Lie Group and Lie Algebra

Guangyue Ji (棘广跃)1 1International Center for Quantum Materials, Peking University, Beijing 100871, China 2

I. GROUP

A. The deﬁnition of group

A group is a set, G, together with an operation • (called the group law of G) that combines any two elements a and b to form another element, denoted a • b or ab. To qualify as a group, the set and operation, (G, •), must satisfy four requirements known as the group axioms:

• Closure

For all a, b in G, the result of the operation, a • b, is also in G.

• Associativity

For all a, b and c in G, (a • b) • c = a • (b • c).

• Identity element

There exists an element e in G such that, for every element a in G, the equation e • a = a • e = a holds. Such an element is unique, and thus one speaks of the identity element.

• Inverse element

For each a in G, there exists an element b in G, commonly denoted a−1 (or −a, if the operation is denoted "+"), such that a • b = b • a = e, where e is the identity element.

B. Some familiar matrix groups

1. General linear group

• General complex linear groups GL(n, C). Its deﬁnition is

GL(n, C) = {A|A is a n × n complex matrix, and det A 6= 0} . (1)

The complex matrix of GL(n, C) has 2n2 independent matrix elements.

• General real linear groups GL(n, R). Its deﬁnition is

GL(n, R) = {A|A is a n × n real matrix, and det A 6= 0} . (2)

The real matrix of GL(n, R) has n2 independent matrix elements.

2. Special linear group

• Special complex linear groups SL(n, C). Its deﬁnition is

SL(n, C) = {A|A is a n × n complex matrix, and det A = 1} . (3)

The group SL(n, C) has 2(n2 − 1) independent matrix elements.

• Special real linear groups SL(n, R). Its deﬁnition is

SL(n, R) = {A|A is a n × n real matrix, and det A = 1} . (4)

The group SL(n, R) has (n2 − 1) independent matrix elements. 3

3. Unitary group and special unitary group

• Unitary group U(n). Its deﬁnition is

† † U(n) = U|U ∈ GL(n, C), and U U = UU = en . (5)

The unitary group U(n) has n2 independent matrix elements. • Special unitary group SU(n). deﬁnition

SU(n) = {U|U ∈ U(n), and det U = 1} . (6)

The special unitary group SU(n) has n2 − 1 independent matrix elements.

4. Orthogonal group and special orthogonal group

• Orthogonal group O(n). deﬁnition

T O(n, C) = O|O ∈ GL(n, C), and O O = en . (7)

The orthogonal group O(n) has n(n − 1) independent matrix elements. • Special orthogonal group SO(n). deﬁnition

SO(n) = {O|O ∈ O(n), and det U = 1} . (8)

5. Symplectic group

For 2n × 2n-dimensional matrix M and vector X and Y     x1 y1  x2   y2       .   .   .   .       xn   yn  X =   , Y= . (9)  xn+1   yn+1       xn+2   yn+2       .   .   .   .  x2n y2n

Under the transformation X ⇒ X0 = MX, Y ⇒ Y 0 = MY , if

n X (xiyn+i − yixn+i) = const. (10) i=1

The sets of matrices M2n×2n consist of the symplectic group Sp(2n, C). The symplectic group Sp(2n, C) has 2n(2n+1) independent matrix elements. The sympletic group can also be deﬁned by

RT JR = J (11) with J 0 I J = n×n . (12) −In×n 0 4

C. 3-D harmonic oscillator

Ref: Howard Georgi. Chapter 14. The Hamiltonian is 1 mω2 Hˆ = ~p2 + ~r2 (13) 2m 2 3 = ω a† a + , (14) k k 2 where 1 ak = √ (mωrk + ipk), (15) 2mω † 1 a = √ (mωrk − ipk). (16) k 2mω

If the ground state is |0i, satisfying

ak |0i = 0 (17) then the energy eigenstates are

a† ··· a† |0i (18) k1 kn with energy 3 ω(n + ). (19) 2

The degeneracy of these states is interesting—it is the number of symmetric combinations of the n indices, k1 ··· kn, which is (n + 1)(n + 2) . (20) 2 Just the dimension of the (n, 0) representation of SU(3). This suggests that the model has a SU(3) symmetry.

D. SU(2) → SU(1, 1)

Ref: PIERRE RAMOND. Chapter 5. The SU(2) Lie algebra has yet another interesting property: one can reverse the sign of two of its elements without changing the algebra. For example

T 1 → −T 1, (21) T 2 → −T 2, (22) T 3 → T 3. (23)

This operation is called an involutive automorphism of the algebra. It is an operation which forms a ﬁnite group of two elements. Its eigenvalues are therefore ±1, and it splits the Lie algebra into even and odd subsets. One can use this to obtain a closely related algebra by multiplying all odd elements by i. This generates another Lie algebra with three elements LA, with

L1 ≡ iT 1, (24) L2 ≡ iT 2, (25) L3 ≡ T 3. (26) 5

Note that L1,2 are no longer hermitian. The new algebra is

L1,L2 = −iL3, (27) L2,L3 = iL1, (28) L3,L1 = iL2. (29)

One can go further and absorb the i on the right-hand side of the commutators, and generate the algebra in terms of the real matrices

1 0 0 1 0 −1 , , . (30) 0 −1 1 0 1 0

Its representation theory is far more complicated than that of its compact coun- terpart. Its simplest unitary representations can be found in the inﬁnite Hilbert space generated by one bosonic harmonic oscillator. To see this, consider the operators

1 L+ ≡ √ a†a†; (31) 2 2 1 L− ≡ √ aa. (32) 2 2

The harmonic oscillator space splits into two representations of SO(2, 1) of even and occupation number states.

E. 1-D translation group

1-D translation group T (a) is

T (a) = x + a. (33)

We choose the wavefunctions xm as basis of the spcae. For example, in the case of 2 base vectors, the result of transformation T (a) is

1 1 T (a) = . x x + a

The entry in the column are base vectors. The matrix representation of T (a) is

1 1 0 1 1 T (a) = = . x a 1 x x + a 6

The ﬁnite dimension representation of T (a) is not unitary. If we continue to add the dimension of the Hilbert space, the representation of T (a) is     1  1 0 0 0 ...  1  x  a 1 0 0 ...  x   2     2  T (a)  x  =  a2 2a 1 0 ...   x   3     3  x  3 2  x    a 3a 3a 1 ...     .   .  ......    1 0 0 0 ...   0 0 0 0 ...   0 0 0 0 ...  1  0 1 0 0 ...   1 0 0 0 ...   0 0 0 0 ...   x      2    2  =  0 0 1 0 ...  + a  0 2 0 0 ...  + a  1 0 0 0 ...   x         x3   0 0 0 1 ...   0 0 3 0 ...   0 3 0 0 ...     .  ......  1   x  a d  2  = e dx  x   3   x   .  .  1   x + a   2  =  (x + a)  .  3   (x + a)   .  .

F. Peter-Weyl theorem:

• 1, the representations of a compact group are ﬁnite dimension and completely irreducible and are isomorphism to unitary representations.

• 2, the representations of non-compact group are not completely irreducible. There are only two unitary representations, namely trival representation and inﬁnite-dimension representation.

II. LIE GROUP AND LIE TRANSFORMATION GROUP

A. Structure function

For a list of parameters (α1, α2, . . . , αn), every list of parameters corresponds to an element of the group. For example, A(α1, . . . , αn), B(β1, . . . , βn), and C(γ1, . . . , γn) = AB, where γ = ϕ(α, β) is the function of (α, β) and satisﬁes:

0 0 • There is an identity element I(α1, . . . , αn) = I(0,..., 0) satisfying

IA = AI = A, ϕ(0, α) = ϕ(α, 0) = α. (34)

• There is an inverse element A−1 = A(¯α) for every A(α) satisfying

A−1A = AA−1 = I, (35) ϕ(¯α, α) = ϕ(α, α¯) = 0. (36)

• Associativity: 7

(AB)C = A(BC), (37) ϕ(ϕ(α, β), γ) = ϕ(α, ϕ(β, γ)). (38) When A(α) acts on the space V , X0 = A(α)X, A(α) is called the Lie transformation group. Specially, if A(α) is V , A(α) is called the Lie group.

B. The inﬁnitesimal generator of Lie group

Ref: Yuxin Liu. Chapter 1. For transformation A(α), its eﬀect is

X0 ⇒ X = A(α)X0 = f(α, X0). (39) If V is multi-dimensional space, for every i, we have i i i X = f (α, X0) = f (0,X). (40) If we continue to change the group parameter by δα, we have i i i X + dX = f (α + dα, X0) (41) = f i(δα, X) (42) i i ∂f (β, X)) σ = f (0,X) + σ δα + ... (43) ∂β β=0 i i σ = X + Uσ(X)δα . (44) Namely, i i σ dX = Uσ(X)δα . (45) In the parameter space, we have αµ + dαµ = ϕµ(δα, α) (46) i µ ∂ϕ (β, X)) σ = ϕ (0, α) + σ δα + ... (47) ∂β β=0 µ µ σ = α + Vσ δα . (48) Namely, σ −1σ µ σ µ δα = V µ dα = Λµ(α)dα . (49) Finally, we have i i σ µ dX = Uσ(X)Λµ(α)dα . (50) i σ We call Uσ(X)Λµ(α) the infinitesimal operator. For the function F (X) ∂F (X) dF (X) = U i (X)Λσ(α)dαµ (51) ∂Xi σ µ ∂ = U i (X)Λσ(α) F (X)dαµ. (52) σ µ ∂Xi i σ ∂ We call Uσ(X)Λµ(α) ∂Xi the infinitesimal operator and ∂ (X) = U i (X) (53) Xσ σ ∂Xi the infinitesimal generator of Lie transformation group. Similarily, ∂ (ξ) = V µ(ξ) (54) Xσ σ ∂ξµ is the infinitesimal generator of Lie group. 8

C. SO(2)

We take the SO(2) group as an example to understand the inﬁnitesimal generator. Under the transformation of SO(2),

x0 cos θ − sin θ x = , (55) y0 sin θ cos θ y namely X0 = f(θ, X). The structure function is ϕ(θ, θ0) = θ + θ0. In the neighborhood of the identity (θ = 0, δθ → 0),

x0 1 −δθ x = , (56) y0 δθ 1 y Then i i ∂f (β, X)) Uσ(X) = σ (57) ∂β β=0 −y = . (58) x

D. 1-D scale and translation group

Under the transformation of 1-D scale and translation group

0 X → X = α1X + α2. (59) Under another transformation β, we obtain

0 00 0 X → X = β1X + β2 (60)

= β1α1X + β1α2 + β2 (61)

= γ1X + γ2. (62) Then i i ∂f (β, X)) Uσ(X) = σ (63) ∂β β=0 = (X, 1). (64) µ µ ∂ϕ (β, α)) Vσ (X) = σ (65) ∂β β=0 α α = ( 1 2 ). (66) 0 1 The inﬁnitesimal generator of Lie transformation group is ∂ = X , (67) X1 ∂X ∂ = . (68) X2 ∂X The inﬁnitesimal generator of Lie group is ∂ ∂ X1 = α1 + α2 (69) ∂α1 ∂α2 ∂ X2 = . (70) ∂α2 I think that δ → d and d → ∆is more appreciate. 9

E. The basic properties of Lie group

1. The Lie’s ﬁrst theorem

We set γµ = ϕµ(β, α) the structure function of Lie group and it is continuously diﬀerentiable about α and β, then

∂γµ = V µ(γ)Λσ(β). (71) ∂βν σ ν

2. The Lie’s second theorem

The inﬁnitesimal generators of Lie group satisfy

µ [Xσ, Xρ] = CσρXµ, (72)

µ where Cσρ is the structure constant.

3. The Lie’s third theorem

µ µ Cσρ = −Cρσ, (73)

η κ η κ η κ Cµν Cηρ + CνρCρµ + CρµCην = 0. (74)

III. LIE ALGEBRA

A. Deﬁnition

1. Deﬁnition of Lie algebra

A Lie algebra is a vector space g over some ﬁeld F together with a binary operation [·, ·]: g × g → g called the Lie bracket that satisﬁes the following axioms:

• Bilinearity,

[ax + by, z] = a[x, z] + b[y, z] (75) [z, ax + by] = a[z, x] + b[z, y] (76) for all scalars a, b in F and all elements x, y, z in g.

• Alternativity, [x, x] = 0 for all x in g.

• The Jacobi identity, [x, [y, z]] + [z, [x, y]] + [y, [z, x]] = 0 for all x, y, z in g.

2. Homomorphism and isomorphism

0 0 0 For two Lie algebras g1 and g2, if there is a correspondence of X in g1 to X in g2, aX + bY in g1 to aX + bY in g2 0 0 and [X,Y ] in g1 to [X ,Y ] in g2, then we say g1 is homomorphic to g2 and write it as g1 ∼ g2. If the correspondence ∼ is one-to-one, we say that they are isomorphic and write it as g1 = g2. 10

B. Adjoint representation

Deﬁning a set of matrices Ta

[Ta]bc ≡ −ifabc. (77)

We need deﬁne the scalar product on the linear space of the generators in the adjoint representation, to turn it into a vector space. A good choice is the trance in the adjoint representation

Tr(TaTb). (78)

We can change its form by making a linear transformation on the Xa, which in turn, induces a linear transformation on the structure constants.

0 Xa → Xa = LabXb. (79)

Then

0 0 [Xa,Xb] = iLadLbefdecXc (80) −1 = iLadLbefdegLgh LhcXc (81) −1 0 = iLadLbefdegLgc Xc. (82)

0 −1 fabc → fabc = LadLbefdegLgc . (83)

Suppose we have done this, so that

Tr(TaTb) = λδab. (84)

In this basis, the structure constants are completely antisymmetric, because we can write

−1 fabc = −iλ Tr([Ta,Tb]Tc) (85) −1 = −iλ Tr(ifabdTdTc) (86) −1 = fabdλ λδdc (87)

= fabc. (88)

fabc = fbca. (89)

C. Lie group and Conservation charge

The function is φα where α = 1, . . . , n. The tranformation operator is U(g) = eiakXk . This is to say

φ → φ0 = U(g)φ = eiakXk φ. (90)

Deﬁne

∂ 0 α β Dkφ = φ (a, φ) = i (Xk)β φ . (91) ∂ak a=0 11

Then

∂L α ∂L α DkL = α Dkφ + α Dk (∂µφ) (92) ∂φ ∂ (∂µφ) ∂L = D φα + ΠµD (∂ φ)α (93) ∂φα k α k µ µ α µ α = (∂µΠα)Dkφ + ΠαDk (∂µφ) (94) µ α β µ α β = (∂µΠα)i (Xk)β φ + Παi (Xk)β ∂µφ (95) h µ α β µ α βi = i (∂µΠα)(Xk)β φ + Πα (Xk)β ∂µφ (96)

h µ α βi = i∂µ Πα (Xk)β φ (97) = 0. (98) where we have used the equation of motion

∂L ∂ Πµ = . (99) µ α ∂φα

We deﬁne the current as

h µ α βi µ ∂µ Πα (Xk)β φ = ∂µ [jk ] = 0. (100)

The charge is Z 3 0 α β d xΠα (Xk)β φ = Q(Xk). (101)

The commutation relation of the ﬁeld φ and the charge Q is Z c c 3 0 α β [φ (x),Q(X)] = φ (x), d yΠα(y)(X)β φ (y) (102) Z 3 c 0 α β = d x φ (x), Πα(y) (X)β φ (y) (103) Z 3 c (3) α β = d xiδαδ (x − y)(X)β φ (y) (104)

c β = i (X)β φ (x). (105)

Theorem: invariant operators satisfy

[−iQ(X), −iQ(Y)] = −iQ ([X, Y]) . (106)

Therefore, we can choose invariants that commuate with each other to describe the physical system.

µ µ [Q(X), j (Y)] = ij ([X, Y]) . (107)

IV. SO(3) AND SU(2) GROUP

A. Euler parameterization

The Euler parameterization is

00 0 00 0 −iγJz −iβJy −iαJz R(α, β, γ) = Rγ RβRα = e e e (108)

= RαRβRγ . (109) 12

B. The relation of SU(2) and SO(3)

They are local isomorphism. The concrete correspondence is 1 R = Tr σ uσ u† . (110) kj 2 k j Proof: Under a SU(2) transformation U, the vector ~x changes to be

~x0 · ~σ = U~x · ~σU †. (111)

We multiply σj with both sides and take the trace 1 x0 = x Tr σ Uσ U † (112) j 2 i j i We can write a vector in the matrix form X = ~x · ~σ (113) z x − iy = . (114) x + y −z

This matrix is Hermitian and traceless. The determinant of the matrix is − x2 + y2 + z2. Under a SU(2) transformation U, the vector changes to be X0 = U †XU (115) = U †~x · ~σU (116) z x − iy = U † U. (117) x + y −z We can see that X0 is still a Hermitian and tracelss matrix and this process can be seen as the rotation of the vector ~x. The transformation matrix U is

i φ nˆ·~σ φ φ e 2 = I cos + i(ˆn · ~σ) sin . (118) 2 2 Then we can prove

† j U3 (α)σiU3(α) = σjR3(α)i , (119) † j U2 (α)σiU2(α) = σjR2(α)i . (120) Therefore

† † † † U (α, β, γ)σiU(α, β, γ) = R3(γ)R2(β)R3(α)σiR3(α)R2(β)R3(γ) (121) l k j = σlR3(γ)kR2(β)j R3(α)i . (122) When the angle of the rotation is ψ and the direction is nˆ, we have U †(ψ)~σU(ψ) = ~σ cos ψ + ~n (~σ · ~n) (1 − cos ψ) + (~σ × ~n) sin ψ. (123) ~x0 = ~x cos ψ + ~n (~x · ~n) (1 − cos ψ) + (~n × ~x) sin ψ. (124)

V. LORENTZ GROUP

A. Lorentz group O(3, 1)

The Lorentz group is the set of Λ which satisfy

ρ σ ΛµηρσΛν = ηµν (125) 13 where η = diag (1, −1, −1, −1). There are six generators Ki and Ji. The concrete forms are ∂ ∂ J = i y − x , (126) z ∂x ∂y ∂ ∂ K = t + x . (127) x ∂x ∂t 0 To be convenient, we deﬁne Kj ≡ −iKj. We have the following commutation relations

[Ji,Jj] = iijkJk (128) 0 0 Ji,Kj = iijkKk (129) 0 0 Ki,Kj = −iijkJk. (130) The group element is ~ ~ ~ 0 ~ Λ = ei(J·θ+K ·φ). (131)

To be convenient, we can deﬁne Mµν : 1 J = M (j, k = 1, 2, 3) , (132) i 2 ijk jk Ki = M0i. (133) The commutation relations are

[Mµν ,Mρσ] = −i (ηµρMνσ − ηνρMµσ + ηνσMµρ − ηµσMνρ) . (134) ± 1 We can also deﬁne Ni = 2 (Ji ± iKi) and they are not Hermitian. The commutation relations are + + + Ni ,Nj = iijkNk (135) − − − Ni ,Nj = iijkNk (136) + − Ni ,Nj = 0. (137) Therefore SO(3, 1) ' SU(2) ⊗ SU(2). (138)

B. The relation of SO(3, 1) ∼ SL(2,C)

Let us deﬁne X as µ X = σµx . (139) µ where σµ = (σ0, ~σ) and σ = σµ = (σ0, −~σ) and Tr(σµσν ) = 2gµν . Conjugate transformation X0 = AXA† (140) where α β A = (141) γ δ and det A=αδ − βγ=1. We can prove that 1 x0µ = Tr[σµAσ A†]xν (142) 2 ν and 1 1 Λ0 = Tr[σ0Aσ A†] = Tr[AA†] > 0. (143) 0 2 0 2 Therefore ∼ SL(2,C)/Z2 = SO(3, 1). (144) 14

C. SL(2,C)

For A ∈ SL(2,C), A†A satisﬁes

det[A†A] = 1, (145) (A†A)† = A†A, (146) and the eigenvalues are bigger than 0. Therefore, we can deﬁne

√ P = A†A (147) U = AP −1 (148) with U is unitary and unimodular P is unimodular, Hermitian and positive deﬁnite.

D. Poincare group

0µ µ ν µ x = Λν x + a . (149)

Pauli-Lubanski pseudovector decribes the spin states of moving particles. It is the generator of the little group of the Poincare group, that is the maximal subgroup (with four generators) leaving the eigenvalues of the four-momentum vector Pµ invariant. 1 W = ε J νρP σ. (150) µ 2 µνρσ The commutator relations,

[P µ,W ν ] = 0, (151) [J µν ,W ρ] = i(gρν W µ − gρµW ν ), (152) ρ σ [Wµ,Wν ] = −iµνρσW P . (153)

Note that

W0 = J~ · P,~ (154) W~ = EJ~ − P~ × K.~ (155)

E. Massive ﬁelds:

µ In the case of a massive ﬁeld, the Cassimir invariant WµW describes the total spin of the particle, with eigenvalues

2 µ 2 W = WµW = −m s(s + 1), (156) where s is the spin quantum number of the particle and m is its rest mass. In the rest frame of the particle, the above commutator acting on the particle’s state amounts to

[Wj,Wk] = ijklWlm, (157) so that the little group amounts to the rotation group. It is also customary to take W3 to describe the spin projection along the third direction in the rest frame. 15

In moving frames, decomposing W = (W0, W~ ) into components (W1,W2,W3), with W1 and W2 orthogonal to P~ , and W3 parallel to P~ , the Pauli-Lubanski vector may be expressed in terms of the spin vector S~ = (S1,S2,S3) as E W = PS ,W = mS ,W = mS ,W = S . (158) 0 3 1 1 2 2 3 c2 3

The tranverse components W1, W2 and S3 satisfy the following commutator relations ih E P [W ,W ] = (( )2 − ( )2)S , (159) 1 2 2π c2 c 3 ih [W ,S ] = W , (160) 2 3 2π 1 ih [S ,W ] = W . (161) 3 1 2π 2

F. Massless ﬁelds

In general, in the case of non-massive representations, two cases may be distinguished. For massless particles,

2 µ W = WµW (162) 2 2 2 = −E ((K2 − J1) + (K1 + J2) ) (163) ≡ −E2(A2 + B2). (164)

1. Continuous spin representations

In the more general case, the components of W~ transverse to P~ may be non-zero, thus yielding the family of representations refered to as the cylindrical luxons, their identifying property being that the components of W~ form a Lie subalgebra isomorphic 2-dimensional Euclidean group ISO(2), with the longitudinal component of W~ playing the role of the rotation generator, and the transverse components the role of translation generators. This amounts to a group contraction of SO(3), and leads to what are known as the continuous spin representations. However, there are no known physical cases of fundamental particles or ﬁelds in this family. It can be proved that continuous spin states are unphysical.

2. Helicity representations

In a special case, W~ is parallel to P~ . For this family, W 2 = 0 and W µ = λP µ; the invariant is, instead, (W 0)2 = (W 3)2, where

W 0 = −J~ · P.~ (165) so the invariant is represented by the helicity operator

W 0/P. (166)

Therefore, the projection of spin of any massless particle with spin S in any direction can have only two, namely +S and −S.

VI. THE TENSOR REPRESENTATION OF GL(N,C)

A. The covariant tensor representation of GL(N,C)

We set ψ a covariant vector. Under the transformation of GL(N,C): 0 ˆ j ψi = ψjD(g)i . (167) 16

For general covariant tensor Fijk···m:

0 ˆ i0 ˆ j0 ˆ m0 Fij···m = D(g)i D(g)j ··· D(g)m Fi0j0···m0 . (168)

There is an important property that the symmetry of the indices of a covariant tensor is invariant under the transformation of GL(N,C). Therefore, we can use the symmetry of the indices to decompose a general tensor space into irreducible subspaces. For example

⊗ = ⊕ . (169) For the case of n indices, the number of ways of index symmetrization equals to the number of the standard Young diagrams. A Young diagram is denoted by [λ1, λ2, ··· , λN ]. The standard Young tableaux satisﬁes:

• In the same row, nleft ≤ nright .

• In the smae line, nup < ndown .

B. The contravariant and mixed tensor representation of GL(N,C)

We set ψ a contravariant vector. Under the transformation of GL(N,C):

h ii ψ0i = ψj Dˆ −1(g) (170) j h iT j = ψj Dˆ −1(g) . (171) i Therefore, the contravariant vector representation is the complex conjugate representation of the covariant vector. For GL(N,C), these two representations are diﬀerent. ijk···m The decomposition of contravariant tensor F is similar to the covariant tensor Fijk···m. The mixed tensors that support the irreducible representation of GL(N,C) satisfy:

• The upper indices have deﬁnite symmetry,

• The down indices have deﬁnite symmetry,

• The contraction of any pair of upper and down indices is zero. (traceless)

The notation is [λ1, λ2, ··· , λN ] is for covariant tensor and [−λN · · · − λ2, −λ1] is for contravariant tensor.

C. The dimension of an irreducible representation of GL(N,C)

The product of occupied number D = (172) hook number

N N + 1 N + 2 N + 3 (173) N − 1 N (174) − − − − − − − − (175) 5 4 2 1 (176) 2 1 (177) 17

VII. THE TENSOR REPRESENTATION OF SU(N)

For SU(N) group, we only need to consider the covariant tensor representation because a contravariant tensor representation is equivalent to a contravariant tensor representation. For the tensor representation of SU(N) group, we have

[λ1, ··· , λN ] = [λ1 − s, ··· , λN − s] . (178) The Levi-Civita tensor is invariant under the SU(N) transformation. Therefore, we can use the Levi-Civita tensor to raise and lower the indices.

A. The decomposition of direct product of irreducible representations of SU(N)

⊗ (179) 1 1 ⊗ (180) 2 and satisfy • In the same line, the number can’t be the same. • From right-up to left-down, the number of a is not less than the number of b if a < b.

B. The branding rule

We can decompose an irriducible representation of G according to the irriducible representation of its subgroup. The general principle is

X (1) (1) (1) UN (f1, f2, ··· , fN ) = ⊕UN−1 f1 f2 ··· fN−1 (181) (1) (1) (1) f1 f2 ···fN−1

(1) (1) (N−1) with f1 ≥ f1 ≥ f2 ≥ · · · ≥ fN−1 ≥ fN−1 ≥ fN . We can continue to do this until the f1 .

VIII. THE TENSOR REPRESENTATION OF SO(N)

For the SO(N) group, the covariant and contravariant presentations are the same. Therefore, the irreducible representation need to satisfy the traceless condition.

A. SO(2M + 1)

B. SO(2M)

We can deﬁne 1 F ± = G + G˜ (182) µν 2 µν µν 1 ˜ ˜ where 2! µνρσGµν = Gµν . The dual tensor Fµν is 1 X F˜± = F (±) = ±F (±). (183) µν 2! µνρσ ρσ µν ρσ

+ − Therefore, Fµν are self-dual tensor and Fµν are self-anti-dual tensor. 18

IX. CARTAN’S SUBALGEBRA

A. The generators of SU(N)

There are N 2 − 1 independent generators. For j 6= k, there are

|jihk| + |kihj| , (184) |jihk| − |kihj| .

For j = k, there are

|1ih1| , |1ih1| − |2ih2| , (185) |1ih1| + |2ih2| − 2 |3ih3| ··· .

They are the Cartan’s generators. They are traceless matrices and commute with each other. Therefore, we can choose the common eigenfunctions of the Cartan’s generators as the Hilbert space.

B. The Cartan’s subalgebra

If the Lie group has N generators and M generators which are Hermitian and commute with each other. The M generators form the Cartan’s subalgebra. We can write Cartan’s subalgebra as Hi i = 1,...,M. M is the rank of the group. We can write non-Cartan’s subalgebra as Ei i = 1,...,N − M. The Cartan’s generators act on the eigenstates

i i H |ji = tj |ji , (186)

i where tj is the weight. ~ 1 M The weight vector tj = tj , . . . , tj . The number of weight vectors equals the dimension of the Hilbert space. The dimension of weight vectors equals the number of Catan’s generators and the rank of the group.

C. The representation of Cartan’s subalgebra

Consider the Lie group representation Dn

|ji → |Dn : tji . (187)

D. Adjoint representation

In the adjoint representation, [Ta]bc = −ifabc. The dimension of the representation is n × n with n is the number of generators. Because the rows and columns of the matrices are labeled by the same index that labels the generators, the states of the adjoint representation correspond to the generators themselves. We will denote the state in the adjoint representation corresponding to an arbitrary generator Xa as |Xai. Linear combinations of these states correspond to linear combinations of the generators

α |Xai + β |Xbi = |αXa + βXbi . (188)

A convenient scalar product on this space is the following:

−1 † hXa|Xbi = λ Tr XaXb . (189) 19

We can compute the action of a generator on a state, as follows:

Xa |Xbi = |Xci hXc| Xa |Xbi (190)

= |Xci [Ta]cb (191) = |Xci (−ifacb) (192)

= ifabc |Xci (193)

= |[Xa,Xb]i . (194)

Root is the weight in the adjoint representation. Root vector is the weight vector in the adjoint representation. The commutation relation are:

[Hi,Hj] = 0 (195)

[Hi,Eα] = αiEα (196)

[Eα,E−α] = α · H. (197)

[Eα,Eβ] = NαβEα+β.(if α + β 6= 0) (198)

X. SU(3) GROUP

A. The Gell-mann matrices

The Gell-mann matrices are

 0 1 0   0 −i 0   1 0 0  λ1 =  1 0 0  λ2 =  i 0 0  λ3 =  0 −1 0  0 0 0 0 0 0 0 0 0  0 0 1   0 0 −i   0 0 0  λ4 =  0 0 0  λ5 =  0 0 0  λ6 =  0 0 1  (199) 1 0 0 i 0 0 0 1 0  0 0 0   1 0 0  λ = 0 0 −i λ = √1 0 1 0 . 7   8 3   0 i 0 0 0 −2

The Gell-Mann matrices, developed by Murray Gell-Mann, are a set of eight linearly independent 3 × 3 traceless Hermitian matrices used in the study of the strong interaction in particle physics. They span the Lie algebra of the SU(3) group in the deﬁning representation. Gell-Mann’s generalization further extends to general SU(N). For their connection to the standard basis of Lie algebras, see the Weyl–Cartan basis.

• The trace of the pairwise product results in the ortho-normalization condition

tr (λiλj) = 2δij. (200)

• The 8 inﬁnitesimal generators of the Lie algebra are indexed by i satisfy the commutation relations

X ijk [λi, λj] = 2i f λk, (201) k 4 X {λ , λ } = δ I + 2 dijkλ , (202) i j 3 ij k k with the structure constants 1 f ijk = − itr (λ [λ , λ ]) , (203) 4 i j k 1 dijk = tr (λ {λ , λ }) . (204) 4 i j k 20

• Fierz completeness relation 1 1 δαδγ = δαδγ + λα · λγ , (205) β δ 3 δ β 2 δ β 16 1 λα · λγ = δαδγ − λα · λγ . (206) β δ 9 δ β 3 δ β

• Casimir operators and invariants. The squared sum of the Gell-Mann matrices gives the quadratic Casimir operator, a group invariant,

8 X 16 C = λ λ = I. (207) i i 3 i=1

ˆ 1 A slightly differently normalized standard basis consists of the F -spin operators, which are defined as Fi = 2 λi for the 3, and are utilized to apply to any representation of this algebra. The Cartan–Weyl basis of the Lie algebra of SU(3) is obtained by another change of basis, where one defines

I± = F1 ± iF2 (208)

I3 = F3 (209)

V± = F4 ± iF5 (210)

U± = F6 ± iF7 (211) 2 Y = √ F8. (212) 3 The cummutation algebra of the generators. The standard form of generators of the SU(3) group satisﬁes the commutation relations given below,

[Y,I3] = 0 (213)

[Y,I±] = 0 (214)

[Y,U±] = ±U± (215)

[Y,V±] = ±V± (216)

[I3,I±] = ±I± (217) 1 [I ,U ] = ∓ U (218) 3 ± 2 ± 1 [I ,V ] = ± V (219) 3 ± 2 ± [I+,I−] = 2I3 (220) 3 [U ,U ] = Y − I (221) + − 2 3 3 [V ,V ] = Y + I (222) + − 2 3 [I+,V−] = −U− (223)

[I+,U+] = V+ (224)

[U+,V−] = I− (225)

[I+,V+] = 0 (226)

[I+,U−] = 0 (227)

[U+,V+] = 0. (228)

The representations of the group lie in the 2-dimensional I3 − Y plane. Here, I3 stands for the z-component of Isospin and Y is the Hypercharge, and they comprise the (abelian) Cartan subalgebra of the full Lie algebra. The maximum number of mutually commuting generators of a Lie algebra is called its rank: SU(3) has rank 2. The remaining 6 generators, the ± ladder operators, correspond to the 6 roots arranged on the 2-dimensional hexagonal lattice of the ﬁgure. 21

In the case of SU(3) group, by contrast, two independent Casimir operators can be constructed, a quadratic and a cubic: they are,

(2) X C = FkFk, (229) k (3) X C = djklFjFkFl. (230) jkl These Casimir operators serve to label the irreducible representations of the Lie group algebra SU(3), because all states in a given representation assume the same value for each Casimir operator, which serves as the identity in a space with the dimension of that representation. This is because states in a given representation are connected by the action of the generators of the Lie algebra, and all generators commute with the Casimir operators. For example, for the triplet representation, D(1, 0), the eigenvalue of C(2) is 4/3, and of C(3), 10/9. More generally, from Freudenthal’s formula, for generic D(p, q), the eigenvalue of C(2) = (p2 + q2 + 3p + 3q + pq)/3. (231) The eigenvalue ("anomaly coeﬃcient") of C(3) = (p − q)(3 + p + 2q)(3 + q + 2p)/18. (232) It is an odd function under the interchange p ↔ q. Consequently, it vanishes for real representations p = q, such as the adjoint, D(1, 1).

B. Representations of the SU(3) group

The representations are labeled as D(p, q), with p and q being non-negative integers, where in physical terms, p is the number of quarks and q is the number of antiquarks. Mathematically, the representation D(p, q) may be constructed by tensoring together p copies of the standard 3-dimensional representation and q copies of the dual of the standard representation, and then extracting an irreducible invariant subspace. Still another way to think about the parameters p and q is as the maximum eigenvalues of the diagonal matrices  1 0 0   0 0 0  H1 =  0 −1 0  ,H2 =  0 1 0  . (233) 0 0 0 0 0 −1 The representations have dimension 1 d(p, q) = (p + 1)(q + 1)(p + q + 2). (234) 2 An SU(3) multiplet may be completely specified by five labels, two of which, the eigenvalues of the two Casimirs, are common to all members of the multiplet. Since [I3,Y ] = 0 , we can label different states by the eigenvalues of I3 and Y operators, |t, yi, for a given eigenvalue of the isospin Casimir. The action of operators on this states are

I± |t, yi = α |t ± 1, yi (239) 1 U± |t, yi = β t ± , y ± 1 (240) 2 1 V± |t, yi = γ t ∓ , y ± 1 . (241) 2 22

The center of coordinate system is in the center of the hexagon.

XI. SU(3) ALGEBRA

The date of content is 2019.05.10.

A. Foundamental representation of the SU(3)

1 The generators Ta = 2 λa, a = 1,..., 8. The common eigenstates of T3 and T8 are

 1  v1 =  0  , (242) 0  0  v2 =  1  , (243) 0  0  v3 =  0  . (244) 1

The generators T3 and T8 are the Catan’s subalgebra of su(3).

1 1 T3v1 = v1,T8v1 = √ v1; (245) 2 2 3 1 1 T3v2 = − v1,T8v1 = √ v2; (246) 2 2 3 1 T3v3 = 0,T8v1 = −√ v3. (247) 3

Therefore,

1 2 v1 ∼ t1 = √1 ; (248) 2 3 1 − 2 v2 ∼ t2 = √1 ; (249) 2 3 0 v ∼ t = . (250) 3 3 − √1 3

The non-Catan’s generators can do vi ←→ vj for i 6= j. 23

B. The relation of fijk and dijk

• The Jacobi relation is [D, [A, B]] + [B, [D,A]] + [A, [B,D]] = 0. (251) Namely,

ifabcifdce = ifdaeifcbe + ifdbciface. (252)

We can deﬁne (Fc)de = ifdce and the above equation can be written as

[Fa, Fb] = ifabcFc. (253) • [D, [A, B]] = {{D,A} ,B} − {A, {D,B}} (254) Namely, 2 2 if if = (d d + δ δ ) − (d d + δ δ ). (255) abc dce dac cbe 3 da be dbe ace 3 db ae

We can deﬁne (Xab)de = δadδbe − δbdδae, and (Da)de = ddac and the above equation can be written as 2 if ( ) = [ , ] + ( ) . (256) abc Fc de Da Db de 3 Xab de • {D, [A, B]} = [{D,A} ,B] − {A, [D,B]} (257) Namely,

ifabcddce = ddacifcbe − ifdbcdace. (258) Namely,

ifabcDc = [Da, Fb] . (259)

XII. WEIGHT OF SU(3)

A. Triality

The triality of irreducible representation is τ Triality = , (260) 3 τ = 0, 1, 2. (261) For the irreducible representation D(p, q): Mod(p − q, 3) Y = . (262) M 3

• The foundamental(quark) representation is D(1, 0) and its YM = 1/3.

• The complex conjugate (anti-quark) representation is D(0, 1) and its YM = 2/3.

• All the irreducible representations of the SU(3) can be constructed by 3 and 3¯. 24

B. Weight

The relative magnitude of two weights is appointed as

0 0 (T3,Y ) > (T3,Y ) (263)

0 0 0 if T3 > T3 or T3 = T3 and Y > Y .

• T3 and Y are additive conserved quantities. Namely,

X i T3 = T3, (264) i X Y = Y i. (265) i

• The decomposition of direct product of irreducible representations has the same triality.

For example,

3 ⊗ 3 = 6 ⊕ 3¯, (266) D(1, 0) ⊗ D(1, 0) = D(2, 0) ⊕ D(0, 1). (267)

For example,

3 ⊗ 3¯ = 8 ⊕ 1, (268) D(1, 0) ⊗ D(0, 1) = D(1, 1) ⊕ D(0, 0). (269)

For example,

3 ⊗ 3 ⊗ 3 = (6 ⊕ 3)¯ ⊗ 3 (270) = (6 ⊗ 3) ⊕ (3¯ ⊗ 3) (271) = (10 ⊕ 8) ⊕ (8 ⊕ 1) (272) = 10 ⊕ 8 ⊕ 8 ⊕ 1. (273) D(1, 0) ⊗ D(1, 0) = D(3, 0) ⊕ D(1, 1) ⊕ D(1, 1) ⊕ D(0, 0). (274)

For example,

8 ⊗ 8 = 27 ⊕ 10 ⊕ 10¯ ⊕ 8 ⊕ 8 ⊕ 1, (275) D(1, 1) ⊗ D(1, 1) = D(2, 2) ⊕ D(3, 0) ⊕ D(0, 3) ⊕ D(1, 1) ⊕ D(1, 1) ⊕ D(0, 0). (276)

In summary, there is a formula to get the decomposition of the direct product

min(n,m0) min(n0,m) X X (n, m) ⊗ (n0, m0) = ⊕ (n − i, n0 − j; m − j, m0 − i) (277) i=0 j=0 min(n,n0) X (n, n0; m, m0) ≡ (n + n0, m + m0) ⊕ (n + n0 − 2i, m + m0 + i) (278) i=1 min(m,m0) X ⊕ (n + n0 + j, m + m0 − 2j) . (279) j=1 25

XIII. THE TENSOR REPRESENTATION OF SU(3)

A. Lower and upper indices

We can label the states of the 3 as 1 1 , = |1i , (280) 2 3 1 1 − , = |2i , (281) 2 3 2 0, − = |3i . (282) 3 In the weight diagram, this is to say 2 1

3 . We can deﬁne the generators as 1 [T ]i = [λ ] . (283) a j 2 a ij

Then, the triplet of states, |ii, transforms under the algebra as

j Ta |ii = |ji [Ta]i . (284) Similarly, we can label the states of 3¯ as 1 1 1 , − = , (285) 2 3 1 1 2 − , − = , (286) 2 3 2 3 0, = . (287) 3 In the weight diagram, this is to say 3

1 2 .

i Then, the triplet of states, , transforms under the algebra as

i j i Ta = − [Ta]j . (288) Now we can deﬁne a state in the tensor product of n 3s and m 3¯s.

i1···im = i1 ··· im | i | i . (289) j1···jn j1 jn It transforms as n E n E i1···im X i1···im k X i1···il−1kil+1···im il Ta = [Ta] − [Ta] . (290) j1···jn j1···jl−1kjl+1···jn jl j1···jn k l=1 l=1 26

B. Tensor components and wave function

Now consider an arbitrary state in this tensor product space |vi = i1···im vj1···jn . (291) j1···jn i1···im v is called a tensor. A tensor is just a wave function. The tensor v is characterized by its tensor components, vj1···jn . i1···im Now, we can think of the action of the genetors on |vi as an action on the tensor components, as follows:

Ta |vi = |Tavi (292) where n m X j X k T vj1···jn = [T ] l vj1···k···jn − [T ] vj1···jn . (293) a i1···im a k i1···im a il i1···k···im l=1 l=1

C. Irreducible representations and symmetry

We can now use the highest weight procedure to pick out the states in the tensor product corresponding to the 2 irreducible representation (n, m). Because |1i is the highest weight of the (1, 0) representation, and the state is the highest weight of the (0, 1) representation, the state with the highest weight in (n, m) is 222··· 111··· . (294)

It corresponds to the tensor vH with components v j1···jn = Nδ ··· δ δ ··· δ . (295) H i1···im j11 jn1 i12 im2

Now, we can construct all the states in (n, m) by acting on the tensor vH with lowering operators. The important point is that vH has two properties that are preseved by the transformation vH → TavH .

• vH is symmetric in the upper indices, and symmetric in the lower indices.

• vH satisﬁes δi1 v j1···jn = 0. (296) j1 H i1···im

D. Invariant tensors

The δi1 is called an invariant tensor. There are two other invariant tensors in SU(3) the completely antisymmetric j1 ijk tensors, and ijk.

E. Clebsch-Gordan decomposition

We can use tensors to decompose tensor products explicitly. i ···i i0 ···i0 i0 ···i0 1 n 1 p i1···in 1 p [u ⊗ v] 0 0 = u v 0 0 . (297) j1···jmj1···jq j1···jm j1···jq For example, 3 ⊗ 3. If we have two 3s, ui and vj, we can write the product as 1 1 uivj = uivj + ujvi + ijk ulvm. (298) 2 2 klm The ﬁrst term, 1 uivj + ujvi (299) 2 transforms like a 6. This contains the highest weight state u1v1. We have added the ujvi term to make it completely symmetric in the two upper indices, and thus irreducible. Next 3 ⊗ 3¯ 1 1 uiv = uiv − δi ukv + δi ukv . (300) j j 3 j k 3 j k 27

F. Matrix elements and operators

The bra state hv| is

hv| = vj1···jn∗ i1···im . (301) i1···im j1···jn

The bra transforms under the algebra an extra minus sign. Namely,

j Ta |ii = |ji [Ta]i . (302)

We can deﬁne

v¯i1···im ≡ vj1···jn∗. (303) j1···jn i1···im

So that

hv| =v ¯i1···im i1···im . (304) j1···jn j1···jn

Therefore, the inner product becomes

hu|vi =u ¯i1···im vj1···jn . (305) j1···jn i1···im

XIV. SU(3) → SU(2) × U(1)

Young tableaux can do many other useful things for us. One that we can already explore in SU(3) is to help us understand how a representation of SU(3) decomposes into representations of its subgroups. An example of this which has some phenomenological importance is the decomposition of SU(3) representations√ into their representations of SU(2) × U(1), and particularly, of isospin and hypercharge (T8 , remember, is just 3Y/2). The three components of the deﬁning representation, the 3, decompose into a doublet with hypercharge 1/3 and a singlet with hypercharge −2/3.

A. 6

→ (306) ( ·)32/3 (307) ( )2−1/3 (308) (· )1−4/3 (309)

B. 3¯

→ (310) · 12/3 (311) ( )2−1/3 (312) 28

XV. SU(N)

A. Generalized Gell-Mann matrices

There are a couple of diﬀerent useful bases for the SU(N) generators. We will start with a generalization of the Gell-Mann matrices, in which we build up from the SU(N) to SU(N + 1) generators one step at a time. We will normalize 1 Tr (T T ) = δ . (313) a b 2 ab

The√ generators of the raising and lowering operators, we can take to have a single non-zero oﬀ-diagonal element, 1/ 2. The group is rank N − 1, because there are N − 1 independent traceless diagonal real matrices. We can choose the N − 1 Cartan generators as follows

m ! 1 X [Hm]ij = p δikδjk − mδi,m+1δj,m+1 . (314) 2m(m + 1) k=1

The weights are N − 1 dimensional vectors,

m ! 1 X νj = [H ] = δ − mδ . (315) m m jj p jk j,m+1 2m(m + 1) k=1 They satisfy,

N − 1 ν2 = . (316) j 2N For i < j 1 ν · ν = − . (317) i j 2N The weights all have the same length, and the angles between any two distinct weights are equal.

B. The branding rule

The canonical group chain:

SU(N) ⊃ SU(N − 1) ⊗ U(1) ⊃ ... ⊃ SU(2) ⊗ [U(1)]N−2 . (318)

For the irreducible representation of SU(N), there are N − 1 Cassimir operators:

{C2,C3,...,CN } . (319)

They can determine the unique irreducible representation. Futhermore, we can use

n (1) (1) (1) (1) (2) (2) (2) (2) (N−2) o C2 ,C3 ,...,CN−1,Y ; C2 ,C3 ,...,CN−2,Y ; ... ; C2 ,I3 (320) to determine the unique state.

XVI. THE CLASSIFICATION OF LIE ALGEBRA

• Positive root: The ﬁrst non-zero component is greater than 0. Before, we need to set the order of Cartan’s generators. 29

• Simple root: The positive root can not be written as the summation of other positive roots. We can prove that all the simple roots are linearly independent.

• The hierarchy of positive roots: X β = kiαi, (321) i X Hierarchy = ki, (322) i

where αi are simple roots.

π • For two simple roots α, β, the angle between them is equal to or bigger than 2 .

A. SO(N)

1. Case1: N = 2l.

The rank of SO(2l) is l. We choose

H1 = (1, −1, 0, 0, ··· , 0, 0) , (323) H2 = (0, 0, 1, −1, ··· , 0, 0) , (324) .. .. (325) Hl = (0, 0, 0, 0, ··· , 1, −1) , (326) as the Cartan’s subalgebra. The root vectors are

±ei ± ej (i < j). (327)

The positive roots are

ei ± ej (i < j). (328)

The simple roots are

ei−1 − ei (For i = 2, ··· , l) (329) el−1 + el. (330)

2. Case2: N = 2l + 1.

The rank of SO(2l) is l. We choose

H1 = (1, −1, 0, 0, ··· , 0, 0, 0) , (331) H2 = (0, 0, 1, −1, ··· , 0, 0, 0) , (332) .. .. (333) Hl = (0, 0, 0, 0, ··· , 1, −1, 0) , (334) as the Cartan’s subalgebra. The root vectors are

±ei ± ej(i < j) (335) ±ei. (336) 30

The positive roots are

ei ± ej(i < j) (337) ±ei (338)

The simple roots are

ei−1 − ei (For i = 2, ··· , l) (339) el. (340)

B. SU(N)

The rank of SU(N) is l = N − 1. We choose

1 H1 = √ (1, −1, 0, 0, ··· , 0, 0, 0) , (341) 2 1 H2 = √ (1, 1, −2, 0, ··· , 0, 0, 0) , (342) 6 .. .. (343) 1 Hl = (1, 1, 1, 1, ··· , 1, 1, −l) , (344) pl(l + 1) as the Cartan’s subalgebra. The root vectors are

ωm − ωn(m 6= n) (345)

The positive roots are

ωm − ωn(m < n) (346)

The simple roots are

ωi−1 − ωi (For i = 2, ··· , n) . (347)

C. Sp(2n, R)

The sympletic group satisﬁes

RT JR = J (348) with J 0 I J = n×n . (349) −In×n 0

There are n(2n + 1) group element parameters. The generators are

iA ⊗ I2×2 (350)

S1 ⊗ σ1 (351)

S2 ⊗ σ2 (352)

S3 ⊗ σ3 (353) where A is any real n × n antisymmetrization matrix and Si any real n × n symmetrization matrix. 31

The rank of Sp(2n, R) is n. We choose

ui 0 Hi = ui ⊗ σ = (354) 3 0 −ui with ui = diag (0, 0, . . . , i, . . . , 0)as the Cartan’s subalgebra. The root vectors are

±ei ± ej (355) ±2ei (356)

The positive roots are

ei ± ej (357) 2ei (358)

The simple roots are

ei−1 − ei (For i = 2, ··· , l) (359) 2el. (360)

D. Exceptional group

E6,E7,E8,F4,G2.

E. Roots vector

The diﬀerent roots satisfy

(α, β)(α, β) mn = . (361) (α, α)(β, β) 4