Lecture 3 SU(2) January 26, Lecture 3

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1 Lecture 3 SU(2) January 26, 2 Lecture 3

2 A Little Group Theory A group is a set of elements plus a compostion rule, such that:. Combining two elements under the rule gives another 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 = E E = I 4. The composition rule is associative A (B C) = (A B) C Lecture 3

E E = E 2. There is an indentity element E I = I E = E 3.

3 The U() Group The set of all functions U(θ) = e iθ form a group. E(θ) U(θ ) = e i(θ+θ ) = E(θ + θ ) I = E() E (θ) = E( θ) ( E(θ ) E(θ 2 ) ) U(θ 3 ) = E(θ ) (E(θ 2 ) E(θ 3 ) ) This is the one dimensional unitary group U() Lecture 3 2

4 Lie Groups If the elements of a group are differentiable with respect to their parameters, the group is a Lie group. U() is a Lie group. de dθ = ie For a Lie group, any element can be written in the form ( n ) E(θ, θ 2,, θ n ) = exp iθ i F i i= The quantities F i are the generators of the group. The quatities θ i are the parameters of the group. They are a set of i real numbers that are needed to specify a particular element of the group. Note that the number of generators and parameters are the same. There is one generator for each parameter. Lecture 3 3

generators of the group. The quatities θ i are the parameters of the group.

5 U() The group U() is the set of all one dimensional, complex unitary matrices. The group has one generator F =, and one parameter, θ. It simply produces a complex phase change. E(θ) = e iθf = e iθ Since the generator F, commutes with itself, the group elements also commute. E(θ )E(θ 2 ) = e iθ e iθ 2 = e iθ 2 e iθ = E(θ 2 )E(θ ) Such groups are called Abelian groups. Lecture 3 4

It simply produces a complex phase change.

6 U(2) The group U(2) is the set of all two dimensional, complex unitary matrices. An complex n n matrix has 2n 2 real parameters. The unitary condition constraint removes n 2 of these. The group U(2) then has four generators and four parameters. E(θ, θ, θ 2, θ 3 ) = e iθ jf j where j =,, 2, 3 The generators are: F i = σ i /2 F = 2 ( ) F = 2 ( ) F 2 = 2 ( ) i i F 3 = 2 ( ) Lecture 3 5

The unitary condition constraint removes n 2 of these.

7 SU(2) The operators represented by the elements of U(2) act on two dimensional complex vectors. The operations generated by F = σ /2 simply change the complex phase of both components of the vector by the same amount. In general we are not so interested 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 more parameter. The group SU(2) then has three generators and three parameters. E(θ, θ 2, θ 3 ) = e iθ jf j where j =, 2, 3 The generators of SU(2) are a set of three linearly independent, traceless 2 2 Hermitian matrices: F = ( ) F 2 2 = ( ) i F 2 i 3 = ( ) 2 Since the generators do not commute with one another, this is a non Abelian group. Lecture 3 6

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

8 SO(3) The group SO(3) is the set of all three dimensional, real orthogonal matrices with unit determinant. [Requiring that the determinant equal and not, excludes reflections.] A real n n orthogonal matrix has n(n )/2 real parameters The group SO(3) then has three parameters. The group SO(3) represents the set of all possible rotations of a three dimensional real vector. For example, in terms of the Euler angles R(α, β, γ) = cos γ sin γ cos β sin β cos α sin α sin γ cos γ A A sin α cos α A sin β cos β Lecture 3 7

] A real n n orthogonal matrix has n(n )/2 real parameters The group SO(3) then has three parameters.

9 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 parametrized by: ( ) a b b a a 2 + b 2 = cos θeiα sin θe iγ sin θe iγ cos θe iα In fact, there is a one-to-one correspondence between SU(2) and SO(3) such that SU(2) represents the set of all possible rotations of two dimensional complex vectors (spinors) in a real three dimensional space. Well not quite. There are actually two SU(2) rotations for every SO(3) rotation. That s because, for SU(2), the rotation with θ i +2π is not the same as the rotation with θ i. There is a sign difference. For SU(2), the range of θ i is: θ i 4π. The set with θ i 2π corresponds to the complete set of SO(3) rotations. Lecture 3 8

(spinors) in a real three dimensional space. Well not quite. There are actually two SU(2) rotations for every SO(3) rotation.

10 The Group SU(2) As we have seen, for the case of j = /2, rotations are represented by the matrices of the form e iθ σ ˆθ/2 = cos(θ/2)i i( σ ˆθ) sin(θ/2) These matrices are 2 2 complex unitary matrices with unit determinant. The determinant is unity because det(e iθ σ ˆθ/2 ) = e Tr( iθ σ ˆθ/2) = e = 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 real parameter (θ x, θ y, θ z ) ( ) σ ˆθ ˆθz ˆθx iˆθ = y ˆθ x + iˆθ y ˆθ z This set of matrices are then elements of a three dimension real vector space that can be identified as the space of physics vectors and the three generators of rotations, σ, σ 2, σ 3 can be identified with the three components of a physical vector S x = h 2 σ S y = h 2 σ 2 S z = h 2 σ 3 Lecture 3 9

The determinant is unity because det(e iθ σ ˆθ/2 ) = e Tr( iθ σ ˆθ/2) = e = The set of all of these matrices forms the group SU(2) under the operation of matrix multiplication.

11 SU(2) and Rotations e iθ σ ˆθ/2 Any normalized( element of a complex two dimensional α vector space is also described by three real β) parameters, the real and imaginary parts of α and β with the constraint α 2 + β 2 = Any normalized element of the two dimensional vector ( space can be obtained by a rotation of the state ) = cos(θ/2) = «i sin(θ/2) ˆθ z ˆθ x + iˆθ y cos θ 2 iˆθ z sin θ 2 ( ˆθ y iˆθ x ) sin θ 2 (ˆθ y iˆθ x ) sin θ 2 cos θ 2 + iˆθ z sin θ 2 ˆθx iˆθ y ˆθ z ( e iθ σ ˆθ/2 = ) cos θ 2 iˆθ z sin θ ( 2 α = (ˆθ y iˆθ x ) sin β) θ 2 ( α Conversely, for every state there is some direction β) ˆn such that: ( α S ˆn β) = h 2 ( α β)! Lecture 3

) = cos(θ/2) = «i sin(θ/2) ˆθ z ˆθ x + iˆθ y cos θ 2 iˆθ z sin θ 2 ( ˆθ y iˆθ x ) sin θ 2 (ˆθ y iˆθ x ) sin θ 2 cos θ 2 + iˆθ z sin θ 2 ˆθx iˆθ y ˆθ z ( e iθ σ

12 A Check on Consistency Under a π/2 rotation about the x-axis, the expectation valuer of Ŝy for the rotated system should equal the expectation value of Ŝz for the non-rotated system. ψ Ŝy ψ = ψ Ŝz ψ Let s check it. ψ Ŝz ψ = ψ e iπσ x/4 e iπσ x/4 Ŝ z e iπσ x/4 e iπσ x/4 ψ = ψ e iπσ x/4 Ŝ z e iπσ x/4 ψ Now, does e iπσx/4 iπσ Ŝ z e x/4 = Ŝy Lecture 3

13 Generalization of the Anticommutation Relations σ i σ j = σ j σ i for i j σ i σ n j = ( ) n σ n j σ i = ( σ j ) n σ i Then for any analytic function of σ j, i.e. and function that can be expanded as a power series, we have σ i f(σ j ) = f( σ j )σ i Using this result we have e iπσ x/4 Ŝ z e iπσ x/4 = h 2 eiπσ x/4 σ z e iπσ x/4 = h 2 eiπσ x/4 e iπσ x/4 σ z = h 2 eiπσ x/2 σ z = h 2 (cos π 2 + iσ x sin π 2 )σ z = h 2 iσ xσ z = h 2 σ y = Ŝy It checks. Lecture 3 2

14 3-Dimensional Representation of SU(2) The structure of a group is defined by the algebra of its generators.for SU(2) this is: [ σi [F i, F j ] = iɛ ijk F k 2, σ ] j 2 = iɛ ijk σ k 2 We can find a set of three 3 3 complex, traceless, Hermitian matrices that satisfy this same algebra. F = 2 A F 2 = i 2 A F 3 = A These generate the three dimensional representation of SU(2). Note that they do not represent all possible rotations of a three dimensional, normalized complex vector. That would require at least five parameter. For example in the homework you show that you cannot rotate the state jm =, into, Lecture 3 3

satisfy this same algebra. F = 2 A F 2 = i 2 A F 3 = A These generate the three dimensional representation of SU(2).

15 Isospin We are used to SU(2) in spin space Transformations are physical rotations in 3- dimensional space Generators are angular momentum operators Now we want to consider SU(2) in a new isospin space. Complete analogy with SU(2) in spin space. But, transformations are not physics rotations but rather rotations in the abstract isospin space. ψ = e ik x c c «{z } spin cu c d «{z } isospin Lecture 3 4

Complete analogy with SU(2) in spin space.

16 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 = 8 parameters and generators. E(θ, θ 2,, θ 8 ) = e iθ jf j where j =, 2,, 8 The generators of SU(3) are a set of eight linearly independent, traceless 3 3 Hermitian matrices: Since there are eight generators, the e SU(3) elements represent rotations of complex three component vectors in an eight dimensional space. Lecture 3 5

E(θ, θ 2,, θ 8 ) = e iθ jf j where j =, 2,, 8 The generators of SU(3) are a set of eight linearly independent,

17 SU(3) Generators λ = A λ 2 = i i A λ 3 = A λ 4 = A λ 5 = i i A λ 6 = A λ 7 = i i A λ 8 = 3 2 A The structure of SU(3) is: [λ a, λ b ] = 2if abc λ c f 23 = f 458 = f 678 = 3 2 f 47 = f 56 = f 246 = f 257 = f 345 = f 637 = 2 f abc is totally antisymmetric Lecture 3 6

Chapter 9 Unitary Groups and SU(N)

Chapter 9 Unitary Groups and SU(N) The irreducible representations of SO(3) are appropriate for describing the degeneracies of states of quantum mechanical systems which have rotational symmetry in three