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Symmetry in Physics

Kihyeon Cho

January 29, 2009High Energy Physics Phenomenology

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

such as these crabs such as these crabs

and this starfish. and this starfish.

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Animals that have Line Animals that have Line SymmetrySymmetry

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

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THESE MASKS HAVE THESE MASKS HAVE SYMMETRYSYMMETRY

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

If an object is reflected in water it is considered to

have line symmetry along the waterline.

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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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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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quark anti quark quark anti quark …… decay

Time

We are all the children of Broken symmetry

Just tiny deviation from perfect symmetry seems to have been enough

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Why matter dominant world?

• Baryon number violation• CP violation• Start from thermal equilibrium

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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

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C, P, T violation?

Since early universe…

“Alice effect”

Intuitively…

Boltzmann and S=kln

• C is violated

• P is violated

• T is violated

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Parity Quantum NumberParity Quantum Number

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Charge-conjugation Quantum NumberCharge-conjugation Quantum Number

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Charge conjugate and Parity

CP is the product of two symmetries: C for charge conjugation, which transforms a particle into its antiparticle, and P for parity, which creates the mirror image of a physical system.

C

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C P

CP

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C and P violation!C and P violation!

Experiments show that only circled ones exist in Nature C and P are both maximally violated!

But, CP and T seems to be conserved, or is it?

CP

We can test this in 1st generation meson system: Pions

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Mirror symmetry Parity P

All events should occur in exactly the same way whether they are seen directly or in mirror. There should not be any difference between left and right and nobody should be able to decide whether they are in their own world or in a looking glass world

Charge symmetry Charge C

Particles should behave exactly like their alter egos, antiparticles,

which have exactly the same properties but the opposite charge

Time symmetry Time T

Physical events at the micro level should be equally independent whether they occur forwards or backwards in time.

There are 3 different principles of symmetry in the basic theory

for elementary particles

Three principles of symmetry

Violation in 1956 1957

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Mirror symmetry Parity P

All events should occur in exactly the same way whether they are seen directly or in mirror. There should not be any difference between left and right and nobody should be able to decide whether they are in their own world or in a looking glass world

Charge symmetry Charge C

Particles should behave exactly like their alter egos, antiparticles,

which have exactly the same properties but the opposite charge

Time symmetry Time T

Physical events at the micro level should be equally independent whether they occur forwards or backwards in time.

There are 3 different principles of symmetry in the basic theory

for elementary particles

Three principles of symmetry

Violation in 1964 1980

Violation in 1956 1957

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CP and T violation!CP and T violation!

For 37 years, CP violation involve Kaons only! Is CP violation a general property of the SM or is it

simply an accident to the Kaons only?

CP violationT violation

K0 +-

We can test this in 2nd generation meson system: Kaons

Need 3rd generation system: B-mesons and B-factories

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노벨 물리학상 2008

Citation: 5483

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표준모형 (Standard Model)

What does world made of?– 6 quarks

u, d, c, s, t, b Meson (q qbar) Baryon (qqq)

– 6 leptons e, muon, tau e, ,

bsd

bsd

tbtstd

cbcscd

ubusud

VVVVVVVVV

'''

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bsd

bsd

tbtstd

cbcscd

ubusud

VVVVVVVVV

'''

Vub

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The CKM Matrix

Unitary with 9*2 numbers 4 independent parameters Many ways to write down matrix in terms of these

parameters

ud us ub

cd cs cC b

td ts tb

KM

V V V

= V V V

V V V

V

37F. Di Lodovico, ICHEP 2008

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Heavy Flavor PhysicsHeavy Flavor Physics

정밀측정으로 표준모형의 검증 => 새로운 물리 현상

Foundation New Physics

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CP and T violation!CP and T violation!

For 37 years, CP violation involve Kaons only! Is CP violation a general property of the SM or is it

simply an accident to the Kaons only?

CP violationT violation

K0 +-

We can test this in 2nd generation meson system: Kaons

Need 3rd generation system: B-mesons and B-factories

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Symmetry violation

1972, Makoto Kobayashi , Toshihide Maskawa found that

why this symmetry was broken in 3X3 matrix,so called CKM matrix

quark anti quark quark anti quark …… decay

Time

If this exchange of identity which double broken symmetry was to take

place between matter and antimatter, a further quark family was needed

1974, c quark J/

1977, b quark

1995, t quark

2008

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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.

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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

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Conservation of electric charge and Conservation of electric charge and gauge invariancegauge invariance

Conservation of electric charge: Qi=QfEvidence for conservation of electric charge: Consider reaction e-ve whichviolates charge conservation but not lepton number or any other quantum number.If the above transition occurs in nature then we should see x-rays from atomictransitions. The absence of such x-rays leads to the limit:e > 2x1022 years

There is a connection between charge conservation, gauge invariance,and quantum field theory.Recall Maxwell’s Equations are invariant under a gauge transformation:

tc

AA

1:potentialscalar

:potentialvector

A Lagrangian that is invariant under a transformation U=ei is saidto be gauge invariant.

There are two types of gauge transformations:local: =(x,t)global: =constant, independent of (x,t)

1. Conservation of electric charge is the result of global gauge invariance2. Photon is massless due to local gauge invariance <= Maxwell eq.

Maxwell’s EQs are locally gauge invariant

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AssessmentAssessment

Global gauge invariance implies the existence of a conserved current, according to Noether’s theorem.

Local gauge invariance requires the introduction of massless vector bosons, restricts the form of the interactions of gauge bosons with sources, and generates interactions among the gauge bosons if the symmetry is non-Abelian.

- Chris Quigg P63

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Gauge invariance, Group Theory, and StuffGauge invariance, Group Theory, and StuffConsider a transformation (U) that acts on a wave function ():ULet U be a continuous transformation then U is of the form:

U=ei is an operator.1. If is a hermitian operator (=*T) then U is a unitary transformation:

U=eiU+=(ei)*T= e-i*T = e-iUU+= eie-i=1 Note: U is not a hermitian operator since UU+

2. In the language of group theory is said to be the generator of UThere are 4 properties that define a group:1) closure: if A and B are members of the group then so is AB2) identity: for all members of the set I exists such that IA=A3) Inverse: the set must contain an inverse for every element in the set AA-1=I4) Associativity: if A,B,C are members of the group then A(BC)=(AB)C

3. If = (1, 2, 3,..) then the transformation is “Abelian” if:U(1)U(2) = U(2)U(1) i.e. the operators commute

4. If the operators do not commute then the group is non-Abelian.The transformation with only one forms the unitary abelian group U(1)The Pauli (spin) matrices generate the non-Abelian group SU(2)

10

01

0

0

01

10zyx i

i S= “special”= unit determinantU=unitaryn=dimension (e.g.2)

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Global Gauge Invariance and Charge Global Gauge Invariance and Charge ConservationConservation

The relativistic Lagrangian for a free electron is:

ii ee

is the electron field (a 4 component spinor)m is the electrons massu= “gamma” matrices, four (u=0,1,2,3) 4x4 matrices that satisfy uv+ vu =2guv

utxyzLet’s apply a global gauge transformation to L

1 cmiL uu

LmiL

eemeeiL

eemeeiL

miL

uu

iiuu

ii

iiiuu

i

uu

constantaisλsince

By Noether’s Theorem there must be a conserved quantity associatedwith this symmetry!

This Lagrangian gives the Dirac equation:

0 mci uu

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The Dirac Equation on One PageThe Dirac Equation on One Page0 mci u

u

0][ 3210

mczyxt

i

0

0

0

0

1000

0100

0010

0001

]

0010

0001

1000

0100

000

000

000

000

0001

0010

0100

1000

1000

0100

0010

0001

[ mczy

i

i

i

i

xti

0

0

10

010

i

ii

])(/(exp[])(/(exp[

)(

0

1

/)|(|

2

2

2 rpEtiUrpEti

McE

ippcMcE

cpcmcE

yx

z

A solution (one of 4, two with +E, two with -E) to the Dirac equation is:

0)( Umcpuu

The function U is a (two-component) SPINOR and satisfies the following equation:

The Dirac equation:

Spinors are most commonly used in physics to describe spin 1/2 objects. For example:

)2/(downspinrepresents1

0while/2)(upspinrepresents

0

1

Spinors also have the property that they change sign under a 3600 rotation!

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E-L equation in 1D

Global Gauge Invariance and Charge ConservationGlobal Gauge Invariance and Charge ConservationWe need to find the quantity that is conserved by our symmetry.In general if a Lagrangian density, L=L(, xu) with a field, is invariantunder a transformation we have:

u

u

xx

LLL

0

For our global gauge transformation we have:

uuu xi

xxii

and)1(

Plugging this result into the equation above we get (after some algebra…)

u

u

u

uu

u x

L

xx

L

x

L

xx

LLL )(0

The first term is zero by the Euler-Lagrange equation.The second term gives us a continuity equation. 0)(

x

L

dt

d

x

L

Result fromfield theory

)1( iei

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Global Gauge Invariance and Charge ConservationGlobal Gauge Invariance and Charge Conservation

The continuity equation is:

u

u

u

u

u

u

u

u

x

LiJ

x

Ji

x

L

xx

L

xwith0

Recall that in classical E&M the (charge/current) continuity equation has the form:

0

Jt

Also, recall that the Schrodinger equation give a conserved (probability) current:

])([and2

***22

ciJcVmt

i

If we use the Dirac Lagrangian in the above equation for L we find:

uuJThis is just the relativistic electromagnetic current density for an electron.The electric charge is just the zeroth component of the 4-vector:

xdJQ

0

Therefore, if there are no current sources or sinks (J=0) charge is conserved as:

000

t

J

x

J

u

u

(J0, J1, J2, J3) =(, Jx, Jy, Jz)

Result fromquantum field theory

Conserved quantity

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Local Gauge Invariance and PhysicsLocal Gauge Invariance and PhysicsSome consequences of local gauge invariance:a) For QED local gauge invariance implies that the photon is massless.

b) In theories with local gauge invariance a conserved quantum number impliesa long range field.

e.g. electric and magnetic fieldHowever, there are other quantum numbers that are similar to electric charge(e.g. lepton number, baryon number) that don’t seem to have a long range forceassociated with them!

Perhaps these are not exact symmetries! evidence for neutrino oscillation implies lepton number violation.c) Theories with local gauge invariance can be renormalizable, i.e. can useperturbation theory to calculate decay rates, cross sections, etc.

Strong, Weak and EM theories are described by local gauge theories.U(1) local gauge invariance first discussed by Weyl in 1919SU(2) local gauge invariance discussed by Yang&Mills in 1954 (electro-weak)e i(x,t) is represented by the 2x2 Pauli matrices (non-Abelian)SU(3) local gauge invariance used to describe strong interaction (QCD) in 1970’s

e i(x,t) is represented by the 3x3 matrices of SU(3) (non-Abelian)

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Local Gauge InvarianceLocal Gauge Invariance

Strong, Weak and EM theories are described by local gauge theories.

U(1) local gauge invariance first discussed by Weyl in 1919SU(2) local gauge invariance discussed by Yang&Mills in 1954

(electro-weak)e i(x,t) is represented by the 2x2 Pauli matrices (non-

Abelian)SU(3) local gauge invariance used to describe strong interaction

(QCD) in 1970’s e i(x,t) is represented by the 3x3 matrices of SU(3) (non-Abelian

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Local Gauge Invariance and QEDLocal Gauge Invariance and QED

Consider the case of local gauge invariance, =(x,t) with transformation:),(),( txitxi ee

The relativistic Lagrangian for a free electron is NOT invariant under this transformation.1 cmiL u

u The derivative in the Lagrangian introduces an extra term:

)],([),(),( txiee utxiutxi

We can MAKE a Lagrangian that is locally gauge invariant by addingan extra piece to the free electron Lagrangian that will cancel the derivative term.

We need to add a vector field Au which transforms under a gauge transformation as: AuAu+u(x,t) with (x,t)=-q(x,t) (for electron q=-|e|)

The new, locally gauge invariant Lagrangian is:u

uuvuvu

u AqFFmiL 16

1

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The Locally Gauge Invariance QED LagrangianThe Locally Gauge Invariance QED Lagrangianu

uuvuvu

u AqFFmiL 16

1

Several important things to note about the above Lagrangian:1) Au is the field associated with the photon.

2) The mass of the photon must be zero or else there would be a term in the Lagrangian of the form:

mAuAu

However, AuAu is not gauge invariant!

3) Fuv=uAv- vAu and represents the kinetic energy term of the photon.

4) The photon and electron interact via the last term in the Lagrangian. This is sometimes called a current interaction since:u

uu

u AJAq In order to do QED calculations we apply perturbation theory (via Feynman diagrams) to JuAu term.

5) The symmetry group involved here is unitary and has one parameter U(1)

e-e-

Au

Ju

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ReferencesReferences

I.S.Cho’s talk (2008)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)