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Lecture 5 – Symmetries and Isospin

● What is a symmetry ? ● Noether’s theorem and conserved quantities● Isospin

Obs! Symmetry lectures largely covers sections of the material in chapters 5, 6 and 10 in Martin and Shaw though not always in the order used in the book.

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0.An experiment measures the reaction rate for at

If the experiment was moved to the same rate would be observed.

This is equivalent to not moving the original experiment but red

R p n x

x a R

'

efining the -axis through

a translation: instead.

The laws of physics are invariant to a transformation of a translation in space - symmetry.

To put it another way, we can't know if our experim

x

x x a

ent took place at a certain place just

by looking at the measured reaction rate.

Definition of a symmetry in particle physics: under a transformation one or more observables

will be unchanged/"invariant to the transformation".

Symmetry – an intuitive example and a definition

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Types of symmetry● Continuous symmetries

Depend on some continuous variable, eg linear translation which can be built up out of infinitesimal steps

Eg linear spatial translation x’=x+x, translation in time t’=t+t

● Discrete symmetries (subsequent lectures) Two possibilities (all or nothing) Take a process, eg A+B C and measure it!

● Does this process happen at the same rate if Eg. The mirror image is considered (parity – move from left-handed to

right-handed co-ordinate system) Eg. All particles are transformed to their anti-particles: A+B C

(charge conjugation)

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Conserved quantities in quantum mechanics

ˆ

ˆ

Consider observable system described by time-independent Hamiltonian.

Wave function of the system satisfies Schrödingers equation: (1.18)

Consider operator for observable and its expectat

Ψψ i HΨ

t

Q

ˆ

ˆˆ ˆ ˆ| |

ˆ ˆ| |

ion value's time-dependence.

(5.01)

(5.02)

(assume no time dependence on the ope

Q ψ |Q |ψ

d Q d Qψ |Q |ψ Q ψ | ψ Q

dt dt t t t

Q Qt t

1 1ˆ ˆˆ ˆ| |

ˆ ˆˆ ˆ ˆ| |

1 ˆ ˆ,

ˆˆ , 0

rator)

(5.03)

is Hermitian

(5.04)

If (5.05) i.e. the operator for an observable commutes with the Ham

d QHΨ Q Ψ QH

dt i i

H HΨ Q Ψ HQ

d QQ H

dt i

H Q

iltonian

then that observable is a conserved quantity.

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Conserved quantities – why we need them and how we find them

0.

In an ideal world in which we can write down the equations for all of the

fundamental forces in all circumstances and calculate that, eg

We don't have such a theory so we need conserved qu

e

antities/conservation laws

to provide important constraints on what can and can't occur. How do we find

such laws and how can we interpret them in the light of our incomplete knowledge

of nature ?

We d ˆ

ˆ ˆˆ, ,

on't know the form of for every situation so we can't try all

commutator combinations: eg , Lepton number etc. and

see which pairs of operators commute.

There is a way to find out i

H

H p H

ˆ

ˆ

f commutes with another operator without knowing or asking

what is.

H

H

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Translational invariance

All positions in space are physically indistinguishable.

If a closed system of particles (no external forces acting) is moved

from one position to another its properties are unaltered.

Consider shift:

2 2 2'1 12 2 2

1 1 1

'

ˆ ˆ '

ˆ

ˆ ˆ'

Hamiltonian becomes:

This is unchanged by the translation, eg free particle

insensitive to

(5.06) - generally true.

i i i

i i

i i

r r r r

H r H r

H x x xx y z

H r H r

x

x1x2

x1+x x2+x

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( )

ˆ ( ) ( )

ˆ( ) ( ) ( ) 1 ( )

ˆ

ˆ ˆ1

Consider single particle and 1-dimension ( )

Displacement operator (5.07) Small displacement

x

x

x x

D x x x x

xx x x x x i x i x i xp x

x x

p ix

D i xp

ˆ ˆ ˆ ˆ'( ) ( ) '( ) ( )

ˆ ˆ ˆ ˆ ˆ'( ) '( ) ( ) ( ) ( )

ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) , 0

ˆ ˆˆ ˆ1 , 0 , 0

(5.08)

Define

From (5.07): = =

(5.09)

(5

x

x x

x H x x D x DH x x

D x x x H x x x x H x x x H x D x

DH x x H x D x D H

i xp H p H

.10) Linear momentum conservation!!

Can generalise to many particles, 3-dimensions and finite displacements.

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What have we done ?

ˆˆ: , 0

(1) Derived the quantum mechanical version of a conservation law:

for an observable (5.05)

(2) Realised it was impossible just to take a Hamiltonian and try out

various observables to

Q H Q

ˆ

see which commute.

(3) Took an example of a displacement in space. Saw that if the Hamiltonian of a system

is invariant to the displacement, then the transformation operator commutes with the

Hamilt

D

ˆ ˆ, 0.

ˆ ˆ ˆ1 .

ˆˆ , 0

onian: (5.09)

(4) is related to the linear momentum operator: (5.08) This led to

(5.10): linear momentum conservation.

The invariance of the Hamiltonian to a

x

x

D H

D D ip x

p H

displacement in space leads to the conservation

of linear momentum. Symmetry!!

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Noether’s theorem

Symmetry Conservation Law

Translation in space Linear momentum

Translation in time Energy

Rotation in space Angular momentum

Gauge transformations Electric, weak and colour charge

Previous slides showed a quantum mechanical ”version” of Noether’s theorem. If a system of particles shows a symmetry, eg its Hamiltonian is invariant to a continuous transformation then there is a conserved quantity.

Discrete symmetries also lead to conserved quantities (to come)

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Charge conservation – (not for lecture/exam)

,

.Electron in an electromagnetic field: Is nature insensitive to a local phase shift (gauge transformation)

depending in position and time: ? (5.11)

Yes, but the solution is subtle. The el

i Q r te

'

ectron is charged and interacts via the electromagnetic force.

( scalar potential) ; vector potential . (5.12)

Gauge freedom with potentials (from electrodynamics):

E B A A

t

2

0 0

0

,

' , ,

ˆ ˆ ˆ2

ˆ .

'

; arbitrary function. (5.13)

Electron in an electrostatic potential: (5.14)

Schrödinger's equation for electron:

i r t Q

A A r t r t

H H e Hm

i H et

e

0

0

' ˆ '

' ˆ ' '

- for simplicity only consider a time variation.

Is it possible to have the same form for Schrödinger's equation: ? (5.15)

should also work (gauge freedom).

i

i H et

i H et

i et

2, ,

2, ,

2

2

: ,2

2

set - ok, since are abitrary.

(5.16) From 5.15 and 5.16, and

r t Q i r t Q

i r t Q i r t Q

e e em t

e i Q e e e Q et t m t t t

i et m

'

.

would behave the same way!

The long range electromagnetic field cancels the local phase shift Charge must be conserved since the cancellation

would not work if the electron lost its charge for a p

eriod of time and couldn't interact with the electromagnetic field.

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' cos sin 1

' sin cos 1

, ,

Transformation is an infinitesimal rotation about the -axis with angle

(5.17)

z

x ε ε x ε x

y - ε ε y -ε y

x x x y y y y x z z z

r x y z r

3

, , , ,0

ˆ ( , , ) ( , , )

( , , )

ˆ ˆ ˆ ˆ1

ˆ ˆ1

(5.18)

(5.19)

(5.20)

y x x y

x y z r r y x

DΨ x y z Ψ x y y x z

Ψ ΨΨ x y z y x

x y

i xp yp Ψ p i p ix y

D ( iεL )

ˆ ˆ ˆ -

ˆ ˆ ˆ ˆ'( ) ( ) '( ) ( )

ˆ ˆ ˆ ˆ ˆ'( ) '( ) ( ) ( ) (

operator for component of orbital angular momentum. (5.21)

Define

From (5.19): = =

z y xL xp yp z

r H r r D r DH r r

D r r r H r r r r H r r r H r D

3 3

)

ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) , 0

ˆ ˆ ˆ ˆ1 , 0 , 0

(5.22)

(5.23) Orbital angular momentum conservation!!

Angular momentum conservation is demanded if we require the laws of ph

r

DH r r H r D r D H

iεL H L H

ysics are invariant to a rotation.

Angular Momentum (no spin)

x

x’

y’y

xp

xp’

yp

’yp

P

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Angular momentumA rotation in space (around -axis) was considered for a situation

involving no spin.

The Hamiltonian was invariant to it symmetry.

The observable quantity which is conserved is orbital angular moment

z

ˆ ˆ ˆ ˆ ˆ, 0

ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ, 0 , 0

um:

(5.23) ; (5.21)

Considering rotations around and axes would similarly give:

, ; ,

z

z z y x

y y x z z z

L

H L L xp yp

x y

H L L zp xp H L L

ˆ ˆ ˆˆ ˆ ,

ˆ ˆ ˆ ˆ ˆ ˆ, ; ,

ˆ

; = (5.24)

Generalise to include spin :

Total angular momentum: ; = = (5.25)

Total angular momentum conserved:

y x i j ijk k

i j ijk k i j ijk k

xp yp L L iL

S

J L S S S iS J J iJ

H

ˆ ˆ ˆ ˆ ˆ, , , 0 (5.26)x y zJ H J H J

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Spin • Simplest case for studying angular momentum.• Algebraically a carbon copy of orbital angular momentum:

• Useful to use matrices to represent states and operators.• A spin ½ particle can have a spin-up or spin-down projection

along an arbitary z-axis.

• General state - linear combination:

• Operators with 2x2 matrices.

ˆ ˆ ˆ, (5.25)i j ijk kS S i S

0 1 0 1 01 1 1ˆ ˆ ˆ1 0 0 0 12 2 2

1

20 1 0 1

1 0 0

(5.29)

Drop and they become the Pauli spin matrices

x y z

x y z

i S S S

i

i

i

0

0 1

(5.30)

2 21 0| | | | 1

0 1 (5.28)

1 01 1 1 1| |

0 12 2 2 2 (5.27)

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Transformations Intuitively easy to understand how the components of a vector

change under a rotation.

Consider how a two component spinor is affected by a rotation of a co-ordinate system.

3ˆ /2'( ) ( ) 1 ..

' 2! 3

ˆ( ) cos sin2 2

|

(5.31) is a matrix:

(5.32)

- direction angle of rotation (right-hand sense), | angle magnitude.

identity

i A A AU U e U e

U I i

I

* 1 *

, ,

( )

matrix , Pauli matrices.

is a 2 2 matrix belonging to the set of all such matrices: SU(2)

Unitary 2x2 matrices with determinant 1.

Unitary ; (5.33)

x y z

U

U U I U U

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Question Find the matrix representing a rotation by around the y-axis.

Show it converts a spin-up to spin-down particle.

Can transform between possible states – equivalent to rotating co-ordinate axes. Obs! – in an experiment a rotation would affect all states and there wouldn’t be observable effect in the physics measurement – nature doesn’t care where your axes are.

ˆ( ) cos sin cos sin2 2 2 2

0 0 1

0 1 0

0 1 1

1 0 0

(5.34)

(5.35)

(5.36)

1Spin-up particle (in direction): (5.37)

0

y y

y

U I i U I i i

i U i

i

z -

U

0

1 (5.38)

z

y

spin-downspin-up

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QuestionRepeat the procedure in the previous question and converts a

spin-down to a spin-up particle.

0

1

0 1 0 1 1 1

1 0 1 0 2 2

Spin-up particle (in direction): (5.39)

(5.40)

Obs! - sign. Unimportant (for the most part).

The amplitude squared is the important quantity

z -

U

.

z

y

spin-downspin-up

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Addition of angular momentum● Two spin ½ particles can form multiplets● A triplet and a singlet can be formed (n=2S+1)

● If nature chooses to use a multiplet, it must use all ”members”. Eg deuteron (np) – 3 x spin 1 states (SZ=-1,0,1) No stable spin 0 np state

2

1

2

1|

2

1

2

1|

2

1

2

1|

2

1

2

1|

2

100|

2

1

2

1|

2

1

2

1|11|

2

1

2

1|

2

1

2

1|

2

1

2

1|

2

1

2

1|

2

110|

2

1

2

1|

2

1

2

1|11|

triplet

singlet

(5.41)

(5.42)

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SU(2) - isospin ● Proton and neutron have similar masses:

● Heisenberg postulated that proton and neutron are two states of the same particle – isospin doublet. Mass differences due to electromagnetic effects.

● Proton and neutron have different projections in internal ”isospin space”.

● Strong force invariant under rotations in isospin space.

● Isospin conserved for strong interactions (Noether’s theorem).

0.9383 0.9396 p nm GeV m GeV

3

3

1 1 1 1| |2 2 2 2

|

(5.34)

isospin quantum number,

quantum number for 3rd component of isospin.

p n

I I I

I

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Testing isospin

● Best way to understand anything is to look at physical situations● Two approaches to testing the hypothesis that isospin is a

symmetry of the strong force. Isospin invariance

● If a strong reaction/decay takes place then the reaction/decay of the isospin-rotated particles must also happen at the same rate.

● If a particle within an isospin multiplet is found in nature then the other multiplet members must also exist since they correspond to different projections in isospin space and the strong force is invariant to a rotation in isospin space.

Isospin conservation ● Isospin quantum numbers must ”add up” when considering reactions/decays.

● Both approaches are complementary (Noether’s theorem)

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p (9383 MeV)

n (9396 MeV)

MeV)MeV)

I3

MeV)

Multiplicity of states: N=2I+1 (5.43)

Isospin multiplets

If the electromagnetic field could be turned off the masses of the particles within the isospin multiplets would be the same according to isospin symmetry.

1 1

2 2

1 1

1 1

2 2

1 0 1 1

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(5.44)

(5.45)

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Rotation in isospin spaceSlightly more interesting than rotations in real space.

180 ' '

.

A rotation by around the -axis in isospin space

converts a proton neutron and

Can measure:

(a) and (b)

(a) and (b) are the 'same' reaction as seen

by the strong force

o y

p p d n n d

if isospin is a good symmetry.

Strong reaction rates are measured to be the same.

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Question0

3

.

Consider reactions (a), (b) and (c)

(a) and (b) (c)

(i) Verify that is conserved for each reaction.

(ii) Estimate the ratio of the reaction rate (a):(b):(c) for identi

p p d n n d p n d

I

3

1 1 1 1 1 1: 0 1 0 1 0 0

2 2 2 2 2 2

cal experimental conditions

i.e. same energies, momenta etc, the only differences are the interacting particles.

(i) Sum (a) (b) - - (c)

Ok, processes can happen but

I

1 1 1 1( ) ( ) 1 1 1 1

2 2 2 2

1 1 1 1( ) 1 1 1 1

2 2 2 2

1 1 1 1 11 1

2 2 2 2 2

at what relative rates ?

(LHS) 0 0 1 1 (RHS)

- - - (LHS) 0 0 1 -1 - (RHS)

(c) - 0 0 0 (LHS) 0 0 1 0 0 (

ii a

b

1

1: : 1:1: .

2

RHS)

(When it gets complicated use Clebsch-Gordan co-efficients to do the above)

Only the piece is important since isospin must be conserved.

Ratio of amplitudes:

Ratio of rates/

a b c

I

M M M

2 2 2 1: : 1:1: .

2cross section: a b cM M M

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Gell-Mann-Nishijima Formula

3

3

1

2

,

rd

For all hadrons and quarks: (5.36)

charge ; 3 component of isospin ; Baryon number

strangeness charmness "bottomness" (4.01)

(no need to conss s c c bb

Q I (B S C B)

Q I B

S n n , C n n B n n

ider "top" hadrons - they don't exist.)

I3 I3-½ ½ ½ -½ +1-1 +1-1

Meson nonet (spin 0) Baryon octet (spin ½)

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Question0

.

The decay takes place with a rate typical of the strong force.

The isosinglet has a quark content .

Deduce the quark content and values of the quantum numbers of

Does it poss

c c

c

c

udc

, 1, 1, 0

ess any isospin partners ?

If so, what is their quark content ?

Verify that isospin is conserved in the above decay.

Strong rate flavour isn't violated!!

has content

From 5.36

c udc B C B S

3

0

3

0

1

2

0 0 , 1 0

1: 1 1 1 0 1 1 0

2

1 0 , 1 -1 , 1 1

is an isosinglet :

,

Part of a Multiplet:

Mass 2520 (from tables)

c

c

c c c

Q I (B S C B)

I I

udc ddc uuc

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Conserved quantities/symmetries

Quantity Strong Weak Electromagnetic

Energy

Linear momentum

Angular momentum

Baryon number

Lepton number

Isospin - -

Flavour (S,C,B) -

Charges (em, strong and weak forces)

Parity (P) -

C-parity (C) -

G-parity (G) - -

CP -

T -

CPT

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Summary● Symmetry: i.e. under a given transformation certain

observable aspects of a system are invariant to the transformation.

● If the Hamiltonian of a system is invariant to a symmetry operation a conservation law is obtained. Noether’s theorem

● Isospin is an algebraic copy of spin but covering an internal ”isospin space”

● Isospin symmetry a powerful way to understand the masses of particles and their reactions. Isospin violation/conservation can be studied by considering

a rotation in isospin space or counting a conserved quantum number.