Notes on Groups S O(3) and S U(2)

Ivan G. Avramidi

New Mexico Institute of Mining and Technology

October, 2011

Author: IGA File: su2.tex; Date: November 29, 2011; Time: 10:07

1 Group S O(3)

1.1 Algebra so(3)

The generators Xi = Xⁱ, i= 1, 2, 3, of the algebra so(3) are 3×3 real antisymmetric matrices defined by

X₁ =



















0 0 0

0 0 1

0 −1 0



















, X₂=



















0 0 −1

0 0 0

1 0 0



















, X₃ =



















0 1 0

−1 0 0

0 0 0



















, (1.1)

or

(Xi)^kj = −ε^{ki j} = ε^{ik j} (1.2) Raising and lowering a vector index does not change anything; it is done just for convenience of notation. These matrices have a number of properties

XiXj = Eji− Iδi j, (1.3) and

εi jkX^k = E^{i j} − Eji (1.4) where

(Eji)^kl = δ^kjδ^l_i (1.5) Therefore, they satisfy the algebra

[Xi, Xj]= −ε^{ki j}Xk (1.6) The Casimir operator is defined by

X² = XⁱXi (1.7)

It is equal to

X² = −2I (1.8)

and commutes with all generators

[Xi, X²]= 0 . (1.9)

A general element of the group S O(3) in canonical coordinates yⁱhas the form exp (X(y)) = I cos r + P(1 − cos r) + X(y)sin r

r (1.10)

= P + Π cos r + X(y)sin r

r , (1.11)

where

r= p

yiyⁱ, θⁱ = yⁱ

r (1.12)

X(y)= Xiyⁱ, and the matricesΠ and P are defined by

Pⁱj = θⁱθj (1.13)

Πⁱj = δⁱj−θⁱθj (1.14)

In particular, for r= π we have

exp (X(y)) = P − Π (1.15)

Notice that

tr P= 1, trΠ = 2 (1.16)

and, therefore,

tr exp (X(y))= 1 + 2 cos r . (1.17) Also,

detsinh X(y)

X(y) = sin r r

!2

(1.18)

1.2 Representations of S O(3)

Let Xibe the generators of S O(3) in a general irreducible representation satisfying the algebra

[Xi, Xj]= −εi jkXk (1.19) They are determined by symmetric traceless tensors of type ( j, j) (with j a positive integer)

(Xi)^l1...ljk1...kj = − jε^(l1i(k1δ^l_k²

2· · ·δ^l_k^j)

j) (1.20)

Then

X²= − j( j + 1)I (1.21)

where

I^l1...ljk₁...kj = δ^(l_(k¹₁· · ·δ^l_k^j)

j) (1.22)

Another basis of generators is (J₊, J−, J3), where

J₊ = −iX1+ X2 (1.23)

J− = −iX1− X₂ (1.24)

J3 = −iX3 (1.25)

so that

J^†₊= J⁻, J₃^†= J3. (1.26) They satisfy the commutation relations

[J₊, J−]= 2J3, [J3, J₊]= J+, [J3, J−]= −J⁻. (1.27) The Casimir operator in this basis is

X² = −J+J−− J²₃+ J3 = −J−J₊− J²₃− J₃ (1.28) From the commutation relations we have

J₃J₊ = J₊(J₃+ 1) (1.29)

J₃J− = J−(J₃− 1) (1.30)

J−J₊ = −X²− J²₃− J₃ (1.31) J₊J− = −X²− J²₃+ J3 (1.32) Since J₃ commutes with X² they can be diagonalized simultaneously. Since they are both Hermitian, they have real eigenvalues. Notice also that since Xi are anti-Hermitian, then

X²≤ 0 . (1.33)

Let |λ, mi be the basis in which both operators are diagonal, that is,

X²|λ, mi = −λ|λ, mi (1.34)

J₃|λ, mi = m|λ, mi (1.35)

There are many ways to show that the eigenvalues m of the operator J₃are integers.

Now, since

X₁²+ X₂² = X²− X₃²= X²+ J₃²≤ 0 (1.36)

then the eigenvalues of J3 are bounded by a maximal integer j,

|m| ≤ j (1.37)

that is, m takes (2 j+ 1) values

− j, − j + 1, · · · , −1, 0, 1, · · · , j − 1, j (1.38)

¿From the commutation relations above we have

J₊|λ, mi = const |λ, m + 1i (1.39) J−|λ, mi = const |λ, m − 1i (1.40) Let |λ, 0i be the eigenvector corresponding to the eigenvalue m= 0, that is

J₃|λ, 0i = 0 . (1.41)

Then for positive m > 0

|λ, mi = aλmJ₊^m|λ, 0i (1.42)

|λ, −mi = aλ,−mJ₋^m|λ, −mi (1.43) Then

J₊|λ, ji = 0 (1.44)

J−|λ, − ji = 0 (1.45)

Therefore,

0= J⁻J₊|λ, ji = 

−X²− J₃²− J₃

|λ, ji (1.46)

0= J+J−|λ, − ji = 

−X²− J₃²+ J3

|λ, − ji (1.47) Thus,

λ²− j²− j

|λ, ji = 0 (1.48)

So, the eigenvalues of the Casimir operator X²are labeled by a non-negative inte- ger j ≥ 0,

λj = j( j + 1) . (1.49)

These eigenvalues have the multiplicity

dj = 2 j + 1 . (1.50)

In particular, for j = 1 we get λ1 = 2 as expected before.

From now on we will denote the basis in which the operators X² and J₃ are diagonal by

|λj, mi =   

j m

. (1.51)

The matrix elements of the generators J±can be computed as follows. First of all,

 j m   

J₃   

j m⁰

= δmm⁰m (1.52)

We have

j( j+ 1) = j m

J₊   

j m −1

  j m −1

J−

   j m

+ m²− m (1.53) and

 j m

J₊   

j m −1

= j m −1

J−

j m

(1.54) This gives

 j m

J₊   

j m⁰

 = δm⁰,m−1p

( j+ m)( j − m + 1) (1.55)

 j m

J−

j m⁰

 = δm⁰,m+1p

( j − m)( j+ m + 1) (1.56) The matrix elements of the generators Xiare

 j m

X₁   

j m⁰

 = δm⁰,m−1i 2

p( j+ m)( j − m + 1) +δm⁰,m+1i

2

p( j − m)( j+ m + 1) (1.57)

 j m

X2

j m⁰

 = δ^m0,m−11 2

p( j+ m)( j − m + 1)

−δm⁰,m+11 2

p( j − m)( j+ m + 1) (1.58)

 j m

X₃   

j m⁰

 = δmm⁰im (1.59)

Thus, irreducible representations of S O(3) are labeled by a non-negative inte- ger j ≥ 0. The matrices of the representation j in canonical coordinates yⁱ will be denoted by

D_mm^j 0(y)=  j m   

expX(y)

j m⁰

(1.60) where X(y) = Xiyⁱ.

The characters of the irreducible representations are χ^j(y)=

j

X

m=− j

D_mm^j (y)=

j

X

m=− j

e^imr = 1 + 2

j

X

m=1

cos(mr) (1.61)

2 Group S U(2)

2.1 Algebra su(2)

Pauli matrices σi = σⁱ, i= 1, 2, 3, are defined by

σ1 =











 0 1 1 0













, σ2 =











 0 −i

i 0













, σ3 =













1 0

0 −1













. (2.1)

They have the following properties

σ^†_i = σⁱ (2.2)

σ²_i = I (2.3)

det σi = −1 (2.4)

tr σi = 0 , (2.5)

σ₁σ₂ = iσ3 (2.6)

σ₂σ₃ = iσ1 (2.7)

σ3σ1 = iσ2 (2.8)

σ1σ2σ3 = iI (2.9)

(2.10) which can be written in the form

σiσj = δi jI+ iεi jkσk. (2.11)

They satisfy the following commutation relations

[σi, σj]= 2iεi jkσk (2.12) The traces of products are

tr σiσj = 2δi j (2.13)

tr σiσjσk = 2iεi jk (2.14) Also, there holds

σiσi = 3I . (2.15)

The Pauli matrices satisfy the anti-commutation relations

σiσj+ σ^jσi = 2δ^{i j}I (2.16) Therefore, Pauli matrices form a representation of Clifford algebra in 3 dimen- sions and are, in fact, Dirac matrices σi = (σiA

B), where A, B= 1, 2, in 3 dimen- sions. The spinor metric E = (E^AB) is defined by

E =













0 1

−1 0













. (2.17)

Notice that

E = iσ2 (2.18)

and the inverse metric E⁻¹= (E^AB) is

E⁻¹= −E (2.19)

Then

σ^T_i = −EσiE⁻¹ (2.20)

This allows to define the matrices Eσi = (σ^{i AB}) with covariant indices Eσ₁ = σ3=













1 0

0 −1













, (2.21)

Eσ₂ = iI =











 i 0 0 i













, (2.22)

Eσ₃ = −σ1 =













0 −1

−1 0













. (2.23)

Notice that the matrices Eσi are all symmetric

(Eσi)^T = Eσⁱ (2.24)

Pauli matrices satisfy the following completeness relation σi

A Bσi

C

D = 2δ^ADδ^CB−δ^ABδ^CD (2.25) which implies

σi(A (BσiC)

D) = δ^(A(Bδ^C)D) (2.26) This gives

σi ABσi CD = 2EADECB− EABECD (2.27) In a more symmetric form

σi ABσi CD = E^ADECB+ E^BDECA (2.28) A spinor is a two dimensional complex vector ψ= (ψ^A), A= 1, 2 in the spinor space C². We use the metric E to lower the index to get the covariant components

ψA = EABψ^B (2.29)

which defines the cospinor ˜ψ = (ψ^A)

ψ = Eψ˜ (2.30)

We also define the conjugated spinor ¯ψ = ( ¯ψ^A) by

ψ¯A = ψ^A∗ (2.31)

On the complex spinor space there are two invariant bilinear forms

hψ, ϕi = ¯ψϕ = ψ^A∗ϕ^A (2.32)

and

ψϕ = ψ˜ ^Aϕ^A = E^ABψ^Bϕ^A = ψ¹ϕ²−ψ²ϕ¹ (2.33) Note that

hψ, ψi = ¯ψψ = ψ^A∗ψ^A = |ψ¹|²+ |ψ²|² (2.34) and

ψψ = ψ˜ Aψ^A = 0 (2.35)

The matrices

Xi = i

2σi (2.36)

are the generators of the Lie algebra su(2) with the commutation relations

[Xi, Xj]= −εi jkXk (2.37) The Casimir operator is

X² = XⁱXⁱ (2.38)

It commutes with all generators

[Xi, X²]= 0 . (2.39)

There holds

X² = −3

4I (2.40)

tr XiXj = −1

2δi j (2.41)

Notice that

X²= − j( j + 1)I (2.42)

with j= ¹₂, which determines the fundamental representation of S U(2).

An arbitrary element of the group S U(2) in canonical coordinates yⁱ has the form

exp

i 2σjy^j

 = I cosr

2 + iσjθ^jsinr

2, (2.43)

This can be written in the form

exp (X(y)) = I cos(r/2) + X(y)sin(r/2)

r/2 . (2.44)

where X(y) = Xiyⁱ. In particular, for r= 2π

exp (X(y)) = −I. (2.45)

Obviously,

tr exp (X(y))= 2 cos(r/2) (2.46)

2.2 Representations of S U(2)

We define symmetric contravariant spinors ψ^{A1...A2 j} of rank 2 j. Each index here can be lowered by the metric EAB. The number of independent components of such spinor is precisely

2 j

X

i=0

1= (2 j + 1) (2.47)

Let Xi be the generators of S U(2) in a general irreducible representation sat- isfying the algebra

[Xi, Xj]= −εi jkXk (2.48) They are determined by symmetric spinors of type (2 j, 2 j)

(Xk)^{A1...A2 j}B₁...B2 j = i jσk(A1

(B1δ^A_B²

2· · ·δ^A_B^{2 j)}

2 j) (2.49)

Then

X²= − j( j + 1)I (2.50)

where

I_B^{A1...A2 j}

1...B2 j = δ^(A_(B¹₁· · ·δ^A_B^{2 j)}

2 j) (2.51)

Another basis of generators is

J₊ = −iX1+ X2 (2.52)

J− = −iX1− X₂ (2.53)

J3 = −iX3 (2.54)

They have all the properties of the generators of S O(3) discussed above except that the operator J₃ has integer eigenvalues m. One can show that they can be either integers or half-integers taking (2 j+ 1) values

− j, − j + 1, . . . , j − 1, j , |m| ≤ j (2.55) with j being a positive integer or half-integer. If j is integer, then all eigenvalues mof J3 are integers including 0. If j is a half-integer, then all eigenvalues m are half-integers and 0 is excluded.

We choose a basis in which the operators X² and J₃ are diagonal. Let ψ = ψ^{A1...A2 j} be a symmetric spinor of rank 2 j. Let e1 = (e1A

) and e2 = (e2A

) be the basis cospinors defined by

e₁^A = δ^A₁, e₂^A = δ₂^A. (2.56)

Let

e_j,m^{A1...Aj+mB1...Bj−m} = e1(A1· · · e₁^Aj+me₂^B1· · · e₂^Bj−m)

= δ^(A₁ ¹· · ·δ₁^Aj+mδ^B₂¹· · ·δ₂^Bj−m) (2.57) Then this is an eigenspinor of the operator J₃with the eigenvalue m

J3ej,m= mej,m (2.58)

Then the spinors

j m

= (2 j)!

( j+ m)!( j − m)!

!1/2

e_j,m (2.59)

form an orthonormal basis in the space of symmetric spinors of rank 2 j.

The matrix elements of the generators Xiare given by exactly the same formu- las

 j m

X1

j m⁰

 = δ^m0,m−1i 2

p( j+ m)( j − m + 1) +δ^m0,m+1i

2

p( j − m)( j+ m + 1) (2.60)

 j m

X₂   

j m⁰

 = δm⁰,m−11 2

p( j+ m)( j − m + 1)

−δm⁰,m+11 2

p( j − m)( j+ m + 1) (2.61)

 j m

X₃   

j m⁰

 = δmm⁰im (2.62)

but now j and m are either both integers or half-integers.

Thus, irreducible representations of S U(2) are labeled by a non-negative half- integer j ≥ 0. The matrices of the representation j in canonical coordinates yⁱ are

D_mm^j 0(y)=  j m   

expX(y)

j m⁰

(2.63) where X(y) = Xiyⁱ. The representations of S U(2) with integer j are at the same time representations of S O(3). However, representations with half-integer j do not give representations of S O(3), since for r = 2π

D_mm^j 0(y)= −δmm⁰. (2.64)

The characters of the irreducible representations are: for integer j χ^j(y)=

j

X

m=− j

D_mm^j (y)=

j

X

m=− j

e^imr = 1 + 2

j

X

m=1

cos(mr) (2.65)

and for half-integer j

χ^j(y)=

−¹

2+ j

X

k=−¹₂− j

D^j₁

2+k,¹2+k(y)=

−¹

2+ j

X

k=−¹₂− j

exp

"

i 1 2 + k

! r

#

= 2

j

X

k=1

cos" 1 2 + k

! r

#

(2.66) Now, let Xiand Yj be the generators of two representations and let

Gi = Xi ⊗ IY + IX⊗ Yi (2.67) Then Gigenerate the product representation X ⊗ Y and

exp[G(y)]= exp[X(y)] exp[Y(y)] (2.68) Therefore, the character of the product representation factorizes

χ^G(y)= χ^X(y)χ^Y(y) (2.69)

2.3 Double Covering Homomorphism S U(2) → S O(3)

Recall that the matrices Eσi = (σ^{i AB}) are symmetric and the matrix E = (E^AB) is anti-symmetric. Let ψ^ABbe a symmetric spinor. Then one can form a vector

Ai = σ^{i BA}ψ^AB = tr (Eσⁱψ) . (2.70) Conversely, given a vector Ai we get a symmetric spinor of rank 2 by

ψAB = 1

2Aⁱσi AB (2.71)

or

ψ = 1

2AⁱσiE⁻¹ (2.72)

More generally, given a symmetric spinor of rank 2 j we can associate to it a symmetric tensor of rank j by

Ai₁...ij = σi₁A₁B₁· · ·σi_jA_jB_jψ^{A1...AjB1...Bj} (2.73)

One can show that this tensor is traceless. Indeed by using the eq. (??) and the fact that the spinor is symmetric we see that the tensor A is traceless. The inverse transformation is defined by

ψA1...AjB1...Bj = 1

2^jA^i1...ijσi1A1B1· · ·σijAjBj (2.74) The double covering homomorphism

Λ : S U(2) → S O(3) (2.75)

is defined as follows. Let U ∈ S U(2). Then the matrices UσiU⁻¹ satisfy all the properties of the Pauli matrices and are therefore in the algebra su(2). Therefore,

UσiU⁻¹= Λ^ji(U)σj. (2.76) That is,

Λⁱj(U)= 1

2tr σⁱUσjU⁻¹ (2.77)

The matrixΛ(U) depends on the matrix U and is, in fact, in S O(3). For infinites- imal transformations

U = exp(Xky^k)= I + i

2σky^k+ · · · (2.78) the matrixΛ is

Λi j = δi j+ εi jky^k+ · · · =h

exp(Xky^k)i

i j (2.79)

It is easy to see that

Λ(I) = Λ(−I) = I . (2.80)

So, in short

Λ(exp(Xiyⁱ))= exp(Λ(Xi)yⁱ) (2.81) where

[Λ(σk)]ⁱ j = 2iεⁱk j (2.82)

3 Invariant Vector Fields

Let yⁱbe the canonical coordinates on the group S U(2) ranging over (−π, π). The position of the indices on the coordinates will be irrelevant, that is, yi = yⁱ. We introduce the radial coordinate r = |y| = pyiyⁱ, so that the geodesic distance

from the origin is equal to r, and the angular coordinates (the coordinates on S²) θⁱ = ˆyⁱ = yⁱ/r .

Let us consider the fundamental representation of the group S U(2) with gen- erators Ta = iσ^a, where σaare the Pauli matrices satisfying the algebra

[Ta, Tb]= −2ε^cabTc. (3.83) Let T (y) = Tiyⁱ. Then

T(y)T (p)= −(y, p) − T(y ∧ p) (3.84) where (y, p)= yipⁱ, (y ∧ p)k = εi jkyⁱp^j and

[T (y)]² = −|y|²I (3.85)

Therefore,

exp[rT (ˆy)]= cos r + T(ˆy) sin r (3.86) where ˆy is a unit vector. Next, we compute

exp[T (F(r ˆq, ρ ˆp))] = cos r cos ρ − (ˆq, ˆp) sin r sin ρ + cos r sin ρT( ˆp) + cos ρ sin rT(ˆq)

− sin r sin ρT ( ˆq ∧ ˆp) (3.87) This defines the group multiplication map F(q, p) in canonical coordinates.

This map has a number of important properties. In particular, F(0, p) = F(p, 0) = p and F(p, −p) = 0. Another obvious but very useful property of the group map is that if q = F(ω, p), then ω = F(q, −p) and p = F(−ω, q). Also, there is the associativity property

F(ω, F(p, q))= F(F(ω, p), q) (3.88) and the inverse property

F(−p, −ω)= −F(ω, p). (3.89)

This map defines all the properties of the group. We introduce the matrices L^αb(x) = ∂

∂y^bF^α(y, x) 

_y=0, (3.90)

R^αb(x) = ∂

∂y^bF^α(x, y)  

_y=0, (3.91)

Let X and Y be the inverses of the matrices L and R, that is,

LX = XL = RY = YR = I . (3.92)

Also let

D= XR , D⁻¹= YL (3.93)

so that

L= RD⁻¹, X = DY (3.94)

Let Cabe the matrices of the adjoint representation defined by

(Ca)^bc = −2ε^bac (3.95)

and C = C(x) be the matrix defined by

C = Cax^a. (3.96)

Then one can show that

X= exp(C) − I

C , Y = I − exp(−C)

C , (3.97)

L= C

exp(C) − I, R= C

I − exp(−C). (3.98)

It is easy to see also that

x^a= L^abx^b= R^abx^b = X^abx^b= Y^abx^b. (3.99) and

X = Y^T, L= R^T (3.100)

and

XY = YX = sinh(C/2) C/2

!2

, (3.101)

LR= RL = C/2 sinh(C/2)

!2

, (3.102)

as well as

D= XR = RX = exp(C) , D^T = YL = LY = exp(−C) . (3.103)

Note that

C² = −4r²Π , (3.104)

where r = |x| and

Πⁱj = δⁱj−θⁱθj (3.105) with θⁱ = xⁱ/r. Therefore,

C²ⁿ = (−1)ⁿ(2r)²ⁿΠ , C²ⁿ⁺¹= (−1)ⁿ(2r)²ⁿC (3.106) Therefore, the eigenvalues of the matrix C are (2ir, −2ir, 0), and for any analytic function

f(C) = f (0)(I − Π) + Π1

2 f (2ir)+ f (−2ir) + 1

4irC f (2ir) − f (−2ir

and, therefore,

tr f (C) = f (0) + f (2ir) + f (−2ir) (3.107) det f (C) = f (0) f (2ir) f (−2ir) (3.108) This enables one to compute

Y = I − Π + sin r

r cos rΠ − 1 2

sin²r

r² C (3.109)

X = I − Π + sin r

r cos rΠ + 1 2

sin²r

r² C (3.110)

R = I − Π + r cot rΠ + 1

2C (3.111)

L = I − Π + r cot rΠ − 1

2C (3.112)

D = I − Π + Π cos(2r) +sin(2r)

2r C (3.113)

Then one can show that

∂

∂x^µF^α(x, y) = L^αγ(F(x, y))X^γµ(x) , (3.114)

∂

∂x^µF^α(y, x) = R^αγ(F(y, x))Y^γ_µ(x) . (3.115) These matrices satisfy important identities

∂_µX^a_ν−∂_νX^a_µ = −2ε^abcX^b_µX^c_ν, (3.116)

∂_µY^a_ν−∂_νY^a_µ = 2ε^abcY^b_µY^c_ν. (3.117)

and

L^µa∂µL^νb− L^µb∂µL^νa = 2ε^cabL^γc, (3.118) R^µa∂µR^νb− R^µb∂µR^νa = −2ε^cabR^νc. (3.119) Let us define the one-forms

σ^a₋= X^aµ(x)dx^µ, σ^a₊= Y^aµ(x)dx^µ. They satisfy important identities

dσ^a₋ = −ε^abcσ^b₋∧σ^c₋, (3.120) dσ^a₊ = ε^abcσ^b₊∧σ^c₊. (3.121) Next, we define the vector fields.

K_a⁻ = L^µa(x)∂_µ, K_a⁺ = R^µa(x)∂_µ.

These are the right-invariant and the left-invariant vector fields on the group. They are related by

K_a⁻ = Dab

K_b⁺ (3.122)

They form two representations of the group S U(2) and satisfy the algebra

[K_a⁺, K_b⁺] = −2ε^cabK_c⁺ (3.123) [K_a⁻, K_b⁻] = 2ε^cabK_c⁻ (3.124)

[K_a⁺, K_b⁻] = 0 (3.125)

That is, the left-invariant vector fields commute with the right-invariant ones.

Now, let Ta be the generators of some representation of the group S U(2) sat- isfying the same algebra. Then, of course,

exp[T (ω)] exp[T (p)]= exp[T(F(ω, p))] (3.126) First, one can derive a useful commutation formula

exp[T (ω)]Tbexp[−T (ω)]= D^abTa, (3.127) where the matrix D = (D^ab) is defined by D= exp C .

More generally,

exp[T (ω)] exp[T (p)] exp[−T (ω)]= exp[T(D(ω)p)] . (3.128)

Next, one can show that

exp[−T (ω)] ∂

∂ωⁱ exp[T (ω)]= Y^aiTa, (3.129) This means that

∂

∂ωⁱ exp[T (ω)]= Y^aiexp[T (ω)]Ta = Y^iaTaexp[T (ω)] , (3.130) The Killing vectors have important properties

K_a⁺exp[T (ω)]= (exp[T(ω)])Ta (3.131) K_a⁻exp[T (ω)]= Ta(exp[T (ω)]) (3.132) This immediately leads to further important equations

exp[K⁺(p)] exp[T (ω)]= exp[T(ω)] exp[T(p)] (3.133) and

exp[K⁻(q)] exp[T (ω)]= exp[T(q)] exp[T(ω)] (3.134) where K^±(p) = p^aK_a^±. More generally,

exp[K⁺(p)+ K⁻(q)] exp[T (ω)]= exp[T(q)] exp[T(ω)] exp[T(p)] (3.135) and, therefore,

exp[K⁺(p)+ K⁻(q)] exp[T (ω)]

_ω=0 = exp[T(q)] exp[T(p)] = exp[T(F(q, p))]

(3.136) In particular, for the adjoint representation this formulas take the form

DCb = D^abCaD (3.137)

K_a⁺D= DCa (3.138)

K_a⁻D= C^aD (3.139)

exp[K⁺(p)]D(x) = D(x)D(p) (3.140) exp[K⁻(q)]D(x)= D(q)D(x) (3.141) exp[K⁺(p)+ K⁻(q)]D(x)= D(q)D(x)D(p) (3.142)

Then for a scalar function f (ω) the action of the right-invariant and left- invariant vector fields is simply

exp[K⁺(p)] f (ω)= f (F(ω, p)), exp[K⁻(q)] f (ω)= f (F(q, ω)). (3.143) Since the action of these vector fields commutes, we have more generally

exp[K⁺(p)+ K⁻(q)] f (ω)= f (F(q, F(ω, p))). (3.144) Therefore,

exp[K⁺(p)+ K⁻(q)] f (ω)  

_ω=0 = f (F(q, p)). (3.145) The bi-invariant metric is defined by

gαβ = δabY^aαY^bβ = δabX^aαX^bβ (3.146) g^αβ = δ^abL^αaL^βb = δ^abR^αaR^βb, (3.147) that is,

g= sinh[C/2]

C/2

!2

= I − Π + sin²r

r² Π . (3.148)

The Riemannian volume elements of the metric g is defined as usual

dvol = g^1/2(x)dx¹∧ dx²∧ dx³ (3.149) where

g^1/2 = (det gµν)^1/2 = sin²r

r² . (3.150)

Thus the bi-invariant volume element of the group is dvol (x)= sin²r

r² dx= sin²r dr dθ (3.151) where dx = dx¹dx²dx³, and dθ is the volume element on S². The invariance of the volume element means that for any fixed p

dvol (x)= dvol (F(x, p)) = dvol (F(p, x)) (3.152) Notice also, that |F(p, q)| is nothing but the geodesic distance between the points

pand q.

We will always use the orthonormal basis of the right-invariant vector fields K_a⁺and one-forms σ^a₊and denote the covariant derivative with respect to K_a⁺sim- ply by ∇a. The Levi-Civita connection ∇ of the bi-invariant metric in the right- invariant basis is defined by

∇_aK_b⁺ = ε^cbaK_c⁺ (3.153) so that the coefficients of the affine connection are

ω^abc = σ^a₊(∇cK_b⁺)= ε^abc. (3.154) so that for an arbitrary vector

∇_aV =h

(K_a⁺V^b)+ ε^bcaV^ci

K_b⁺ (3.155)

Then

∇_aK_d⁻= M^bdaK_b⁺ (3.156) where

M^bda = Ka⁺Ddb+ ε^bcaDdc

= −2ε^bcaDdc+ ε^bcaDdc = −ε^bcaDdc

(3.157) so that

∇_aK_d⁻ = −ε^bcaDdc

K_b⁺ (3.158)

Now, the curvature tensor is

R^abcd = −ε^fcdε^af b. (3.159) The Ricci curvature tensor and the scalar curvature are

Rab= 2δ^ab, R= 6 . (3.160)

The scalar Laplacian is

∆0 = δ^abK_a⁻K_b⁻= δ^abK_a⁺K_b⁺. (3.161) One can show that

∆0

|ω|

sin |ω| = |ω|

sin |ω| (3.162)

Further,

∆0exp[T (ω)]= (exp[T(ω)])T² = T²exp[T (ω)] (3.163)

where T²= T^aT^a. This means that

exp[t∆²₀] exp[T (ω)]= exp(tT²) exp[T (ω)] (3.164) The Killing vector fields K_a^± are divergence free, which means that they are anti-self-adjoint with respect to the invariant measure dvol (ω). that is, Thus the Laplacian∆0is self-adjoint with respect to the same measure as well.

Let ∇ be the total connection on a vector bundle V realizing the representation G. Let us define the one-forms

A= σ^a₊Ga = G^aY^aµdx^µ. (3.165) and F = dA + A ∧ A. Then, by using the above properties of the right-invariant one-forms we compute

F = 1

2F_abσ^a₊∧σ^b₊= 1

2F_abY^aµY^bνdx^µ∧ dx^ν (3.166) where

F_ab = ε^cabGc (3.167)

Thus A is the Yang-Mills curvature on the vector bundle V with covariantly con- stant curvature.

The covariant derivative of a section of the bundle V is then (recall that ∇a =

∇_K+ a)

∇_aϕ = (Ka⁺+ Ga)ϕ (3.168)

and the covariant derivative of a section of the endomorphism bundle End(V) along the right-invariant basis is then

∇_aQ= Ka⁺Q+ [Ga, Q] (3.169) Therefore, we find

∇_aGb = ε^cbaGc+ [Ga, Gb]= 0 . (3.170) Then the derivatives along the left-invariant vector fields are

∇_K⁻

aϕ = (Ka⁻+ Ba)ϕ (3.171)

∇_K⁻_aQ= Ka⁻Q+ [Ba, Q] (3.172) where

Ba = Dab

Gb (3.173)

The Laplacian takes the form

∆ = ∇^a∇^a = (Ka⁺+ G^a)(K_a⁺+ G^a)= ∆0+ 2G^aK_a⁺+ G² (3.174) where G²= GiGⁱ.

We want to rewrite the Laplacian in terms of Casimir operators of some rep- resentations of the group S U(2). The covariant derivatives ∇ado not form a rep- resentation of the algebra S U(2). The operators that do are the covariant Lie derivatives. The covariant Lie derivatives along a Killing vector ξ of sections of this vector bundle are defined by

L_ξ = ∇ξ− 1

2σ^a₊(∇bξ)ε^bcaGc (3.175) By denoting the Lie derivatives along the Killing vectors K_a^±by K_a^±this gives for the right-invariant and the left-invariant bases

K_a⁺ = LK⁺_a = ∇a+ Ga = Ka⁺+ 2Ga. (3.176) K_a⁻ = LK⁻_a = ∇K_a⁻− Ba= Ka⁻. (3.177) It is easy to see that these operators form the same algebra S U(2) × S U(2)

[K_a⁺, K_b⁺] = −2ε^cabK_c⁺ (3.178) [K_a⁻, K_b⁻] = 2ε^cabK_c⁻ (3.179)

[K_a⁺, K_b⁻] = 0 (3.180)

Now, the Laplacian is given now by the sums of the Casimir operators

∆ = K²− G² (3.181)

where

K²= 1

2K₊²+ 1

2K₋² (3.182)

and K_±²= Ka^±K_a^±.

4 Heat Kernel on S

Our goal is to evaluate the heat kernel diagonal. Since it is constant we can eval- uate it at any point, say, at the origin. We use the geodesic coordinates yⁱ defined

above. That is why, we will evaluate the heat kernel when one point is fixed at the origin. We will denote it simply by U(t; y). We obviously have

exp(t∆) = exp

−tG² exp

tK² . (4.1)

Therefore, the heat kernel is equal to U(t, y)= exp

−tG² Ψt 2, y

(4.2) whereΨ(t, y) = exp(2tK²)δ(y) is the heat kernel of the operator 2K².

4.1 Scalar Heat Kernel

Let

Φ(t, ω) = (4πt)^−3/2e^t

∞

X

n=−∞

|ω| + 2πn

sin |ω| exp −(|ω|+ 2πn)² 4t

!

(4.3) One can show by direct calculation that this function satisfies the equation

∂tΦ = ∆²0Φ (4.4)

with the initial condition

Φ(0, ω) = δS³(ω) (4.5)

Therefore, this is the scalar heat kernel on the group S U(2) and, therefore, on S³. Now, let us compute the integral

Ψ(t) = Z

S U(2)

dvol (ω)Φ(t, ω) exp[T(ω)] (4.6) Obviously,Ψ(0) = 1. Next, we have

∂tΨ = Z

S U(2)

dvol (ω) exp[T (ω)]∆0Φ(t, ω) (4.7) Now, by integrating by parts we get

∂tΨ = Z

S U(2)

DωΦ(t, ω)∆0exp[T (ω)] (4.8) which gives

∂tΨ = ΨT² (4.9)

Thus, we obtain a very important equation exp(tT²)=Z

S U(2)

dvol (ω)Φ(t, ω) exp[T(ω)] (4.10) This is true for any representation of S U(2).

We derive two corollaries. First, we get exp(tT²)= Z

S U(2)

dvol (ω)Φ(t, ω) exp[T(p)] exp[T(ω)] exp[−T(p)] (4.11) which means that for any p

Φ(t, ω) = Φ(t, F(−p, F(ω, p)). (4.12) or

Φ(t, F(p, q)) = Φ(t, F(q, p)). (4.13) Also, we see that

exp[(t+ s)T²]= Z

S³×S³

dvol (q)dvol (p)Φ(t, q)Φ(s, p) exp[T(q)] exp[T(p)] (4.14)

By changing the variables z= F(q, p) and p 7→ −p we obtain exp[(t+ s)T²]= Z

S U(2)

dvol (z) Z

S U(2)

dvol (p)Φ(t, F(p, z))Φ(s, p) exp[T(z)]

(4.15) which means that

Z

S U(2)

dvol (p)Φ(t, F(p, z))Φ(s, p) = Φ(t + s, z) (4.16) In particular, for z= 0 and s = t this becomes

Z

S U(2)

dvol (p)Φ(t, p)Φ(t, p) = Φ(t, 0) (4.17)