .

. . . .

. . . .|

⋅ 𝑚

𝜓)𝜓*𝜓,𝜓Z

CovariantformofDirac’sequation

• Diracequation:𝑖 >>?𝜓 = −𝑖�� ⋅ 𝛻 + 𝛽𝑚 𝜓

• MultiplyDirac’seq.fromtheleftby𝛽;thenitbecomes:• 𝑖𝛽 >

>?𝜓 + 𝑖𝛽�� ⋅ 𝛻 − 𝑚 𝜓 = 0

• IntroducenowtheDirac𝛾-matrices:𝛾] ≡ 𝛽, 𝛽�� (vectorof4matrices!)• CovariantformofDiraceq:

• Realise thatDirac’sequationisasetof4coupleddifferentialequations.

𝑖𝛾]𝜕] − 𝑚 𝜓 = 0

DiracGammaMatrices

𝛾% =

1 0 0 00 1 0 00 0 −1 00 0 0 −1

𝛾) =

0 0 0 10 0 1 00 −1 0 0−1 0 0 0

𝛾* =

0 0 0 −𝑖0 0 𝑖 00 𝑖 0 0−𝑖 0 0 0

𝛾, =

0 0 1 00 0 0 −1−1 0 0 00 1 0 0

• Thereissomefreedomtoimplement: 𝛾], 𝛾} = 2𝑔]} in4x4matrices.• WewillusetheDirac-Paulirepresentation

Or:𝛾% = 𝛽 = 𝟙* 00 −𝟙*

and𝛾~ = 𝛽𝛼~ =0 𝜎~−𝜎~ 0

Notetheindices:(confusing!)

𝜇, 𝜈 = 0,1,2,3 aretheLorentzindicesinspace-time:

Diracmatrixindices:1,2,3,4Havetodowiththerowandcolumnindicesofthematrix(andspinors)

• Note:althoughthegammamatricesindicesareLorentz-indices(“space-time”,thegamma-matricesarenot4-vectors!

Exercise– 13:DiracAlgebra

• Diracalgebra:• Writetheexplicitformofthe 𝛾-matrices• Showthat: 𝛾], 𝛾} ≡ 𝛾]𝛾} + 𝛾}𝛾] = 2𝑔]}

• Showthat: 𝛾% * = 𝟙Z ; 𝛾) * = 𝛾* * = 𝛾, * = −𝟙Z• Useanti-commutationrulesof𝛼 and𝛽 toshowthat:𝛾]x = 𝛾%𝛾]𝛾%

• Define𝛾� = 𝑖𝛾%𝛾)𝛾*𝛾, andshow:𝛾�� = 𝛾� ; 𝛾� * = 𝟙Z ; 𝛾�, 𝛾] = 0

Exercise– 14:SolutionsoffreeDiracequation

a) ShowthatthefollowingplanewavesaresolutionstoDirac’sequation

𝜓) =

10

𝑝n 𝐸 +𝑚⁄𝑝E + 𝑖𝑝m 𝐸 +𝑚⁄

𝑒D 7⋅EFG? ;𝜓* =

01

𝑝E − 𝑖𝑝m 𝐸 +𝑚⁄−𝑝n 𝐸 +𝑚⁄

𝑒D 7⋅EFG?

𝜓, =

𝑝n 𝐸 −𝑚 ⁄𝑝E + 𝑖𝑝m 𝐸 −𝑚⁄

10

𝑒D 7⋅EFG? ;𝜓Z =

𝑝E − 𝑖𝑝m 𝐸 −𝑚⁄−𝑝n 𝐸 −𝑚⁄

01

𝑒D 7⋅EFG?

b) WritetheDiracequationforparticleinrest(choose�� = 0)andshowthat𝜓) and𝜓*arepositiveenergy solutions:𝐸 = + 𝑝* +𝑚*� whereas𝜓, and𝜓Zarenegativeenergysolutions: 𝐸 =− 𝑝* +𝑚*� .

c) Considerthehelicity operator�� ⋅ 𝑝 = 𝜎E𝑝E + 𝜎m𝑝m + 𝜎n𝑝n andshowthat𝜓) correspondstopositivehelicitysolutionand𝜓* tonegativehelicity.Similarlyfor𝜓,and 𝜓Z.

SeeGriffithforaderivationofthesolutions

SpinandHelicity– hintforexercise14c)

• Foragivenmomentum𝑝 therestillisatwo-folddegeneracy:whatdifferentiatessolutions𝜓) from𝜓*?

• DefinethespinoperatorforDiracspinors:Σ = �� 00 ��

,where �� arethethree2x2Paulispinmatrices• Definehelicity 𝜆 asspin“up”/”down”wrt directionofmotionoftheparticle

𝜆 = 12Σ ⋅ �� ≡

12�� ⋅ �� 00 �� ⋅ �� =

12 𝑝

𝜎E𝑝E + 𝜎m𝑝m + 𝜎n𝑝n

• SplitofftheEnergyandmomentumpartofDirac’sequation: 𝑖𝛾]𝜕] − 𝑚 𝜓 = 0

• Exercise:Trysolutions𝜓) and𝜓* toseetheyarehelicityeigenstateswith𝜆 = +1/2 and𝜆 = −1/2• Diracwantedtosolvenegativeenergiesandhefoundspin-½fermions!

𝐼 00 −𝐼 𝐸 − 0 𝜎D

−𝜎D 0 𝑝D − 𝐼 00 𝐼 𝑚 𝜓�

𝜓�= 0

Antiparticles

• Diracspinorsolutions

• Sinceweworkwithantiparticles,insteadofnegativeenergyparticlestravellingbackwardsinsteadintime,antiparticlesolutionsaredefined

• Wherenowtheenergyoftheantiparticlesolutions𝑣) and𝑣* ispositive:𝐸 > 0

• Explicit:𝑣) =𝑝E − 𝑖𝑝m 𝐸 +𝑚⁄−𝑝n 𝐸 +𝑚⁄

01

and𝑣* =

𝑝n 𝐸 +𝑚⁄𝑝E + 𝑖𝑝m 𝐸 +𝑚⁄

10

• Where𝐸 and�� arenowtheenergyandmomentumoftheantiparticle

𝜓D 𝑥] = 𝜓D 𝑡, �� = 𝑢D 𝐸, �� 𝑒D 7EFG? = 𝑢D 𝑝] 𝑒FD7^E^

with𝑖 = 1,2,3,4

𝑢, −𝐸,−�� 𝑒D F7 EF FG ? = 𝑣* 𝐸, �� 𝑒FD 7EFG? = 𝑣* 𝑝] 𝑒D7^E^

𝑢Z −𝐸,−�� 𝑒D F7 EF FG ? = 𝑣) 𝐸, �� 𝑒FD 7EFG? = 𝑣) 𝑝] 𝑒D7^E^

Adjoint spinors• Adjoint spinors• SolutionsoftheDiracequationarecalledspinors• Currentdensityandcontinuityequationrequireadjoints insteadofcomplexconjugates

𝑖𝛾%𝜕𝜓𝜕𝑡

+ 𝑖 t 𝛾~𝜕𝜓𝜕𝑥~

− 𝑚𝜓 = 0�

~�),*,,

−𝑖𝜕𝜓x

𝜕𝑡𝛾% − 𝑖 t

𝜕𝜓x

𝜕𝑥~−𝛾~ − 𝑚𝜓x = 0

�

~�),*,,

• Diracspinor: 𝜓 =

𝜓)𝜓*𝜓,𝜓Z

,adjoint Diracspinor:𝜓� = 𝜓), 𝜓*, 𝜓,, 𝜓Z

• Diracequation:𝑖𝛾]𝜕]𝜓 −𝑚𝜓 = 0 ;adjoint Diracequation:𝑖𝜕]𝜓�𝛾] − 𝑚𝜓� = 0

• Theminussignin −𝛾~ disturbstheLorentzinvariantform• Restorebydefiningadjoint spinor:

𝜓� = 𝜓x𝛾%𝛾%� = 𝛾%; 𝛾~� = −𝛾~

DiracCurrentdensityandconservedcurrent

• Applyasimilartrickasbefore:• Multiplyadjoint Diraceq fromfromrightby𝜓 andmultiplyDiraceq.fromleftby 𝜓�

• Probability:Zero-th componentofthecurrent:

𝑗% = 𝜓�𝛾%𝜓 = 𝜓x𝜓 = t 𝜓D *Z

D�)

• Thisalwaysgivesapositiveprobability,whichwasthemotivationofDirac.

𝑖𝜕]𝜓�𝛾] + 𝑚𝜓� 𝜓 = 0𝜓� 𝑖𝜕]𝜓𝛾] −𝑚𝜓 = 0

𝜓� 𝜕]𝛾]𝜓 + 𝜕]𝜓�𝛾] 𝜓 = 0+

Definethe4-veccurrent:𝑗] = 𝜓�𝛾]𝜓

Satisfiesthecontinuityequation:𝜕]𝑗] = 0

Diracinsummary

• Diracwaslookingforanexplanationforpositiveandnegativeenergysolutionsbylinearising Klein-Gordonequation• Hefoundthathissolutionsdescribedspin-½particles• Hepredicted,basedonsymmetry,thatforeachparticlethereshouldexistanantiparticle(thenegativeenergysolution).

• Wehadrelativisticfields:• Spin-0:Klein-Gordon:e.g.pionparticles• Spin-1/2:Dirac:e.g.quarksandleptons• Howaboutforces?Spin=1

Part1WaveEquationsandProbability

1c)Spin-1

Griffiths§7.1– §7.3

TheElectromagneticField– includingexercise15• Maxwellequationsdescribeelectricandmagneticfieldsinducedbychargesandcurrents:(usedHeavyside-Lorentzunits:𝑐 = 1, 𝜖% = 1, 𝜇% = 1)

1.Gauss’law:𝛻 ⋅ 𝐸 = 𝜌

2.Nomagneticcharges:𝛻 ⋅ 𝐵 = 0

3.Faraday’slawofinduction:𝛻×𝐸 + >�>?= 0

4.ModifiedAmpère’slaw:𝛻×𝐵 − >G>?= 𝚥

From1.and4.derivecontinuity𝛻 ⋅ 𝚥 = − >K

>?

è chargeconservationThiswasthemotivationforMaxwelltomodifyAmpère’slaw

• DefineaLorentzcovariant4-vectorfield𝐴] = 𝑉, 𝐴 asfollows:𝐵 = 𝛻×𝐴 (thenautomatically2.follows)

𝐸 = −>�>?− 𝛻𝑉 with𝑉 = 𝐴% (thenautomatically3.follows)

a) ShowMaxwellequationscanbesummarizedincovariantform:𝜕]𝜕]𝐴} − 𝜕}𝜕]𝐴] = 𝑗} (Deriveexpressionsfor𝜌 and𝚥 anduse:𝛻× 𝛻×𝐴 = −𝛻*𝐴 + 𝛻 𝛻 ⋅ 𝐴

Griffiths§7.4

GaugeInvariance(includingexercise15)b) Field 𝐴] isjustintroducedasamathematicaltool• Chooseany𝐴] aslongas 𝐸 and𝐵 fieldsdon’tchange

• Exercise:showthisexplicitly!

c) ChoosetheLorentzgaugecondition:𝜕]𝐴] = 0• Exercise:showthatwecanchoseagaugefieldsuchthatthisispossible

• MaxwellequationinLorentzgaugebecomes:𝜕]𝜕]𝐴} = 𝑗} also:𝐴} = 𝑗}• VerysimilartoKlein-Gordonequation𝜕]𝜕]𝜙 +𝑚*𝜙 = 0• Butnowmassofthephoton=0.• Alsonow4-equationsà polarizationsstatesofthephotonfield

• Photonfieldsolutions:𝐴] 𝑥 = 𝑁𝜀] 𝑝 𝑒FD7�E�

• Agaugetransformationimplies:𝜀] → 𝜀�] = 𝜀] + 𝑎𝑝]• Differentpolarizationvectorswhichdifferbymultipleof𝑝] describesamephoton

𝐴] → 𝐴�] = 𝐴] + 𝜕]𝜆𝑉 → 𝑉� = 𝑉 +

𝜕𝜆𝜕𝑡

𝐴 → 𝐴′ = 𝐴 − 𝛻𝜆

Exercise– 16Antisymmetrictensor𝐹]}

• Maxwell’sequation𝜕]𝜕]𝐴} − 𝜕}𝜕]𝐴] = 𝑗} canbefurthershortenedbyintroducingtheantisymmetrictensor:𝐹]} ≡ 𝜕]𝐴} − 𝜕}𝐴]:

• ShowthatMaxwell’sequationsbecome:𝜕]𝐹]} = 𝑗}

• Hint:derivetheexpressionsforcharge(𝑞 = 𝑗%)andcurrent(𝐼 = 𝚥)separately.Usetheidentity:𝛻× 𝛻×𝐴 = −𝛻*𝐴 + 𝛻 𝛻 ⋅ 𝐴 .Rememberthedefinitions:

A] = 𝐴%, −𝐴 ; 𝜕] =>>?, 𝛻 ; 𝑔]} = 𝑔]} = diag 1, −1,−1,−1

𝐹]} =

0 −𝐸E −𝐸m −𝐸n𝐸E 0 −𝐵n 𝐵m𝐸m 𝐵n 0 −𝐵E𝐸n −𝐵m 𝐵E 0

Part2GaugeTheory

2a)Variational CalculusandLagrangians

Griffithschapter10

LagrangeFormalismclassical• ClassicalMechanics:TheLagrangian leadstoequationsofmotion• 𝐿 𝑞D, 𝑞D = 𝑇 − 𝑉 where𝑞D and𝑞Darethegeneralizedcoordinatesandvelocities.• ThepathofaparticleisfoundfromHamilton’sprincipleofleastaction

FromthistheEulerLagrangeequationsfollowandprovidetheequationsofmotion:𝑑𝑑𝑡

𝜕𝐿𝜕𝑞D

=𝜕𝐿𝜕𝑞D

𝑆 = ¢ 𝑑𝑡𝐿 𝑞, �� = 0?9

?£𝛿𝑆 = 0

• Example:Ballfallsfromheight𝑦 = ℎ :𝑞 = 𝑦, �� = 𝑑𝑦/𝑑𝑡 = 𝑣m• 𝐸7¦? = 𝑇 = 𝑚𝑔𝑞• 𝐸~D§ =

)*𝑚��*

• EulerLagrange:𝑑𝐿/𝑑𝑞 = 𝑚𝑔 ;𝑑𝐿/𝑑�� = 𝑚��• ¨

¨?>©>ª{

= >©>ª{

gives𝑚�� = 𝑚𝑔 à �� = 𝑔𝑡 + 𝑣% à 𝑞 = 𝑦 = )*𝑔𝑡* + 𝑣%𝑡 + 𝑦%

Griffiths§10.1

See:https://en.wikipedia.org/wiki/Lagrangian_mechanics

Exercise– 17:LagrangeFormalismclassical• Exampleofvariational calculusandleastactionprinciple:whatistheshortestpathbetweentwopointsinspace?• Distanceoftwoclosepoints:

𝑦 = 𝑑𝑥* + 𝑑𝑦*� = 𝑑𝑥* 1 + ¨m¨E

*�= 1 − 𝑦�*� 𝑑𝑥 with𝑦� = 𝑑𝑦/𝑑𝑥

• Totallengthfrom 𝑥%, 𝑦% to 𝑥), 𝑦) :

𝑙 = ¢ 𝑑𝑙E£

E­= ¢ 1 + 𝑦�*� 𝑑𝑥

E£

E­= ¢ 𝑓 𝑦, 𝑦′ 𝑑𝑥

E£

E­• Taskistofindafunction𝑦 𝑥 forwhich𝑙 isminimal

• Ingeneralassumethepathlengthisgivenby:𝐼 = ∫ 𝑓 𝑦, 𝑦′ 𝑑𝑥E£E­

• Variational principle:shortestpathisstationary:𝛿𝐼 = 0a) Write𝛿𝑓 𝑦, 𝑦′ = >°

>m𝛿𝑦 + >°

>m±𝛿𝑦� where𝛿𝑦� = 𝛿 ¨m

¨E= ¨

¨E𝛿𝑦

Showusingpartialintegrationthat𝛿𝐼 = 0 leadstotheHamiltonLagrangeequation>°>m− ¨

¨E>°>m±

= 0

b) Herefortheshortestpathwehave𝑓 𝑦′ = 𝑙 = 1 + 𝑦�*�

.Then𝜕𝑓/𝜕𝑦 = 0 and𝜕𝑓/𝜕𝑦� = 𝑦�/ 1 + 𝑦�*�

Showthatthevariational principleleadstoastraightlinepath: ¨¨E

m�)"m±9� = 0 orthat𝑦� isaconstant:

𝑑𝑦/𝑑𝑥 = 𝑎 ;𝑦 = 𝑎𝑥 + 𝑏

LagrangeFormalisminfieldtheory

• RelativisticFieldtheory:fieldsreplacethegeneralizedcoordinates• Alsotimeandspacewillbetreatedsymmetric• Replace𝐿 𝑞, �� byaLagrangedensityℒ 𝜙 𝑥 , 𝜕𝜙 𝑥 intermsoffieldsandgradientssuchthat𝐿 ≡ ∫𝑑,𝑥ℒ 𝜙, 𝜕𝜙�

�

• Principleofleastactionsbecomes:

• EulerLagrangeEquationsofmotionbecomes:

S=∫ 𝑑Z𝑥?9?£

ℒ 𝜙 𝑥 , 𝜕𝜙 𝑥 andagain𝛿𝑆 = 0𝑡), 𝑡*areendpointsofthepath

𝜕ℒ𝜕𝜙 𝑥

= 𝜕]𝜕ℒ

𝜕 𝜕]𝜙 𝑥

Griffiths§10.2

Exercise– 18:Lagrangians andwaveequations

• ScalarField(“pion”)a) ShowthattheEuler-Lagrangeequationsforℒ = )

*𝜕]𝜙 𝜕]𝜙 −𝑚*𝜙*

resultsintheKlein-Gordonequation

• DiracField(Fermion)b) ShowthattheEuler-Lagrangeequationsforℒ = 𝑖𝜓�𝛾]𝜕]𝜓 −𝑚𝜓�𝜓

resultsintheDiracequation

• Electromagneticfield(photon)c) Showthatℒ = −)

Z𝜕]𝐴} − 𝜕}𝐴] 𝜕]𝐴} − 𝜕}𝐴] − 𝑗]𝐴]

resultsinMaxwell’sequations

Griffiths§10.2

TheGaugePrinciple:Interactions

• globalgaugeinvariance:thephaseofthewavefunctionisnotobservable:Changingthewavefunction𝜓 𝑥 → 𝜓� 𝑥 = 𝑒Dz𝜓 𝑥 shouldnotchangetheLagrangian foranelectron• LookatDiracLagrangian:𝑖𝜓�𝛾]𝜕]𝜓 −𝑚𝜓�𝜓• Itshouldnotchangefor𝜓 → 𝜓� and𝜓� → 𝜓′ = 𝜓�x𝛾% ;𝜓� = 𝑒"Dz𝜓� è OK.

• local gaugeinvariance:invarianceunderchaging phasesinspaceandtime• Anelectronwavefunctioncanhaveadifferentphasesatdifferentplacesandtimes• 𝜓 𝑥 → 𝜓� 𝑥 = 𝑒Dz E 𝜓 𝑥 and𝜓� 𝑥 → 𝜓� 𝑥 = 𝑒FDz E 𝜓� 𝑥

• CheckthisfortheDiracLagrangianProblemintheterm:𝜕]𝜓 𝑥 → 𝜕]𝜓� E = 𝑒Dz E 𝜕]𝜓 𝑥 + 𝑖𝜕]𝛼 𝑥 𝜓 𝑥

• ItseemsthattheLagrangian willchange,butthisisnotallowed!

Griffiths§10.3

Part2GaugeTheory

2b)LocalGaugeInvariancei)QED

Griffithschapter10

Exercise– 18:CovariantDerivative• WeinsistthattheLagrangian doesnotchangeandinventa“covariant”derivative:• Replacein𝑖𝜓�𝛾]𝜕]𝜓 −𝑚𝜓�𝜓 thederivativeby:𝜕] → 𝐷] ≡ 𝜕] + 𝑖𝑞𝐴]• Requirethatthevectorfield𝐴] transformstogetherwiththeparticlewave𝜓

• è Exercise:checkthattheLagrangian nowisinvariant!

• Whathavewedone?• Weinsist theelectroncanhavealocalphasefactor𝛼 𝑥 withoutchangingthephysics• Wethenmust atthesametimeintroduceaphoton,whichcouplestocharge!è Gaugeinvarianceimpliesinteractions!

• RemembergaugetransformationsEMfield:𝐴] → 𝐴�] = 𝐴] + 𝜕]𝜆• 𝜆iscoupledtothephaseofthewavefunctionoftheelectrons

• Thesameprinciplecanalsobeusedforweakandstronginteractions:implementothersymmetries

𝜓 𝑥 →𝜓� 𝑥 = eDªz E 𝜓 𝑥

𝐴] 𝑥 → 𝐴�] 𝑥 = 𝐴] 𝑥 −1𝑞𝜕]𝛼 𝑥

Griffiths§10.3

QuantumElectrodynamics(QED)

• ThefreeDiracLagrangian is: ℒ = 𝑖𝜓�𝛾]𝜕]𝜓 −𝑚𝜓�𝜓

• Resultingin:

𝜕] → 𝐷] ≡ 𝜕] + 𝑖𝑞𝐴]• Introducingelectromagnetismimplies:

ℒ = 𝑖𝜓�𝛾]𝐷]𝜓 −𝑚𝜓�𝜓

ℒ = 𝑖𝜓�𝛾]𝜕]𝜓 −𝑚𝜓�𝜓 − 𝑞𝜓�𝛾]𝐴]𝜓

ℒ = ℒ¶·¸¸ − ℒ¹º» with ℒ¹º» = −𝐽]𝐴] and 𝐽] = 𝑞𝜓�𝛾]𝜓

• Rememberthattheprobabilitycurrentwas𝜓�𝛾]𝜓suchthatwenowhaveachargecurrent:𝐽] = 𝑞𝜓�𝛾]𝜓

• ThesystemisdescribedasfreeLagrangian plusaninteractionLagrangianoftheform:“current× field”ℒ¹º» = −𝐽]𝐴]

Part2GaugeTheory

2b)LocalGaugeInvarianceii)Yang-Millstheories* (Weak,Strong)

Griffiths§10.4- §10.5

*Note:thisisamoretechnicalpart:focusontheconceptinvolved;theprecisemathematicsislessimportantfornow

YangMillsTheories

• QEDiscalledaU(1)symmetry.Itmeansthata1-dimensionalunitarytransformation(thephasefactor)doesnotchangethephysics.• Theunitarysymmetrycouplestothechargequantumnumber

• Letusrequirethattheweakinteractioncannotdifferentiatebetweenanupandadownquark• ℒ = 𝑢� 𝑖𝛾]𝜕] − 𝑚 𝑢 + �� 𝑖𝛾]𝜕] − 𝑚 𝑑 where 𝑢 and𝑑 arespinorwaves

• Rewriteitasℒ = 𝜓� 𝑖𝛾]𝐼𝜕] − 𝐼𝑚 𝜓 with𝜓 = 𝑢𝑑 and𝐼 = 1 0

0 1• Wethinkofthe“up”and“down”directionsinweakisospinspace

SU2GaugeInvariance• Werequiregaugeinvariance:• 𝜏 = 𝜏), 𝜏*, 𝜏, arethePauliMatrices• Thisisnowarotationinisospinspacegeneratedby2x2Paulimatrices!

• JustlikeQEDthereistheproblemthattheLagrangian doesnotautomaticallystayinvariant(justwriteitout),because:

• TosolveacorrespondingcovariantderivativemustbeintroducedtokeeptheLagrangian invariant:

• 𝑔 isthecouplingconstantthatreplacescharge𝑞 inQEDand𝐵] isnoweanewvectorforcefieldthatreplaces𝐴] ofQED.

• Theobject𝐵] isnowa2x2matrix:

𝑏] = 𝑏), 𝑏*, 𝑏, arenowthreenewgaugefields• Weneed3insteadofone,becausetherearethreegeneratorsof2x2rotations

• Wenowgetthedesiredbehaviour if:

𝜓 𝑥 → 𝜓� 𝑥 = 𝐺 𝑥 𝜓 𝑥 with𝐺 𝑥 = exp D*𝜏 ⋅ �� 𝑥

𝜕]𝜓 𝑥 → 𝜕]𝜓� 𝑥 = 𝐺 𝑥 𝜕]𝜓 + 𝜕]𝐺 𝜓

𝐼𝜕] → 𝐷] = 𝐼𝜕] + 𝑖𝑔𝐵]

𝐵] =12𝜏 ⋅ 𝑏] =

12𝜏)Â𝑏]Â =

12

𝑏, 𝑏) − 𝑖𝑏*𝑏) + 𝑖𝑏* −𝑏,

𝐷]𝜓 𝑥 → 𝐷′]𝜓� 𝑥 = 𝐺 𝑥 𝐷]𝜓

𝐼 = 1 00 1

Gaugetransformationfor𝐵] field– (forexperts)

• Wegetthedesiredbehaviour if:

• Theleftsideofthisequationis:

• Whiletherighthandsideis:

• Sotherequiredtransformationofthefieldis:

• Multiplytheequationby𝐺F) ontheright(andomitting𝜓):

• Comparethistothecaseofelectromagnetismwhere𝐺Ã: = 𝑒Dz E gives:

… whichisexactlywhatwehadbefore.

𝐷]𝜓 𝑥 → 𝐷′]𝜓� 𝑥 = 𝐺 𝑥 𝐷]𝜓

𝐷′]𝜓� 𝑥 = 𝜕] + 𝑖𝑔𝐵]� 𝜓′= 𝐺 𝜕]𝜓 + 𝜕]𝐺 𝜓 + 𝑖𝑔𝐵]� 𝐺𝜓

𝐺 𝐷]𝜓 = 𝐺 𝜕]𝜓 + 𝑖𝑔𝐺𝐵]𝜓

𝑖𝑔𝐵]� 𝐺𝜓 = 𝑖𝑔𝐺 𝐵]𝜓 − 𝜕]𝐺 𝜓

𝐵]� = 𝐺𝐵]𝐺F) +DÄ 𝜕]𝐺 𝐺F)

𝐴]� = 𝐺Ã:𝐴𝐺Ã:F) +DÄ 𝜕]𝐺Ã: 𝐺Ã:F) = 𝐴] −

)ª𝜕]𝛼

Interpretation• Wetrytodescribeaninteractionwithasymmetrybetweentwostates:• “up”and“down”stateswithinvarianceunderSU2rotations

• Todothisrequirestheexistenceofthreeforcefields,relatedtothegaugefield:𝐵]• Whatarethey?• Theymustbethreemasslessbosons,similartothephoton,thatcoupleto“up”and“own”states.• Theyarethe𝑊F, 𝑍%,𝑊" bosons.• Howcometheyhaveamass(unlikethephoton?à Higgsmechanism

• AgaintheinteractionLagrangian willbeoftheform“currentxfield:”𝐽]𝑏],wherethecurrentisnow:𝐽] =

Ä*𝜓𝛾]𝜏𝜓

• The“up”and“down”statesare𝜓 = 𝑢𝑑 and𝜓 = 𝜈

𝑒 andwedescribetheweakinteraction.

• Howaboutthestronginteraction?

Thestronginteraction

• The“charge”ofthestronginteractionis“colour”• Thewavefunctionofaquarkhasthreecomponents:

• 𝜓 =𝜓Å𝜓Ä𝜓Æ

;Requireasymmetrygeneratedby3x3rotationsin3-dimcolorspace:SU(3)

• Thereare8generatormatrices𝜆D andasaconsequencethereare8vectorfieldsneededtokeeptheLagrangian invariant• Thereexist8gluons,relatedto:

𝜆) =0 1 01 0 00 0 0

𝜆* =0 −𝑖 0𝑖 0 00 0 0

𝜆, =1 0 00 −1 00 0 0

𝜆Z =0 0 10 0 01 0 0

𝜆� =0 0 −𝑖0 0 0𝑖 0 0

𝜆Ç =0 0 00 0 10 1 0

𝜆È =0 0 00 0 −𝑖0 −𝑖 0

𝜆É =13�

0 0 00 1 00 0 −2

TheStandardModel

• TheStandardModelappliesgaugeinvarianceatthesametimeto• Electromagnetism(U(1)symmetrytransformations)à 1photon• Weakinteraction(SU(2)symmetrytransformations)à 3weakbosons• Stronginteraction (SU(3)symmetrytransformations)à 8gluons

• TheSMgaugegroupis𝑆𝑈 3 ⊗ 𝑆𝑈 2 ⊗𝑈(1)

• Foranexactsymmetrytheforceparticlesshouldbemasslessfor• 𝑆𝑈 3 isexact.• 𝑆𝑈 2 ⊗ 𝑈(1) isanapproximate(ie “broken”)symmetry.• ItisbrokenintheHiggsmechanismsuchthatthereremainsonemasslessbosonandthreemassiveparticles.

StandardModel