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Lecture 3

SU(2)

January 26, 2011

Lecture 3

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A Little Group Theory

A group is a set of elements plus a compostion rule,

such that:

1. Combining two elements under the rule givesanother element of the group.

E E = E

2. There is an indentity element

E I = I E = E

3. Every element has a unique inverse

E E1

= E1

E = I4. The composition rule is associative

A (B C) = (A B) C

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The U(1) Group

The set of all functions U() = ei form a group.

E() U() = ei(+) = E( + )I = E(0)

E1() = E()

E(1)E(2)U(3) = E(1)E(2)E(3)This is the one dimensional unitary group

U(1)

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Lie Groups

If the elements of a group are differentiable with

respect to their parameters, the group is a Lie group.

U(1) is a Lie group.

dE

d= iE

For a Lie group, any element can be written in the form

E(1, 2, , n) = exp

ni=1

iiFi

The quantities Fi are the generators of the group.

The quatities i are the parameters of the group.They are a set of i real numbers that are needed tospecify a particular element of the group.

Note that the number of generators and parameters arethe same. There is one generator for each parameter.

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U(1)

The group U(1) is the set of all one dimensional,

complex unitary matrices.

The group has one generator F = 1, and oneparameter, .

It simply produces a complex phase change.

E() = eiF = ei

Since the generator F, commutes with itself, thegroup elements also commute.

E(1)E(2) = ei1ei2 = ei2ei1 = E(2)E(1)

Such groups are called Abelian groups.

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U(2)

The group U(2) is the set of all two dimensional,

complex unitary matrices.

An complex n n matrix has 2n2 real parameters.The unitary condition constraint removes n2 of these.

The group U(2) then has four generators and four

parameters.

E(0, 1, 2, 3) = eijFj where j = 0, 1, 2, 3

The generators are: Fi = i/2

F0 =1

2

1 00 1

F1 =

1

2

0 11 0

F2 =1

2

0 ii 0

F3 =

1

2

1 00 1

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SU(2)

The operators represented by the elements of U(2)

act on two dimensional complex vectors.The operations generated by F0 = 0/2 simply changethe complex phase of both components of the vectorby the same amount. In general we are not sointerested in these operations.

The group SU(2) is the set of all two dimensional,complex unitary matrices with unit determinant.

The unit determinant constraint removes one moreparameter. The group SU(2) then has threegenerators and three parameters.

E(1, 2, 3) = eijFj where j = 1, 2, 3

The generators of SU(2) are a set of three linearlyindependent, traceless 2 2 Hermitian matrices:

F1 =

1

20 1

1 0

F2 =

1

20 ii 0 F3 = 12 1 00 1

Since the generators do not commute with one another,this is a non Abelian group.

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SO(3)

The group SO(3) is the set of all three dimensional,

real orthogonal matrices with unit determinant.[Requiring that the determinant equal 1 and not 1,excludes reflections.]

A real n n orthogonal matrix has n(n 1)/2 realparameters

The group SO(3) then has three parameters.

The group SO(3) represents the set of all possiblerotations of a three dimensional real vector.

For example, in terms of the Euler angles

R(, ,) =0

@cos sin 0sin cos 0

0 0 1

1

A

0

@cos 0 sin

0 1 0

sin 0 cos

1

A

0

@cos sin 0sin cos 0

0 0 1

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SU(2) and SO(3)

Both the groups SU(2) and SO(3) have three real

parameters

For example the SU(2) elements can be parametrizedby:

a b

b a |a|

2 + |b|2 = 1

cos ei sin ei sin ei cos ei

In fact, there is a one-to-one correspondence betweenSU(2) and SO(3) such that SU(2) represents the setof all possible rotations of two dimensional complexvectors (spinors) in a real three dimensional space.

Well not quite. There are actually two SU(2) rotations

for every SO(3) rotation. Thats because, for SU(2),the rotation with i+2 is not the same as the rotationwith i. There is a sign difference. For SU(2), therange of i is: 0 i 4. The set with 0 i 2corresponds to the complete set of SO(3) rotations.

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The Group SU(2)

As we have seen, for the case of j = 1/2, rotationsare represented by the matrices of the form

ei/2 = cos(/2)I i( )sin(/2)These matrices are 22 complex unitary matrices withunit determinant. The determinant is unity because

det(ei

/2

) = eTr(

i

/2)

= e0

= 1

The set of all of these matrices forms the group SU(2)under the operation of matrix multiplication.

These elements are described by a set of three realparameter (x, y, z)

=

z x iyx + iy z

This set of matrices are then elements of a threedimension real vector space that can be identified as

the space of physics vectors and the three generatorsof rotations, 1, 2, 3 can be identified with the threecomponents of a physical vector

Sx =h

21 Sy =

h

22 Sz =

h

23

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SU(2) and Rotations

Any normalized element of a complex two dimensional

vector space

is also described by three realparameters, the real and imaginary parts of and with the constraint ||2 + ||2 = 1

Any normalized element of the two dimensional vector

space can be obtained by a rotation of the state1

0

ei/2 = cos(/2)

1 00 1

i sin(/2)

z x iy

x + iy z

!

=cos

2 iz sin 2 (y ix)sin 2

(y ix)sin 2 cos 2 + iz sin 2

ei/2

10

=

cos 2 iz sin 2(y ix)sin 2

=

Conversely, for every state

there is some direction

n such that:

S n

=

h

2

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A Check on Consistency

Under a /2 rotation about the x-axis, theexpectation valuer of

Sy for the rotated system shouldequal the expectation value of Sz for the non-rotated

system.|Sy| = |Sz|

Lets check it.

|Sz| = |eix/4eix/4Szeix/4eix/4|

= |eix/4Szeix/4|

Now, does

eix/4Szeix/4 = Sy

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Generalization of the Anticommutation

Relations

ij = ji for i = j

inj = (1)nnj i = (j)ni

Then for any analytic function of j, i.e. and function

that can be expanded as a power series, we haveif(j) = f(j)i

Using this result we have

eix/4

Szeix/4

=

h

2 eix/4

zeix/4

=h

2eix/4eix/4z =

h

2eix/2z

=h

2

(cos

2

+ ix sin

2

)z =h

2

ixz =h

2

y = Sy

It checks.

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3-Dimensional Representation of SU(2)

The structure of a group is defined by the algebra of

its generators.For SU(2) this is:

[Fi, Fj] = iijkFk

i2

,j2

= iijk

k2

We can find a set of three 3 3 complex, traceless,Hermitian matrices that satisfy this same algebra.

F1 =1

2

0@ 0 1 01 0 1

0 1 0

1A F2 = i

2

0@ 0 1 01 0 1

0 1 0

1A

F3 = 0@1 0 0

0 0 00 0 1

1AThese generate the three dimensional representationof SU(2).

Note that they do not represent all possible rotations ofa three dimensional, normalized complex vector. Thatwould require at least five parameter. For example inthe homework you show that you cannot rotate thestate |jm = |1, 1 into |1, 0

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Isospin

We are used to SU(2) in spin space

Transformations are physical rotations in 3-dimensional space

Generators are angular momentum operatorsNow we want to consider SU(2) in a new isospinspace.

Complete analogy with SU(2) in spin space.

But, transformations are not physicsrotations but rather rotations in theabstract isospin space.

= eikxcc

| {z }

spin

cucd

| {z }

isospin

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SU(3)

The group SU(3) is the set of all three dimensional,

complex unitary matrices with unit determinant.

This set has 2(3)2 (3)2 1 = 8 parameters andgenerators.

E(1, 2, , 8) = eijFj where j = 1, 2, , 8

The generators of SU(3) are a set of eight linearlyindependent, traceless 3 3 Hermitian matrices:

Since there are eight generators, the e SU(3) elementsrepresent rotations of complex three component vectorsin an eight dimensional space.

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SU(3) Generators

1 =0@ 0 1 01 0 0

0 0 0

1A 2 = 0@ 0 i 0i 0 00 0 0

1A

3 =

0

@1 0 00 1 00 0 0

1

A4 =

0

@0 0 10 0 0

1 0 0

1

A5 =

0@ 0 0 i0 0 0

i 0 0

1A 6 =

0@ 0 0 00 0 1

0 1 0

1A

7 =

0@

0 0 0

0 0 i0 i 0

1A 8 =

13

0@

1 0 0

0 1 0

0 0 2

1A

The structure of SU(3) is:

[a, b] = 2ifabcc

f123 = 1 f458 = f678 =

3

2

f147 = f516 = f246 = f257 = f345 = f637 =1

2

fabc is totally antisymmetric