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Invariant Principles and Conservation laws

Kihyeon Cho

April 26, 2011HEP

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ContentsContents

SymmetryParity (P)Gauge invarianceCharge (C)CP Violation

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Line SymmetryLine SymmetryShape has line symmetry when one half

of it is the mirror image of the other half.

Symmetry exists all around us and many people see it as being a thing of beauty.

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Is a butterfly Is a butterfly symmetrical?symmetrical?

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Line Symmetry exists in Line Symmetry exists in nature but you may not nature but you may not

have noticed.have noticed.

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At the  beach there are a At the  beach there are a variety of shells with line variety of shells with line

symmetry. symmetry.

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Under the Under the seasea there are also there are also many symmetrical objects many symmetrical objects

such as these crabssuch as these crabs

and this starfish. and this starfish.

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Animals that have Line SymmetryAnimals that have Line Symmetry

Here are a few more great examples of mirror image in the animal kingdom.

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THESE MASKS HAVE SYMMETRYTHESE MASKS HAVE SYMMETRY

These masks have a line of symmetry from the forehead to the chin.  

The human face also has a line of symmetry in the same place.

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Human SymmetryHuman Symmetry

The 'Proportions of Man' is a famous work of art by Leonardo da Vinci that

shows the symmetry of the human form.

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REFLECTION IN WATERREFLECTION IN WATER

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The Taj MahalThe Taj Mahal

Symmetry exists in architecture all around the world.  The best known example of this is the Taj Mahal.

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This photograph shows 2 lines of symmetry. One vertical, the other along the waterline.

(Notice how the prayer towers, called minarets, are reflected in the water and side to side).

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22D Shapes and SymmetryD Shapes and Symmetry

After investigating the following shapes by cutting and folding, we found:

 

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an equilateral triangle has 3 internal angles and 3 lines of symmetry. 

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    a square has 4 internal angles and 4 lines of symmetry. 

 

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   a regular pentagon has 5 internal angles and 5 lines of symmetry.

 

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a regular hexagon has 6 internal angles and 6 lines of symmetry .

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     a regular octagon has 8 internal angles and 8 lines of

symmetry.

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Symmetry in Physics Symmetry is the most crucial

concepts in Physics. Symmetry principles dictate the

basic laws of Physics, and define the fundamental forces of Nature.

Symmetries are closely linked to the particular dynamics of the system:

E.g., strong and EM interactions conserve C, P, and T. But, weak interactions violate all of them.

Different kinds of symmetries: Continuous or Discrete Global or Local Dynamical Internal

Examples of Symmetry Examples of Symmetry OperationsOperations

Translation in SpaceTranslation in TimeRotation in SpaceLorentz TransformationReflection of Space (P)Charge Conjugation (C)Reversal of Time (T)Interchange of IdenticalParticlesChange of Q.M. Phase Gauge TransformationsWe focus on this

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Conservation RulesConserved

QuantityWeak Electromagnetic Strong

I(Isospin) No

(I=1 or ½)

No Yes

(No in 1996)

S(Strangeness) No

(S=1,0)

Yes Yes

C(charm) No

(C=1,0)

Yes Yes

P(parity) No Yes Yes

C(charge) No Yes Yes

CP No Yes Yes

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Conserved Quantities and SymmetriesConserved Quantities and SymmetriesEvery conservation law corresponds to an invariance of the Hamiltonian (or Lagrangian) of the system under some transformation.We call these invariances symmetries. There are 2 types of transformations: continuous and discontinuousContinuous give additive conservation laws

x x+dx or +d examples of conserved quantities:

electric chargemomentumbaryon #

Discontinuous give multiplicative conservation lawsparity transformation: x, y, z (-x), (-y), (-z)charge conjugation (particleantiparticle): e- e+

examples of conserved quantities: parity (in strong and EM)charge conjugation (in strong and EM)parity and charge conjugation (strong, EM, almost always in weak)

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Conserved Quantities and SymmetriesConserved Quantities and SymmetriesExample of classical mechanics and momentum conservation.In general a system can be described by the following Hamiltonian: H=H(pi,qi,t) with pi=momentum coordinate, qi=space coordinate, t=timeConsider the variation of H due to a translation qi only.

dt

dppwith

q

Hp

p

Hq i

ii

ii

i

dtt

Hdp

p

Hdq

q

HdH

ii

iii

i

3

1

3

1

For our example dpi=dt=0 so we have:

3

1ii

i

dqq

HdH

Using Hamilton’s canonical equations:

We can rewrite dH as:

3

1

3

1 iii

ii

i

dqpdqq

HdH

If H is invariant under a translation (dq) then by definition we must have:

03

1

3

1

iii

ii

i

dqpdqq

HdH

This can only be true if:00

3

1

3

1

ii

ii p

dt

dp or

Thus each p component is constant in time and momentum is conserved.

ReadPerkins: Chapters 3.1

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Conserved Quantities and Quantum MechanicsConserved Quantities and Quantum MechanicsIn quantum mechanics quantities whose operators commute with theHamiltonian are conserved.Recall: the expectation value of an operator Q is:

),,(and),(with* txxQQtxxdQQ How does <Q> change with time?

xdt

Qxdt

QxdQ

txdQ

dt

dQ

dt

d ***

*

Recall Schrodinger’s equation:

Ht

iHt

i **

and

Substituting the Schrodinger equation into the time derivative of Q gives:

xdQHi

xdt

QxdQH

ixdQ

dt

dQ

dt

d **** 11

H+= H*T= hermitian conjugate of H

Since H is hermitian (H+= H) we can rewrite the above as:

xdHQit

QQ

dt

d)],[

1(*

So then <Q> is conserved. 0],[ and 0

HQt

Q

ReadPerkins: Chapters 3.1

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Three Important Discrete Symmetries

• Parity, P– Parity reflects a system through the origin. Converts

right-handed coordinate systems to left-handed ones.– Vectors change sign but axial vectors remain unchanged

• x x L L

• Charge Conjugation, C– Charge conjugation turns a particle into its anti-particle

• e e K K

• Time Reversal, T– Changes the direction of motion of particles in time

• t t

• CPT theorem– One of the most important and generally valid theorems in quantum

field theory.– All interactions are invariant under combined C, P and T

transformations.– Implies particle and anti-particle have equal masses and lifetimes

Discrete SymmetriesAn example of a discrete transformation is the operation of inverting all angles: -In contrast a rotation by an amount is a continuous transformation. 

Reminder: Discrete symmetries give multiplicative quantum numbers.Continuous symmetries give additive quantum numbers.

 

The three most important discrete symmetries are:Parity (P) (x,y,z) (-x,-y,-z)Charge Conjugation (C) particles anti-particlesTime Reversal (T) time -time

Other not so common discrete symmetries include G parity: G=CR=C exp (iπI2) = (-1) l+S+I

G parity is important for pions under the strong interaction. 

Note: discrete transformations do not have to be unitary transformations !P and C are unitary transformationsT is not a unitary transformation, T is an antiunitary operator!

 

ReadPerkins: Chapters 3.3, 3.4

ReadPerkins: Chapters 4.5.1

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ReadPerkins: Chapters 3.3, 3.4

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Parity Quantum NumberParity Quantum Number

ParityLet us examine the parity operator (P) and its eigenvalues. The parity operatoracting on a wavefunction is defined by:

P(x, y, z) = (-x, -y, -z)P2(x, y, z) = P(-x, -y, -z) = (x, y, z)

Therefore P2 = I and the parity operator is unitary.If the interaction Hamiltonian (H) conserves parity then [H,P]=0, and:

P(x, y, z) = (-x, -y, -z) = n(x, y, z) with n = eigenvalue of PP2(x, y, z) = PP(x, y, z) = nP(x, y, z) = n2(x, y, z)

(x, y, z) = n2(x, y, z) n2 = 1 so, n=1 or n=-1.The quantum number n is called the intrinsic parity of a particle.

If n= 1 the particle has even parity.If n= -1 the particle has odd parity.

In addition, if the overall wavefunction of a particle (or system of particles)contains spherical harmonics (YL

m) then we must take this into account toget the total parity of the particle (or system of particles).The parity of YL

m is: PYLm

= (-1)L YLm.

For a wavefunction (r, , )=R(r)YLm (, ) the eigenvalues of the parity

operator are:P(r, , )=PR(r)YL

m (, ) = (-1)LR(r)YLm (, )

The parity of the particle would then be: n(-1)L

Note: Parity is a multiplicative quantum number

M&S pages 88-94

ReadPerkins: Chapters 3.3, 3.4

ParityThe parity of a state consisting of particles a and b is:

(-1)Lnanb

where L is their relative orbital momentum and na and nb are the intrinsicparity of each of the two particle. Note: strictly speaking parity is only defined in the system where the total momentum (p) =0 since the parity operator (P) and momentum operator anticommute, (Pp=-p).

How do we know the parity of a particle ? By convention we assign positive intrinsic parity (+) to spin 1/2 fermions:

+parity: proton, neutron, electron, muon (-)

Anti-fermions have opposite intrinsic parity-parity: anti-proton, anti-neutron, positron, anti-muon (+)

Bosons and their anti-particles have the same intrinsic parity. What about the photon?Strictly speaking, we can not assign a parity to the photon since it is never at rest. By convention the parity of the photon is given by the radiation field involved:

electric dipole transitions have + parity magnetic dipole transitions have - parity

We determine the parity of other particles (, K..) using the above conventionsand assuming parity is conserved in the strong and electromagnetic interaction.Usually we need to resort to experiment to determine the parity of a particle.

ReadPerkins: Chapters 3.3, 3.4

ParityExample: determination of the parity of the using -dnn.For this reaction we know many things:a) s=0, sn=1/2, sd=1, orbital angular momentum Ld=0, => Jd=Ld+sd=0+1=1b) We know (from experiment) that the is captured by the d in an s-wave state. Thus the total angular momentum of the initial state is just that of the d (J=1).c) The isospin of the nn system is 1 since d is an isosinglet and the - has I=|1,-1> note: a |1,-1> is symmetric under the interchange of particles. (see below)d) The final state contains two identical fermions and therefore by the Pauli Principle the wavefunction must be anti-symmetric under the exchange of the two neutrons.Let’s use these facts to pin down the intrinsic parity of the .i) Assume the total spin of the nn system =0. Then the spin part of the wavefunction is anti-symmetric: |0,0> = (2)-1/2[|1/2,1/2>|1/2-1/2>-|1/2,-1/2>|1/2,1/2>] To get a totally anti-symmetric wavefunction L must be even (0,2,4…) Cannot conserve momentum (J=1) with these conditions! ( since J=L+s => 10+0,2,)ii) Assume the total spin of the nn system =1. Then the spin part of the wavefunction is symmetric: |1,1> = |1/2,1/2>|1/2,1/2>

|1,0> = (2)-1/2[|1/2,1/2>|1/2-1/2>+|1/2,-1/2>|1/2,1/2>] |1,-1> = |1/2,-1/2>|1/2,-1/2> ( since J=L+s => 1=1+1)

To get a totally anti-symmetric wavefunction L must be odd (1, 3, 5…) L=1 consistent with angular momentum conservation: nn has s=1, L=1, J=1 3P1

The parity of the final state is: nnnn(-1)L= (+)(+)(-1)1= -The parity of the initial state is: nnd(-1)L= n(+)(-1)0 = n

Parity conservation gives: nnnn(-1)L = nnd(-1)L n= -

ReadPerkins: Chapters 3.3.1

ParityThere is other experimental evidence that the parity of the is -:

the reaction -dnn 0 is not observedthe polarization of ’s from 0

Some use “spin-parity” buzz words:buzzword spin parity particlepseudoscalar 0 - , kscalar 0 + higgs (none observed)vector 1 - pseudovector 1 + A1(axial vector)

How well is parity conserved? Very well in strong and electromagnetic interactions (10-13) not at all in the weak interaction!The puzzle and the downfall of parity in the weak interactionIn the mid-1950’s it was noticed that there were 2 charged particles that had (experimentally)consistent masses, lifetimes and spin = 0, but very different weak decay modes:

++ 0 ++ - + The parity of += + while the parity of += -Some physicists said the +and + were different particles, and parity was conserved.Lee and Yang said they were the same particle but parity was not conserved in weak interaction!Lee and Yang win Nobel Prize when parity violation was discovered.

Note: ++ is now known as the K+.

M&S pages 240-248

ReadPerkins: Chapters 3.3, 3.4

Discrete Symmetries, ParityParity and nature:

The strong and electromagnetic interactions conserve parity.The weak interaction does not.

Thus if we consider a Hamiltonian to be made up of several pieces:H = Hs + HEM + HW

Then the parity operator (P) commutes with Hs and HEM but not with HW .

The fact that [P, HW] 0 constrains the functional form of the Hamiltonian.

What does parity do to some common operations ?vector or polar vector x - x or p - p.axial or pseudo vectors J = x p J.time (t) t t.

name form parityscalar r•r +pseudoscalar x•(y z) -vector r -axial vector r x p +Tensor Fuv indefinite

According to special relativity, the Hamiltonian or Lagrangian of any interaction must transform like a Lorentz scalar.

ReadPerkins: Chapters 3.5

Parity Violation in -decay

+ +

60CoJ=5

60Ni*J=4

60Ni*J=4

B-fieldJz=1 Jz=1

pv

pvpe-

pe-

YESNO

Classic experiment of Wu et. al.(Phys. Rev. V105, Jan. 15, 1957)looked at spectrum from:

eeNiCo *6028

6027

followed by:)33.1()17.1(*60

28*60

28 NiNi

Note: 3 other papers reportingparity violation published withina month of Wu et. al.!!!!!

detector

detector

detector

counting ratedepends on <J>pe

which is – under aparity transformation

ReadPerkins: Chapters 3.5

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ReferencesReferences

Class P720.02 by Richard Kass (2003)B.G Cheon’s Summer School (2002)S.H Yang’s Colloquium (2001)Class by Jungil Lee (2004)PDG home page

(http://pdg.lbl.gov)