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Lecture 4

SU(2)

February 1, 2010

Lecture 4

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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 E 1

= E 1

E = I 4. 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)

E 1() = 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 ) = expn

i=1

ii F i

The quantities F i 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) = ei 1ei 2 = ei 2ei 1 = 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) = ei j F j where j = 0 , 1, 2, 3

The generators are: F i = i / 2

F 0 =12

1 00 1 F 1 =

12

0 11 0

F 2 =12

0 ii 0 F 3 =12

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 F 0 = 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) = ei j F j where j = 1 , 2, 3

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

F 1 =12

0 11 0 F 2 =

12

0

i

i 0 F 3 =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 reections.]A real n n orthogonal matrix has n(n 1)/ 2 realparametersThe 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 11A0@

cos 0 sin 0 1 0

sin 0 cos

1A0@

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) rotationsfor 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

= 1The 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 iy

x + iy zThis set of matrices are then elements of a threedimension real vector space that can be identied as

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

S x =h2

1 S y =h2

2 S z =h2

3

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SU(2) and Rotations

Any normalized element of a complex two dimensional

vector space is also described by three real

parameters, the real and imaginary parts of and with the constraint | |2 + | |2 = 1Any normalized element of the two dimensional vector

space can be obtained by a rotation of the state10

ei / 2 = cos (/ 2) 1 00 1 i sin( / 2) z x i yx + i y 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 =

h2

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A Check on Consistency

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

S y for the rotated system shouldequal the expectation value of S z for the non-rotated

system. |S y| = |S z |

Lets check it.

|S z | = |ei x / 4ei x / 4S z ei x / 4ei x / 4|

= |ei x / 4S z ei x / 4|Now, does

ei x / 4S z ei x / 4 = S y

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Generalization of the AnticommutationRelations

i j = j i for i = j

i nj = ( 1)n nj i = ( j )n i

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

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

Using this result we have

ei x / 4

S z ei x / 4

=h2 e

i x / 4

z ei x / 4

=h2

ei x / 4ei x / 4z =h2

ei x / 2z

=h

2(cos

2+ ix sin

2)z =

h

2ix z =

h

2y = S y

It checks.

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3-Dimensional Representation of SU (2)

The structure of a group is dened by the algebra of

its generators.For SU (2) this is:

[F i , F j ] = i ijk F ki2

,j2

= i ijkk2

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

F 1 =1

2 0@0 1 01 0 10 1 0

1A F 2 =i

2 0@0 1 01 0 10 1 0

1AF 3 = 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 of a three dimensional, normalized complex vector. Thatwould require at least ve 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.

= eik xcc | {z }spin

cucd | {z }isospin

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SU (3) Generators

1 = 0@0 1 01 0 00 0 0 1A 2 = 0@

0

i 0i 0 00 0 0 1A

3 = 0@

1 0 00 1 00 0 0

1A

4 = 0@

0 0 10 0 01 0 0

1A 5 = 0@

0 0 i0 0 0i 0 0

1A 6 = 0@0 0 00 0 10 1 0

1A 7 = 0

@0 0 00 0 i0 i 0

1A

8 =1

30@

1 0 00 1 0

0 0 21A

The structure of SU (3) is:

[ a , b] = 2if abc c

f 123 = 1 f 458 = f 678 =32

f 147 = f 516 = f 246 = f 257 = f 345 = f 637 =12

f abc is totally antisymmetric