Chapter6
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¡
6.1Introduction
6.2InvariancePrincipleReminder
§  6.2.1InvarianceinClassicalMechanics
§  6.2.2InvarianceinQuantumMechanics
§  6.2.3ContinuousTransformations:TranslationsandRotations
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6.3Spin-StatisticsConnection
6.4Parity
6.5Spin-ParityoftheπMeson
§  6.5.1Spinoftheπ Meson
§  6.5.2Parityoftheπ Meson
§  6.5.3Particle–AntiparticleParity
6.6ChargeConjugation
§  6.6.1ChargeConjugationinElectromagneticProcesses
§  6.6.2ViolationofCintheWeakInteraction
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6.7TimeReversal
6.8CPandCPT
6.9ElectricChargeandGaugeInvariance
¡
Symmetriesandinvariancepropertiesofunderlying
interactions
§  playimportantroleinphysics
§  oftenleadtouniversalconservationlaws
▪  (Space)translationalinvarianceàmomentumconservation
▪  (Time)translationalinvarianceàenergyconservation
▪  Rotationalinvarianceàangularmomentumconservation
§
§
Gaugeinvariancerestrictsformoffundamentalinteractions
Discretesymmetries
▪  Parity,Chargeconjugation,Timereversalàveryusefulforclassificationand
whenwewanttoknowwhetheraprocessisallowedornot…
¡
Symmetriesaresoimportantthatevenbrokenonesareuseful
¡  ElectroweakSymmetryBreaking,CP-violation,…
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Twoimportantaspectsinphysicsare
§  theinvariance(orsymmetry)oftheequationsdescribingthebehaviorofa
systemunderagiventransformation
§  theconservationofsomephysicalquantities
¡
Invariancepropertiesare
§  intimatelyconnectedtoconservationlaws
▪  conservationofangularmomentumrelatedtoinvarianceunderspatialrotations
▪  conservationofenergyrelatedtoinvarianceundertimetranslation
▪  conservationofthelinearmomentumrelatedtohomogeneityofspace.
¡
Noether’stheorem:
§  aphysicalconservedquantitycorrespondstoeachinvariant,andviceversa
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Equationsrulingdynamicevolutionofasystem(SchrödingerorLagrange
equations)
§  Firstorderdifferentialequationsintimeandsecondorderinspacecoordinates
§  Everyfirstintegralofmotiongivesrisetoaconservationlaw
¡
Eachfundamentalinteractionobeysvariousconservationlaws
§  Interactionformalismobeysinvariancerequirements,whichlimit
mathematicalformofinteraction
¡
Transformations
§  Continuous
▪  canbeachievedbyapplyingsuccessiveinfinitesimaltransformations(rotation,…)
▪  Correspondtoadditiveconservationlaws
§  Discrete
▪  mirrorreflectioninspace(Parity),Chargeconjugation,…
▪  Correspondtomultiplicativeconservationlaws
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Lagrangeequations
§  LagrangianL=T-V=Kineticenergy–potentialenergy
§  ngeneralisedcoordinatesqi,nconjugatedmomentapi
¡
pi =
∂L
∂q!i
MotionofasystemisdescribedbytheEuler-Lagrangeequation
∂L d # ∂L &
− % ( =0
∂qi dt $ ∂q!i '
dpi ∂L
−
=0
dt ∂qi
∂L
dp
=0 ⇒ i =0 ⇒ pi = constant
∂qi
dt
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∂L d # ∂L &
− % ( =0
∂qi dt $ ∂q!i '
¡
Translationalongx
§  Linearmomentumconserved
¡
Roatation
§  Angularmomentum
conserved
¡
L=
dpi ∂L
−
=0
dt ∂qi
1
∂L
L = T −V = mx! 2 ⇒ px =
= mx! = constant
2
∂x!
1
1
∂L
mϕ! 2 r 2 = mv 2 ⇒ pϕ =
= mvr = constant
2
2
∂ϕ!
Timetranslation
§  Energyconserved
¡
Poincareinvariance(Lorentz+space-timetranslation)
§  Lagrangianrelativisticallytransformsasscalar
¡
Noether’stheorem
§  inaLagrangianfieldtheory,aconservedquantityisassociatedtoa
continuoussymmetry(andviceversa)
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Hamiltonequations
§  HamiltonianH=T+V
q!i =
∂H
;
∂pi
p! i = −
∂H
∂qi
§  InvariancemeansthatHdoesnotchangeunderagiventransformation
¡
Spacetranslations
§  Infinitesimaltranslationalongx
§  Ifmomentumisconserved,thenH
isinvariant
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x → x + dx
∂H
= −dx p! x
∂x
p! x = 0 ⇒ dH =0
dH = dx 8
! t) = T (q)
! −U(q)
L(q, q,
1
= m q! 2 −U(q)
2
T:kinetic energy; U:potential energy (U = mgq for gravity)
Euler − Lagrange equations lead to equationsof motion:
∂L d # ∂L &
dU
− % ( =0
⇒
F=
= ma
∂q dt $ ∂q! '
dq
Classically,
In QM or QFT, dealing with wavefunctions or fields,
define Lagrange density £ as a functional of the field Φ(x µ ).
The integral over the 3-dimensional space leads to: L =
∫ £(Φ,∂ Φ)
⎛ ∂£ ⎞
∂£
⎟⎟ =0
The Euler-Lagrange equations become:
−∂µ ⎜⎜
∂Φ
⎝ ∂(∂µ Φ) ⎠
µ
Lagrangiandensityofafreeelectronintermsofitswavefunctionorfield
§  itistrivialtogofromtheLagrangiantotheequationsofmotionsà
¡
For scalar fields (neutral pion, Higgs),
1
1 2 2
£ = (∂µΦ)(∂µ Φ) − M Φ
2
2
leads to the Klein-Gordon equation
2
!2
1 ∂
2
µ
µ
(∂µ∂ + M )Φ = 0; where ∂µ∂ = 2 2 − ∇
c ∂t
¡
1
Dirac field (electron)
2
£ = Ψ(iγ µ∂µ − m) Ψ
For a spin
leads to the Dirac equation :
(iγ µ∂µ − m) Ψ(x)=0
Thesituationismorecomplexincaseofelementaryparticlesininteraction
§  equationsingeneralnotknown
§  whenknown,theyaredifficulttosolve
àTodescribeelementaryparticleinteractions,itisnecessarytoproposeanexpressionfor
theLagrangianoftheinteractingquantumfields
¡
Averagevalue(orexpectationvalue)ofanoperatorQ
§  associatedwithanobservedquantityq,actingona
wavefunctionψ
§  Q†=Q(hermiticity)
¡
∗
q = ∫ ψ Q ψ dτ ≡ ψ Q ψ
τ
Timeevolutionof<q>describedeitherbySRorHR
§  ψ(t)àSchrödingerrepresentation
§  Q(t)àHeisenbergrepresentation
▪  IfQdoesnotexplicitelydependontime,qandassociated
quantumnumbersareconservedifQcommuteswithH
i
∂
ψ s (t) = H sψ s (t)
∂t
dQ ∂Q
−i
=i
+ [Q, H ]
dt
∂t
§  Relationbetweenthe2representationsàseebook
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¡
∂ψ (x)
ψ (x ') = ψ (x + dx) = ψ (x) + dx
Infinitesimaltranslationalongx:x´=x+dx
∂x
⎛
∂⎞
= ⎜1+ dx ⎟ψ (x) = dDxψ (x)
dDx:operatorgeneratinginfinitesimal
⎝
∂x ⎠
translation
∂
i
dDx = 1+ dx = 1+ px dx
∂x
!
Δx = n dx, n → ∞ : FinitetranslationΔx=seriesof
Translations
§
§
§
infinitesimaltranslationsdx
§  pxisgeneratorofunitaryoperatorDx,
associatedwithspatialtranslationalongx
¡
Equivalentstatements:
1.
2.
3.
n
⎛ i
⎞
⎛i
⎞
Dx = lim ⎜1+ px dx ⎟ = exp ⎜ px Δx ⎟
n→∞ ⎝
⎠
⎝!
⎠
!
Dx† = Dx
Hinvariantunderspacetranslations[Dx,H]=0
[px,H]=0
pxconserved
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¡
Rotationsinspace
φ → φ + dφ
∂
i
dRz = 1+ dφ
= 1+ Lz dφ
∂φ
!
§  Infinitesimalrotationaroundzaxis
§  dRz:operatorassociatedwith
infinitesimalrotations
§  Finiterotation:
Lz = −
i ∂
! ∂φ
Δφ = n dφ, n → ∞
§  LzisgeneratorofunitaryoperatorRzassociatedwithspatialrotationaroundz
n
⎛ i
⎞
⎛i
⎞
Rz = lim ⎜1+ Lz dφ ⎟ = exp ⎜ Lz Δφ ⎟
n→∞ ⎝
⎠
⎝!
⎠
!
¡
Equivalentstatements:
1.
2.
3.
Hinvariantunderspacerotations[Rz,H]=0
[Lz,H]=0
AngularmomentumLzconserved
13
¡
Sameformalismforspin(andIsospin–wewillseelater)
#
# yp − zp &
% L̂x = ŷp̂z − ẑp̂y
z
y
%
(
! ! !
%
L = r × p = % zpx − xpz ( ⇒ L̂ = % L̂y = ẑp̂x − x̂p̂z
%
(
%% L̂ = x̂p̂ − ŷp̂
% xpy − ypx (
y
x
$
'
$ z
.
&
0
(
0
(
( with /
0
((
0
'
1
&
(
(
L̂2 = L̂x2 + L̂y2 + L̂z2 ⇒ '
(
(
)
&
(
L̂+ = L̂x + iL̂y
(
with '
L̂− = L̂x − iL̂y
(
(
)
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* L̂ , L̂ , = i"L̂
z
+ x y* L̂ , L̂ , = i"L̂
x
+ y z* L̂ , L̂ , = i"L̂
y
+ z x-
" L̂2 , L̂ $ = 0
x%
#
" L̂2 , L̂ $ = 0
y%
#
" L̂2 , L̂ $ = 0
z%
#
" L̂2 , L̂ $ = 0
±%
#
" L̂ , L̂ $ = ± L̂
±
# z ±%
L̂2 = L̂− L̂+ + L̂z + L̂z2
14
¡
Pictorialrepresentationofthe2l+1statesofl=2
L̂z , L̂2 → common eigenstates l, m
L̂z l, m = m l, m
−l ≤ m = −l, −l +1,..., +l −1, +l
L̂2 l, m = l(l +1) l, m
L̂+ l, m = l(l +1) − m(+1) l, m +1
L̂− l, m = l(l +1) − m(−1) l, m −1
¡
Couplingof2am/spins
§
Clebsh-Gordoncoefficients
From Thomson
! ! !
%l = l + l
1
2
'
l1, m1 ⊕ l2 , m2 → l, m & l1 − l2 ≤ l ≤ l1 + l2 '
( m = m1 + m2
l, m =
∑ C(m , m ;l, m) l , m
1
2
1
1
l2 , m2
m1,m2
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¡
Couplingof½X½,1X½
§  Spinmultiplicity:2l+1
§  Symmetric,anti-symmetric
andmixedconfigurations
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1
2
1
× : 2 ⊗ 2 = 3⊕ 1
2
1
1× : 3 ⊗ 2 = 4 ⊕ 2
2
1 1
1
2
2
× × : 2 ⊗ 2 ⊗ 2 = 4S ⊕ 2 MS ⊕2 MA
2
16
¡
Particles
§  ½integerspin(1/2,3/2,…),Fermi-Dirac-statisticsàfermions
§  Integerspin(1,2,…),Bose-Einsteinstatisticsàbosons
¡
StatisticsfixsymmetrypropertiesofWFforapairofidentical
particleswrttheirexchange
I (1, 2) → (2,1)
§  OperatorIreversespositionof2particles
§  Eigenvalues:I2=1;I=±1
I ψ (1, 2) = ψ (2,1)
I 2ψ (1, 2) = I ψ (2,1) = ψ (1, 2)
I ψ (1, 2) = ±ψ (1, 2)
⇒ ψ (2,1) = ±ψ (1, 2)
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¡
Underexchange,WFfor2identical
§  Bosonsmustbesymmetric
§  Fermionsmustbeanti-symmetric
¡
1↔ 2 : ψ (1, 2) = ψ (2,1)
1↔ 2 : ψ (1, 2) = −ψ (2,1)
TotalWFproductof2functions
§  Spatialfunctionα
▪  describesorbitalmotionofparticlewrttotheother
àsphericalharmonics
§  Spinfunctionβ
▪  Symmetricifthe2spinsareparallel
▪  Anti-symmetricifanti-parallel
¡
Yl m (θ , φ ) → (−1)l
Identical
§  bosonsmusthavebothαandβsymoranti-symmetric
§  fermionsmusthaveαsymandβanti-symmetricorvice-versa
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¡
Behaviorofastateunderaspatial
reflection
§  Preversesspatialcoordinatesrandp
§  Left-Rightsymmetry
§  Applicationonwavefunction
! P̂
! ! P̂ !
t !!
→ t ' = t ; p !!
→ − p; J ! !
→J
!
!
P̂ ψ (r,t) ≡ P ψ (−r,t)
P̂
!
!
!
P̂ 2ψ (r,t) = PP̂ψ (−r,t) = P 2ψ (r,t)⇒ P = ±1
Parityappliedtwice
¡
!
!
r !P̂!
→ −r
§  Eigenvalueequation
P̂ψ = Pψ = ±1ψ
ExamplesofWFwith
¡
§  Positiveparity:cosx
Negativeparity:sinx Undefinedparity:sinx+cosx
¡
Particleatrest(p=o)iseigenstateofparitywitheigenvalueP=±1(intrinsicparity)
¡
Partityconservedà[H,P]=0
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ψ nlm (r, θ , ϕ ) = χ nl (r)Yl m (θ , φ )
n, l, m : principal, orbital, magnetic QNs
Y00 =
1
4π
Yl m : spherical harmonics , Pl m : Legendre polynomials
Y10 =
3
cosθ
4π
Yl m (θ , φ ) =
(2l +1)(l − m)! m
Pl (cosθ )eimφ
4π (l + m)!
Y1±1 = ∓
3
sin θ e ∓iφ 8π
x = r sin θ cos φ ! r → r
#
y = r sin θ sin φ " θ → π − θ
# φ → π +φ
z = r cosθ
$
Spaceinversion
"$
P̂Yl m (θ , φ ) = (−1)l Yl m (θ , φ )
#
P̂ Pl m (cosθ ) = (−1)l+m Pl m (cosθ )$%
!
!
!
l
⇒ P̂ ψlmn (r) = Pψlmn (−r) = P(−1) ψlmn (r)
P̂ eimφ = (−1)m eimφ
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¡  π
Spin
§  Principleofdetailedbalance
¡  π
Parity
¡  Particle-Antiparticleparity
§  Dirac
▪  P(fermion-antifermion)=-1
▪  P(boson-antiboson)=+1
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pp → π + d
+
σ ( pp → π d) =
¡
π + d → pp
vs
2
M if
(2sπ +1)(2s d +1)
matrix element
ppßàπ+d
π vi v f
2
pπ
phase space & flux factor
§  Principleofdetailedbalance
§  Assumestronginteractioninvariantunder
vi = v p − v p' ; v f = vd − vπ
timereversalandparity
¡
Deuteron
§  d=pn=3S1
§  lpn=0;Swave;sd=1;2sd+1=3
¡
Factor½foridenticalppinfinalstate
¡
Crosssectionmeasurementsatsame
energy
§  Velocityfactorsequal
§  Measuremomenta
¡
¡
Deducesπ+=0
Pions(π+, π-,π0)producedwithsame
abundanceine+e-,ppbar,ppcolliders
(Ecm>10GeV)èsπ=0
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2
2 (2s p +1)
1
2
σ (π d → pp) = M fi
p
p
2
π v'f v'i
+
F.Ould-Saada
same Ecm
⇒ v 'f v 'i = vi v f
detailed balance principle ⇒
2
M if = M fi
2
σ ( pp → π + d)
(2sπ +1)(2sd +1) pπ2
=2
+
σ (π d → pp)
(2s p +1)2
p 2p
3
pπ2
= (2sπ +1) 2
2
pp
22
π − d → nn
¡
Pioncaptureindeuteriumatvery
lowenergy
§  lπd=0
¡
WFdescribingthennsystemψtot
§  Mustbeantisymmetricunder
!
! ! !
π d : sd = 1, sπ = 0 , Ji = sπ + sd + Lπ d = 1
!
!
!
nn : J f = snn + Lnn = 1
−
ψtot = α space β spin
exchangeof2identicalfermions
1↔ 2 ⇔ r → r, θ → π − θ , φ → π + φ
Yl m (θ , φ ) ⎯1↔2
⎯⎯
→(−1)l Yl m (θ , φ )
§  (-1)lgivessymmetryofαspaceunderexchange1-2
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§  βspincombinationofβ1andβ2
1
1
1
1
β spin : β1 (s1 = , s1z = ± ) ; β2 (s2 = , s2 z = ± )
2
2
2
2
" β (1,1) = β1 (1 / 2, +1 / 2)β2 (1 / 2, +1 / 2)
$
s = s1 + s2 = 1; sz = 0, ±1# β (1, 0) = 1 / 2 [ β1 (1 / 2, +1 / 2)β2 (1 / 2, −1 / 2) + β2 (1 / 2, +1 / 2)β1 (1 / 2, −1 / 2)]
$
% β (1, −1) = β1 (1 / 2, −1 / 2)β2 (1 / 2, −1 / 2)
{
s = 0; sz = 0 β (0, 0) = 1 / 2 [ β1 (1 / 2, +1 / 2)β2 (1 / 2, −1 / 2) − β2 (1 / 2, +1 / 2)β1 (1 / 2, −1 / 2)]
¡
¡
Symmetryofβspin:(-1)s+1combination
Symmetryofψtot
:(-1)l+s+1=-1
Symmetryarguments
2.  Angularmomentumconservation
1.
nninp-wavestate3P1
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ψtot "1↔2
""
→(−1)l+s+1ψtot = −ψtot
1. l + s +1 odd , l + s even
2. J = 1
!l = 0
!l = 1
!l = 1
!l = 2
#
#
#
#
J = 1 " s = 1
"s = 0
‹" s = 1
"s = 1
#l + s = odd #l + s = odd #l + s = even #l + s = odd
$
$
$
$
F.Ould-Saada
24
¡
P(nn) = P(n)⋅ P(n)⋅ (−1)l = (+1)(+1)(−1)1 = −1
Parityoffinalstatenn
§  Parityconservedinstronginteractions
P(π − d) = P(nn) = −1
P(d) = P( p)⋅ P(n)⋅ (−1)l=0 = +1
P(π − d) = P(π − )⋅ P(d)⋅ (−1)l=0 = P(π − ) = −1
¡
Pionπ-hasnegativeparity
P(π − ) = P(π + ) = P(π 0 ) = −1
§  N-pionsystemhasparity(-1)N
P(N π − ) = (−1) N
¡
Variousmesonfamilies
§  Pionsarepseudo-scalars:JP=0-
§  Scalarmesons:JP=0+
§  Vectormesons:JP=1-
§  Axialmesons::JP=1+
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¡
¡
Parity–multiplicativeQN SI,EMinvariantunderParity
àP(ψ1ψ2)=P1.P2
à[H,P]=0
¡
InvarianceunderparityofDiracequationà P(e e ) = −1
¡
¡
Convention:P=+1forleptonsandquarksandP=-1foranti-fermions
Parityofphoton:-1(seeSlide30)
Intrinsicparitiesofhadronsfollowstructureintermsofquarks+orbitallbetween
constituentquarks:
l
¡
+ −
P(−1)
¡
¡
¡
¡
P( f f ) = −1
WIviolatesParity(maximally)
Meson=quark-antiquark:P=(-1)(-1)l=(-1)l+1
Pion(l=0):P=-1
Proton(uud,l=0):P=+1
neutron(udd,l=0):P=+1
Notobserved
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26
¡
¡
¡
¡
Cq=-q
Cψ(q)=ψ (-q)
c2=1àc=±1
EffectofCparityon
protonand
electron
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27
¡
¡
¡
Operationchangingparticleàantiparticle
MultiplicativeQNconservedinSI,EM–not
inWI
Distinguishcaseswhere
§  (a)Particle=antiparticle:γ,π0à
Ĉ π 0 = ± π 0
EMfieldsproducedbymovingelectriccharges,
whichchangesignunderC,soCγ=-1
#
%
n
Ĉ nγ = (−1) nγ %
$ ⇒ Cπ 0 = +1
0
π → γγ (99%) %
%
Cγ = −1
&
Ĉ γ = − γ
Ĉ a, ψ a = Ca a, ψ a
Ca = ±1 : C − parity
§  (b)Particlediff.antiparticle:π+àπ-,nà anti-n
–onlylinearcombinationsarerelevant
Ĉ b, ψ b = Cb b, ψ b
Ĉ π + = π −
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Ĉ π + = π −
F.Ould-Saada
/ γγγ C − invariance
π0 →
π 0 → γγγ
−8
<
3.1×10
π 0 → γγ
π 0 → J PC = 0 −+
28
¡
(b)mesons(spin-less):π+àπ-
§  Interchangingpositionofparticlesàreverses
Ĉ π +π − ; L = (−1)L π +π − ; L
relativepositioninWFà(-1)L
¡
(b)fermions–anti-fermions
§  Interchangingpositionsà(-1)L
Ĉ f f ; L, S
= (−1)L+S f f ; L, S
§  Interchangefermion-antifermionà(-1)
§  Interchangingspinsà(-1)S+1
¡
↑1↑ 2
S z = +1
(b)mesonswithspin:0,1,2
1
↑ 1 ↓ 2+ ↓ 1 ↑ 2
2
↓1↓ 2
)
Ĉ m + m − ; L, S = (−1)L+S m + m − ; L, S
1
↑ 1 ↓ 2− ↓ 1 ↑ 2
2
)
Ĉ ( pp → π +π −π +π −...) = ( pp → π −π +π −π +...)
Ĉ ( e+e− → π +π −π +π −...) = ( e−e+ → π −π +π −π +...)
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(
(
Sz = 0
S z = −1
Sz = 0
⎫
⎪⎪
⎬ S =1
⎪
⎪⎭
S =0
Averagenumberandenergyspectra
ofπ+andπ-mustbeequal
àverifiedexperimentally
29
¡
(Classical)
Poisson’s
equation:
! ! !
)
1 !
∇⋅ E ( r ,t) = ρ ( r ,t) : P − invariant +
! !
! !
ε0
* ⇒ E ( r ,t) →− E (− r ,t)
§  ElectricfieldE
! P !+
vschargeρ
!P !
! P
!
r →− r ⇒ ρ ( r ,t) →ρ(−r ,t) ; ∇ →− ∇,
density
!
)
!
!
! !
∂A
! ! i( k!r! −Et )
¡  EMfieldA
E = (−∇φ ) − ‚
A( r ,t) = Nε ( k )e
+
∂
t
*Pγ = −1
§  Parity:Pγ=-1;C! ! P ! !
! ! P ! ! +
!P !
parity:Cγ=-1
r →− r ⇒ A( r ,t) →Pγ A(−r ,t) ; E ( r ,t) →Pγ E (−r ,t) ,
¡  π0=q-qbar
C
! ! C ! !
)
q →− q ⇒ A( r ,t) →Cγ A( r ,t)
+
§  Pπ0=-1
*Cγ = −1
C
C
! ! C ! !
§  JPC=0-+
!
!
q →− q ⇒ E ( r ,t) →− E ( r ,t); φ ( r ,t) →− φ ( r ,t) +,
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30
¡
SIandEMinvariantunderCoperation
§  Ceigenvaluesareconservedquantumnumbers
¡
WIviolateC(andP)…butconserveCPtoagoodprecision
§  neutrinos
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31
¡
T-SymmetryofSI&EMàInvarianceundertransformationbutviolatedinWI
§  Noconservedquantumnumberassociatedtotimereversal(neglectWI),unlikePandC
!
! T ! ! T
! ! T
t ⎯⎯
→ t ' = −t ; r ⎯ ⎯
→ r ; p ⎯⎯
→ − p; J ⎯ ⎯
→−J
T
¡
IfsysteminvariantunderT,probability
2
! 2 T
!
2
' !
ψ (r,t) ⎯ ⎯
→ ψ (r,t) = ψ (r,−t)
¡
SchrödingerequationnotinvariantunderT
"
" "
"
∂ψ (r, t)
i!
= H (r, p)ψ (r, t)
∂t
!
Ψ(r,t) = e
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!!
( p⋅r −Et )
i
"
!
!
⇒ T ψ (r,t) = ψ ' (r,t) = e
!!
( p⋅r −Et )
−i
"
32
¡
IntroduceT-operatorbyanalogywithP:
!
!
!
!
ψ (r,t) ⎯T⎯
→ψ ' (r,t) = ψ * (r,−t) ≡ T̂ ψ (r,t)
§  ThenSchrödingerequationinvariant!
"
" "
"
∂ψ * (r, t)
−i!
= H (r, p)ψ * (r, t)
∂t
"
" " * "
∂ψ * (r, −t)
t → −t ⇒ i!
= H (r, p)ψ (r, −t)
∂t
!
ψ (r,t) = e
¡
!!
( p⋅r −Et )
i
"
!
* !
⇒ T̂ ψ (r,t) = ψ (r,−t) = e
SameformasforΨ
!!
( p⋅r +Et )
−i
"
=e
!!
(− p⋅r −Et )
i
"
Time-reversedwavefunctiondescribesaparticlewithmomentum-p
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33
¡
QuantumMechanicsoperatorscorrespondingtophysicalobservablesmustbe
Linear
Oˆ (α1ψ1 + α 2ψ 2 ) = α1 Oˆ ψ1 + α 2 Oˆ ψ 2
( ) (
∫ dx (Oˆ ψ ) ψ = ∫ dxψ (Oˆ ψ )
§  toensuresuperpositionprincipleholds
¡
andHermitian
*
§  eigenvalues(observedvalues)arereal
1
*
1
2
)
2
!
!
!
T
Ψ(r ,t) #
#
→ Ψ* ( r , − t) ≡ TˆΨ(r ,t)
Tˆ (α1ψ1 + α 2ψ 2 ) = α *1 Tˆψ€1 + α *2 Tˆψ 2 ≠ α1 Tˆψ1 + α 2 Tˆψ 2
( ) ( )
∫ dx€(Tˆψ ) ψ ≠ ∫ dxψ (Tˆψ )
*
1
2
*
1
( )
( )
2
Timereversaloperatordoesnotcorrespondtoaphysicalobservable
€ NoobservableconservedasaconsequenceofTinvariance
¡
¡
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34
¡
¡
T-invarianceleadstoarelationbetweenprocessanditstime-reversed
Reactionsandtime-reversedcounterpartsarerelated
!
!
!
!
a ( pa , ma ) + b( pb , mb ) → c( pc , mc ) + d ( pd , md )
⎫
!
!
!
!
⎬
c(− pc ,−mc ) + d (− pd ,−md ) → a (− pa ,−ma ) + b(− pb ,−mb )⎭
¡
ReactionsproceedwithequalratesifWIneglected
§  mi:magneticquantumnumber
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35
Timereversal&Parity
¡
CombinationofT&P
§  SamerateofreactionsifPandTinvariancehold(neglectWI!)
!
!
!
!
a ( pa , ma ) + b ( pb , mb ) → c ( pc , mc ) + d ( pd , md )
⇓T
!
!
!
!
c (− pc , −mc ) + d (− pd , −md ) → a (− pa , −ma ) + b (− pb , −mb )
⇓ P̂
!
!
!
!
c ( pc , −mc ) + d ( pd , −md ) → a ( pa , −ma ) + ( pb , −mb )
§  Ifaveragedoverallpossiblespinprojections
¡
Principleofdetailedbalance
§  ConfirmedexperimentallyinavarietyofStrongandEMprocesses
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mi = −si , −si +1,... si (i = a, b, c, d)
!
!
!
!
i ≡ a ( pa ) + b ( pb ) ↔ c ( pc ) + d ( pd ) ≡ f
36
¡
IfPorTisconserved,
Hamiltonianofinteractionmust
notcontaintermsthatchange
signaftertheoperation
§  Magneticdipolemoments
allowed
§  NotelectricDM!µΕn≠0would
implyèPandTviolated
§  Longitudinalpolarizationonly
throughWI
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37
¡
CPviolationplayedanimportantroleintheearliestmomentsoftheUniverse.
§  Itisbelievedthatatthebeginning,allquantumnumbersoftheUniversewereequalto
zero,withanequalnumberofparticlesandantiparticles.
§  Probablyaftert~10-35s,aphasetransitiontookplace,afterwhichparticlesbeganto
decaywithasmallCPviolation
▪  Leadingtoaslightpredominanceofparticleswithrespecttoantiparticles(<1/109).
§  Whenlaterparticle–antiparticleannihilated,thatlittleexcessofparticlesproducedthe
matter-dominatedUniversewhichweobservetoday.
§  ThesmallamountofCPviolationobservedintheweakinteraction(BandKmeson
decays)seemsnotenoughtofullyexplainthisscenario.
¡
TheCPviolationinvolvesaviolationofTimereversalaswellbecauseall
interactionsareinvariantunderCPTinanyorderthethreetransformationsare
applied.
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38
¡  01/03/16
F.Ould-Saada
39
¡
AlthoughC&Pviolatedinweakinteractions(100%)
§  CPviolatedinsomeweakprocesses(~0.1%)
§  Tviolatedinsomeweakprocesses(~0.1%)
§  thereisageneralresult
¡
CPT(Lüders)theorem:“AnyQuantumTheorythat(i)obeysthepostulatesof
SpecialRelativity,(ii)admitsastatewithminimumenergyand(iii)respects
causalityisinvariantunderCPT”
§  Causalityofphysicaleventsrequiresthatthefieldsobeycommutationoranti-
commutationrelations,implyingthecorrectstatisticsaccordingtothespinof
particles:
▪  Fermi-DiracstatisticsforfermionsandBose-Einsteinstatisticsforbosons
¡
CPTinvariancepredictsthatparticlesandantiparticlesmusthaveexactly
samemassesandlifetimes,oppositemagneticmoments,…
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40
¡
ConsequencesofCPTinvariance
§  Particle&antiparticlehavesamemass,lifetime,oppositemagneticmoments,…
§  Particleinstate|a>=|m,τ,…>
[CPT, H ] = 0!#
"⇒
2
(CPT ) = 1 #$
qp
mp
qp
mp
τ µ+
τ µ−
01/03/16
2
a H a = a H (CPT ) a = a CPT H CPT a
a H a = a H a ⇒ ma = ma
< 0.99999999991± 00000000009 ,
< 1.00002 ± 0.00008 ,
F.Ould-Saada
"#m + − m − $%
e
e
< 8⋅10 −9
me
"µ + − µ − $
# e
e %
< (−0.5 ± 2.1) ⋅10 −12
µe
41
GaugeinvariancediscoveredinEMinteraction
¡
§  Deeplycorrelatedtoelectriccharge(Q)conservation
§  Extensiontolocalgaugeinvariance
Classicalelectrodynamics
¡
§  Potentialφdefineduptoanarbitraryconstant
§  Measurablequantities(E)dependonpotentialdifference,notonabsolutevalue
E,Bfieldexpressedintermsofscalarandvectorpotentials
¡
§  Invariantundertransformationofscalarandvectorpotentialoftype
⎧! ! !
∂Λ
1 ∂Λ
A → A' (x ' ) = A (x ) +
⎨ A' = A + ∇Λ ; φ ' = φ −
⎩
∂xµ
c ∂t
µ
¡
µ
µ
µ
!
%!
1 ∂A
' E = −∇ϕ −
&
c ∂t
!
!
!
' B = ∇ × A (
µ
Asconsequenceofthissymmetry
§  Photonismassless!
¡
ConservationofQ
§  àinvarianceforlocalgroupofgaugetransformations
§  àgaugefieldA(quantum=photon)requiredwhichcouplestoQ
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Moreinchapter11
42