Applications of group theory to quantummechanics

Lecture 3: The group SU(2)

Hubert de Guise

Department of Physics, Lakehead University

Guadalajara 2013

Plan of Lecture 3

Continuous groupsThe example of U(1)∼ SO(2)General considerations

The group SU(2)and its algebra su(2)The group SU(2)su(2) and angular momentum theoryConstructing representations of su(2)Schur-Weyl dualityThe SU(2) transformations and Wigner D-functions

A derivation using polynomials

Application to 2-port interferometry

An example of a simple continuous group

A continuous group is one for which the elements in the group aregiven in terms of continuous parameters.

I The set eiα with α a real number. Since e2πni = 1 for anyinteger n it is clear that the range of α can be restricted to0 ≤ α ≤ 2π.

I When the range of every parameter is finite, the group is said to becompact.

I If the range of one or more parameter is not restricted, the group issaid to be non–compact.

I It is easily verified that the set of matrices of the form

R(θ) =

(cos θ − sin θsin θ cos θ

), 0 ≤ θ ≤ 2π .

form a group. Here, θ is taken modulo 2π. The unit element isR(0). The inverse of R(θ) is R(−θ).

I The two groups in the above examples are isomorphic, i.e. thereis a one-to-one correspondence between the elements in the seteiα and the matrices R(θ).

Correspondences between some results

I For finite groups every representation is equivalent to a unitaryrepresentation.

I True for compact groups; false for non-compact.I For finite groups every finite–dimensional representation is

completely reducible.I True for compact groups; false for non-compact.

I For finite groups every irreducible representation is finitedimensional.

I True for compact groups; false for non-compact.I For finite groups there is a finite number of conjugacy classes

and thus a finite number of representations.I This is false for compact and non-compact groups.

I For finite group, each irreducible representation is contained inthe regular representation a number of times equal to itsdimension.

I True for compact groups; not always true for non-compact.I For finite groups, the matrices D are orthogonal when summed

over the group elements.I True for compact groups; not always true for non-compact.

Lie groups

Lie groups are:I Continuous groups whose elements R are given in terms of

continuous differentiable parameters a = (a1, a2, . . . , an),I The parameters can be chosen to the identity element is a = 0.I The product R(a)R(b) is another element R(c) by the group

axiom.I Generally one write c = ϕ(a, b) or ci = ϕi(a1, . . . , an; b1, . . . , bn).I The connection between c and a, b is usually complicated (example

to follow) and need not be linear.I The group has n infinitesimal generators given by

Xj ≡(∂R(a1, . . . , an)

∂aj

)a=0

I For “sufficiently small” ai: R(a1, . . . , an) ≈ 1l +∑k akXk.

Examples

I For the group with elements eiα we have one parameter andthus one generator. It is given by

X =deiα

dα

∣∣∣α=0

= i

and eiα ≈ 1 + iα for small α.I For the group R(θ) we have one parameter and thus one

generator, given by

X =

(d

dθ

(cos θ − sin θsin θ cos θ

))θ=0

=

(− sin θ − cos θcos θ − sin θ

)θ=0

= =

(0 −11 0

)

Infinitesimal generators and their properties

I Theorem: The finite elements of a continuous group can bereached by successive applications of infinitesimal elements:

R(a) = exp(∑k

akXk)

= 1l + (∑k

akXk) + 12 (∑k

akXk)2 + 13! (∑k

akXk)3 + . . .

for some ak.I Corollary: The (matrix) representation of all elements of a

continuous group are uniquely determined by therepresentations of its generators.

I Theorem: The set of infinitesimal generators close undercommutation, i.e. the commutator of every pair of generators is alinear combination of generators:

[Xi, Xj ] =∑k

cijkXk

The group SU(2)

The group SU(2) is defined by the set of 2×2 matrices with theproperties:

I They are Unitary, i.e. if T ∈SU(2) then T−1 = (T t)∗ ≡ T †.

I The are Special, i.e. their determinant is +1.I They are of the general form(

a −bb∗ a∗

), a, b ∈ C , aa∗ + bb∗ = 1

Let us show such matrices are a group:I Suppose T1 and T2 are in SU(2). Then look at T3 = T2T1.

I ((T3)t)∗ = (T t1Tt2)∗ = (T t1)∗(T t2)∗ and

T †3T3 = T †1T†2T2T1 = T †1 1lT1 = T †1T1 = 1l .

I Det(T3)=Det(T2T1)=Det(T2)Det(T1)=1I The unit is 1l2×2.I Clearly the matrix product is associative.

The Euler parametrization

I SU(2) matrices are commonly parametrized by 3 angles (α, β, γ):

R(α, β, γ) =

e−12 i(α+γ) cos

(β2

)−e−

12 i(α−γ) sin

(β2

)e

12 i(α−γ) sin

(β2

)e

12 i(α+γ) cos

(β2

) =

(e−

iα2 0

0 eiα2

)cos(β2

)− sin

(β2

)sin(β2

)cos(β2

) (e−iγ2 0

0 eiγ2

)

I The first two generators are

Xz =∂R(α, β, γ)

∂α= i

2

(−1 00 1

),

(e−

iα2 0

0 eiα2

)= exp(αXz)

Xy =∂R(α, β, γ)

∂α= 1

2

(0−11 0

) cos(β2 )− sin(β2 )

sin(β2 ) cos(β2 )

= exp(βXy)

su(2) and angular momentum theoryThe familiar Pauli matrices:

σz = 2iXz =

(1 00 −1

), σy = 2iXy =

(0 −ii 0

),

σx = 2i[Xy, Xz] =

(0 11 0

)satisfy the commutation relations

[σx, σy] = iσz , [σy, σz] = iσx , [σz, σx] = iσy .

I A representation Γ of the su(2) algebra is any triple of d×dmatrices Γ(Lx),Γ(Ly),Γ(Lz) which satisfy the samecommutation relations as the Pauli matrices.

I Given a representation of the su(2) algebra by d×d matrices, wecan construct a representation of the group SU(2) by d×dmatrices by taking

DΓ(α, β, γ) = e−iαΓ(Lz)e−iβΓ(Ly)e−iγΓ(Lz)

I If the representation Γ of su(2) is irreducible, so it therepresentation DΓ(α, β, γ) obtained through its exponentiation.

Constructing representations of su(2)

I We take the (abstract) algebra su(2) to be spanned byLx, Ly, Lz.

I Choose (in an arbitrary way but otherwise conventional) as basisof states |αM〉 for the vector space on which the matrices acteigenstates of Lz:

Lz|αM〉 = M |αM〉.

The meaning of the label α will be clarified later.I Take the complex linear combinations of su(2) (technically calledA1 by the mathematicians)

L+ = Lx + iLy, L− = Lx − iLy.

I The set Lz, L+, L− satisfies the non-zero commutationrelations

[Lz, L±] = ±L±, [L+, L−] = 2Lz.

The laddering action of L+ and L−

I Now look at

LzL+|αM〉 = (LzL+ − L+Lz) |αM〉+ L+Lz|αM〉= L+|αM〉+ML+|αM〉= (M + 1)L+|αM〉.

I L+|αM〉 has eigenvalue M + 1 and so must be proportional to|α,M + 1〉.

I We can continue this way looking at Lz (L+)n |αM〉 to show that

(L+)n |αM〉 must be proportional to |αM + n〉.

I For a finite-dimensional representation, the sequence musteventually end, i.e. there is a maximum value of M , which wedenote by L, for which

L+|αL〉 = 0.

I We identify the label α with L, this largest eigenvalue of Lz,.I Starting from |LL〉, we recover |LM〉 by successive application ofL−, i.e. |LM〉 ∼ (L−)

L−M |LL〉.

Matrix elements

I The set |LM〉,M = −L, ..., L is a basis for a representation ofsu(2) of dimension 2L+1.

I We say that states |LM〉 have angular momentum L.I The usual argument shows that the operatorsLz, L+, L− acting

on the states |LM〉 satisfy

Lz|LM〉 = M |LM〉,L±|LM〉 =

√(L∓M)(L+ 1)|L,M + 1〉,

I The eigenvalue M of Lz associate to the eigenvector |LM〉 iscalled the weight of the state.

I With this nomenclature, the operators L+ and L− are calledrespectively raising and lowering operators for obvious reasons.

I The state |LL〉 with eigenvalue M = L is killed by the raisingoperator and so is called the highest weight state.

I Likewise, the state |L,−L〉 with eigenvalue M = −L is killed bythe lowering operator and called the lowest weight state.

A polynomial representationI We introduce two dummy variables ξ and η and three operators

12

(ξ ∂∂ξ − η

∂∂η

), ξ ∂∂η , η ∂

∂ξ .

I Note that, acting on an arbitrary function f(ξ, η), they have thesame commutation relations as Lz, L+, L− v.g.

[ξ∂

∂η, η

∂

∂ξ]f(ξ, η) = ξ

∂

∂η

(η∂

∂ξf(ξ, η)

)− η ∂

∂ξ

(ξ∂

∂ηf(ξ, η)

),

= ξ∂

∂ξf(ξ, η) + ξη

∂2

∂η∂ξf(ξ, η)

−η ∂∂ηf(ξ, η)− ηξ ∂2

∂η∂ξf(ξ, η) ,

= 2[

12 (ξ ∂∂ξ − η

∂∂η )]f(ξ, η)

is like [L+, L−] = 2Lz.I The highest weight state |LL〉 satisfiesL+|LL〉 = 0 , Lz|LL〉 = L|LL〉. This corresponds to ξ2L since

ξ∂

∂ηξ2L = 0 , 1

2

(ξ ∂∂ξ − η

∂∂η

)ξ2L = Lξ2L .

A polynomial representation

I We know |LM〉 ∼ (L−)L−M |LL〉. Repeated action of η ∂∂ξ on ξ2L

gives

(η∂

∂ξ)L−Mξ2L ∼ ηL−MξL+M

with12 (ξ ∂∂ξ − η

∂∂η )(ηL−MξL+M

)= M

(ηL−MξL+M

)I Thus, |LM〉 7→ NMηL−MξL+M , where NM is a normalization

constant.I To find NM note that 〈LM | 7→ NM (∂η)

L−M(∂ξ)

L+M .I Thus the states |LM〉 are mapped to polynomials in ξ, η :

|LM〉 → 1√(L+M)!(L−M)!

ξL+MηL−M ,

〈LM | → 1√(L+M)!(L−M)!

(∂

∂ξ

)L+M (∂

∂η

)L−M.

Connection with Young diagrams

For definiteness we consider a system of three spin-1/2 particles.

I We can map|+〉i → ξi , |−〉k → ηk

I We then have

L+ → ξ1∂

∂η1+ ξ2

∂

∂η2+ ξ3

∂

∂η3,

L− → η1∂

∂ξ1+ η2

∂

∂ξ2+ η3

∂

∂ξ3,

Lz → 12

(ξ1

∂

∂ξ1+ ξ2

∂

∂ξ2+ ξ3

∂

∂ξ3− η1

∂

∂η1− η2

∂

∂η2− η3

∂

∂η3

)I Note that L±, Lz are invariant under S3, i.e. they carry the irrep

.

Connection with Young diagrams

I Look at S3/23/2(ξ1, ξ2, ξ3) = ξ1ξ2ξ3 ↔ |+〉1|+〉2|+〉3.

I First have

L+S3/2

3/2 = 0 , LzS3/2

3/2 = 32S

3/2

3/2 ⇒ S3/2

3/2 ↔ | 32

32〉

I This polynomial also is unchanged under permutations in S3, v.g.

P12S3/2

3/2 = ξ2ξ1ξ3 = S3/2

3/2 , P13S3/2

3/2 = ξ3ξ2ξ1 = S3/2

3/2

and so transforms by irrep .

I Now consider L−S3/23/2 ≡

√3S

3/21/2 to get

√3S

3/21/2 = η1ξ2ξ3 + ξ1η2ξ3 + ξ1ξ2η3 , LzS

3/21/2 = 1

2S3/21/2

Thus, S3/21/2 ↔ |

32

12 〉.

I Again P12S3/21/2 = S

3/21/2 etc so again S3/2

1/2 carries the irrep .

I The L’s carry the irrep ; since Γ ⊗ Γλ = Γλ ∀ partitions λ,L±, Lz do not mix states associated with different Young diagram.

Connection with Young diagrams

I Next look at the tableau1 23 , its symmetrizer and associated

state:√

6S1/21/2(ξ1, ξ2, ξ3) = Θ(12)3ξ1ξ2η3 = s12a13ξ1ξ2η3

= 2ξ1ξ2η3 − η1ξ2ξ3 − ξ1η2ξ3

↔ 2|+〉1|+〉2|−〉3 − |−〉1|+〉2|+〉3 − |+〉1|−〉2|+〉3.

I Again noteL+S

1/2

1/2 = 0 , LzS1/2

1/2 = 12S

3/2

3/2

Thus: S1/2

1/2 → | 12

12〉

I Next, look at L−√

2S1/21/2 = L−s12a13ξ1ξ2η3.

I Since L− is invariant under permutation:

L−s12a13ξ1ξ2η3 = s12a13L−ξ1ξ2η3

will be proportional to the state S1/2

−1/2 for the tableau1 23 .

Connection with Young diagrams

What of P23Θ(12)3ξ1ξ2η3? This is the second basis state for the

2-dimensional irrep1 23 .

I Set√

2T1/21/2 = P23Θ(12)3ξ1ξ2η3 = 2ξ1η2ξ3 − ξ1ξ2η3 − η1ξ2ξ3

↔ 2|+〉1|−〉2|+〉3 − |+〉1|+〉2|−〉3 − |−〉1|+〉2|+〉3

I This also a highest weight state since

L+T1/21/2 = 0 , LzT

1/21/2 = 1

2T1/21/2

I Obvisouly T 1/21/2 6= S

1/21/2

I Because L− commutes with any permutation, L−T1/21/2 6= L−S

1/21/2 .

Connection with Young diagrams

I What of the tableau1 32 ? We have s13a12ξ1ξ2η3 = 0 so we get

nothing new here.

I Maybe try the symmetrizer for1 32 starting with ξ1η2ξ3? This

gives back T 1/21/2 .

I What is the meaning of the T 1/2 states?I When we couple three spin-1/2 states together, the possible

resulting values of s are 3/2, 1/2 and again 1/2.I The states S1/2

M and T 1/2M are the two basis states (there is one

such basis for each M ) that transform one into the other under theelements of S3.

I Because as constructed the permutation operators commutewith the angular momentum operators, states |LM〉 of fixed Lalso carry a representation of S3.

Schur-Weyl dualityA representation of SU(2) is completely specified by giving a Youngdiagram. A diagram with λ1 + λ2 and λ2 boxes in the first and secondrow respectively is associated with the SU(2) irrep of dimensionλ1 + 1 and angular momentum value s = 1

2λ1.Thus, for instance:

Young diagram s

12

1

0

32

12

2

1

0

Wigner D-functions using polynomials

I The Wigner D-function is defined as

DLM ′M (α, β, γ) ≡ 〈LM ′|e−iαLze−iβLy e−iγLz |LM〉

= e−iαM′dLM ′M (β)e−iγM

dLM ′M (β) ≡ 〈LM ′|e−iβLy |LM〉

I For L = 1/2, we already know

d1/2(β) =

cos(β2

)− sin

(β2

)sin(β2

)cos(β2

) or

d1/21/2,1/2 = cos

(β2

)d

1/21/2,−1/2 = − sin

(β2

)d

1/2−1/2,1/2 = sin

(β2

)d

1/2−1/2,−1/2 = cos

(β2

)I What of other values of L?

Wigner D-functions using polynomials

We observe that, under the identification:

| 12 ,12 〉 ↔ ξ, | 12 ,−

12 〉 ↔ η,

the transformations of the kets

Ry(β)| 12 ,12 〉 = cos

(β2

)| 12 ,

12 〉+ sin

(β2

)| 12 ,−

12 〉,

Ry(β)| 12 ,−12 〉 = − sin

(β2

)| 12 ,

12 〉+ cos

(β2

)| 12 ,−

12 〉

imply the transformation of the dummy operators

Ry(β)ξ = cos(β2

)ξ + sin

(β2

)η,

Ry(β)η = − sin(β2

)ξ + cos

(β2

)η.

Wigner D-functions using polynomials

Hence, from the identification

|LM〉 → ξL+MηL−M√(L+M)!(L−M)!

we infer

Ry(β)|LM〉

→ Ry(β)

(ξL+MηL−M√

(L+M)!(L−M)!

)=

(Ry(β)ξ)L+M

(Ry(β)η)L−M√

(L+M)!(L−M)!

=

(cos(β2

)ξ + sin

(β2

)η)L+M (

− sin(β2

)ξ + cos

(β2

)η)L−M

√(L+M)!(L−M)!

Wigner D-functions using polynomials

I Expanding, we get

Ry(β)|LM〉 → 1√(L+M)!(L−M)!

×∑x,y

(−1)L−M−y cos(β2 )L+M−x+y sin(β2 )L−M−y+xξ2L−x−yηx+y.

I Now recall:

〈LM ′| → 1√(L+M ′)!(L−M ′)!

(∂

∂ξ

)L+M ′ (∂

∂η

)L−M ′I Thus we get the final form:

dLM ′M (β) = 〈LM ′|Ry(β)|LM〉

=∑x

(−1)M′−M+x

√(L+M ′)!(L−M ′)!(L+M)!(L−M)!

(L+M − x)!x!(L−M ′ − x)!(M ′ −M + x)!

× cos

(β

2

)2L+M−M ′−2x

sin

(β

2

)M ′−M+2x

.

Examples

D11,0(α, β, γ) −e−iα sin(β)√

2

D3/21/2,1/2(α, β, γ) − 1

2 e−12 i(α−γ) sin

(12β)

(3 cos(β) + 1)

D7−4,2(α, β, γ) 1

2

√112 e4iα−2iγ sin6

(12β)

cos2(

12β)

×(91 cos3(β) + 78 cos2(β) + 3 cos(β)− 4

)D8

5,0(α, β, γ) − 364

√1001

2 e−5iα sin5(β) cos(β)(5 cos(2β) + 3)

Some propertiesI Wonderful Orthogonality Theorem. If dΩ = sinβdαdβdγ,∫

dΩDJ1M ′1M1

(Ω)(DJ2M ′2M2

(Ω))∗

=8π2

2J + 1δJ1J2δM ′1M1

δM ′2M2

I Compare with the result for finite groups∑p

DΓkµν (Rp)

(D

Γ`βα(Rp)

)∗=

n

dim(Γ`)δΓkΓ`δµβδνα .

I Two rotations by the same angles through a different axis ofrotation are in the same class.

I χL(β) =∑M dLMM (φ) =

sin[(L+12 )φ]

sin(φ) .I Composition: If Ω = Ω1 Ω2,∑

M

DLM1M (Ω1)DL

MM2(Ω2) = DL

M1M2(Ω)

I Relation to spherical harmonics:

Y`m(ϑ, ϕ) =

√2L+ 1

4πD`

0m(α, ϑ, ϕ) =

√2L+ 1

4πD`∗m0(ϕ, ϑ, α)

The Schwinger realizationSuppose we have a two-mode system, where |n1, n2) is the quantumstate with n1 particles in mode 1, and n2 is the number of particles inmode 2.

I We note that 12 (a†1a1 − a†2a2), a2a

†1, a1a

†2 have the commutation

relations as Lz, L+, L−, v.g.

[a2a†1, a1a

†2]|n1, n2) = a2a

†1a1a

†2|n1, n2)− a1a

†2a2a

†1|n1, n2)

= n1(n2 + 1)|n1, n2)− (n1 + 1)n2|n1, n2)

= (n1 − n2)|n1, n2)

is like [L+, L−] = 2Lz.I The highest weight is |N, 0) since this satisfies a†1a2|N, 0) and

12 (a†1a1 − a†2a2)|N, 0) = 1

2N |N, 0), with N = n1 + n2.I The identification

Lz ↔ 12 (a†1a1 − a†2a2) ,

L+ ↔ a†1a2 , L− ↔ a†2a1

is called the Schwinger realization of su(2).

Characterization of a two-port lossless interferometerLet a1,in, a2,in denote the annihilation operators for one quantum oflight in inputs 1 and 2 of a 2-port lossless interferometer, respectively.

I The interferometer will scatter photon so the annihilationoperators at output are related to those at the input by(

a1,outa2,out

)=

(U11 U12

U21 U22

)(a1,ina2,in

)I This in turn implies

(a†1,out, a†2,out) = (a†1,in, a

†2,in)

(U11 U12

U21 U22

)†I If the interferometer is lossless, the number of photons at input

and output must be the same, i.e.

Nout = a†1,outa1,out + a†2,outa2,out = (a†1,out, a†2,out)

(a1,outa2,out

)= (a†1,in, a

†2,in)U†U

(a1,outa2,out

)= Nin

or U†U = 1l, i.e. the matrix U is a 2 × 2 unitary matrix.

Interferometry and angular momentum theory

The matrix U can always be written as U = eiϕR(α, β, γ), with eiϕ anoverall phase and

R(α, β, γ) =

(e−

iα2 0

0 eiα2

)cos(β2

)− sin

(β2

)sin(β2

)cos(β2

) (e−iγ2 0

0 eiγ2

)

the D1/2 SU(2) matrix.I The factorization of R(α, β, γ) = Rz(α)Ry(β)Rz(γ) has the

following interpretation:I Rz(γ) represents a phase shifter that introduces a relative phase

e−iγ between the two path of the interferometer.I The same interpretation holds for Rz(α).I Ry(β) representes a beam splitter. Photons entering port 1 have a

probability |U11|2 = (cos 12β)2 = (d

1/2

1/2,1/2(β))2 of being transmitted.Photon entering port 2 have the same probability of transmission.|U12|2 = |U21|2 is the probability of being reflected into the otherport.

Interferometry and angular momentum theory

Optical elements can be combined through matrix multiplication.

(cos(

12s2) − sin

(12s2

)sin(

12s2

)cos(

12s2

) )(e−i(φ2−φ1)

2 0

0 ei(φ2−φ1)

2

)(cos(

12s1) − sin

(12s1

)sin(

12s1

)cos(

12s1

) )

is an SU(2) matrix so can be written as R(α, β, γ) for some α, β, γ.

Interferometry and angular momentum theory

Now:

R(α, β, γ)|n1n2) = R(α, β, γ)(a†1)n1(a†2)n2

√n1!n2!

|0, 0)

=(R(α, β, γ)a†1)n1(R(α, β, γ)a†2)n2

√n1!n2!

|0, 0)

so that, repeating the argument given in details for the polynomialrepresentation:

(n′1n′2|R(α, β, γ)|n1n2) = DJ

M1M2(α, β, γ)

with J = 12 (n1 +n2) = 1

2 (n′1 +n′2), M1 = 12 (n′1−n′2),M2 = 1

2 (n1−n2).Thus:

(1, 2|R(α, β, γ)|3, 0) = D3/2−1/2,3/2(α, β, γ) =

1

2

√3 e

12 i(α−3γ) sin

(12β)

sin(β)

Summary of this LectureI Two types of continuous groups: compact and non-compact.I Many of the results for finite groups are applicable to compact

continuous groups provided adjustments are made.I For Lie groups, all the information is contained in the infinitesimal

generators.I The generators form an algebra,I Representations of the algebra “exponentiate” to representations of

the group.I There are many ways of constructing representations of the

su(2) generators, v.g.I Polynomials are useful because it’s easy to take derivatives and

multiply by variables,I The Schwinger representation uses harmonic oscillator creation

and destruction operators.I Representations of SU(2) can be labeled by Young diagrams.I Schur-Weyl duality ties this Young diagram with permutation

symmetry of the corresponding N-body problem.I A lossless passive interferometer is an SU(2) system, and the

scattering matrix for N photons is given in terms of groupfunctions Dj , where j = N/2.