Group Theory

Group theory for physicists

Literature:

• I.V. Schensted, A course on the applications of group theory to quantum mechanics

• W.-K. Tung, Group theory in physics

• H. Boerner, Darstellungen von Gruppen

• R. Gilmore, Lie groups, Physics, and Geometry

• H.F. Jones, Groups, representations and physics

• J.F. Cornwell, Group theory in physics

• S. Sternberg, Group theory and physics

• R. Gilmore, Lie groups, Lie algebras, and some of their applications

• D.B. Želobenko, Compact Lie groups and their representations

Contents

1. Introduction 51.1. Why group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2. Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3. Examples and general properties . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4. Invariant subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5. The symmetric (or permutation) group . . . . . . . . . . . . . . . . . . . . . . . 111.6. The action of a group on a set . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.7. Equivalence classes and invariant subgroups . . . . . . . . . . . . . . . . . . . . 131.8. Cosets and factor groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.9. Direct product of two groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.10. Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.11. Example: Homomorphism between S`(2,C) and the Lorentz group . . . . . . . 20

2. Matrix representations 232.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2. Equivalent representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4. Reduction of representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4.1. Reducibility of invariant spaces and representations . . . . . . . . . . . 262.4.2. OA operators using the example of D3 . . . . . . . . . . . . . . . . . . . 282.4.3. Four theorems and their consequences . . . . . . . . . . . . . . . . . . . 312.4.4. Characters of representations . . . . . . . . . . . . . . . . . . . . . . . . 32

2.5. The regular representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.6. Product representations and Clebsch-Gordan coefficients . . . . . . . . . . . . . 362.7. Subduced and induced representations . . . . . . . . . . . . . . . . . . . . . . . 40

3. Applications in quantum mechanics 413.1. Selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2. The symmetry group of the Hamiltonian and degeneracies . . . . . . . . . . . . 423.3. Perturbation theory and lifting of degeneracies . . . . . . . . . . . . . . . . . . 44

4. Expansion in irreducible basis vectors 484.1. Irreducible basis vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2. Projection operators on irreducible bases . . . . . . . . . . . . . . . . . . . . . . 484.3. Irreducible operators and the Wigner-Eckart theorem . . . . . . . . . . . . . . 514.4. Left ideals and idempotents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5. Representation of the symmetric group and Young diagrams 575.1. Why Sn is important . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.2. 1-dimensional and associated representations of Sn . . . . . . . . . . . . . . . . 575.3. Young diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.4. Symmetrizers and antisymmetrizers of Young tableaux . . . . . . . . . . . . . . 605.5. Irreducible representations of Sn . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2

Contents 3

5.6. More applications of Young tableaux . . . . . . . . . . . . . . . . . . . . . . . . 62

6. Lie groups 656.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.2. Examples of Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.3. Invariant integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.4. Properties of compact Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . 696.5. Generators of Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706.6. SO(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.7. SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.7.1. Angle-and-axis parametrization . . . . . . . . . . . . . . . . . . . . . . . 756.7.2. Euler angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.7.3. Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.7.4. Irreps of SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.8. SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.8.1. Parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.8.2. Relationship between SO(3) and SU(2) . . . . . . . . . . . . . . . . . . 806.8.3. Inavriant integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.8.4. Orthogonality and completeness relations . . . . . . . . . . . . . . . . . 82

7. Lorentz and Poincaré group 837.1. Relativistic kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837.2. Generators and Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837.3. Finite-dimensional representations of the Lorentz group . . . . . . . . . . . . . 847.4. Unitary irreps of the Poincaré group . . . . . . . . . . . . . . . . . . . . . . . . 85

7.4.1. One-particle states and Casimir operators . . . . . . . . . . . . . . . . . 857.4.2. Little group and induced representations . . . . . . . . . . . . . . . . . . 877.4.3. Massive particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887.4.4. Massless particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887.4.5. Tachyons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897.4.6. Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.5. Parity and time reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

8. The tensor method to construct irreps of Gl(m) and its subgroups 908.1. Tensors and tensor spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908.2. Action of the symmetric group on the tensor space . . . . . . . . . . . . . . . . 918.3. Decomposition of the tensor space into irreducible subspaces under Sn and Gl(m) 92

8.3.1. Symmetry classes in tensor space . . . . . . . . . . . . . . . . . . . . . . 928.3.2. Totally symmetric and totally antisymmetric tensors . . . . . . . . . . . 938.3.3. Tensors with mixed symmetry . . . . . . . . . . . . . . . . . . . . . . . 948.3.4. Complete reduction of the tensor space . . . . . . . . . . . . . . . . . . 95

8.4. Irreps of U(m) and SU(m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 968.5. Complex conjugate representation . . . . . . . . . . . . . . . . . . . . . . . . . 978.6. Reduction of the product of two irreps . . . . . . . . . . . . . . . . . . . . . . . 988.7. Applications in (particle) physics . . . . . . . . . . . . . . . . . . . . . . . . . . 100

8.7.1. The Goldstone theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008.7.2. SU(2) isospin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008.7.3. SU(2) flavor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1018.7.4. SU(3) flavor and the quark model . . . . . . . . . . . . . . . . . . . . . 104

4 Contents

A. Appendix 108A.1. Proof of Cayley’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108A.2. Proofs of the four theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108A.3. The Kronecker product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

1. Introduction

1.1. Why group theory

• group theory = description of symmetries and their consequences(symmetry = invariance under a particular transformation)

• important in many fields of physics

• one can learn a lot about a system by analyzing its symmetries (without knowing thedynamics)

• examples of symmetries in physics:1) continuous space-time symmetries:

a) homogeneity of space = invariance under spacial translations

~x→ ~x+ ~a

→ conservation of momentumb) homogeneity of time = invariance under temporal translations

t→ t+ t0

→ conservation of energyc) isotropy of space = invariance under rotations

~x→ A~x

→ convervation of angular momentumd) space-time symmetry (special relativity) = invariance under Poincaré transfor-

mations

(~x, t)→ Λ(~x, t) + ~a

→ see Sec. 72) discrete space-time symmetries:

a) parity transformation: ~x→ −~xb) time reversal: t→ −tc) discrete translations on a latticed) discrete rotational symmetries on a lattice

3) permutation symmetries (systems of several identical particles)4) gauge invariance in electrodynamics (classical and QED)→ conservation of charge

5) internal symmetries in nuclear and particle physics (spin, isospin, color, etc.)→ particle spectrum (e.g., degeneracies in the spectrum)

5

6 1. Introduction

1.2. Basic definitions

• group: a group G is a set of elements for which we can define a "multiplication" andwhich staisfies the following four properties:1) if A and B are group elements, then AB is also a group element (closure of the

group)2) the multiplication is associative: A(BC) = (AB)C3) the group contains one and only one identity element I:

for all A ∈ G we have AI = IA = A

4) for each A ∈ G there is one and only one inverse element A−1 ∈ G such thatAA−1 = A−1A = I

remarks:– the group is defined by its elements and the multiplication rule, i.e., there can

be two different groups with the same elements but with a different multiplica-tion rule

– in general the multiplication is not commutative, i.e., AB 6= BA; if the multi-plication is commutative, the group is called Abelian

– if the number of group elements is finite: finite group; thenorder of the group = number of group elementsotherwise: infinite group which can be discrete (e.g., translations on an infi-nite lattice) or continuous (e.g., continuous rotations)

• subgroup: if a subset H ⊆ G satisfies the four group properties (with the multiplicationrule for G) it is called a subgroup of G– each group has two trivial subgroups: I and G– all other subgroups are called non-trivial

the order of G is divisible by the order of H (proof later)

• for a finte group (with n elements) the structure of the group is completely specified bythe multiplication table (with n2 elements):

I A B C . . .

I I A B C . . .A A A2 AB AC . . .B B BA B2 BC . . .C C CA CB C2 . . ....

......

...... . . .

• theorem: All elements in a row (or column) of the multiclication table must be distinct(proof in exercices)this leads to the rearrangement lemma: if we multiply all elements of the groupI, A,B,C, . . . by one of the elements we again obtain all the elements of the group,but in a different order

• isomorphism: two groups G and G′ are isomorphic, if there exists a map f from G toG′ which is one-to-one and onto and which preserves the group multiplication, i.e., forA,B ∈ G and A′, B′ ∈ G′ we have:if A′ = f(A) and B′ = f(B), then A′B′ = f(AB)

1.3. Examples and general properties 7

– if two groups are isomorphic, they have the same multiplication table (and thus areidentical up to the names of the group elements)

– if the map f between the elements of the two groups is onto but not one-to-one,the two groups are homomorphic→ see Sec. 1.10

1.3. Examples and general properties

• a group structure I, a, a2, a3, . . . , an−1, an = I is called cyclic group Cn. The smallestnon-cyclic group is of order 4. The smallest non-Abelian group is of order 6.

• group with two elements: Z2 = I, Aconstruction of the multiplication table (22 = 4 elements):

I2 = I, IA = A, AI = A, A2 =? (A or I?)

use the theorem from Sec. 1.2→ A2 = I, and thus the multiplication table for Z2 is:

I A

I I AA A I

this is the only possibility for the multiplication table→ all groups of order 2 are isomorphic to Z2

• examples for groups isomorphic to Z2:1) consider the following two transformations from R3 to R3

I : (x, y, z)→ (x, y, z)P : (x, y, z)→ (−x,−y,−z) (parity)

multiplication is defined as consecutive transformations

→ I2 = I, IP = P, PI = P, P 2 = I

→ isomorphic to Z2

2) instead of the two spacial transformation we now consider operators acting upon(real or complex) functions of ~x = (x, y, z):

OIf(~x) = f(~x)OP f(~x) = f(−~x)

→ O2I = OI , OIOP = OP , OPOI = OP , O2

P = OI

→ isomorphic to Z2the operators OI and OP are linear, i.e., O(αf + βg) = αOf + βOg

3) permutation of two objects A and B:

P1(AB) = AB, P2(AB) = BA

→ isomorphic to Z2

8 1. Introduction

4) consider operators acting on the complex wave function of two particles (whichdepends on the spatial and spin coordinates):

OEψ(~x1, σ1; ~x2, σ2) = ψ(~x1, σ1; ~x2, σ2)OSψ(~x1, σ1; ~x2, σ2) = ψ(~x2, σ2; ~x1, σ1)

→ OE and OS form a group isomorphic to Z2

• at first sight Z2 looks trivial, but we can learn many of the concepts of group theoryfrom Z2

• now consider example 2 and two functions fe and fo that satisfy

OP fe(~x) ≡ fe(−~x) = fe(~x) (fe has even parity)OP fo(~x) ≡ fo(−~x) = −fo(~x) (fo has odd parity)

(e.g., fe = x2yz, fo = xyz)fe and fo have special properties under the group OI , OP :

– fe is invariant under OP– fo only changes sign under OP

• one of the aims of group theory is the classification of objects that have special symmetryproperties under a certain group (in general this is much more difficult than in this simpleexample)

• assume that we know the structure of a certain group G

– in general there are many groups isomorphic to G (e.g., G′)

– G′ contains operators that act on certain objects

– using the structure of G we can immediately classify these objects with respect totheir symmetry properties under G′

• the relation∞∫−∞

∞∫−∞

∞∫−∞

dx dy dzf∗e fo = 0

is an example of an "orthogonality relation" between objects with special symmetryproperties (a.k.a. "selection rule" in QM)

• an arbitrary function can be written as a sum of an even and an odd function:

f = fe + fo with fe = 12[f(~x) + f(−~x)]

fo = 12[f(~x)− f(−~x)]

this is an example of an "expansion theorem": objects without special symmetry proper-ties can be expressed as linear combinations of (i.e., "expanded in") objects with specialsymmetry properties

1.4. Invariant subspaces 9

1.4. Invariant subspaces

• first two definition we need later on:– let V be a vector space of dimension m with basis v1, . . . , vm and W be a vector

space of dimension n with basis w1, . . . , wn;furthermore assume V ∩W = 0

– the direct sum V ⊕W is the space with basis v1, . . . , vm, w1, . . . , wn∗ we have dim(V ⊕W ) = m+ n

∗ example: R2 = R1 ⊕ R1

– the tensor product V ⊗W is the space with basis vectors vi⊗wj (i = 1, . . . ,mand j = 1, . . . , n)∗ the notation vi ⊗ wj is purely abstract (the basis vectors "consist" of one basisvector each from V and W )∗ we have dim(V ⊗W ) = mn

• consider a function f1 without special symmetry properties under OI , OP and define

f2(~x) = OP f1(~x) = f1(−~x)→ OP f2(~x) = f1(~x)

– f2 is called the image of f1 under the operation OP (and vice versa)– the pair f1, f2 is called an image pair under OP– examples: (x2 + y + z), (x2 − y − z) or (xy2 + z2), (−xy2 + z2)

• let S be the set of all functions

α1f1 + α2f2

with arbitrary coefficients α1, α2

– this set is a two-dim. linear function space, which is spanned by f1 and f2 (proofof "S = space" in exerices)

– f1 and f2 form a basis of S

• the space S is invariant under OI , OP , i.e., if one of the operators acts on an f ∈ Sthe result is still in S

proof : OIf = f ∈ SOP f = OP (α1f1 + α2f2) = α1OP f1 + α2OP f2 = α1f2 + α2f1 ∈ S

• however, the space S has smaller invariant subspaces– to see this define new basis functions

f1 = 12(f1 + f2), f2 = 1

2(f1 − f2)

– these also span S since

f1 = f1 + f2, f2 = f1 − f2

f = α1f1 + α2f2 = α1(f1 + f2) + α2(f1 − f2) = (α1 + α2)f1 + (α1 − α2)f2

10 1. Introduction

– f1 and f2 have well-defined parity:

OP f1 = 12OP f1 + 1

2OP f2 = 12f2 + 1

2f1 = f1

OP f2 = 12OP f1 −

12OP f2 = 1

2f2 −12f1 = −f2

– the one-dim. spaces S1 and S2 that are spanned by f1 and f2 are separately invariantunder OI , OP :

OP (αf1) = αf1 ∈ S1

OP (αf2) = −αf2 ∈ S2

• we say that the space S is reducible w.r.t. the action of OI , OP , i.e., it can be splitinto invariant subspaces of smaller dim.symbolically: S = S1 ⊕ S2S1 and S2 are irreducible, i.e., they cannot be split into smaller invariant subspaces

• example: consider the two image pairs

h1 = x2 + y + z h2 = x2 − y − zg1 = e−xyz g2 = exyz

– the four products h1g1, h1g2, h2g1, h2g2 span a 4-dim. space Sh ⊗ Sg, i.e.,

f = ah1g1 + bh1g2 + ch2g1 + dh2g2

– this space is invariant under OI , OP :

OP (h1g1) = (OPh1)(OP g1) = h2g2

OP (h1g2) = (OPh1)(OP g2) = h2g1

OP (h2g1) = (OPh2)(OP g1) = h1g2

OP (h2g2) = (OPh2)(OP g2) = h1g1

→ OP f = ah2g2 + bh2g1 + ch1g2 + dh1g1 ∈ Sh ⊗ Sg

– h1g1 and h2g2 are an image pair→ they span a 2-dim. invariant subspace Sω

– h1g2 and h2g1 are an image pair→ they also span a 2-dim. invariant subspace Sδ

– every function in Sh⊗Sg can be written in a unique way as a sum of two functionsin Sω and Sδ:

f = fω + f δ with fω = ah1g1 + dh2g2 ∈ Sω

f δ = bh1g2 + ch2g1 ∈ Sδ

symbolically: Sh ⊗ Sg = Sω ⊕ Sδ

1.5. The symmetric (or permutation) group 11

– the spaces Sω and Sδ can be further reduced if we introduce basis functions withwell-defined parity:

f1 = 12(h1g1 + h2g2) f2 = 1

2(h1g1 − h2g2)

f3 = 12(h1g2 + h2g1) f4 = 1

2(h1g2 − h2g1)

→ f = (a+ d)f1 + (a− d)f2 + (c+ b)f3 + (c− b)f4

the subspaces Si spanned by the fi are invariant under OI , OP (proof as above)

→ Sh ⊗ Sg = S1 ⊕ S2 ⊕ S3 ⊕ S4

this notation means the following:∗ the space on the LHS and the subspaces on the RHS are separately invariantunder the group∗ every function in the space on the LHS can be expressed uniquely as a sum offunctions, each of which lies in one of the subspaces on the RHS

– in this example all irreducible invariant subspaces are 1-dim., but in general theyare of higher dimension (for order(G) ≥ 6)

1.5. The symmetric (or permutation) group

• symmetric group Sn = group of permutation of n objects

• there are n! permutations → order(Sn) = n!

• a typical group element is

p =(

1 2 . . . np1 p2 . . . pn

)

this notation means: "move the first object to position p1, the second object to positionp2, etc."e.g., for n = 6: (

1 2 3 4 5 66 4 1 2 5 3

)applied to [a, b, c, d, e, f ]

gives [c, d, f, b, e, a]

• a permutation can be split into disjoint cycles (which have no elements in common), e.g.,

(1 2 3 4 5 66 4 1 2 5 3

)= (163)(24)(5)

3-cycle

↑2-cycle

1-cycle

– the 3-cycle means "1 goes to 6, 6 goes to 3, 3 goes to 1"– within a cycle the numbers can be shifted cyclically:

(163) = (613) = (316) but 6= (136)

12 1. Introduction

– the order of the disjoint cycles does not matter– the 1-cycles are usually not written explicitly– every `-cycle (` > 2) can be written as a product of 2-cycles (also called transposi-

tions), e.g.,

(163) = (13)(16)

convention: the right-most cycle is applied firstnote: (163) 6= (16)(13), but this is not in contradiction to the above statementabout the order, which referred to disjoint cycles

• Cayley’s theorem: Every group of order n is isomorphic to a subgroup of Sn.

• example: S3

– group elements:element I (12) (13) (23) (123) (321)

applied to [a,b,c] [a,b,c] [b,a,c] [c,b,a] [a,c,b] [c,a,b] [b,c,a]– multiplication table (exercises):

I (12) (13) (23) (123) (321)I I (12) (13) (23) (123) (321)

(12) (12) I (321) (123) (23) (13)(13) (13) (123) I (321) (12) (23)(23) (23) (321) (123) I (13) (12)(123) (123) (13) (23) (12) (321) I(321) (321) (23) (12) (13) I (123)

→ S3 is a non-Abelian group (true for all Sn with n > 2)– subgroups:∗ I and S3 (trivial)∗ I, (12), I, (13), I, (23) (isomorphic to Z2)∗ I, (123), (321) (isomorphic to C3)

1.6. The action of a group on a set

• let G be a group and M be a setthe action of G on M is given by a map

G×M →M : (g,m) 7→ gm for g ∈ G and m ∈M

that satisfies:

g1(g2m) = (g1g2)m (associativity)Im = m ∀m ∈M

this implies that the action is one-to-one and onto, i.e., M is mapped in a one-to-onefashion onto itself (m′ = gm → m = g−1m′)

• example: M = R3, G = rotation group, action = rotation of vectors

1.7. Equivalence classes and invariant subgroups 13

• the orbit of a point m ∈M under the action of G on M is

G ·m ≡ gm | g ∈ G

The orbit of "typical" points has n = order(G) elements.The orbit of "special" points has less than n elements.Example: G = D3 = symmetry group of an equilateral triangle (isomorphic to S3 =permutation of the corners of the triangle)– identity– 2 rotations (of 120 and 240)– 3 reflections (axes through the center and one of the corners)

M = all points in R2 with origin = center of the triangle

× ×

×

××

×

special point (orbithas 1 element)

typical points (orbithas 6 elements)

special points (orbithas 3 elements)

• the set of group elements that map an m ∈M to itself, i.e.,

Gm = g ∈ G | gm = m

is called isotropy group (or stabilizer or stationary subgroup or little group) ofm.– this set is a group (proof in exercises)– in our example:∗ the isotropy group of × is I∗ the isotropy group of is D3

∗ the isotropy group of • is I,R ' Z2 where R is the reflection about the axisthrough •.

– in all these cases we have(number of elements of orbit) · order(isotropy group) = order(group)this is true in general for all finite groups (proof in exercises)

1.7. Equivalence classes and invariant subgroups

• a group element y is called conjugate (or equivalent) to a group element x if thereexists a group element g such that

y = gxg−1

notation: y ∼ x

• examples:

14 1. Introduction

1) for S3: (13) ∼ (12) since (23)(12)(23)−1 = (13)2) rotation group in 3 dimensions:

– Rn(ϕ) = rotation about an axis n and an angle ϕ– for arbitrary R we have RRn(ϕ)R−1 = Rn′(ϕ) with n′ = Rn→ rotations about the same angle, but different axes, are equivalent.

• physical interpretation: equivalent group elements correspond to the same operation,but in a different basis

• an equivalence class (or simply class) is a set of group elements that are equivalentto each other.Properties (proofs in exercises):– the identity forms a class by itself– in general the elements of a class don’t form a group– if the group is Abelian, each elements forms a class by itself– every element of G belongs to one and only one class– the order of the group is divisible by the number of elements in a class– the number of classes is equal to the number of non-equivalent irreducible represen-

tations of the group (later)

• for S3: the first class is Inow conjugate (12) with all elements of S3

I(12)I = (12)(12)(12)(12) = (12)(13)(12)(13) = (13)(321) = (23)(23)(12)(23) = (23)(123) = (13)

(123)(12)(321) = (123)(13) = (23)(321)(12)(123) = (321)(23) = (13)

→ (12), (13) and (23) form a classfor the remaining two elements we have

I(123)I = (123)(12)(123)(12) = (12)(13) = (321)

→ (123) and (321) are equivalent and thus in the same class, i.e., there are three classes:

c1 = I , c2 = (12), (13), (23) , c3 = (123), (321)

→ two elements of S3 are equivalent if they have the same cycle structure (this is truein general for Sn)

• a subgroup K ⊆ G is called conjugate to a subgroup H ⊆ G if there exists a g ∈ Gsuch that

K = gHg−1 ≡ ghg−1 | h ∈ H

e.g., the subgroup K = I, (13) is conjugate to the subgroup H = I, (12) since(23)I(23) = I and (23)(12)(23) = (13)

1.8. Cosets and factor groups 15

• if ghg−1 ∈ H for all h ∈ H and all g ∈ G, then it is an invariant (or normal orselfconjugate) subgroup of G

• every group has two trivial invariant subgroups: I and G– a group is called simple if it does not have any non-trivial invariant subgroups– a group is called semi-simple if it does not have any non-trivial Abelian invariant

subgroups– for S3 the only nontrivial invariant subgroup is H2 = I, (123), (321)→ S3 is neither simple nor semi-simple since H2 is Abelian

1.8. Cosets and factor groups

• consider a subgroup H ⊆ G with order(H) = nH and elements hi and a group elementg ∈ G– the set

gH ≡ ghi | i = 1, . . . , nH

is called left coset of H w.r.t. the element g– similarly the set

Hg ≡ hig | i = 1, . . . , nH

is called right coset of H w.r.t. g– hence a coset is a subset of G– if g ∈ H, then gH = Hg = H (rearrangement lemma); therefore some authors

require g /∈ H when defining cosets, but we allow for g ∈ H (and thus H is also acoset)

– in the following we mainly consider left cosets (right cosets analogous)

• properties:– two cosets g1H and g2H are either identical (if g−1

1 g2 ∈ H) or disjoint, i.e., theyhave no elements in common (if g−1

1 g2 /∈ H)proof:∗ suppose they have an element in common: g1h1 = g2h2 = g

∗ then g2 = g1h1h−12 → g2H = g1h1h

−12 H = g1H, since h1h

−12 ∈ H and

hH = H for h ∈ H (rearrangement lemma)∗ in this case we have g−1

1 g2 = h1h−12 ∈ H

∗ converse analogous– number of coset elements = order(H)→ the group G is split into disjoint cosets of H, with order(H) elements each:

G = H ∪ g1H ∪ g2H ∪ · · ·

→ order(G) is divisible by order(H) (see Sec. 1.2)– every group element g belongs to one and only one of the different cosets of H

16 1. Introduction

– for S3: let H1 = I, (12) (not invariant), and H2 = I, (123), (321) (invariant).The left and right cosets of H1 are:

IH1 = I, (12)(12)H1 = (12), I(13)H1 = (13), (123)(23)H1 = (23), (321)

(123)H1 = (123), (13)(321)H1 = (321), (23)

H1I = I, (12)H1(12) = (12), IH1(13) = (13), (321)H1(23) = (23), (123)H1(123) = (123), (23)H1(321) = (321), (13)

→ for H1 the left and right cosets are different, and (e.g.)

S3 = H1 ∪ (13)H1 ∪ (23)H1

The cosets of H2 are

IH2 = I, (123), (321)(12)H2 = (12), (23), (13)(13)H2 = (13), (12), (23)(23)H2 = (23), (13), (12)

(123)H2 = (123), (321), I(321)H2 = (321), I, (123)

H2I = I, (123), (321)H2(12) = (12), (23), (13)H2(13) = (13), (12), (23)H2(23) = (23), (13), (12)H2(123) = (123), (321), IH2(321) = (321), I, (123)

→ for H2 the left and right cosets are identical, and (e.g.)

S3 = H2 ∪ (12)H2

• in general: if H is an invariant subgroup, then left and right cosets are identical, since

gHg−1 = H

ghig−1 = hi with i = 1, . . . , order(H)

(in general the elements on the LHS and RHS are in a different order)

→ ghi = hig→ gH = Hg

in this case the partitioning of G into cosets is unique→ there is a natural "factorization" of G based on this partitioning.

• if H is invariant, we can view its cosets as elements of a new group:– multiplication is defined by

(g1H) · (g2H) ≡ g1hig2hj | hi, hj ∈ H= g1g2hk with hk = (g−1

2 hig2)︸︷︷︸∈H since H is invariant

hj ∈ H

= (g1g2)H

→ closure property is satisfied

1.9. Direct product of two groups 17

– other group properties:∗ associativity:

g1H · (g2H · g3H) = g1H · (g2g3H) = (g1g2g3)H == (g1g2)H · g3H = (g1H · g2H) · g3H

∗ identity element = H, since gH ·H = gH = H · gH∗ inverse element of gH = g−1H, since gH · g−1H = H = g−1H · gH

• hence we have a theorem: if H is invariant, then the set of all cosets, i.e., gH | g ∈ G,is a new group with multiplication law

(g1H) · (g2H) = (g1g2)H

this group is called G/H = factor group, with order(G/H) = order(G)order(H)

• if H is not invariant the construction above doesn’t work anymore, and the set of all(left or right) cosets doesn’t form a group (convince yourself using the example of H1)

• example: for S3 the subgroup H2 = I, (123), (321) is invariant, and the factor groupS3/H2 has two elements:

I, (123), (321) and (12), (13), (23)

this is isomorphic to Z2 (exercises)

1.9. Direct product of two groups

• consider two groups A and B with elements a ∈ A and b ∈ B.The direct product A⊗B consists of all pairs (a, b) with the multiplication law

(a1, b1) · (a2, b2) = (a1a2, b1b2)

for finite groups we have order(A⊗B) = order(A)·order(B)

• typical examples:– suppose A acts on a vector space U and B acts on a vector space V– then A⊗B acts on the product space W = U ⊗ V , and it does so as follows (withu ∈ U and v ∈ V ):

(a, b)(u⊗ v) = (au)⊗ (bv)

• another example:– A and B act on the same vector space (or on the same objects)– suppose∗ ab = ba for all a ∈ A and b ∈ B (but A and B could be non-Abelian)∗ A and B have only the identity in common

– then G = A ⊗ B is the set of products g = ab = ba, and for every g ∈ G thisrepresentation is unique

18 1. Introduction

• let G = A⊗B– then A and B are invariant subgroups of G

proof:

gakg−1= g(ak, I)g−1 = (ai, bj)(ak, I) (ai, bj)−1︸︷︷︸

(a−1,b−1)

= (aiaka−1i , bjIb

−1j ) = (a`, I) = a` ∈ A

and analogous for B– A is isomorphic to G/B (and B to G/A):

G/B = (ai, bj)B = (ai, B) rearrangement lemma

and this group has the same multiplication law as A (see Sec. 1.8)

• however, the converse is not true:if H is an invariant subgroup of G, then in general G 6= H⊗(G/H) since in general G/His not an invariant subgroup.example: S3 has the subgroupsH1 = I, (12) andH2 = I, (123), (321). H2 is invariantS3/H2 ' Z2 ' H1, but S3 6= H1 ⊗H2 since H1 is not an invariant subgroup (and sincethe elements of H1 and H2 don’t commute)

1.10. Homomorphisms

• a homomorphism from a group G to another group G′ is a map f from G to G′ thatpreserves the group multiplication, i.e.,

f(g1g2) = f(g1) · f(g2) ∀g1, g2 ∈ G

• if this map is one-to-one and onto it is called isomorphism

• example: the following map from S3 to Z2 is a homomorphism

f

G = S3 G′ = Z2

I(123) (321)

(12) (13)(23)

I′

A

subset H

subset M

(check yourself usingthe multiplication ta-ble)

• the image of the homomorphism f : G→ G′ is the image set of G under f :

image(f) = f(G) = f(g) | g ∈ G

in our example: image(f) = I ′, A

• the kernel (or center) of the homomorphism is the preimage of the identity of G′, i.e.,the set of all elements of G that are mapped to the identity of G′:

ker(f) = f−1(I ′) = g ∈ G | f(g) = I ′

in our example: ker(f) = I, (123), (321)

1.10. Homomorphisms 19

• from the definition of the homomorphism it follows that– I is mapped to I ′: f(I) = I ′

– inverses are mapped to each other: f(g−1) = f(g)−1 ∀g ∈ G

• if G has an invariant subgroup H, there is a natural homomorphism from G to G/H:

G 3 g 7→ gH ∈ G/H

turning this around yields a theorem:Let f be a homomorphism from G to G′ and onto. Let K be the kernel of f . Then Kis an invariant subgroup of G. Furthermore, the factor group G/K is isomorphic to G′.Schematically:

I

ab

I ′K f

G G′homomorphism G→ G′

pK

qK q′

I ′

p′K

G/K G′isomorphism G/K ' G′

proof:– group properties:∗ if a, b ∈ K, then f(ab) = f(a)f(b) = I ′ · I ′ = I ′ and thus ab ∈ K∗ associativity is clear∗ f(I) = I ′ (see above) implies I ∈ K∗ for a ∈ K and arbitrary g ∈ G we have

f(gag−1) = g′I ′(g−1)′ = g′I ′(g′)−1 = I ′

→ gag−1 ∈ K, i.e., K is invariant.– the elements of the factor group G/K are the cosets gK. Define the map

ρ : G/K → G′ , gK 7→ g′ = f(g)

this map is unique, since g1K = g2K implies f(g1) = f(g2):

g1K = g2K → g−11 g2 = k ∈ K → g2 = g1k

f(g2) = f(g1k) = f(g1) f(k)︸︷︷︸I′

= f(g1)

we still have to show that ρ is an isomorphism∗ the group multiplication is preserved:

ρ(pK)ρ(qK) = p′q′ = f(p)f(q) = f(pq) = (pq)′ = ρ(pqK)

20 1. Introduction

∗ the map ρ is one-to-one: if ρ(pK) = ρ(qK), then

ρ(q−1pK

)= ρ

(q−1K · pK

)= ρ

(q−1K

)ρ (pK) = [ρ(qK)]−1ρ(pK) = I ′

→ q−1pK = K → pK = qK

∗ since f is onto, ρ is also onto

1.11. Example: Homomorphism between S`(2,C) and theLorentz group

• let M be Minkowski space, i.e., M = R4 with Lorentz metric

‖x‖2 = x20 − x2

1 − x22 − x2

3 , where x = (x0, x1, x2, x3)

is called a four-vector.

• a homogeneous Lorentz transformation (LT) Λ is a linear map from M to M that pre-serves the Lorentz metric, i.e.,

‖Λx‖2 = ‖x‖2 ∀x ∈M

• the Lorentz group L = O(3, 1) is the group of all homogeneous LT’s

• now identify each point in M with a Hermitian 2×2 matrix:

X =(x0 + x3 x1 − ix2x1 + ix2 x0 − x3

)→ det(X) = x2

0 − x21 − x2

2 − x23 = ‖x‖2

• the set of all Hermitian 2×2 matrices is a four-dimensional real vector space (but not agroup). A basis of this space is given by I and the three Pauli matrices:

I =(

1 00 1

), σ1 =

(0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

)X = x0I + x1σ

1 + x2σ2 + x3σ

3

• now let A be an arbitrary complex 2×2 matrix. Define the action of A on X by

X 7→ AXA†

this induces an action of A on the four-vector x:

x 7→ ϕ(A)x

• we have (AXA†)† = AXA†, i.e., AXA† is Hermitian, and thus ϕ(A)x is again a realfour-vector. Furthermore,

det(AXA†

)= | detA|2 detX

1.11. Example: Homomorphism between S`(2,C) and the Lorentz group 21

• S`(2,C) is the group of complex 2×2 matrices with determinant 1.If A ∈ S`(2,C), then det(AXA†) = detX and thuswwϕ(A)x

ww2 = ‖x‖2

i.e., ϕ(A) represents a LT

• we also have

(AB)X(AB)† = ABXB†A† = A(BXB†)A†

→ ϕ(AB)x = ϕ(A)ϕ(B)x

i.e., ϕ is a homomorphism between S`(2,C) and the Lorentz group

• however, ϕ is not an isomorphism since ϕ(A) = ϕ(−A), i.e., the matrices A and -Arepresent the same LT

• examples (see exercises):1) for the matrix

Uθ =(

e−iθ 00 eiθ

)

ϕ(Uθ) is a rotation about the x3-axis by an angle of 2θ2) for the matrix

Vα =(

cos(α) − sin(α)sin(α) cos(α)

)

ϕ(Vα) is a rotation about the x2-axis by an angle of 2α3) for the matrix

Mr =(r 00 1/r

)

ϕ(Mr) is a Lorentz boost in the direction x3 with parameter 2 ln r

• the homomorphism from S`(2,C) to L is not onto:– all A ∈ S`(2,C) are continuously connected to I– there are different types of LT’s:∗ proper LT’s: det Λ = +1improper LT’s: det Λ = −1∗ orthochronous LT’s: Λ00 ≥ 1non-orthochronous LT’s: Λ00 ≤ −1∗ only the proper and orthochronous LT’s are continuously connected to I: "con-tinuous LT’s", these form a group L0

– the homomorphism ϕ only maps S`(2,C) to the continuous LT’s→ the image of the homomorphism is

image(ϕ) = ϕ(S`(2,C)) = L0

22 1. Introduction

• homomorphism between SU(2) and O(3):– SU(2) ⊂ S`(2,C) is the group of unitary 2×2 matrices with determinant 1, i.e.,AA† = I and detA = 1

– now let A ∈ SU(2) and e0 = (1, 0, 0, 0), i.e., E0 = I

E0 7→ AE0A† = AIA† = I = E0

i.e., ϕ(A)e0 = e0

– O(3) is the group of real and orthogonal 3×3 matrices, i.e., RRT = I

– for LT of the form

Λ =(

1 00 R

)with R ∈ O(3) (∗)

we have

Λe0 = e0 (∗∗)

conversely one can show that all LT’s satisfying (∗∗) have the form (∗)→ O(3) is the subgroup of L for which Λe0 = e0

→ ϕ is a homomorphism between SU(2) and O(3). Again the map is two-to-one,since ϕ(A) = ϕ(−A)

– similarly to the discussion above, SU(2) is only mapped to elements of O(3) thatare continuously connected to I, i.e., elements with determinant 1

→ image(ϕ) = ϕ(SU(2)) = SO(3)

2. Matrix representations

2.1. Definitions

• consider a λ-dimensional space S that is spanned by linearly independent functionsf1, . . . , fλ. Suppose S is invariant under a group G of linear operators A1, . . . , An, i.e.,Ajfi is again in S:

Ajfi =λ∑k=1

Γ(Aj)kifki = 1, . . . , λj = 1, . . . , n (∗)

the Γ(Aj)ki are constant coefficients

• we can view Γ(Aj) as a matrix (k = row index, i = column index)– Γ(Aj) defines the action of the operator Aj on the basis functions– the i-th column of Γ(Aj) contains the components of the function Ajfi in the basis

of the functions f1, . . . , fλ

– for every operator Aj there is one matrix Γ(Aj), hence altogether we have n suchmatrices

– every Γ has dimension λ– the set of these matrices is called Γ(G)

• we now show that the Γ’s satisfy the multiplication law of the group, i.e., Γ(Am)Γ(Aj) =Γ(AmAj)proof:

AmAjfi = Am∑k

Γ(Aj)kifk =∑k

Γ(Aj)kiAmfk =

=∑k

Γ(Aj)ki∑l

Γ(Am)lkfl =∑l

∑k

Γ(Am)lkΓ(Aj)ki

︸︷︷︸

[Γ(Am)Γ(Aj)]li

fl =

!=∑l

Γ(AmAj)lifl

→ Γ(AmAj) = Γ(Am)Γ(Aj)

• a set of matrices satisfying the group multiplication law is calledmatrix representationof the group

• terminology options for (∗):– the functions fi form a basis of the representation Γ(G)– the fi transform under G in the representation Γ(G)– the fi furnish the representation Γ(G)

23

24 2. Matrix representations

• S is called carrier space of the representation Γ

• the dimension of the representation = dimension λ of the Γ matrices

• Γ(I) = 1λ → dim. of the representation = Tr Γ(I)

• from the definition of the inverse we have Γ(A−1j ) = Γ−1(Aj)

• a representation is called faithful if the homomorphism between the group elements andthe representation matrices is one-to-one, i.e., different group elements are representedby different matrices

• every group has a trivial representation, in which

Γ(Aj) = 1 ∀j

this representation is not faithful

• an important theorem (proof in exercises):1) If a group G has a nontrivial invariant subgroup H, then a representation of the

factor group G/H is also a representation of G. This representation is not faithful.2) Converse: If Γ(G) is an unfaithful representation of G, then G has at least one

invariant subgroup H such that Γ defines a faithful representation of the factorgroup G/H

2.2. Equivalent representations

• consider the same space S as in Sec. 2.1, but with another basis f1, . . . , fλ. The twobases are related by a transformation S

fi =λ∑k=1

Skifk (i = 1 . . . , λ)

or fk =λ∑i=1

S−1ik fi (k = 1, . . . , λ)

the matrix S must be invertible, i.e., detS 6= 0

• the functions fi furnish another representation Γ(A1), . . . , Γ(An) of the group G

• the relation between these two representations is:

Aj fi = Aj∑k

Skifk =∑k

SkiAjfk =∑k

Ski∑l

Γ(Aj)lkfl =

=∑k

Ski∑l

Γ(Aj)lk∑m

S−1ml fm =

∑m

∑lk

S−1mlΓ(Aj)lkSki

︸︷︷︸

[S−1Γ(Aj)S]mi

fm =

!=∑m

Γ(Aj)mifm

⇒ Γ(Aj) = S−1Γ(Aj)S same S ∀j

the representations Γ(G) and Γ(G) are called equivalent

• equivalent representations derscribe the same action of the group elements in differentbasis systems (e.g., rotated coordinate systems)

2.3. Examples 25

2.3. Examples

• consider again example 2 from Sec. 1.3

• the space spanned by an even function fe (e.g., fe = x2yz) has dimension 1 (see Sec.1.4). From OIfe = fe and OP fe = fe we have

Γ(1)(I) = 1 , Γ(1)(P ) = 1

– this is a representation of Z2 (and all groups isomorphic to Z2)– every function with even parity transforms under OI , OP in this representation– if we consider another group isomorphic to Z2, then all functions that are invariant

under the two operators of the group also transform in this representation.Example 4 from Sec. 1.3: the function x1x2 + y1y2 transforms under OI , OP inthe representation Γ(1).

• the space spanned by an odd function fo (e.g., fo = xyz) also has dimension 1. FromOIfo = fo and OP fo = −fo we have

Γ(2)(I) = 1 , Γ(2)(P ) = −1

– this is another representation of Z2 (and all groups isomorphic to Z2)– all odd functions transform under OI , OP in this representation– isomorphic groups: functions that change sign under the operator corresponding toA also transform in this representation.Example 4 from Sec. 1.3: the function x1y1 − x2y2 transforms under OE , OS inthe representation Γ(2)

• consider two functions f1 and f2 that form an image pair under OI , OP :

OIf1 = f1 OIf2 = f2

OP f1 = f2 OP f2 = f1

(f1 and f2 span a two-dimensional space)From these equations we immediately obtain the representation of OI , OP in the basisof f1 and f2:

Γ(3)(I) =(

1 00 1

), Γ(3)(P ) =

(0 11 0

)

under a group isomorphic to Z2 all image pairs transform in this representation.Example 4 from Sec. 1.3: the functions x2

1y2 and x22y1 transform under OE , OS in the

representation Γ(3)

• basis transformation (see Sec 1.4)

f1 = 12(f1 + f2) , f2 = 1

2(f1 − f2)

→ OI f1 = f1 , OI f2 = f2 , OP f1 = f1 , OP f2 = −f2

from this we immediately obtain the representation of OI , OP in this basis

Γ(4)(I) =(

1 00 1

), Γ(4)(P ) =

(1 00 −1

)


Γ(4) is equivalent to Γ(3)

Γ(4) = S−1Γ(3)S with S = 12

(1 11 −1

)

• now consider a 4-dim. space spanned by the functions

f1 = cosh(xyz) + x2 + y + z f2 = x2 + y + z

f3 = sinh(xyz) + x2 − y − z f4 = x2 − y − z

action of OP :

OP f1 = f1 − f2 + f4 OP f2 = f4

OP f3 = f2 − f3 + f4 OP f4 = f2

→ the space is invariant under OI , OP .The representation of OI , OP in this basis is

Γ(5)(I) =

1 0 0 00 1 0 00 0 1 00 0 0 1

, Γ(5)(P ) =

1 0 0 0−1 0 1 10 0 −1 01 1 1 0

• for all representations we have Γ2(P ) = 1, as expected (since O2

P = I)

2.4. Reduction of representations

2.4.1. Reducibility of invariant spaces and representations

• some representations are more "fundamental" than others. The functions that furnishthese representations have special symmetry properties under the group

• conversely, the symmetry properties of a function can be deduced from the representationof the group in which the function transforms

• a λ-dim. space S that is invariant under G is called reducible if there exists a set oflinearly independent functions f1, . . . , fλ spanning S, and if this set can be decomposedinto two subsets f1, . . . , fλa and fλa+1, . . . , fλ such that each of these subsets spans aspace that is again invariant under G.Let

Sa = span(f1, . . . , fλa

)Sb = span

(fλa+1, . . . , fλ

),

then every function f =∑λi=1 αifi ∈ S can be written as the sum of two functions

fa =λa∑i=1

αifi ∈ Sa and fb =λ∑

i=λa+1αifi ∈ Sb

symbolically: S = Sa ⊕ Sb

2.4. Reduction of representations 27

• what does this imply for the matrix representation Γ(G) furnished by the functionsf1, . . . , fλ?– Aj fi for i ≤ λa doesn’t contain components in Sb– Aj fi for i > λa doesn’t contain components in Sa→ the matrices Γ(Aj) are block diagonal:

Γ(Aj) =(

Γa(Aj) 00 Γb(Aj)

)j = 1, . . . , n (∗)

with dim(Γa(Aj)

)= λa and dim

(Γb(Aj)

)= λ− λa

• both of the sets Γa(G) and Γb(G) are matrix representations of the group G

• a matrix representation of form (∗) and every representation equivalent to it is calledreducible– if all matrices are already block diagonal we say that the matrix representation is

already in the reduced form– if representation Γ(G) is not in reduced form we have to check if there exists a

transformation S such that Γ(Aj) = S−1Γ(Aj)S is block diagonal for all j = 1, . . . , n(same S for all j!). This is an important problem in representation theory.

• example: consider the representations Γ(3) and Γ(4) of Z2 from Sec. 2.3

Γ(3)(I) =(

1 00 1

), Γ(3)(P ) =

(0 11 0

)

Γ(4)(I) =(

1 00 1

), Γ(4)(P ) =

(1 00 −1

)

– Γ(4) is already in reduced form and contains the representations Γ(1) and Γ(2) onthe diagonal

– Γ(3) is equivalent to Γ(4) since

Γ(4)(Aj) = S−1Γ(3)(Aj)S with S = 12

(1 11 −1

)

• a space is called irreducible ("irrep") if it is not reducible

• the action of the group elements on an irreducible space is given by an irreducible matrixrepresentation (in arbitrary basis)

• the representations Γa(G) and Γb(G) in (∗) could again be reducible, in which case theycan be made block diagonal by a suitable basis transformation. This process is continueduntil only irreps remain, i.e., in general a representation consists of several irreps, andone can find a transformation S such that

S−1Γ(Aj)S =

Γa(Aj) 0 0 · · ·

0 Γb(Aj) 0 · · ·0 0 Γc(Aj) · · ·...

...... . . .

(∗∗)

with all Γi(Aj) irreducible→ the irreps are the "building blocks" from which all representations are made


• a representation of the form (∗∗), in which all diagonal blocks are irreducible is calledfully reduced

• the same irrep can occur in (∗∗) more than onceSymbolically:

Γ(G) = Γ1(G)⊕ · · · ⊕ Γ1(G)︸︷︷︸a1-times

⊕Γ2(G)⊕ · · · ⊕ Γ2(G)︸︷︷︸a2-times

⊕ · · · =∑i ⊕

aiΓi(G)

i.e., the representation Γ contains the irrep Γi ai times

• from (∗∗) we see that the invariant space S decomposes into irreducible invariant sub-spaces. The action of group elements on the basis vectors of the subspaces is given byΓi

2.4.2. OA operators using the example of D3

• D3 = symmetry group of an equilateral triangle (isomorphic to S3)

•

• •L1

L2L3

p1=(x1, y1)

p2=(x2, y2)p3=(x3, y3)

x

y

• group elements– I = identity– C = clockwise rotation by 120 = (123)C = counterclockwise rotation by 120 = (321)

– σ1, σ2, σ3 = reflections w.r.t. L1, L2, L3 = (23), (13), (12)

• multiplication table as in Sec. 1.5

• now consider (one-to-one) maps A of the xy-plane onto itself (the 6 elements of D3 areexamples of such maps A)

• let p = (x, y) be a point in the xy-plane

• the point that is mapped by A to the point p will be called A−1p (i.e., A(A−1p) = p)

• with every map A we associate an operator OA that acts on an arbitrary functionf(x, y) = f(p)– define the action of the operator OA on f by

(OAf) (p) = f(A−1p

)– important: the LHS means: "compute the function OAf at point p", i.e., OA acts

on the function f not on the value f(p), i.e., the new function OAf at the point phas the same value as the old function at point A−1p (which is mapped to p by A)


• the 6 operators OA associated with the elements of D3 form a group D3 isomorphic toD3 since (

(OAOB)f)

(p) =(OA(OBf)

)(p) = (OBf) (A−1p) =

= f(B−1A−1p

)= f

((AB)−1p

)= (OABf) (p)

• we now consider the action of these operators on certain functions of (x, y) = p and thusgenerate representations of D3 ' S3

• consider the function

ϕ1(x, y) = e−(x−x1)2−(y−y1)2 = e−‖p−p1‖2

– what is OCϕ1

ϕ2(x, y) = (OCϕ1)(p) = ϕ1(C−1p

)= e−‖C−1p−p1‖2

= e−‖C−1(p−Cp1)‖2= e−‖C−1(p−p2)‖2

= e−‖p−p2‖2

– analogously we get

ϕ3(x, y) = (OCϕ1)(x, y) = e−‖p−p3‖2

– for the reflections we have(Oσ1ϕ1

)(p) = ϕ1

(σ−1

1 p)

= e−wwσ−1

1 p−p1ww2

=

= e−wwσ−1

1 (p−σ1p1)ww2

= e−wwσ−1

1 (p−p1)ww2

= e−‖p−p1‖2 = ϕ1(p)(Oσ2ϕ1

)(p) = e−

wwσ−12 (p−σ2p1)

ww2

= e−wwσ−1

2 (p−p3)ww2

= e−‖p−p3‖2 = ϕ3(p)(Oσ3ϕ1

)(p) = e−

wwσ−13 (p−σ3p1)

ww2

= e−wwσ−1

3 (p−p2)ww2

= e−‖p−p2‖2 = ϕ2(p)

and similarly for ϕ2 and ϕ3

– thus we get the following tableϕ1 ϕ2 ϕ3

OI ϕ1 ϕ2 ϕ3OC ϕ2 ϕ3 ϕ1OC ϕ3 ϕ1 ϕ2Oσ1 ϕ1 ϕ3 ϕ2Oσ2 ϕ3 ϕ2 ϕ1Oσ3 ϕ2 ϕ1 ϕ3

– the space S = span(ϕ1, ϕ2, ϕ3) is invariant under D3

– the functions ϕ1, ϕ2, ϕ3 furnish a 3-dimensional representation of the group

Γ(5)(I) =

1 0 00 1 00 0 1

, Γ(5)(C) =

0 0 11 0 00 1 0

, Γ(5)(C) =

0 1 00 0 11 0 0

Γ(5)(σ1) =

1 0 00 0 10 1 0

, Γ(5)(σ2) =

0 0 10 1 01 0 0

, Γ(5)(σ3) =

0 1 01 0 00 0 1


• Is this representation reducible?Yes, since S is reducible, i.e., there is a basis transformation by which S can be decom-posed into smaller, invariant subspaces

ϕ1 = ϕ1 + ϕ2 + ϕ3 , ϕ2 =√

3(ϕ2 − ϕ3) , ϕ3 = 2ϕ1 − ϕ2 − ϕ3

• ϕ1 is invariant under D3 since the operators OA only interchange the terms in the sum→ ϕ1 spans a 1-dim. invariant subspace (which is irreducible) and furnishes a 1-dim.irrep of the group

Γ(1)(I) = Γ(1)(C) = Γ(1)(C) = Γ(1)(σ1) = Γ(1)(σ2) = Γ(1)(σ3) = 1= trivial representation of D3 (every group has it)

• for ϕ2, ϕ3 we obtain

ϕ2 ϕ3OI ϕ2 ϕ3

OC −12 ϕ2 −

√3

2 ϕ3√

32 ϕ2 − 1

2 ϕ3

OC −12 ϕ2 +

√3

2 ϕ3 −√

32 ϕ2 − 1

2 ϕ3Oσ1 −ϕ2 ϕ3

Oσ212 ϕ2 −

√3

2 ϕ3 −√

32 ϕ2 − 1

2 ϕ3

Oσ312 ϕ2 +

√3

2 ϕ3√

32 ϕ2 − 1

2 ϕ3

• the space span(ϕ2, ϕ3) is invariant under D3, the functions ϕ2, ϕ3 furnish a 2-dim. rep-resentation of the group

Γ(3)(I) =(

1 00 1

), Γ(3)(C) =

−12

√3

2−√

32 −1

2

, Γ(3)(C) =

−12 −

√3

2√3

2 −12

Γ(3)(σ1) =

(−1 00 1

), Γ(3)(σ2) =

12 −

√3

2−√

32 −1

2

, Γ(3)(σ3) =

12

√3

2√3

2 −12

• the representation of the group in the space span(ϕ1, ϕ2, ϕ3) is thus given by

Γ(4)(I) =

1 0 00 1 00 0 1

, Γ(4)(C) =

1 0 00 −1

2

√3

20 −

√3

2 −12

, Γ(4)(C) =

1 0 00 −1

2 −√

32

0√

32 −1

2

Γ(4)(σ1) =

1 0 00 −1 00 0 1

, Γ(4)(σ2) =

1 0 00 1

2 −√

32

0 −√

32 −1

2

, Γ(4)(σ3) =

1 0 00 1

2

√3

20√

32 −1

2

• since Γ(4) is a representation on the same space as Γ(5), only in a different basis, Γ(4) and

Γ(5) are equivalent

Γ(4)(Aj) = S−1Γ(5)(Aj)S with S =

1 0 21√

3 −11 −

√3 −1

∀ Aj ∈ D3


• Γ(4) is already in the reduced form, but Γ(5) is not

• notation:

Γ(4) = Γ(1) ⊕ Γ(3)

Γ(5) = Γ(1) ⊕ Γ(3)

• remaining question: is the 2-dim. representation Γ(3) reducible?

2.4.3. Four theorems and their consequences

• the following theorems are due to Frobenius and Schur, and the proofs are given in theappendix A.2

• Theorem 1: Every representation of a group G by matrices with det 6= 0 can be con-verted to a unitary representation (i.e., the matrices of this represetation are unitary).

• Theorem 2 (Schur’s lemma 1): A matrix which commutes with all matrices of an irrepis proportional to the identity matrix.

• Theorem 3 (Schur’s lemma 2):Suppose we have two irreps of a groupG with elements A1, . . . , An: Γ1(G) with dimensionλ1 and Γ2(G) with dimension λ2.If there exists a λ2 × λ1 matrix M such that

MΓ1(Aj) = Γ2(Aj)M ∀ j = 1, . . . , n

then if λ1 6= λ2 we have M = 0, and if λ1 = λ2 we have either M = 0 or detM 6= 0.If detM 6= 0 the two representations are equivalent since Γ1(G) = M−1Γ2(G)M .

• Theorem 4 (orthogonality relations):Let Γi(G) and Γk(G) be two non-equivalent, unitary irreps of G with order(G) = n.Then we have

n∑j=1

(Γi(Aj)µν

)∗Γk(Aj)µ′ν′ = 0 for all µ, ν, µ′, ν ′

For the matrix elements of a single unitary irrep with dimension λi we haven∑j=1

(Γi(Aj)µν

)∗Γi(Aj)µ′ν′ = n

λiδµµ′δνν′

Combined:n∑j=1

(Γi(Aj)µν

)∗Γk(Aj)µ′ν′ = n

λiδikδµµ′δνν′

• consequence of Theorem 2:All irreps of an Abelian group have dimension 1. (proof in exercises)

• consequence of Theorem 4:– for fixed i, µ, ν we collect the n numbers Γi(A1)µν , . . . ,Γi(An)µν in a vector v(iµν)

with n components


– for every representation Γi there are λ2i such vectors (since µ, ν = 1, . . . , λi)

– Theorem 4 says that such a vector is orthogonal to all vectors V (kµ′ν′) if i 6= k orµ 6= µ′ or ν 6= ν ′

– however, in n dimensions there are at most n vectors that are mutually orthogonal→ λ2

1 + λ22 + · · · ≤ n. In Sec 2.5 we will show that in fact∑

i

λ2i = n

The sum is over the different irreps i, i.e., for a finite group there is a finite numberof non-equivalent irreps each of which has finite dimension

2.4.4. Characters of representations

• the trace

χ(Aj)

=∑k

Γ(Aj)kk

of the matrix representing Aj is called character of Aj in the representation Γ(G)

• from Sec. 2.2: the matrices of a representation transform under a basis transformationS as Γ→ SΓS−1. The trace of a matrix is invariant under such a transformation:

Tr(S−1ΓS

)= Tr

(SS−1Γ

)= Tr (Γ)

i.e., in two equivalent representations every group element has the same character→ the characters are basis-independent while the representing matrices depend on the

choice of basis→ characters contain the essential information on the structure of the irrep (the ma-

trices contain a lot of "irrelevant" information)

• furthermore, all group elements in the same class have the same character, since for twomutually conjugate group elements p and gpg−1 we have

Tr Γ(gpg−1

)= Tr

(Γ(g)Γ(p)Γ(g−1)

)= Tr

(Γ(g−1)Γ(g)Γ(p)

)= Tr Γ(p)

• now set µ = ν and µ′ = ν ′ in Theorem 4 and sum over µ and µ′:n∑j=1

(Γi(Aj)µν

)∗Γk(Aj)µ′ν′

= n


→n∑j=1

(χi(Aj))∗

χk(Aj)

= nδik

this is an orthogonality relation for characters

• since the characters depend only on the class we can also write this as

∑c

nc(χic

)∗χkc = nδik

c labels the class, and nc the number of group elements in class c


• let m be the number of different classes of G– for fixed i we can collect the m numbers χi1 . . . , χim in a vector with m components– in m dimensions there are at most m orthogonal vectors

→ number of irreps ≤ number of classes– in the exercises we show that in fact the equal sign holds

→ number of irreps = number of classes

– the m×m matrix χic (with i, c = 1, . . . ,m) is called character table of the group

• for a reducible representation of the form

Γ(Aj)

=∑i⊕

aiχi(Aj)

we have

χ(Aj)

=∑i

aiχi(Aj)

here χi(Aj) is the character of Aj in the irrep Γi

→n∑j=1

∣∣∣∣χ (Aj)∣∣∣∣2 =∑ik

aiak

n∑j=1

(χi(Aj))∗

χk(Aj)

︸︷︷︸nδik

= n∑i

a2i

• if Γ(G) is irreducible, then one of the ai = 1 and all the others = 0, and thus

n∑j=1

∣∣∣∣χ (Aj)∣∣∣∣2 = n

if Γ(G) is reducible, then one of the ai > 1 or several ai 6= 0 and thus

n∑j=1

∣∣∣∣χ (Aj)∣∣∣∣2 > n

→ looking at the characters one can conclude whether a given representation is reducible

• for Γ(3) from Sec. 2.4.2:∣∣∣χ(3) (I)∣∣∣2 + 2 ·

∣∣∣χ(3) (C)∣∣∣2 + 3 ·

∣∣∣χ(3) (σ1)∣∣∣2 = 4 + 2 + 0 = 6 = order(D3)

→ Γ(3) is irreducible, and the space span(ϕ1, ϕ2) is invariant and irreducible under D3

• so for S3 we have found a 1-dim. and a 2-dim. irrep (λ1 = 1 and λ3 = 2)– from λ2

1 + λ22 + · · · = n we have

1 + λ22 + 4 = 6

we conclude that there is one more irrep with dimension λ2 = 1 (and no otherirreps)


– this remaining irrep can easily be constructed from the multiplication table:

Γ(2)(I) = Γ(2)(C) = Γ(2)(C) = 1Γ(2)(σ1) = Γ(2)(σ2) = Γ(2)(σ3) = −1

• from this we can compute the character table of S3: The classes are I, C, C,σ1, σ2, σ3. The irreps are

Γ(1)(Aj)

= 1 for all j

Γ(2)(Aj)

see above

Γ(3)(Aj)

see Sec. 2.4.2

→ character table:

I C, C σ1, σ2, σ3Γ(1) 1 1 1Γ(2) 1 1 -1Γ(3) 2 -1 0

• if we know the characters of all irreps of a group, then for a given representation (whichin general is reducible) we can compute how many times the various irreps are containedin it:

χ(Aj)

=∑i

aiχi(Aj)

j = 1, . . . , n

n∑j=1

(χk(Aj))∗

χ(Aj)

=∑i

ai

n∑j=1

(χk(Aj))∗

χi(Aj)

︸︷︷︸nδik

= nak

→ ak = 1n

n∑j=1

(χk(Aj))∗

χ(Aj)

= 1n

∑c

nc(χkc

)∗χc

• for the reducible representation Γ(5) of D3 we have:

a1 = 16(1 · 1 · 3 + 2 · 1 · 0 + 3 · 1 · 1) = 1

a2 = 16(1 · 1 · 3 + 2 · 1 · 0 + 3 · (−1) · 1) = 0

a3 = 16(1 · 2 · 3 + 2 · (−1) · 0 + 3 · 0 · 1) = 1

thus Γ(5) = Γ(1) ⊕ Γ(3), as we found earlier

2.5. The regular representation

• a group algebra is a set of elements that form a linear vector space, in which both,an addition and a multiplication are defined such that the group properties are satisfied(with one exception: the zero element of the algebra doesn’t have an inverse groupelement)

2.5. The regular representation 35

• for a group G with elements gi (i = 1, . . . , n) the linear combination∑ni=1 cigi with

coefficients ci form an algebra with multiplication law n∑i=1

cigi

n∑j=1

djgj

=n∑

i,j=1cidjgigj

– because gigj ∈ G the product is again an element of the algebra– the group elements are the basis vectors of the vector space– dim(vector space) = order(G)

• the multiplication law gigj = gk can also be written as

gigj =n∑

m=1gm(∆i)mj

with (∆i)mj = 1 for m = k and (∆i)mj = 0 for m 6= k (and i, j are fixed)

• the n × n matrices ∆i (i = 1, . . . , n) form a representation of G, the so-called regularrepresentation (∆i is the matrix representing gi)proof: Let ga, gb, gc ∈ G with gagb = gc. Then,

gagbgj =∑m

gagm(∆b)mj =∑m,k

gk(∆a)km(∆b)mj

gcgj =∑k

gk(∆c)kj

The LHSs are equal, thus the RHSs are also equal; comparison of the coefficients of gkyields:

(∆c)kj =∑m

(∆a)km(∆b)mj = (∆a∆b)kj

→ ∆c = ∆a∆b

• Theorem: The regular representation contains all irreps of G, and the multiplicity ofan irrep k equals its dimension λk:

∆i ∼m∑

k⊕=1λkΓk(gi) (m = # of classes) (∗)

=

1Γ2

. . .Γ2

. . .

︸︷︷︸λ2 blocks

Γm. . .

Γm︸︷︷︸λm blocks

proof:


– the characters of the regular representation are

χR(gi) =∑m

(∆i)mm

– for the identity we have

Igi =n∑

m=1gm(∆I)mj → (∆I)mj = δmj → χR(I) = n

– for gi 6= I:

gigj =∑m

gm(∆i)mj 6= gj → (∆i)jj = 0 → χR(gi) = 0

– with the formula from Sec. 2.4.4 we thus have

ak = 1n

∑i

(χk(gi)

)∗χR(gi) = 1

n

(χk(I)

)∗n = λk

• set gi = I in (∗):

∆I =∑k⊕

λkΓk(I)∣∣∣∣Tr ·

⇒ χR(I) = n =∑k

λ2k

this is the missing proof of the formula in Sec 2.4.3

2.6. Product representations and Clebsch-Gordan coefficients

• in physical applications we often deal with vector spaces that are tensor products ofsmaller vector spaces.Examples: orbital angular momentum and spin of the electron, systems with severalidentical particles, etc.

• let U and V be two vector spaces with bases ui and vj, and let W = U ⊗ V withbasis wk, where wk = ui ⊗ vj

• with every pair of operators A,B that act1 on U and V ,

Aui =∑i′

ui′Ai′i

Bvj =∑j′

uj′Bj′j

we can associate a product operator D = A⊗B acting on W :

Dwk =∑k′

wk′Dk′k with Dk′k ≡ Ai′iBj′jk = (i, j), k′ = (i′, j′)

often A and B are the same physical operator, but in different spaces (e.g., orbital angularmomentum of two particles)

1A acts on U and B on V .

2.6. Product representations and Clebsch-Gordan coefficients 37

• if Γµ(G) and Γν(G) are two matrix represetations of a group G on the spaces U and V ,respectively, then the matrices

Γµ⊗ν(g) = Γµ(g)⊗ Γν(g) with g ∈ G

form a representation of G on W = U ⊗ V , the so-called product representation(proof follows directly from the definition of Dk′k above)

• for the characters we have

χµ⊗ν(g) = χµ(g)χν(g)

• we have2

(Γµ ⊗ Γν)(i−1)n+k,(j−1)n+` = ΓµijΓνk` with i, j = 1, . . . ,dim(Γµ)

k, ` = 1, . . . ,dim(Γν) = n

• in general the product representation is reducible:

Γµ ⊗ Γν ∼∑λ⊕

aλΓλ with∑λ

aλnλ = nµnν

(nλ is the dimension of Γλ, and the aλ follow from Sec. 2.4.4)i.e., W can be decomposed into a direct sum of irreducible subspaces W λ

α (which areinvariant under G), with dim(W λ

α ) = nλ; the index α = 1, . . . , aλ distinguishes thedifferent subspaces that correspond to the same irrep λ

Γµ ⊗ Γν ∼

Γ1

. . .Γ1

. . .

︸︷︷︸a1 blocks

Γλ. . .

Γλ︸︷︷︸aλ blocks

. . .

• there is a basis transformation from the product basis wk to a new orthonormal basiswλα` in which the representation matrices are block diagonal– ` = 1, . . . , nλ labels the basis vectors of W λ

α

– the basis transformation is written as follows (with k = (i, j) and in Dirac notation):∣∣∣wλα`⟩ =∑i,j

∣∣∣wij⟩ ⟨i, j(µ, ν) α, λ, `⟩︸︷︷︸

Clebsch-Gordan coeff.

(∗)

– the CG coefficients are the matrix elements of the basis transformation, with∗ (i, j) = row index (old basis)

2here we use the Kronecker product of two matrices defined in A.3


∗ (α, λ, `) = column index (new basis)∗ (µ, ν) fixed

• for (∗) we write symbolically (sum over repeated indices):

w = wU or wa = wkUka

from

δaa′ = 〈wa | wa′〉 = 〈wkUka | wk′Uk′a′〉 = U∗kaUk′a′ 〈wk | wk′〉︸︷︷︸δaa′

= U †akUka′ =(U †U

)aa′

we have U †U = 1, i.e., the CG coefficients form a unitary matrix

→ w = wU−1 = wU † or wk = waU†ak = waU

∗ka

now define ⟨α, λ, ` (µ, ν)i, j

⟩≡⟨i, j(µ, ν) α, λ, `

⟩∗→ the converse of (∗)∣∣∣wij⟩ =

∑α,λ,`

∣∣∣wλα`⟩ ⟨α, λ, ` (µ, ν)i, j⟩

(∗∗)

• the CG coefficients satisfy orthonormality and completeness relations (this follows fromUU † = 1 = U †U , see exercises):∑

αλ`

⟨i′, j′(µ, ν) α, λ, `

⟩ ⟨α, λ, ` (µ, ν)i, j

⟩= δi′iδj′j∑

ij

⟨α′, λ′, `′ (µ, ν)i, j

⟩ ⟨i, j(µ, ν) α, λ, `

⟩= δα′αδλ′λδ`′`

• simplification of the notation:–∣∣∣wij⟩ → |i, j〉 and

∣∣∣wλα`⟩ → |α, λ, `〉

– Einstein’s summation convention for row/column indices–⟨i, j(µ, ν) α, λ, `

⟩→ 〈i, j | α, λ, `〉

• let D(g) be the operator that describes the action of a group element g on W = U ⊗ V .Then we have

D(g) |i, j〉 =∣∣∣i′, j′⟩Γµ(g)i′iΓν(g)j′j

D(g) |α, λ, `〉 =∣∣∣α, λ, `′⟩Γλ(g)`′`

(since U is carrier space of Γµ, V is carrier space of Γν and W λα is carrier space of Γλ)

→ D(g) |α, λ, `〉 (∗)= D(g) |i, j〉〈i, j | α, λ, `〉 =

=∣∣∣i′, j′⟩Γµ(g)i′iΓν(g)j′j 〈i, j | α, λ, `〉 =

(∗∗)=∣∣∣α′, λ′, `′⟩⟨α′, λ′, `′ ∣∣∣ i′, j′⟩Γµ(g)i′iΓν(g)j′j 〈i, j | α, λ, `〉

=∣∣∣α′, λ′, `′⟩ δα′αδλ′λΓλ(g)`′` (see above)

→ δα′αδλ′λΓλ(g)`′` =⟨α′, λ′, `′

∣∣∣ i′, j′⟩Γµ(g)i′iΓν(g)j′j 〈i, j | α, λ, `〉 (∗ ∗ ∗)

2.6. Product representations and Clebsch-Gordan coefficients 39

– (∗) corresponds to w = wU

– (∗∗) corresponds to w = wU †

– (∗ ∗ ∗) corresponds to Γ = U †Γµ⊗νU– the LHS of (∗ ∗ ∗) shows that in the new basis the matrices are block diagonal

• application in QM:coupling of two spins s1 and s2 to a total spin s– the 3 components of the spin operator are the generators of the group SU(2) (see

Sec. 6.1)– spin s1 = 1

2 → 2 states with z-components m1 = ±12 . The two wave functions

corresponding to these states∗ span a 2-dimensional space that is invariant under SU(2)∗ furnish an irrep of SU(2) with dimension 2

– analogously for s2

– this means we have a product space of dimension 4 with basis states

|s1,m1〉 ⊗ |s2,m2〉

– the two spins can couple to a total spin s = 0 (with m = 0) or s = 1 (withm = −1, 0, 1)→ the product space can be decomposed into two subspaces that are invariant andirreducible under SU(2).Symbolically:

12︸︷︷︸

dim=2

⊗ 12︸︷︷︸

dim=2

= 1︸︷︷︸dim=3

⊕ 0︸︷︷︸dim=1

– the CG coefficients describe the basis transformation from the product basis to thetotal spin basis

– the wave functions in the total spin basis furnish a 1- dimensional and a 3- dimen-sional irrep of SU(2)

– which basis we should use depends on the problem, e.g.:∗ two spins interacting with an external magnetic field ~B = Bz:

H = − (γ1~s1 + γ~s2) · ~B = −(γ1s1z + γ2s2z)B

→ product basis∗ two spins interacting with each other:

H = a~s1 · ~s2 = a

2(s2 − s2

1 − s22

)→ total spin basis

• in Sec. 4 we will learn how to construct the CG coefficients, or more generally, how toconstruct the basis transformation from a reducible representation to a sum of irreps


2.7. Subduced and induced representations

• let H (with order nH) be a subgroup of G (with order nG)

• a representation Γ of G is also a representation of H if we restrict it to the Γ(h) withh ∈ H– this representation is called subduced representation– even if Γ(G) is irreducible, in general Γ(H) is reducible

• conversely, from an irrep Γ of H one can construct (induce) a representation of G:– let f1, . . . , fm be the basis states of Γ(H)– if all operators of G act on this basis, in general we end up with a larger space with

basis f1, . . . , fn (n ≥ m)– this space is invariant under G by construction→ the basis states furnish a representation Γind of G (which in general is reducible)

– we have (without proof):

dim(Γind(G)

)= nGnH

dim(Γ(H)

)• Frobenius’ reciprocity theorem (without proof):

– the irrep Γλ(G) subduces a representation of H (in general reducible). Suppose thissubduced representation contains the irrep Γµ(H) aλµ times

– the irrep Γµ(H) induces a representation of G (in general reducible). Suppose thisinduced representation contains the irrep Γλ(G) bµλ times

– then we have

aλµ = bµλ for all λ, µ

3. Applications in quantum mechanics

3.1. Selection rules

• in the following we show that the orthogonality relations for irreps lead to selection rulesin QM

• scalar products in QM typically have the form

〈f | g〉 =∑· · ·∑∫

· · ·∫

dx1 . . . wf∗g

– integrals over continuous variables (e.g., position or momentum)

– sums over discrete variables (e.g., spin)

– f and g are complex functions (e.g., wave functions)

– w is a real function (e.g., Jacobi determinant of variable transformation)

• an operator A is called unitary if it preserves the scalar product, i.e.,

〈Af | Ag〉 = 〈f | g〉

• Theorem: Let G be a group of linear, unitary operators A1, . . . , An. Suppose thefunctions fν1 , . . . , fνnν transform in the unitary irrep Γν(G) with dim(Γν) = nν , e.g.,

Ajfνα =

nν∑β=1

Γν(Aj)βαfνβ (∗)

Analogously, suppose the functions gν′1 , . . . , gν′nν′

transform in the unitary irrep Γν′(G).(I.e., the f and g have special symmetry properties w.r.t. the group. If ν 6= ν ′, f and ghave different symmetry properties.) We then have

⟨fνα

∣∣∣ gν′α′⟩ = vδνν′δαα′ (∗∗)

where v is independent of α. This implies that two functions with different symmetryproperties are orthogonal.

41

42 3. Applications in quantum mechanics

Proof: Since the Aj are unitary we can write

⟨fνα

∣∣∣ gν′α′⟩ = 1n

n∑j=1

⟨Ajf

να

∣∣∣ Ajgν′α′⟩ =

(∗)= 1n

n∑j=1

⟨nν∑β=1

Γν(Aj)βαfνβ

∣∣∣∣∣∣∣nν′∑β′=1

Γν′(Aj)β′α′gν′β′

⟩=

= 1n

∑ββ′

∑j

Γν(Aj)∗βαΓν′(Aj)β′α′︸︷︷︸⟨fνβ

∣∣∣ gν′β′⟩= n

nνδνν′δαα′δββ′ (see Sec. 2.4.3)

= δνν′δαα′1nν

∑β

⟨fνβ

∣∣∣ gν′β ⟩︸︷︷︸=: v (independent of α)

• furthermore, an arbitrary function f to which we can apply the operators Aj can bewritten as a linear combination of the functions with special symmetry properties (=basis functions).This expansion theorem, together with (∗∗), leads to selection rules of QM:Proof:– an invariant space can be viewed as a direct sum of irreducible invariant spaces→ a function in an invariant space can be written as a linear combination of thebasis functions that span these irreducible spaces

– it remains to be shown that f can be embedded in an invariant space– consider the space R that is spanned by f and all images of f under the operators

of G, i.e., (with A1 = I)

R = span(f,A2f, . . . , Anf)

R contains f and is invariant under G; hence the embedding has been shown. (Wenow have to decompose R into irreducible subspaces. These are carrier spaces ofirreps of G, and these irreps determine what basis functions are needed for theexpansion of f . This is f -dependent.)

3.2. The symmetry group of the Hamiltonian and degeneracies

• let H be the Hamiltonian of a q.m. system and Aj be a unitary operator acting on thesame space as H. We then define

Ht ≡ AjHA−1j

"the transform of H under the operation Aj"

• typically Aj acts on a state |ψ〉 such that Aj |ψ〉 is the state in the new coordinate system.We then have (Aj is unitary):

〈ψ | Hψ〉 =⟨Ajψ

∣∣∣ AjHψ⟩ =⟨Ajψ

∣∣∣ AjHA−1j Ajψ

⟩=⟨Ajψ

∣∣∣ HtAjψ⟩

i.e., observers in the two systems measure the same energy

3.2. The symmetry group of the Hamiltonian and degeneracies 43

• if AjH = HAj , we have Ht = H, i.e., Aj leaves H invariant and thus the Hamiltonianhas the same form in both systems

• the set of all unitary operators A1, . . . , An that commute with H form a group G, thesymmetry group of H (proof is simple)

• now let Aj ∈ G and let |ψ〉 be an eigenstate of H with energy E

H |ψ〉 = E |ψ〉

→ H(Aj |ψ〉

)= AjH |ψ〉 = E

(Aj |ψ〉

)(∗)

i.e., Aj |ψ〉 is an eigenstate of H with energy E

• if E is non-degenerate we have Aj |ψ〉 ∼ |ψ〉.If E is m-fold degenerate, Aj |ψ〉 is a linear combination of the states |ψ1〉 , . . . , |ψm〉 withenergy E.In both cases the space S = span(|ψ1〉 , . . . , |ψm〉) is invariant under the action of thesymmetry group of H.→ degenerate states furnish a representation of G:

Aj |ψi〉 =m∑k=1

Γ(Aj)ki |ψi〉i = 1, . . . ,mj = 1, . . . , n (∗)

this representation could be reducible or irreducible, but in the typical case it is irre-ducible:– all states that transform in the same irrep of G must have the same energy:

H |ψi〉 = Ei |ψ〉(∗)−−→ H

(Aj |ψi〉

)= Ei

(Aj |ψi〉

)(∗∗)−−→

∑k

Γ(Aj)kiH |ψk〉︸︷︷︸Ek|ψk〉

=∑k

Γ(Aj)kiEi |ψk〉(with Γirreducible)

⇒ Γ(Aj)kiEk = Γ(Aj)kiEi (no summation)

now define a diagonal m×m matrix E = diag(E1, . . . , Em)[Γ(Aj)E

]ki

= Γ(Aj)kÈì = Γ(Aj)kiEi[EΓ(Aj)

]ki

= Ek`Γ(Aj)ì = Γ(Aj)kiEk→ Γ(Aj)E = EΓ(Aj) for all j

according to Schur’s lemma 1, E ∼ 1m

→ all Ei are the same– if Γ is reducible and |ψi〉 and |ψk〉 transform in different irreps of G,(

( ) 00 ( )

)← i← k

we have Γ(Aj)ki = 0 for all (Aj), and Schur’s lemma 1 is not applicable→ we cannot conclude Ek = Ei, i.e., there is no reason why |ψi〉 and |ψk〉 shouldbe degenerate


– if states have the same energy even tough they transform in different irreps, wespeak of "accidental degeneracies". This can have two reasons:1) fine-tuning of a parameter in H (very unlikely)2) we have not yet found the full symmetry group (i.e., we have not yet fully

understood the problem)

• from this we learn:– the degenerate sates with the same energy transform in an irrep of the symmetry

group of H, and therefore can be classified using these irreps– number of degenerate states = dimension of the irrep

• example: Hydrogen atom (without spin-orbit coupling)

H = − ~2

2m∇2 − e2

r

– the eigenstates are labeled by the quantum numbers n = 1, 2, . . . (principal quantumnumber), ` = 0, 1, . . . , n − 1 (orbital angular momentum) and m = −`, . . . , ` (z-component of angular momentum)

ψ(~r) = Rn`(r)Y`m(θ, ϕ)

– the Hamiltonian of a central force problem V (~r) = V (r) in three dimensions isinvariant under O(3)∗ because of this symmetry the energy doesn’t depend on m→ (2`+ 1)-fold degeneracy∗ for fixed ` the Y`m furnish a (2`+ 1)-dimensional irrep of O(3) (see Sec. 6.7.4)

– however, the energy doesn’t depend on ` either ("accidental degeneracy")→ n2-fold degeneracy since

∑n−1`=0 (2`+ 1) = n2

– the reason for this is that the symmetry group is larger than O(3):For V (r) ∼ 1/r (Coulomb potential) H is invariant under O(4) (H commutes withthe Runge-Lenz vector)→ energy is independent of `→ n2-fold degeneracy (corresponding to the dimensions of the irreps of O(4))

3.3. Perturbation theory and lifting of degeneracies

• typical problem:

H = H0 + H ′

↑"solvable"

↑"small perturbation"

• let G be the symmetry group of H0, then there are two cases1

1) H ′ is invariant under G2) H ′ is invariant under B ⊂ G

1 There is actually a third case, namely that the perturbation could lead to a symmetry group C of H, suchthat G ⊂ C, but this will not be covered in these lectures.

3.3. Perturbation theory and lifting of degeneracies 45

• in case 1. the perturbation H ′ does not lead to a splitting of the degenerate states of H0

• in case 2. (some of) the degeneracies will be lifted– the exact states of H transform in irreps of B– the degenerate states of H0 transform in irreps of G– for these irreps of G, the matrices corresponding to the elements of B form a

(subduced) representation Γsub(B) of B. In general this representation is reducible,i.e.,

Γsub(B) =r∑

i⊕=1aiΓi(B) with dim(Γi) = λi

– for every irrep of B contained in Γsub(B) there is a new energy level→

∑i ai new energy levels

a1 of them are λ1-fold degenerate,a2 of them are λ2-fold degenerate, etc.

• example 1: H atom as in Sec. 3.2.If we add a small central potential V (r) (but not 1/r), the O(4) symmetry is broken toO(3), and every energy level is split into n levels with different values for `.

9 n = 3531

` = 2` = 1` = 0

4 n = 2 31 ` = 1

` = 0

1 n = 1 1 ` = 0

• example 2: central force problem V (r) 6= 1/r, + small perturbation invariant under theOA operators of D3

– i.e., G = O(3), B = D3 ' S3

– consider in particular the 3-fold degenerate energy level with ` = 1 and with theunperturbed eigenfunctions

R(r)Y1,1(θ, ϕ) , R(r)Y1,0(θ, ϕ) , R(r)Y1,−1(θ, ϕ)

– these 3 functions furnish a 3-dimensional irrep Γ(G) of O(3) and a 3-dimensionalsubduced representation Γsub(B) of D3

– D3 has two 1-dimensional and one 2-dimensional irreps→ Γsub(B) must be reducible

– the characters of the elements of D3 in the representation Γsub(B) are

χ(I) = 3 , χ(C) = χ(C) = 0 , χ(σ1) = χ(σ2) = χ(σ3) = 1

derivation:∗ Γsub(I) = 13 → χ(I) = 3


∗ action of a rotation by an angle α about the z-axis:

r → r , θ → θ , ϕ→ ϕ+ α(OAf

)(p) = f

(A−1p

)(ORαY`m

)= Y`m(θ, ϕ− α) = f(cos θ)eim(ϕ−α) = e−imαY`m(θ, ϕ)

→ Γsub`=1(ORα

)=

e−iα 0 00 1 00 0 eiα

→ χ(C) =

1∑m=−1

e−im 2π3 = 1 + 2 cos(2π/3) = 0

∗ σ1 is a reflection w.r.t. the yz-plane:

x→ −x , y → y , z → z

in spherical coordinates:

x = r sin θ cosϕ , y = r sin θ sinϕ , z = r cos θ

i.e.,

r → r , θ → θ , cosϕ→ − cosϕ , sinϕ→ sinϕOσ1Y1,1 = Oσ1f(cos θ) (cosϕ+ i sinϕ) = f(cos θ) (− cosϕ+ i sinϕ) = Y1,−1

Oσ1Y1,0 = Oσ1f(cos θ) = Y1,0

Oσ1Y1,−1 = Oσ1f(cos θ)(cosϕ− i sinϕ) = f(cos θ)(− cosϕ+ i sinϕ) = −Y1,1

→ Γsub`=1(Oσ1

)=

0 0 −10 1 0−1 0 0

→ χ(σ1) = 1

– character table of D3 from Sec. 2.4.4:I C, C σ1, σ2, σ3

Γ1 1 1 1Γ2 1 1 -1Γ3 2 -1 0

– with the formula ak = 1n

∑c nc(χkc )∗χc from Sec. 2.4.4 we have

a1 = 16[

1 · 1 · 3 + 2 · 1 · 0 + 3 · 1 · 1]

= 1

a2 = 16[

1 · 1 · 3 + 2 · 1 · 0 + 3 · (−1) · 1]

= 0

a3 = 16[

1 · 2 · 3 + 2 · (−1) · 0 + 3 · 0 · 1]

= 1

i.e., the originally 3-fold degenerate level splits into two new levels, one of which is2-fold degenerate

3.3. Perturbation theory and lifting of degeneracies 47

131

1D3O(3) C3

1` = 1 2

– the group C3 = I, C, C is a subgroup of D3

∗ we could add to H an even smaller perturbation that is invariant under the OAoperators of C3

∗ the two degenerate states furnish a 2-dimensional representation of C3

∗ since C3 only has 1-dimensional irreps, the two states are split by the newperturbation

4. Expansion in irreducible basis vectors

4.1. Irreducible basis vectors

• consider

– an operator representation U(G) of a group G on a space V

– an invariant, irreducible subspace Vµ under G with dimension nµ

– an orthonormal basis ∣∣eµi ⟩ (i = 1, . . . , nµ) of Vµ

then we have ∀g ∈ G

U(g)∣∣∣eµi ⟩ =

∣∣∣eµj ⟩Dµ(g)ji (sum over j = 1, . . . , nµ)

where Dµ(G) is an irrep of G

• the∣∣eµi ⟩ are called irreducible basis vectors of the irrep µ

• for two such irreducible bases we have (see Sec. 3.1)

⟨uµi

∣∣∣ vνj ⟩ = δµνδij1nµ

⟨uµk

∣∣∣ vµk⟩

4.2. Projection operators on irreducible bases

• we now learn how an arbitrary vector |x〉 ∈ V can be expanded in irreducible basisvectors. The idea is simple: project |x〉 on the basis vector |ei〉 using the projectionoperator Ei = |ei〉〈ei|:

Ei |x〉 = |ei〉〈ei | x〉 = |ei〉xi

however, the details are more complicated.

• using the definitions of Sec. 4.1 we first define generalized projection operators:

Pµji ≡nµn

∑g∈G

[Dµ(g)−1

]jiU(g)

with n = order(G) and i, j = 1, . . . , nµ

• Theorem: For fixed |x〉 ∈ V and fixed j the nµ vectors Pµji |x〉 (i = 1, . . . , nµ) areeither zero, or they transform in the irrep µ.

48

4.2. Projection operators on irreducible bases 49

Proof:

U(g)Pµji |x〉 = nµn

∑g′

U(g)U(g′) |x〉[Dµ(g′)−1

]ji

=

= nµn

∑g′

U(gg′) |x〉[Dµ(g′)−1

]ji

=

= nµn

∑g′′

U(g′′) |x〉[Dµ(g−1g′′)−1

]ji

=

=

nµn

∑g′′

U(g′′) |x〉[Dµ(g′′)−1

]jk

Dµ(g)ki =

= Pµjk |x〉Dµ(g)ki (no sum over µ)

• this means that, starting from a vector |x〉 ∈ V , we can construct an irreducible subspaceof V corresponding to the irrep µ, with basis Pµji |x〉. The basis vectors are orthogonal,but not automatically normalized.

• by varying µ, j, and |x〉 we can find all irreducible subspaces (of course we only needthose irreps µ that are contained in the representation we want to reduce)

• in the following we will assume that the U and D are unitary → D−1 = D†, U−1 = U †

• example 1: reduction of the space S =span(ϕ1, ϕ2, ϕ3) of Sec. 2.4.2 (invariant underD3 ' S3)

– S3 has two 1-dimensional irreps and one 2-dimensional irrep (Γ1,Γ2,Γ3)

– the generalized projection operators are

P 111 = 1

6(OI +OC +OC +Oσ1 +Oσ2 +Oσ3

)P 2

11 = 16(OI +OC +OC −Oσ1 −Oσ2 −Oσ3

)P 3

11 = 13

(OI −

12OC −

12OC −Oσ1 + 1

2Oσ2 + 12Oσ3

)P 3

12 = 13

(−√

32 OC +

√3

2 OC −√

32 Oσ2 +

√3

2 Oσ3

)

P 321 = 1

3

(√3

2 OC −12OC −

√3

2 Oσ2 +√

32 Oσ3

)

P 322 = 1

3

(OI −

12OC −

12OC +Oσ1 −

12Oσ2 −

12Oσ3

)

– applied to a vector in S, e.g., ϕ1:(see Sec. 2.4.2 for the action of the operators on ϕ1)

∗ µ = 1:

P 111ϕ1 = 1

6(ϕ1 + ϕ2 + ϕ3 + ϕ1 + ϕ3 + ϕ2

)= 1

3(ϕ1 + ϕ2 + ϕ3

)this function is invariant under D3 and transforms in the trivial irrep Γ1.

50 4. Expansion in irreducible basis vectors

∗ µ = 2:

P 211ϕ1 = 1

6(ϕ1 + ϕ2 + ϕ3 − ϕ1 − ϕ3 − ϕ2

)= 0

this had to be zero since Γ2 is not contained in the 3-dimensional representationΓ(5).

∗ µ = 3:first j = 1:

P 311ϕ1 = 1

3(ϕ1 −

12ϕ2 −

12ϕ3 − ϕ1 + 1

2ϕ3 + 12ϕ2

)= 0

P 312ϕ1 =

√3

6(− ϕ2 + ϕ3 − ϕ3 + ϕ2

)= 0

now j = 2:

P 321ϕ1 =

√3

6(ϕ2 − ϕ3 − ϕ3 + ϕ2

)∼ ϕ2 − ϕ3

P 322ϕ1 = 1

3

(ϕ1 −

12ϕ2 −

12ϕ3 + ϕ1 −

12ϕ3 −

12ϕ2

)∼ 2ϕ1 − ϕ2 − ϕ3

the last two functions transform according to Γ3. Thus we have the basistransformation of Sec. 2.4.2.

• example 2: reduction of a product representation– let Dµ⊗ν be a product representation of G on Vµ ⊗ Vν . In general we have Dµ⊗ν ∼∑

λ⊕ aλDλ.

How do we find the irreducible subspaces of Vµ ⊗ Vν?

– start with the product basis |k, `〉 =∣∣∣eµk⟩⊗∣∣∣eν`⟩ and apply the generalized projection

operators P λji to it.

– for fixed λ, j, k, ` the nλ vectors P λji |k, `〉 (i = 1, . . . , nλ) are either all zero, or theyspan an irreducible subspace

– by varying λ, j, k, ` we can find all irreducible subspaces– exercises: reduction of D3⊗3(S3)

• the action of the generalized projection operators on an irreducible basis is

Pµji∣∣eνk⟩ = nµ

n

∑g

U(g)∣∣eνk⟩Dµ(g)†ji = nµ

n

∣∣eν` ⟩∑g

Dν(g)`kDµ(g)†ji︸︷︷︸= nnµδµνδi`δjk

=

=∣∣∣eµi ⟩ δµνδjk (∗)

• from this we can derive more identities (proofs in exercises):

PµjiPν`k = δµνδjkP

µì (property of projection operators)

U(g) =∑µij

PµjiDµ(g)ij (inverse of the original definition)

U(g)P ν`k =∑i

P νìDν(g)ik (operator form of the Theorem)

4.3. Irreducible operators and the Wigner-Eckart theorem 51

• it follows from these identities and from (∗) thatPµi ≡ P

µii is a projection operator on

∣∣eµi ⟩Pµ =

∑i P

µi is a projection operator on Vµ

proof: (no summation over repeated indices)

Pµi Pνk = PµiiP

νkk = Pµiiδµνδik = Pµi δµνδik

PµP ν =∑ik

Pµi Pνk =

∑ik

Pµi δµνδik = δµν∑i

Pµi = δµνPµ

• the projection operators are complete, i.e.,∑µ

Pµ = 1

proof:

Pµ∣∣eνk⟩ =

∑i

Pµii∣∣eνk⟩ (∗)=

∑i

∣∣∣eµi ⟩ δµνδik =∣∣∣eµk⟩ δµν

→∑µ

Pµ∣∣eνk⟩ =

∣∣eνk⟩ ∀ν, k

→∑µ

Pµ = 1

• summary:– decompose the space V into irreducible subspaces: V =

∑µ,α⊕ V

µα

∗ µ labels the irrep∗ α counts how often the irrep µ occurs

– denote the basis of V by |α, µ, k〉 (i = 1, . . . , nµ). We then have

Pµ |α, ν, k〉 = |α, µ, k〉 δµνPµi |α, ν, k〉 = |α, µ, i〉 δµνδikPµji |α, ν, k〉 = |α, µ, j〉 δµνδjk

4.3. Irreducible operators and the Wigner-Eckart theorem

• consider a set of operators Oµi (i = 1, . . . , nµ) acting on a space V . If they transformunder a group G as

U(g)Oµi U(g)−1 = OµjDµ(g)ji

they are called irreducible operators (or irreducible tensors) corresponding to theirrep µ

• consider now a set of irreducible operators Oµi and a set of irreducible vectors ∣∣eνj ⟩.

How do the vectors Oµi∣∣eνj ⟩ transform?

U(g)Oµi∣∣∣eνj⟩ = U(g)Oµi U(g)−1U(g)

∣∣∣eνj⟩ = Oµk∣∣eν` ⟩Dµ(g)kiDν(g)`j

i.e., they transform in the product representation Dµ⊗ν


• this product representation can be reduced (see Sec. 2.6), and the vectors Oµi∣∣eνj ⟩ can

be expanded in the irreducible basis vectors∣∣∣wλα`⟩:

Oµi

∣∣∣eνj⟩ =∑α`λ

∣∣∣wλα`⟩ ⟨α, λ, `(µ, ν)i, j⟩

(we assume here that the operators Oµi are normalized in such a way that the vectorsOµi

∣∣eνj ⟩ are normalized, otherwise the CG coefficients would not form a unitary matrix.)This leads to the Wigner-Eckart theorem:⟨

eλ`

∣∣∣Oµi ∣∣∣eνj⟩ =∑αρm

⟨eνλ∣∣ wραm⟩︸︷︷︸ ⟨α, ρ,m(µ, ν)i, j

⟩= δρλδm`

1nλ

∑k

⟨eλk∣∣ wλαk⟩ (see Sec. 4.1)

⟨eλ`

∣∣∣Oµi ∣∣∣eνj⟩ =∑α

⟨α, λ, `(µ, ν)i, j

⟩〈λ‖Oµ‖ν〉α

with the so-called reduced matrix element

〈λ‖Oµ‖ν〉α ≡1nλ

∑k

⟨eλk

∣∣∣ wλαk⟩which does not depend on i, j, or `.

• very important theorem, since the (many) matrix elements on the LHS can be expressedin terms of (only a few) reduced matrix elements on the RHS (the CG coefficients aretabulated).

• in practice the reduced matrix elements are computed as follows:– compute as many matrix elements on the LHS as there are reduced matrix elements– view the Wigner-Eckart theorem as a system of linear equations for the reduced

matrix elements and solve for the latter

• example: electromagnetic transitions in atoms and nuclei (O(3) symmetry)– the states are |j,m〉 (definite angular momentum)– the operators are "multipole transition opertors" (i.e., the quantized electromagnetic

vector potential ~A(~x, t) expanded in ladder operators for definite angular momen-tum, see Tung Sec. 8.7)

4.4. Left ideals and idempotents

• so far we have learned how to decompose a reducible representation into irreduciblecomponents– to do this we need to know the irreps– question for this section: How to construct the irreps ?

• consider the regular representation (see Sec. 2.5), since it contains all irreps Dµ (withmultiplicity nµ = dim(Dµ))

• the carrier space of the regular representation is the space of the group algebra (orFrobenius algebra), G = span(g1, . . . , gn)

4.4. Left ideals and idempotents 53

• every element p ∈ G is a vector in G (then we call it |p〉), but also an operator on G(then we call it p):

r |q〉 = (rigi)∣∣∣qjgj⟩ =

∣∣∣gigj⟩ riqj ∣∣∣∣gigj = gk(∆i)kj

= |gk〉 ri(∆i)kjqj

this dual role of the elements of G is the reason for the special properties of the regularrepresentation

• G can be decomposed into invariant, irreducible subspaces corresponding to the irrepsof G

• a subspace L of G that is invariant under left multiplication is called left ideal:

L = r with p |r〉 ∈ L ∀p ∈ G

a left ideal always contains the zero element of G

• if L is an irreducible subspace, it is called minimal or irreducible left ideal

• if we have found the irreducible left ideals, we can construct the irreps of G from them(by acting with the group elements on the basis vectors of the left ideal)→ task: find the irreducible left ideals

• denote the projection operator on an irreducible left ideal Lµα by Pµα . This operator musthave the following properties:1) Pµα |r〉 ∈ Lµα ∀r ∈ G, in short: Pµα G = Lµα

2) if |q〉 ∈ Lµα, then Pµα |q〉 = |µ〉3) Pµα r = rPµα ∀r ∈ G

proof: consider an arbitrary s ∈ G. This element can be expanded as |s〉 =∑νβ

∣∣sνβ⟩with sνβ ∈ Lνβ

rPµα |s〉 = rPµα∑νβ

∣∣∣sνβ⟩ = r∣∣sµα⟩ =

∣∣rsµα⟩Pµα r |s〉 = Pµα r

∑νβ

∣∣∣sνβ⟩ = Pµα∑νβ

∣∣∣rsνβ⟩︸︷︷︸∈Lν

β

=∣∣rsµα⟩︸︷︷︸∈Lµα

4) PµαP νβ = δµνδαβPµα

• furthermore we define

Lµ =∑α

Lµα → G =∑µ

Lµ

and first construct the projection operator Pµ on Lµ

• let e be the identity of G. Since e ∈ G there is a unique decomposition e =∑µ eµ with

suitable eµ ∈ Lµ. Then Pµ is given by right multiplication with eµ, i.e.,

Pµ |r〉 ≡∣∣∣reµ⟩ ∀r ∈ G

proof:


– Pµ is a linear operator (proof simple)– for r ∈ G we have on the one hand

r =∑µ

rµ with rµ ∈ Lµ

and on the other hand

r = re = r∑µ

eµ =∑µ

reµwith reµ ∈ Lµ(since Lµ is a left ideal)

→ rµ = reµ = Pµr

this proves properties 1. and 2.– for arbitrary q, r ∈ G we have

Pµr |q〉 = Pµ |rq〉 =∣∣∣rqeµ⟩

rPµ |q〉 = r∣∣∣qeµ⟩ =

∣∣∣rqeµ⟩→ Pµr = rPµ (property 3.)

– the decomposition of eν in invariant components is on the one hand

eν = 0 + · · ·+ eν + · · ·+ 0

and on the other hand

eν = eνe = eν∑µ

eµ = eνe1︸︷︷︸∈L1

+ · · ·+ eνeν︸︷︷︸∈Lν

+ eνeν+1︸︷︷︸∈Lν+1

+ · · ·

since the decomposition is unique we have eµeν = δµνeµ

→ PµP ν = δµνPµ (property 4.)

• this derivation works in the same way for projectors on irreducible left ideals, defined by

Pµα |r〉 ≡∣∣reµα⟩

• elements of the group algebra satisfying

eµeν = δµνeµ

are called idempotents. If there is an additional normalization factor on the RHS, theyare called essentially idempotent.

• the idempotent eµ generates the left ideal Lµ:

Lµ = reµ | r ∈ G

• an idempotent is called irreducible or primitive if it generates an irreducible left ideal

Lµα = reµα | r ∈ G

otherwise it is called reducible and can be written as the sum e1 + e2 of two idempotents(with e1e2 = 0).

4.4. Left ideals and idempotents 55

• How can we find out if an idempotent is primitive?Theorem: An idempotent e is primitive if and only if

ere = λre ∀r ∈ G (λr is a number)

Proof:

1) Suppose e is a primitive idempotent→ L = re | r ∈ G is an irreducible left ideal.Associate with r ∈ G an operator R on G:

R |q〉 ≡ |qere〉 ∈ L

→ R is a projection operator on L.

Rs |q〉 = |sqere〉 = sR |q〉 ∀s ∈ G

→ by Schur’s lemma 1, R is proportional to the identity in L.→ ere = λre

2) Suppose ere = λre ∀r ∈ G and e = e1 + e2 with two idempotents e1 6= 0 and e2 6= 0(and with e1e2 = 0)

ee1 = (e1 + e2)e1 = e1e1 + 0 = e1→ ee1e = · · · = e1 = λe (by assumption ere = λe)

e1 = e1 = λ2ee = λ2e→ λ = λ2

→ λ = 0 or λ = 1→ e1 = 0 or e1 = e , i.e., e2 = 0

which contradicts the assumption e1 6= 0 and e2 6= 0.

• We also need a criterion to decide whether two primitive idempotents generate equivalentor non-equivalent irreps.Theorem: Two primitive idempotents e1 and e2 generate equivalent irreps if and only ife1re2 6= 0 for an r ∈ G.Proof: Let L1 and L2 be the two irreducible left ideals generated by e1 and e2, and D1

and D2 the corresponding irreps.

1) Suppose e1re2 = s 6= 0 for an r ∈ G. Consider the linear map S:

L1 3 q1S7−→ q2 = q1s ∈ L2.

For all p ∈ G we have

Sp |q1〉 = S |pq1〉 = |pq1s〉 = p |q1s〉 = pS |q1〉→ Sp = p

↑operator on L2 since S projects on L2

S as an operator on L1

→ SD1(p) = D2(p)S

by Schur’s lemma 2, D1 and D2 are equivalent.


2) If D1 and D2 are equivalent, there is a transformation S such that SD1(p)S−1 =D2(p) ∀p ∈ G, or Sp = pS as a linear map from L1 to L2

|s〉 ≡ S |e1〉 ∈ L2 → |s〉 = |se2〉|s〉 = S |e1〉 = S |e1e1〉 = Se1 |e1〉 = e1S |e1〉 = e1 |s〉 = |e1s〉

→ s = se2 = e1s → e1se2 = s

• The trivial representation is generated by the primitive idempotent

e1 = 1n

n∑i=1

gi

(this is the generalized projection operator P 111 from Sec. 4.2)

Proof:1) The left ideal generated by e1 is L1 = re1 | r ∈ G with

re1 =

∑j

rjgj

1n

∑i

gi

= 1n

∑j

rj∑i

gjgi︸︷︷︸∑kgk

= ce1 where c =∑i

ri is a number

→ L1 is 1-dimensional and thus irreducible.2) The elements of L1 are invariant under the group:

g · ce1 = c

n

∑i

ggi = c

n

∑k

gk = ce1

→ trivial representation of the group.

• Summary:– The group algebra can be decomposed into left ideals Lµ (µ corresponds to the

non-equivalent irreps of the group)– The Lµ are generated by right multiplication with the idempotents eµ, with eµeν =δµνeµ and

∑µ eµ = e.

– Every Lµ can be decomposed into nµ irreducible left ideals Lµα (α = 1, . . . , nµ).– The Lµα are generated by right multiplication with the primitive idempotents eµα,

with

eµαreµβ = δαβλre

µα ∀r ∈ G

– If we have found all primitive idempotents we can easily construct all irreps of thegroup from them.→ the task is to find all non-equivalent primitive idempotents.

• exercises: reduction of the regular representation of C3.

5. Representation of the symmetric groupand Young diagrams

5.1. Why Sn is important

The representation theory of Sn is the basis for the study of many other groups:

• finite groups of order n are isomorphic to subgroups of Sn

• the primitive idempotents of Sn are used to construct the irreps of the classical continuousgroups, e.g., U(m), O(m), etc.

• for systems of identical particles, the symmetry group of the Hamiltonian H alwayscontains Sn as a factor.→ the eigenstates of H transform in irreps of Sn.

5.2. 1-dimensional and associated representations of Sn

• An ("alternating group") = group of the even permutations (i.e., even number of trans-positions)– An is an invariant subgroup of Sn– the factor group Sn/An is isomorphic to Z2→ Sn has two irreps given by the irreps of Z2 (see Sec. 1.10)

Ds(p) = 1 ∀p ∈ Sn (trivial representation)

Da(p) = (−1)p ≡

1 for even p−1 for odd p

(−1)p is called parity of the permutation p.

• The 1-dimensional irreps also follow from this theorem:The symmetrizer s ≡

∑p p and the antisymmetrizer a ≡

∑p(−1)pp of Sn are essen-

tially idempotent and primitive (proof in exercises).– for all p ∈ Sn we have

spa =↑

rearrangementlemma

sa =∑qr

(−1)rqr =∑q

(−1)q∑r

(−1)q+rqr︸︷︷︸= a

rearrangementlemma

= a∑q

(−1)q︸︷︷︸=0

= 0

→ s and a generate non-equivalent irreps of Sn with basis vectors |ps〉 and |pa〉.– for all p ∈ Sn we have ps = s and pa = (−1)pa→ both irreps are 1-dimensional, with matrix elements 1 or (−1)p.

• Suppose we have a representation Dλ of Sn with dimension nλ. Then Dλ and Dλ =Dλ ⊗Da are called associated representations

57

58 5. Representation of the symmetric group and Young diagrams

– Dλ also has dimension nλ.– Dλ(p) = (−1)pDλ(p)

→∑p

∣∣∣χλ(p)∣∣∣2 =

∑p

∣∣∣χλ(p)∣∣∣2 ?= n!

→ Dλ is irreducible if and only if Dλ is irreducible.– If χλ(p) = 0 for all odd p, Dλ and Dλ are equivalent (because in this case all char-

acters are the same, see Sec. 2.4.4), and Dλ is called self-associated. Otherwisethey are non-equivalent.

• the characters of the irreps of Sn are real.Proof: p−1 is in the same class as p

→ χ(p) = χ(p−1) =↑

representationis unitary

χ(p)∗

this yields a theorem that is important for applications to systems with bosons andfermions.

• Theorem: Let Dλ and Dµ be two irreps of Sn:1) Dλ ⊗Dµ contains Ds exactly once (not at all) if Dλ and Dµ are equivalent (non-

equivalent).2) Dλ ⊗ Dµ contains Da exactly once (not at all) if Dλ and Dµ are associated (not

associated).(proof in exercises)

5.3. Young diagrams

• a partition λ ≡ (λ1, . . . , λr) of an integer n is a sequence of positive integers such that

r∑i=1

λi = n with λi ≥ λi+1

– two partitions λ and µ are equal if λi = µi for all i.– λ > µ(λ < µ) if the first nonzero term in the sequence λi−µi is positive (negative).– a partition is represented graphically by a Young diagram:∗ n boxes arranged in r rows.∗ The i-th row contains λi boxes.

• examples:– for n = 3 there are 3 distinct partitions:

(3) (2,1) (1,1,1)

– for n = 4 there are 5 distinct partitions:

5.3. Young diagrams 59

(4) (3,1) (2,2) (2,1,1) (1,1,1,1)

• every partition of n corresponds to a class of Sn and vice versa:– Every class corresponds to a certain cycle structure.– The i-th row of the diagram can be interpreted as a λi cycle.– Each of the numbers 1, . . . , n occurs in one and only one of the cycles →

∑i λi = n.

→ The number of Young diagrams for n equals the number of classes of Sn, andtherefore the number of irreps of Sn.

• Example: For S3 we have:

e: three 1-cycles → (1,1,1)(12), (13), (23): one 2-cycle, one 1-cycle → (2,1)(123), (321): one 3-cycle → (3)

• a Young tableau is a Young diagram containing the numbers 1, . . . , n in the boxes(each number appearing once).Example:

3 4 12 or

2 413

• in a normal Young tableau the numbers 1, . . . , n appear in ascending order, first fromleft to right, then from top to bottom.Example:

1 2 34 or

1 243

For every Young diagram there is one and only one normal Young tableau.

• In a standard Young tableau the numbers increase (but not necessarily in strict order)in the rows and columns.Example:

1 2 43 or

1 432

• The notation in the literature is sometimes different, e.g., Young graph, Young pattern,Young frame, etc.

• The normal Young tableau of the partition λ will be called Θλ.

• An arbitrary tableau is obtained from Θλ by a permutation p of the n numbers in theboxes: Θp

λ ≡ pΘλ. We have qΘpλ = Θqp

λ .


5.4. Symmetrizers and antisymmetrizers of Young tableaux

• We shall see that for every Young tableau we can define a primitive idempotent thatgenerates an irrep of Sn on the group algebra Sn.

• Given a Young tableau Θpλ we define

– horizontal permutations hpλ which only permute the numbers in the rows of Θpλ.

– vertical permutations vpλ which only permute the numbers in the columns of Θpλ.

• For a Young tableau Θpλ we define

symmetrizer : spλ =∑h

hpλ

antisymmetrizer : apλ =∑v

(−1)vvpλ

irreducible symmetrizer :(or Young operator)

epλ =∑h,v

(−1)vhpλvpλ = spλa

pλ

epλ is very important since it is a primitive idempotent.

• example: Standard tableaux of S3:

– Θ1 = 1 2 3 : all p are h → s1 =∑p p = s (symmetrizer of S3)

only e is a v → a1 = ee1 = s1a1 = s

– Θ2 =1 23 : the h are e and (12) → s2 = e+ (12)

the v are e and (13) → a2 = e− (13)e2 = s2a2 = e+ (12)− (13)− (321)

– Θ3 =123

: only e is a h → s3 = eall p are v → a3 =

∑p(−1)pp = a (antisymmetrizer of S3)

e3 = s3a3 = a

– Θ(23)2 =

1 32 : the h are e and (13) → s

(23)2 = e+ (13)

the v are e and (12) → a(23)2 = e− (12)

e(23)2 = s

(23)2 a

(23)2 = e− (12) + (13)− (123)

• We can learn a lot from this example: (In the following we omit the superscript p tosimplify the notation.)1) For every tableau Θλ, the horizontal and vertical permutations each form a subgroup

of Sn.2) The sλ and aλ are (total) symmetrizers and antisymmetrizers of the corresponding

subgroup.

→ sλhλ = hλsλ = sλ

aλvλ = vλaλ = (−1)vaλ

sλsλ = nλsλ with nλ =r∏i=1

λi!

aλaλ = nλaλ

5.5. Irreducible representations of Sn 61

→ sλ and aλ are essentially idempotent, but in general not primitive.3) The eλ are primitive idempotents (exercises).4) e1 = s and e3 = a generate the two 1-dimensional irreps of S3 (see Sec. 5.2). e2

generates a 2-dimensional left ideal L2 of S3 (by right multiplication with e2):

ee2 = e2

(12)e2 = (12) + e− (321)− (13) = e2

(23)e2 = (23) + (321)− (123)− (12) ≡ r2

(13)e2 = (13) + (123)− e− (23) = −e2 − r2

(123)e2 = (123) + (13)− (23)− e = −e2 − r2

(321)e2 = (321) + (23)− (12)− (123) = r2

i.e., L− 2 = span(e2, r2). Since e2 is primitive, L− 2 is irreducible.→ The Young operators of the normal Young tableaux generate all irreps of thegroup.

5) e(23)2 also generates a 2-dimensional irrep:– This irrep must be equivalent to the irrep generated by e2 (since there is only

one 2-dimensional irrep).

– The irreducible left ideal generated by e(23)2 is

L(23)2 = span

(e

(23)2 , r

(23)2

)with r

(23)2 = (123)− (13) + (23)− (321).

– This is orthogonal to the other left ideals L1 = span(e1), L3 = span(e3) andL2.

6) S3 is the direct sum of the four irreducible left ideals. The decomposition of theidentity is

e = 16e1 + 1

3e2 + 13e

(23)2 + 1

6e3

→ the regular representation of S3 is reduced by the Young operators of the standardYoung tableaux.

5.5. Irreducible representations of Sn

• we now generalize what we learned about S3 in Sec. 5.4 to Sn (some proofs in exercises,some omitted)

• again we omit the superscript p for simplicity; everything is valid for arbitrary Youngtableaux.

• Theorem 1: The symmetrizers of the Young tableau Θλ satisfy

sλraλ = ξreλ ∀r ∈ Sne2λ = ηeλ , with ξr, η numbers, and η 6= 0

→ eλ is essentially idempotent.


• Theorem 2: The Young operator eλ is primitive idempotent and therefore generates anirrep of Sn on Sn.

• Theorem 3: The irreps generated by eλ and epλ (with p ∈ Sn) are equivalent.

• Theorem 4: The Young operators eλ and eµ generate inequivalent irreps if the corre-sponding Young diagrams are different (i.e., if λ 6= µ viewed as partitions).

• Theorem 5: The Young operators of the normal Young tableaux generate all inequivalentirreps of Sn. (I.e., we can identify the different irreps with the different Young diagrams.)

• Theorem 6:

a) The irreducible left ideals generated by the Young operators of the standard tab-leaux are linearly independent.

b) The direct sum of these left ideals generates Sn.

5.6. More applications of Young tableaux

• proofs omitted (too lengthy)

• the dimension of the irrep Dλ corresponding to the Young diagram λ equals the numberof standard tableaux and is given by

nλ = n!∏i<j(ì − `j)∏

i ì!= n!∏

i,k hik

with n! = order of Sni, j = 1, . . . , r (number of rows of the diagram)k = 1, . . . , λi (number of boxes in row i)ì = λi + r − ihik = "hook length" of the box i, k

= number of boxes along a hook from i, k to the right and downwards

example:

•

h23 = 7

• this implies that Sn has only two 1-dimensional irreps (Ds and Da from Sec. 5.2):

Ds= · · ·︸︷︷︸n boxes

Da= ...

n boxes

5.6. More applications of Young tableaux 63

• the representation Dλ associated to an irrep Dλ is obtained by transposing the corre-sponding Young tabeau Θλ, i.e., exchanging rows and columns:

Θλ = Θλ =

• recursive calculation of the characters of the irreps of Sn:– the boundary [λ] of a Young diagram λ = "south-east boundary" = set of boxes

whose right edge or lower edge or right lower vertex is part of the border of thediagram. (Sometimes [λ] is called staircase.)

Example:

123

4567

in general: [λ] = all boxes (i, j) ∈ λ so that (i+ 1, j + 1) /∈ λ.– a skew hook = connected part of the boundary that can be removed to leave a

proper Young diagram.In the example above: 1-2, 1-4, 1-5, 1-7, 2, 2-4, 2-5, 2-7, 4, 4-5, 4-7, 7→ a box at the end of a row is the beginning of a skew hook→ a box at the end of a column is the end of a skew hook

– every hook corresponds to a skew hook and vice versa1. The hook length then equalsthe length of the skew hook.

Example: The skew hook 1-5 corresponds to the hook

123

4567

•

– the leg length of a skew hook is the number of vertical steps (=number of rowscovered by the skew hook - 1)

– let c be a class of Sn with cycle structure c = (a1, a2, . . . , aq) (in any order).What is the character χλc of this class in irrep Dλ?∗ select an arbitrary cycle of c with length ai.∗ denote by c the class of Sn−ai obtained by removing the cycle ai from class c∗ start with the Young diagram of Dλ, determine all skew hooks with length ai,

and denote the Young diagram of Sn−ai obtained by removing such a hook byλ

∗ we then have

χλc =∑λ

±χλc

· + sign for skew hooks with even leg length· - sign for skew hooks with odd leg length

1This should rather be taken as an additional condition for skew hooks.


· use the same algorithm to calculate the χλc → recursion

· if nothing remains, χλ=0() ≡ 1 (and include the sign depending on the leg

length)∗ if there is no skew hook with length ai, then χλc = 0

– to make this method efficient, the order in which the ai are removed should bechosen to minimize the total number of skew hooks to be removed

– example: S13, c = (7, 4, 2), λ = (6, 3, 3, 1) =

∗ there is only one hook with length 7. This hook corresponds to the skew hook(with leg length 2)

∗∗∗∗∗∗∗ → χ(6,3,3,1)(7,4,2) = +χ(2,2,1,1)

(4,2)

∗ now there is only one hook with length 4. This hook corresponds to the skewhook (with leg length 2)

∗∗∗∗ → χ(6,3,3,1)(7,4,2) = +χ(2)

(2) = 1

• there is also a non-recursive (but less efficient) method:– determine all possibilities ρ to "disassemble" the Young diagram λ completely by

successfully removing skew hooks with lengths a1, . . . , aq

– denote by kρ the sum of the leg lengths of the skew hooks occuring in possibility ρ– then

χλc =∑ρ

(−1)kρ

6. Lie groups

6.1. Introduction

• so far we considered finite discrete groups. We now proceed to consider continuous groups

• in this case the group elements are continuous functions of finitely many real parameters:

G 3 g = g(α1, . . . , αn) ≡ g(α) "order n" or "dimension n" or "n parameters"

i.e., a group element is specified by a point on a n-dimensional manifold

• the group elements g(α) and g(α′) are "close to each other" if their "distance" (∑i(αi −

α′i)2)1/2 in parameter space is small

• the parameterization in not unique. We could also parameterize the group elements byn linearly independent well-behaved functions βi(α) (i = 1, . . . , n)

• the multiplication law is now

g(α)g(α′) = g(α′′) withα′′1 = f1(α1, . . . , αn;α′1, . . . , α′n)...α′′n = fn(α1, . . . , αn;α′1, . . . , α′n)

the structure of the group is determined completely by the functions f1, . . . , fn (similarto the multiplication table for finite groups)

• the group must have an identity element I. I corresponds to a certain point α0 =(α0

1, . . . , α0n) in parameter space.

→ we have for all i and α

fi(α1, . . . , αn;α01, . . . , α

0n) = fi(α0

1, . . . , α0n;α1, . . . , αn) = αi

• for every g ∈ G there must be an inverse element g−1. This corresponds to the pointα = (α1, . . . , αn) in parameter space, and from gg−1 = g−1g = I we have

fi(α1, . . . , αn; α1, . . . , αn) = fi(α1, . . . , αn;α1, . . . , αn) = α0i

We only consider groups for which we can invert these equations (at least in principle):

αi = hi(α1, . . . , αn;α01, . . . , α

0n)

• the group is called Lie group if the functions fi and hi are analytic for all i (i.e., allderivatives are continuous)

• a Lie group is called compact if the intervals of the parameters αi are finite and closed;otherwise it is called non-compact

65

66 6. Lie groups

6.2. Examples of Lie groups

1) SO(2)• rotations about a fixed axis (e.g., z)• 1 parameter: rotation angle ϕ• multiplication law: R(ϕ1)R(ϕ2) = R(ϕ1 + ϕ2), i.e., f = ϕ1 + ϕ2 → abelian group• identity: ϕ = 0• inverse element: R(ϕ)−1 = R(−ϕ)

2) Translation group:• translation (for example) in xy-plane• 2 parameter group (i.e., order 2)• multiplication law:

T (x1, y1)T (x2, y2) = T (x1 + x2, y1 + y2)

• identity: x = y = 0• inverse: T (x, y)−1 = T (−x,−y)(Note that if we had parameterized the coordinate system differently, e.g., in polarcoordinates (r, ϕ), the multiplication law would not be quite as straightforward.)

3) General linear group Gl(N,C):• group of non-singular linear homogeneous transformations in an N -dimensional

complex space• the defining or fundamental representation is given by complex N ×N matrices Asuch that detA 6= 0. The vectors of the space transform as

z′i =N∑j=1

Aijzj

• we have 2N2 (real) parameters• the multiplication law is

A′′ = AA′ or A′′ij(A,A′) =∑k

AikA′kj

→ 2N2 equations (for real and imaginary parts of A′′ij)• identity: 1• inverse of A is A−1

All the following examples are subgroups of Gl(N,C).

4) Unitary group U(N):• group of linear unitary transformations in an N -dimensional complex space• unitary means that the quantity∑

i

|zi|2 = |z1|2 + · · ·+ |zN |2

is left invariant

6.2. Examples of Lie groups 67

• this imposes some restrictions on the matrices of U(N):∑i

|z′i|2 =∑ijk

(Uijzj)∗Uikzk =∑ijk

U∗ijUikz∗j zk

!=∑k

|zk|2

i.e., ∑i

U∗ijUik = δjk → U †U = 1N = UU †

this implies:

k = j :∑i

U∗ikUik =∑i

|Uik|2 = 1 N equations

k < j : <∑i

U∗ijUik = 0 =∑i

U∗ijUik = 0 N(N − 1) equations

• the total number of parameters is thus

2N2 − (N +N2 −N) = N2

• also we have

1 = detUU † = |detU |2 ⇒ detU = eiϕ (pure phase)

5) Special unitary group SU(N):• subgroup of U(N)• restriction to unitary matrices with detU = 1• N2 − 1 parameters

6) Orthogonal group O(N):• group of orthogonal transformations in an N -dimensional real space• orthogonal means that ∑

i

x2i

is preserved• the xi transform as

x′i =∑j

Oijxj

thus we have ∑i

(x′i)2 = · · · =∑ijk

OTkiOijxjxk!=∑j

x2j

⇒∑i

OTkiOij = δjk → OTO = 1N = OOT

so that

k = j :∑i

O2ik = 1 N equations

k < j :∑i

OijOik = 0 12N(N − 1) equations

68 6. Lie groups

• the number of parameters is

N2 −(N + 1

2N2 − 1

2N)

= 12N(N − 1)

• here we have

1 = detOOT = (detO)2 ⇒ detO = ±1

→ the parameter space of O(N) decomposes into two disconnected pieces:a) transformations with detO = 1

– continuously connected to the identity– SO(N) which is a subgroup of O(N)

b) transformations with detO = −1– not continuously connected to the identity– they are products of rotations and reflections

• the most important case is O(3)– 3 parameters– homomorphic to SU(2) (see Sec. 1.11)– often parameterized by Euler angles

7) non-compact unitary group U(n,m):• Transformations in an (n+m)-dimensional complex space that leave

|z1|2 + · · ·+ |zn|2 − |zn+1|2 − · · · − |zn+m|2

invariant• i.e., the metric is

diag(1, . . . , 1︸︷︷︸n

,−1, . . . ,−1︸︷︷︸m

)

8) non-compact orthogonal group O(n,m):• transformations in an (n+m)-dimensional real space which leave

x21 + · · ·+ x2

n − x2n+1 − · · · − x2

n+m

invariant (metric is the same as in 7.)

9) Lorentz group O(3,1) (see Sec. 1.11):• inhomogeneous Lorentz group = Poincaré group = O(3,1) + 4-dimensional trans-lations

x′µ = Λνµxν + aµ

• most definitions (subgroups, factor groups etc.) are analogous to those for finitegroups, e.g.,

U(n) = (SU(n)⊗U(1))/Zn , O(n)/SO(n) = Z2

6.3. Invariant integration 69

6.3. Invariant integration

• in many proofs we used the rearrangement lemma in the formn∑i=1

f(Ai) =n∑i=1

f(AiB) =n∑i=1

f(BAi) (∗)

with n = order(G) and Ai, B ∈ G.

• instead of the sum over i we now have integrals over the parameters α1, . . . , αn of thegroup G. In the following we write A = g(α1, . . . , αn) ∈ G. To obtain an analogue of (∗)we need an integration measure

dµ(A) = dµ(α1, . . . , αn) = ρ(α1, . . . , αn)︸︷︷︸density function

dα1 . . . dαn

such that ∫G

dµ(A)f(A) =∫G

dµ(A)f(AB) =∫G

dµ(A)f(BA)

for arbitrary B ∈ G.

• the integration is over the entire parameter space of G:

→∫G

dµ(A)f(AB) A′=AB=

∫G

dµ(A′B−1)f(A′) !=∫G

dµ(A′)f(A′)

we want this to hold for arbitrary functions f and therefore have to require

dµ(A) = dµ(AB) = dµ(BA) ∀B ∈ G

such a dµ(A) is called invariant integration measure or Haar measure

• if we find such a measure, many of the results obtained for finite groups can be carriedover to continuous groups

6.4. Properties of compact Lie groups

• out of the examples in Sec. 6.2, U(N), O(N), and their subgroups are compact

• Theorem: For a compact Lie group there exists an invariant integration measure. (Forconcrete calculations we often do not need to know it explicitly.)

• a compact Lie group has a countably infinite number of irreps, all of finite dimension.

• every matrix representation is equivalent to a unitary representation

• every representation can be decomposed into a sum of irreps

• the matrix elements of the irreps are now functions of the group parameters and satifyorthogonality and completeness relations:∫

G

dµ(A)Dµ(A)∗ijDν(A)k` = δµνδikδj`nµ

∫G

dµ(A)

︸︷︷︸≡vol(G)∑

µij

nµDµ(A)ijDν(A′)∗ij = δ(A−A′)

70 6. Lie groups

– nµ = dim(G)– vol(G) = volume of the group (normalization arbitrary). Typically one normalizes

such that vol(G) = 1, which we will do in the following

• orthogonality relation for characters:∫G

dµ(A)χµ(A)∗χν(A) = δµνvol(G)

• criterion for the reducibility of the representation µ:∫G

dµ(A)|χµ(A)|2 = vol(G)

• for irreducible basis functions we have as in Sec. 4.1:⟨uµi

∣∣∣ vνj ⟩ = δµνδij1nµ

∑k

⟨uµk

∣∣∣ vµk⟩with the following definition of the scalar product:

〈f | g〉 =∫G

dµ(A)f∗(A)g(A)

• Peter-Weyl Theorem: Every well-behaved function f(α1, . . . , αn) on the parameterspace of G can be expanded in the irreducible representation functions of G: (vol(G) = 1)

f(α1, . . . , αn) =∞∑µ=1

nµ∑i,j=1

cµijDµ(α1, . . . , αn)ij

with

cµij = nµ⟨Dµij

∣∣∣ f〉 = nµ

∫G

dµ(A)Dµ(A)∗ijf(A)

6.5. Generators of Lie groups

• many properties of Lie groups follow from the behavior of the group elements near theidentity (infinitesimal transformations). Finite transformations are then obtained bymany successive infinitesimal transformations.

• in the following we choose the parameters such that α0i = 0 for all i, i.e., g(0, . . . , 0) = I

• because of e0 = 1 it is sensible to write an infinitesimal group element as

g(ε1, . . . , εn) = ei∑n

j=1 εjSj

the operators Si are called generators of the group (one generator per parameter)

• for the special groups with det(g) = 1 the generators are traceless:

det(g) = eTr ln g = ei∑

jεjTrSj != 1 → TrSj = 0

6.5. Generators of Lie groups 71

• for unitary groups with gg† = 1 the generators are Hermitian:

g† = g−1 → e−i∑

jεjS†j = e−i

∑jεjSj → Sj = S†j

• the generators are formally defined through derivatives:

iS1 = limε→0

g(ε, 0, . . . , 0)− g(0, . . . , 0)ε

etc.

i.e., Si = −i ∂g∂αi

∣∣∣α=0

here we have to distinguish two cases:1) the group elements A are maps of a coordinate system on itself2) the group elements are OA operators acting on functions (see Sec. 2.4.2)

• example: SO(3), i.e., rotations in 3 dimensions. Here we use the following (widely used)conventions:(a) ε = −ϕ for a mathematically positive rotation angle ϕ (counterclockwise)(b) x′i = Aijxj , i.e., ~x′ = A~x

(c) the OA operators are defined by(OAf

)(~x) = f(A−1~x), as in Sec. 2.4.2

case 1: rotation by an angle ϕ about the z-axis: ~x′ = A~x with

Az(ϕ) =

cosϕ − sinϕ 0sinϕ cosϕ 0

0 0 1

→ Sz = i limϕ→0

Az(ϕ)− 1ϕ

=

0 −i 0i 0 00 0 0

analogously : Sx =

0 0 00 0 −i0 i 0

, Sy =

0 0 i0 0 0−i 0 0

case 2: for an infinitesimal rotation about the z-axis we have

A−1z (ϕ)~x =

1 ϕ 0−ϕ 1 00 0 1

xyz

=

x+ yϕy − xϕz

→

(OAz(ϕ)f

)(~x) = f(A−1

z ~x) = f(x+ yϕ, y − xϕ, z)

= f(x, y, z) + ϕ(y∂x − x∂y)f(x, y, z) + · · ·= (1− iϕSz)f(x, y, z) + · · ·

→ Sz = i(y∂x − x∂y) = Lz

~L is the angular momentum operator:

~L = ~r × ~p = ~r × 1i∇ (with ~ = 1)

analogously : Sx = Lx , Sy = Ly

• physical meaning: the generators are operators corresponding to measurable quantities

• the commutator of two generators is a linear combination of all generators. To see this,consider infinitesimal transformations with one generator each:

72 6. Lie groups

•p

gi gj

g−1ig−1

j

gij

gi = eiεiSi = 1 + iεiSi + ε2i

2 S2i + · · · (no sum over i)

g−1j g−1

i gjgi =(1− iεjSj +

ε2j

2 S2j

)(1− iεiSi + ε2

i

2 S2i

)(1 + iεjSj + ε2

i

2 S2j

)(1 + iεiSi + ε2

i

2 S2i

)+ · · · = · · · =

= 1 + εiεj [Si, Sj ] + · · ·

≡ gij!= 1 + εiεj

∑k

ckijSk + · · ·

since gij must be a group element whose infinitesimal parameters are proportional toεiεj .

→ [Si, Sj ] =n∑k=1

ckijSk

– the coefficients ckij are called structure constants of the Lie group

– the commutation relations of the generators correspond to the multiplication lawof the group near the identity, i.e., they determine the local structure of the group,but not the global (topological) properties

• for the structure constants we have ckij = −ckji and the Jacobi identity

∑m

(cmij c

nmk + cmjkc

nmi + cmkic

nmj

)= 0

• the generators form a vector space of dimension n, the Lie algebra L(G) of the groupG

– the law of addition of the generators follows from the multiplication law of the group

– furthermore a multiplication can be defined:

Si · Sj ≡ [Si, Sj ]

– the Lie algebra is closed under this multiplication

• a subalgebra is a set of generators that is closed under multiplication

• if H is a subgroup of G, L(H) is a subalgebra of L(G)

• a subalgebra L(H) of L(G) is called invariant if [Sg, Sh] ∈ L(H) for all Sg ∈ L(G) andSh ∈ L(H)

• a Lie group is called simple if it does not have any nontrivial abelian invariant Liesubgroups (discrete subgroups are allowed).It is called semi-simple if it does not have any nontrivial abelian invariant Lie subgroups

6.6. SO(2) 73

• the structure constants form a representation of the Lie algebra, the adjoint represen-tation:

Si → ckij

these are n matrices (i = 1, . . . , n) with k = row index and j = column index.Proof: These n matrices satisfy the same commutation relations as the generators.

• if the Lie group is simple, the adjoint representation is irreducible; otherwise it is re-ducible.

• the Killing form is a scalar product of A,B ∈ L(G) defined as:

(A,B) ≡ Tr[Dad(A)Dad(B)

]≡ Tr ad(AB)

• applied to the generators we obtain the Cartan metric

gij ≡ Tr ad(SiSj) =n∑

k,`=1cki`c

`jk (i, j = 1, . . . , n)

det gij 6= 0 is a necessary and sufficient condition for G to be semi-simple

• the number ` of mutually commuting generators is called rank of G:

[Si, Sj ] = 0 for i, j = 1, . . . , `

(every group has at least rank 1)The commuting generators span the Cartan subalgebra and can be diagonalized si-multaneously

• for a group of rank ` there are ` Casimir operators (polynomials in the generators)that commute with all generators.Example: Quadratic Casimir operator C2 =

∑ij gijSiSj (with gij the Cartan metric)

6.6. SO(2)

• SO(2) = group of rotations in the plane about a fixed point

• 1 parameter; natural choice: rotation angle ϕ with 0 ≤ ϕ < 2π.

• "defining" representation: Action of SO(2) on a 2-dimensional vector

xi → Rijxj with R(ϕ) =(

cosϕ − sinϕsinϕ cosϕ

)

• abelian group: R(ϕ1)R(ϕ2) = R(ϕ1 + ϕ2)

• infinitesimal rotation with generator J :

R(dϕ) = e−idϕJ = 1− idϕJ + · · ·→ R(ϕ+ dϕ) = R(ϕ)R(dϕ) = R(ϕ)− idϕR(ϕ)J

= R(ϕ) + dϕdR(ϕ)dϕ + · · ·

→ dR(ϕ)dϕ = −iR(ϕ)J

this is a differential equation with boundary condition R(0) = 1 → solution: R(ϕ) =e−iϕJ (this is true for finite rotations, not just infinitesimal ones)

74 6. Lie groups

• similarly to Sec. 6.5 we obtain

J =(

0 −ii 0

)this gives (with J2 = 1):

R(ϕ) = e−iϕJ = 1− iϕJ − 1ϕ2

2! − iJ(− ϕ3

3!)

+ · · ·

= 1 cosϕ− iJ sinϕ =(

cosϕ − sinϕsinϕ cosϕ

)X

• now consider a vector space V and a representation of SO(2) by unitary operators U(ϕ)on V :– in analogy to the calculation above we find U(ϕ) = e−iϕJ with a Hermitian operatorJ (so that U is unitary)

– since SO(2) is abelian, all irreps are 1-dimensional. Thus we have for a vector |α〉in an irreducible subspace

J |α〉 = α |α〉U(ϕ) |α〉 = e−iϕα |α〉

with α ∈ R (so that U is unitary)– because of R(2π) = R(0) we have e2πiα = 1 → α = m ∈ Z, i.e., we have irrepsUm(ϕ) = e−imϕ that are characterized by integers m:(a) m = 0: R(ϕ)→ U0(ϕ) = 1 (trivial representation)(b) m = 1: R(ϕ) → U1(ϕ) = e−iϕ; this is an isomorphism between SO(2) and the

complex numbers on the unit circle(c) m = −1: R(ϕ)→ U−1(ϕ) = eiϕ; like (b), but in the other direction(d) m = ±2: R(ϕ) → U±2(ϕ) = e∓2iϕ; homomorphism between SO(2) and the

unit circle such that the circle is covered twice, etc. for higher m.Only the irreps with m = ±1 are faithful

• the defining representation is reducible and can be reduced by diagonalizing

J =(

0 −ii 0

)it has eigenvalues ±1 and eigenvectors e± = x± iy:

Je± = ±e± , R(ϕ)e± = e∓iϕe± , i.e., R = U1 ⊕ U−1

• the invariant integration measure is dϕ/2π, since dϕ = d(ϕ + ϕ) = d(ϕ + ϕ) withϕ = const, and vol(SO(2)) = 1

• orthogonality and completeness relations:2π∫0

dϕ2π U

m(ϕ)∗Un(ϕ) =2π∫0

dϕ2π ei(m−n)ϕ = δmn

∞∑n=−∞

Un(ϕ)Un(ϕ′)∗ =∞∑

n=−∞e−in(ϕ−ϕ′) = δ(ϕ− ϕ′)

the Um(ϕ) = e−imϕ are the basis functions of the Fourier expansion

6.7. SO(3) 75

• multivalued representations and global properties of the group:– consider the following map:

R(ϕ) → U1/2(ϕ) = e−iϕ/2 (∗)U1/2(ϕ+ 2π) = e−iϕ/2−iπ = −U1/2(ϕ)U1/2(ϕ+ 4π) = e−iϕ/2−2πi = U1/2(ϕ)

(∗) defines a 1-to-2 map, i.e., with every R(ϕ) we associate two numbers ±e−iϕ/2.This is a two-valued representation of SO(2) in the sense that the group multiplica-tion is preseved if both numbers are acceptable (e.g., if only bilinears like the scalarproduct 〈ψ1 | ψ2〉 are physically relevant, since these are then still invariant undera rotation by 2π)

– more generally we can also define m-valued representations:

R(ϕ) → Un/m(ϕ) = e−iϕn/m

(n and m must not have a common denominator)– in general, continuous groups have multivalued representations only if the parameter

space is multiply connected (i.e., it is topologically nontrivial)– in case of SO(2) there are closed paths in parameter space (0 ≤ ϕ < 2πm) that

wind m times around the circle and therefore cannot be continuously deformed intoeach other

– according to our current knowledge:∗ in classical physics only single-valued representations occur∗ in quantum physics only single- and two-valued representations occur

6.7. SO(3)

6.7.1. Angle-and-axis parametrization

• SO(3) = group of rotations in 3 dimensions; 3 real parameters

• we can parametrize a rotation as Rn(ψ) with rotation angle ψ and rotation axis n:

n =

sin θ cosϕsin θ sinϕ

cos θ

y

z

x

n

θ

ϕ

• intervals of the parameters: 0 ≤ θ ≤ π, 0 ≤ ϕ < 2π, 0 ≤ ψ ≤ π, since Rn(2π − ψ) =R−n(ψ)

• rotations about a fixed axis form a subgroup of SO(3), which is isomorphic to SO(2)

• for an arbitrary rotation R ∈SO(3) we have

RRn(ψ)R−1 = RRn(ψ)

hence all rotations by the same angle are in the same class

76 6. Lie groups

• a rotation can be visualized by a vector ~ψ = ψn. The tips of these vectors fill a ball Bwith radius π:

y

z

x

n

θ

ϕπ

• the parametrization has a redundancy: R−n(π) = Rn(π). Therefore two points that arediametrically opposite on the surface of the ball B can be identified with each other

• this means that there are two types of closed curves in parameter space:(a) curves that can be shrunk to a point by continuous deformations(b) curves for which this is not possible

a

b

•

•

p

p

curve b is also closed!(in parameter space)

these global properties influence the possible representations of the group

6.7.2. Euler angles

• a rotation can also be parametrized by Euler angles:

R = R3(α)R2(β)R3(γ)

with

R2(ψ) = Ry(ψ) =

cosψ 0 − sinψ0 1 0

sinψ 0 cosψ

, R3(ψ) = Rz(ψ) =

cosψ − sinψ 0sinψ cosψ 0

0 0 1

• parameter intervals: 0 ≤ α, γ < 2π , 0 ≤ β ≤ π

• relation to angle-and-axis parameters:

ϕ = 12(π − α+ γ) , tan θ =

tan β2

sin α+γ2

, cosψ = 2 cos2 β

2 cos2 α+ γ

2 − 1

6.7.3. Generators

• from Sec. 6.5 (with S → J):

J1 =

0 0 00 0 −i0 i 0

, J2 =

0 0 i0 0 0−i 0 0

, J3 =

0 −i 0i 0 00 0 0

6.7. SO(3) 77

• this can also be written as

(Jk)ij = −iεkij = −iεijk

with the totally antisymmetric tensor ε

• the generators transform under SO(3) like vectors, i.e.,

RJkR−1 = J`R`K

(proof by matrix multiplication using the Euler angle parametrization)

• the generator of a rotation about the axis n = nkek is

Jn = nkJk (proof in exercises)

• the rotation in both parametrizations can be expressed in terms of the generators:

Rn(ψ) = e−iψJknk

R(α, β, γ) = e−iαJ3e−iβJ2e−iγJ3

• the Lie algebra of SO(3) is [Ji, Jj ] = iεijkJk

• the rank of SO(3) is 1, i.e., there is only one Casimir operator. The structure constantsare ckij = iεijk→ the Cartan metric is gij ∼ δij , and thus

C2 = J2 = J21 + J2

2 + J23 with [Jk, J2] = 0 for k = 1, 2, 3

• if the Hamiltonian of a quantum mechanical system is invariant under rotations, we have

[H,Rn(ψ)] = 0 for all n and ψ→ [H,Jk] = 0 for k = 1, 2, 3

as shown in Sec. 3.2, the states of the system then transform in irreps of SO(3)

6.7.4. Irreps of SO(3)

• every representation of a Lie group is also a representation of the Lie algebra

• a representation of the Lie algebra provides us with a representation of the Lie group ifcertain global (topological) conditions are met

• we first construct irreps of the Lie algebra on a space V (see QM I). To do so we constructirreducible subspaces of V as follows:1) choose a suitable starting vector2) generate an irreducible basis by repeated application of the generators on this vector

• Schur’s lemma 1 implies that J2 ∼ 1 in an irreducible subspace→ all states |m〉 in the irreducible subspace have the same eigenvalue of J2

→ the irreps of the SO(3) algebra can be labeled by the eigenvalues of J2

78 6. Lie groups

• as a starting vector we choose an eigenstate of the commuting operators J2 and J3:

J3 |m〉 = m |m〉

• define

J± = J1 ± iJ2

a short calculation gives

J3(J± |m〉) = (m± 1)(J± |m〉) or 0→ J± |m〉 ∼ |m± 1〉 or 0

• since the irreducible subspace has finite dimension, the sequence must terminate, say form = j at the upper end and for m = ` at the lower end

J3 |j〉 = j |j〉 , J3 |`〉 = ` |`〉J+ |j〉 = 0 , J− |`〉 = 0

this yields

J2 |j〉 = (J23 + J3 + J+J−) |j〉 = j(j + 1) |j〉

J2 |`〉 = (J23 − J3 + J+J−) |`〉 = `(`− 1) |`〉

• since all states in the irreducible subspace have the same eigenvalue of J2, we obtain

j(j + 1) = `(`− 1)

2 solutions: ` = −j and ` = j + 1, but by assuming j ≥ ` we just have ` = −j

• since we go from j to ` = −j in integer steps we obtain

j − (−j) = 2j ∈ N

→ we have irreps Dj of the SO(3) algebra with j = 0, 12 , 1,

32 , 2, . . .

• the number of basis vectors in irrep j is 2j + 1 → dim(Dj) = 2j + 1

• for the orthonormal basis vectors |jm〉 we have

J2 |jm〉 = j(j + 1) |jm〉J3 |jm〉 = m |jm〉J± |jm〉 = [j(j + 1)−m(m± 1)]1/2 |j,m± 1〉

• the irreps of the SO(3) algebra yield irreps of the group SO(3):

U(α, β, γ) |jm〉 =∣∣∣jm′⟩Dj(α, β, γ)m′m

here U is the operator that implements the rotation R(α, β, γ) on V . Multiplying by 〈jn|gives

Dj(α, β, γ)nm = 〈jn|e−iαJ3e−iβJ2e−iγJ3 |jm〉= e−i(αn+γm) 〈jn|e−iβJ2 |jm〉︸︷︷︸

=:dj(β)nm

6.8. SU(2) 79

• for j = 0, 1, 2, . . . these irreps are single-valued, and for j = 12 ,

32 , . . . they are two-valued.

Proof (with α = β = 0 , γ = 2π):

Dj [R3(2π)]nm = e−i2πm 〈jn | jm〉 = (−1)2mδnm∣∣∣ · (−1)2(j−m) = 1

= (−1)2jδnm

for arbitrary n we have Rn(2π) = RR3(2π)R−1 with a suitable R

→ Dj [Rn(2π)] = (−1)2j1 for all n

the two-valued irreps exist since the parameter space is doubly connected (see Sec. 6.7.1)

• however, all these irreps are single-valued irreps of SU(2) (see Sec. 6.8.2)

• relation between representation matrices and spherical harmonics and Legendre func-tions:

Y`m(θ, ϕ) =

√2`+ 1

4π D`(ϕ, θ, 0)∗m0

P`m(cos θ) = (−1)m√

(`+m)!(`−m)! d

`(θ)m0

• characters: since all rotations by the same angle ψ are in one class, it is sufficient toconsider R3(ψ):

χj(ψ) =∑m

Dj [R3(ψ)]mm =j∑

m=−je−imψ =

sin (2j+1)ψ2

sin ψ2

in particular, we have for the defining or fundamental 3-dimensional representation:

χ1(ψ) = 1 + 2 cosψ

• further properties in Sec. 6.8

6.8. SU(2)

6.8.1. Parametrization

• SU(2) = group of unitary 2×2 matrices with determinant 1; 3 real parameters

• let A ∈ SU(2) with

A =(a bc d

).

The conditions AA† = 1 and detA = 1 give 4 equations:

|a|2 + |b|2 = 1 ac∗ + bd∗ = 0|c|2 + |d|2 = 1 ad− bc = 1

the solution of these equations yields the standard parametrization of SU(2):

A =(

cos θeiζ − sin θeiη

sin θe−iη cos θe−iζ

)with 0 ≤ θ ≤ π

20 ≤ η, ζ < 2π

80 6. Lie groups

• the matrix A can also be written as

A =(r0 − ir3 −r2 − ir1r2 − ir1 r0 + ir3

)with ri ∈ Rand detA = r2

0 + r21 + r2

2 + r23 = 1

now interpret the ri as Cartesian coordinates in a 4-dimensional Euclidean space→ the parameter space of SU(2) is the surface of the unit sphere in this space→ the parameter space is compact and simply connected (as opposed to the parameter

space of SO(3))

6.8.2. Relationship between SO(3) and SU(2)

• with every 3-dimensional vector ~x = x1, x2, x3) we can associate a Hermitian, traceless2×2 matrix:

X =3∑i=1

xiσi

with the Pauli matrices

σ1 =(

0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

)

then

−detX = −det(

x3 x1 − ix2x1 + ix2 −x3

)= |~x|2

• now consider a linear map of X that is induced by an A ∈SU(2) (see Sec. 1.11):

X → X ′ = AXA′ (∗)

X ′ is again Hermitian and traceless and can therefore be associated with a 3-dimensionalvector ~x′

• because of detX ′ = detX we have |~x′|2 = |~x|2→ the SU(2) transformation (∗) induces an SO(3) transformation in the 3-dimensionalspace of the vectors ~x

• the two SU(2) matrices ±A yield the same rotation→ the map from SU(2) to SO(3) is 2-to-1. In other words: The kernel of the homomor-phism, i.e., the preimage of the identity of SO(3), consists of the two SU(2) matrices(

1 00 1

)and

(−1 00 −1

)

these form a Z2 subgroup of SU(2). Symbolically: SO(3) ' SU(2)/Z2

• in the parametrization of A by the ri we consider r1, r2, r3 as independent parametersand

r0 = ±√

1− (r21 + r2

2 + r23)

6.8. SU(2) 81

→ A = 1 corresponds to r1 = r2 = r3 = 0. Near the identity we have

A = 1− iσidri

→ the σi form a basis of the Lie algebra of SU(2)

• the three matrices 12σi satisfy the same commutation relations as the generators of SO(3):[

σk2 ,

σ`2

]= iεk`m

σm2

→ SO(3) and SU(2) have the same Lie algebra (but not the same global properties) ifwe replace Jk → σk

2

• a general SU(2) matrix is obtained by exponentiation:

A = e−i~σ2 ·~α

with ~α = nψ we obtain (exercises)

A = 1 cos ψ2 − i~σ · n sin ψ2

• in Sec. 6.7.4 we have constructed all irreps of the Lie algebra of SO(3) (and thus of theLie algebra of SU(2)). Since SU(2) is simply connected, all these irreps are single-valuedirreps of SU(2)

6.8.3. Inavriant integration

• reminder: we require∫G

dµ(A)f(A) =∫G

dµ(A)f(B−1A) =∫G

dµ(BA)f(A)

to be invariant for an arbitrary function f , therefore: dµ(A) != dµ(BA)

• general method to compute the Haar measure (for an arbitrary group):– for a given parametrzation A(ξ), compute ∂A

∂ξi

– express the products A−1 ∂A∂ξi

as linear combinations of the generators:

A−1(ξ)∂A(ξ)∂ξi

=n∑k=1

SkA(ξ)ki

this defines a matrix A(ξ)– the weight function is then

ρA(ξ) = det A(ξ)

and the Haar measure is: dµ(A) = ρA(ξ)∏ni=1 dξi

• proof: see Tung Sec. 8.2

• for SU(2) we obtain in the standard parametrization of Sec. 6.8.1 (exercises):

dµ(A) = 14π2 sin(2θ)dθ dη dζ

82 6. Lie groups

6.8.4. Orthogonality and completeness relations

• the irreducible representation functions of SU(2) and SO(3) satisfy the orthogonalityrelations (with vol(SU(2)) = 1):

(2j + 1)∫

dµ(A)Dj(A)∗mnDj′(A)m′n′ = δjj′δmm′δnn′

• in the parametrization by Euler angles this simplifies to

2j + 12

1∫−1

d(cosβ)dj(β)mndj′(β)mn = δjj′

(no summationover n and m

)

• Peter-Weyl theorem: The irreducible representation functions form a complete basis ofthe space of square-integrable functions of the group parameters, i.e.,

f(A) =∑jmn

f jmnDj(A)mn with f jmn = (2j + 1)

∫dµ(A)Dj(A)∗mnf(A)

• more generally, f(A) need not be a scalar function, but can also be a vector or an operator

7. Lorentz and Poincaré group

7.1. Relativistic kinematics

• in the following ~ = c = 1, gµν = diag(1,-1,-1,-1) for Minkowski space, t = x0, ~x =(x1, x2, x3), xµ = (x0, ~x)

• Lorentz transformations (LT) leave

x2 = xµxµ = gµνx

µxν = x20 − |~x|2

invariant

• space-time splits into 3 distinct regions:

x0

x1

lightcone

space-like space-like

time-like(future)

time-like(past)

– future cone: x2 > 0 and x0 > 0

– past cone: x2 > 0 and x0 < 0

– space cone: x2 < 0

these 3 regions are separated by the light cone (x2 = 0)

7.2. Generators and Lie algebra

• classification of homogeneous LT’s and homomorphism with Sl(2,C): see Sec. 1.11

• in the following we only consider continuous (i.e., proper and orthochronous) LT’s, i.e.,the group L0

• Poincaré group = Lorentz group and 4-dimensional translations

x′µ = Λµνxν + aµ with Λ−1 = ΛT

• the Poincaré group is the symmetry group of all relativistic quantum field theories→ elementary particles transform in irreps of the Poincaré group

• the Poincaré group has 10 generators:– 3 for rotations in 3-dimensional space, e.g.,

Jz = −i(x∂

∂y− y ∂

∂x

)for rotations about the z-axis. (Here and later we choose case 2 of Sec. 6.5, i.e.,OA operators; we could have chosen case 1, e.g., maps of the coordinate system toitself)

83

84 7. Lorentz and Poincaré group

– 3 for boosts, e.g.,

Kx = i(t∂

∂x− x ∂

∂t

)for boosts along the x-direction

– 4 for translations, e.g.,

Px = i ∂∂x

for translations in the x-directionin covariant notation:

Pµ = i ∂

∂xµ, Jµν = i

(xµ

∂

∂xν− xν

∂

∂xµ

)→

J0i = −Ji0 = Ki

Jij = −Jji = εijkJk

• this leads to the Lie algebra

[Pµ, Pν ] = 0

[Pµ, Jρσ] = i(gµρPσ − gµσPρ

)[Jµν , Jρσ] = i

(gνρJµσ − gµρJνσ + gµσJνρ − gνσJµρ

)→ translations form an abelian subgroup→ rotations form a subgroup→ pure boosts do not form a subgroup→ rotations plus boosts form the continuous Lorentz group L0

• introducing the definitions:

Mi = 12(Ji + iKi) and Ni = 1

2(Ji − iKi)

we obtain

[Mi,Mj ] = iεijkMk [Mi, Nj ] = 0[Ni, Nj ] = iεijkNk

→ the Lorentz group has the same algebra as SU(2)⊗SU(2)

7.3. Finite-dimensional representations of the Lorentz group

• the irreps of SU(2)⊗SU(2) are products of irreps of SU(2).Notation: (j1, j2) with j1,2 = 0, 1

2 , 1,32 , 2, . . .

These irreps have finite dimension (2j1 + 1)(2j2 + 1) and are also irreps of L0

• trivial representation: (0,0) (scalars)

• two fundamental representations:1) spinor or Weyl representation: (0,12) with dimension 2

– the representation matrices are the Sl(2,C) matrices A from Sec. 1.11

7.4. Unitary irreps of the Poincaré group 85

– the 2-dimensional objects that transform in (0,12) are called spinors ξ2) conjugate spinor representation: (1

2 , 0)– the representation matrices are A∗

– there is another type of spinor, η, that transforms in (12 ,0)

• these two spinor types transform into each other under a parity transformation: ξ ↔ η

• a Dirac spinor contains both types of spinors, i.e., ψ =(ξη

), and transforms in the

reducible representation (0, 12)⊗ (1

2 , 0) of L0; however, this is an irrep of the full Lorentzgroup if we include parity, since parity mixes ξ and η. The Dirac equation can be derivedby applying Lorentz boosts to a Dirac spinor

• four-vectors transform in the irrep (12 ,

12)

• antisymmetric tensors of rank 2 (6 components) transform in (1,0)⊗(0,1).Example: Electromagnetic field strength tensor Fµν

• symmetric tensors of rank 2 (10 components) can be decomposed into the trace (whichtransforms in (0,0)) and 9 traceless components (which transform in (1,1)).Example: Energy-momentum tensor Tµν

• L0 and SU(2)⊗SU(2) have the same Lie algebra, but:– L0 corresponds to the exponentiation of iJk, iKk, while SU(2)⊗SU(2) corresponds

to the exponentiation of iMk, iNk– the definition of Mk and Nk shows that the sets Jk,Kk and Mk, Nk cannot be

simultaneously Hermitian– thus unitary finite-dimensional irreps of SU(2)⊗SU(2) give non-unitary finite-di-

mensional irreps of L0

• physical states have to transform in unitary irreps since symmetry operations are realizedby unitary transformations (otherwise transition probabilities between states are notpreserved)

• the unitary irreps have infinite dimension since the Lorentz group is non-compact

7.4. Unitary irreps of the Poincaré group

7.4.1. One-particle states and Casimir operators

• let C be a Casimir operators of the group G, i.e., [C, Si] = 0 for all generators Si. Allbasis states of an irrep of G are eigenstates of C with the same eigenvalue (Schur’s lemma1)→ the irreps of G can be labeled by the eigenvalues of the Casimir operators

• in the following we consider the action of the Poincaré group on the Hilbert space ofone-particle states. We will learn:– the unitary irreps are characterized by the particle type (mass and spin/helicity)– for a given particle type (= irrep) the basis states of the irrep correspond to the

possible physical states (four-momentum and spin projection)


• within an irrep the states can be labeled by the eigenvalues of commuting generators

– because of [Pµ, P ν ] = 0 we can choose the states in Hilbert space to be eigenstates|p, σ〉 of four-momentum with

Pµ |p, σ〉 = pµ |p, σ〉

– here, σ is a placeholder for all other indices that are necessary to label a stateuniquely

• one Casimir operator of the Poincaré group is

C1 = PµPµ = P 2

→ C1 |p, σ〉 = p2 |p, σ〉 = (E2 − |~p|2) |p, σ〉 = m2 |p, σ〉

with m = rest mass of the state (i.e., of the particle)→ irreps can be labeled by m (among others)

• a second Casimir operator is

C2 = WµWµ , with Wµ = −12ε

µνρσJνρPσ

the Pauli-Lubanski (pseudo-)vector. (We will look at the eigenstates of C2 later.)

• the states transform under pure translations as follows:

T (a) |p, σ〉 = e−iaµPµ |p, σ〉 = e−iaµpµ |p, σ〉

i.e., in unitary irreps of T4 (with dimension 1 since T4 is abelian)

• pure LTs Λ are implemented on the Hilbert space by unitary operators U(Λ). U(Λ)generates a state with four-momentum p′ = Λp:

Pµ[U(Λ) |p, σ〉

]= U(Λ)

[U †(Λ)PµU(Λ)

]|p, σ〉

= U(Λ)[P ρ(Λ−1) µ

ρ

]|p, σ〉 (see Sec. 4.3)

= Λµρpρ[U(Λ) |p, σ〉

]= p′µ

[U(Λ) |p, σ〉

]→ the state U(Λ) |p, σ〉 is a linear combination of states with four-momentum p′:

U(Λ) |p, σ〉 =∑σ′

∣∣∣p′, σ′⟩D(Λ, p)σ′σ︸︷︷︸unitary representationof the Poincaré group

(∗)

• the Hilbert space is reducible and can be decomposed into irreducible subspaces w.r.t.the Poincaré group (see Sec. 7.4.2), i.e.,

– the matrices D can be brought to block diagonal form

– the states of a particular particle type can be identified with the basis states of anirrep of the Poincaré group

7.4. Unitary irreps of the Poincaré group 87

7.4.2. Little group and induced representations

• four-momenta pµ can be divided into six categories:

category standard vector kµ little group1. p2 = m2 > 0, p0 > 0 massive particles (m, 0, 0, 0) SO(3)2. p2 = m2 > 0, p0 < 0 (−m, 0, 0, 0) SO(3)3. p2 = 0, p0 > 0 massless particles (k, 0, 0, k) E24. p2 = 0, p0 < 0 (−k, 0, 0, k) E23. p2 < 0 tachyons (0, 0, 0, k) SO(2,1)3. pµ ≡ 0 vacuum (0, 0, 0, 0) L0

the cases 2., 4., and 5. are presumably not realized in nature

• within a category we can, for fixed p2, obtain an arbitrary pµ by applying a suitable LTto a common "standard vector" kµ (see table):

pµ = L(p)µνkν or in short p = L(p)k

the states with four-momentum pµ are then defined by:

|p, σ〉 ≡ U(L(p)

)|k, σ〉

• the set of all LTs A that leave a given pµ invariant, i.e.,

Aµνpν = pµ

is a subgroup of the Poincaré group, the so-called little group (or stabilizer or isotropygroup) of pµ

• little groups for the standard vectors: see table (for E2 see Sec. 7.4.4). (Within acategory the little group has the same structure for all pµ, i.e., it is conjugate to the littlegroup of kµ)

• now consider the action of an arbitrary LT Λ (with p′ = Λp) on |p, σ〉:

U(Λ) |p, σ〉 = U(Λ)U(L(p)

)|k, σ〉

= U(ΛL(p)

)|k, σ〉

= U(L(p′)

)U †(L(p′)

)U(ΛL(p)

)|k, σ〉

= U(L(p′)

)U(L(p′)−1ΛL(p)

)|k, σ〉

• L(p′)−1ΛL(p) leaves kµ invariant

L(p)k = p

ΛL(p)k = Λp = p′

L(p′)−1ΛL(p)k = L(p′)p′ = L(p′)−1L(p′)k = k

→ A(Λ, p) = L(p′)−1ΛL(p) is an element of the little group of kµ, i.e.,

U(A) |k, σ〉 =∑σ′

∣∣∣k, σ′⟩D(A)σ′σ︸︷︷︸unitary representationof the little group


and thus

U(Λ) |p, σ〉 = U(L(p′)

)U(A) |p, σ〉

=∑σ′

U(L(p′)

) ∣∣∣k, σ′⟩D(A)σ′σ

=∑σ′

∣∣∣p′, σ′⟩D (A(Λ, p))σ′σ

compare with (∗) in Sec. 7.4.1: the representations of the Poincaré group are inducedby the irreps of the little group (see Sec. 2.7)

• in general, irreps of a subgroup induce reducible representations of the group. However,in our case the representation of the Poincaré group induced by the irreps of the littlegroup are again irreducible. (For proof, see Tung Theorem 10.10 (ii))

7.4.3. Massive particles

• case 1 in Sec. 7.4.2

• the little group is the rotation group SO(3) with irreps labeled by s = 0, 12 , 1, . . .

• C2 = WµWµ applied to the standard vector (in the rest frame of the particle) yields

C2 |k, σ〉 = · · · = −m2 ~J2 |k, σ〉 = −m2s(s+ 1) |k, σ〉

with s = intrinsic angular momentum ("spin") of the particle→ spin is indeed a kind of angular momentum

• the unitary irreps of the Poincaré group are labeled by two parameters: mass m andspin s

• the unitary irreps are infinite-dimensional since, for fixed p2 = m2, p can take on infinitelymany values

• the index σ corresponds to the eigenvalues of the helicity operator ~J · P (= J3 if theboost is in the z-direction), see Tung Sec. 8.4.1

– the operator commutes with the Pµ and yields the spin component in the directionof motion

– under a LT the indices σ are mixed within an irrep

7.4.4. Massless particles

• case 3 of Sec. 7.4.2

• the generators of the little group are A = J1 + K2, B = J2 − K1, and J3. Thesethree generators have the same Lie algebra as the Euclidean group E2 (translations androtations in a plane)

• we have [A,B] = 0 → A and B can be diagonalized simultaneously

A |k, a, b〉 = a |k, a, b〉 , B |k, a, b〉 = b |k, a, b〉

7.5. Parity and time reversal 89

assume that there exists a pair of eigenvalues a, b 6= 0. This would lead to a continuumof eigenvalues, since after a rotation by an angle θ we have

|k, a, b〉θ ≡ U−1[R(θ)] |k, a, b〉

A |k, a, b〉θ = (a cos θ − b sin θ) |k, a, b〉θB |k, a, b〉θ = (a sin θ + b cos θ) |k, a, b〉θ

however, in nature we do not observe massless particles that have such a continuousdegree of freedom θ→ for physical states we require a = b = 0

• the physical states |k, σ〉 are then distinguished by the eigenvalues of J3:

J3 |k, σ〉 = σ |k, σ〉

– as in Sec. 7.4.3 σ corresponds to the helicity = spin projection on the direction ofmotion (recall ~k = (0, 0, k))

– σ can only take on integer and half-integer values (corresponding to single-valuedand double-valued irreps, respectively)

– for massless particles σ is invariant under LT’s

• states with different helicity can in principle be viewed as different particles, but: Paritytransformations exchange states with opposite helicity→ if the theory is invariant under parity, the two states correspond to the same particle(with two helicity states).Examples: photon with σ = ±1

graviton with σ = ±2but: If neutrinos were massless, we would have neutrinos with σ = −1

2and antineutrinos with σ = 1

2 , since the weak interactions violate parity(however, we now know that neutrinos are massive)

7.4.5. Tachyons

see Tung Sec. 10.4.5

7.4.6. Vacuum

see Tung Sec. 10.4.1

7.5. Parity and time reversal

read Tung Sec. 11 and 12further reading: Weinberg, The quantum theroy of fields, Vol. I, Sec. 2

8. The tensor method to construct irrepsof Gl(m) and its subgroups

8.1. Tensors and tensor spaces

• Let Vm be an m-dimensional vector space and g be the set of invertible linear trans-fomations on Vm. The transformations g form a group which is isomorphic to Gl(m,C)

• let |i〉 , i = 1, . . . ,m be a basis of Vm– a matrix representation of Gl(m) is given by

g |i〉 = |j〉 gji (sum over j) , with det(gij) 6= 0

– this is the so-called defining or fundamental representation of Gl(m)– this representation is irreducible, and we simply call it g

• the product space V nm = Vm ⊗ · · · ⊗ Vm (n factors) is called tensor space

• a basis of V nm is given by

|i1 . . . in〉 ≡ |i1〉 ⊗ · · · ⊗ |in〉 with ik = 1, . . . ,m

in short: |i〉n

• an arbitrary element |x〉 ∈ V nm can be written as

|x〉 = |i1 . . . in〉xi1...in

in short: |x〉 = |i〉n xi. The xi are called tensor components of |x〉

• the elements of Gl(m) induce the following transformation on V nm:

g |i〉n = |j〉nD(g)ji with D(g)ji = gj1i1 . . . gjnin

the D(g) form an mn-dimensional representation of Gl(m), the product representationg ⊗ · · · ⊗ g, with carrier space V n

m

• the |x〉 ∈ V nm transform under Gl(m) as follows:

|xg〉 ≡ g |x〉 = g |i〉n xi = |j〉nD(g)jixi!= |j〉n x

gj

→ xgj = D(g)jixi

the objects that transform like |x〉 are called tensors of rank n under Gl(m). Inparticular, vectors are tensors of rank 1

• if we restrict the operators or matrices g to the subgroups U(m), SU(m), or O(m), wecan define tensors under these groups

90

8.2. Action of the symmetric group on the tensor space 91

8.2. Action of the symmetric group on the tensor space

• consider a permutation p =(

1 . . . np1 . . . pn

)∈ Sn and associate with it a linear transfor-

mation on V nm:

|xp〉 ≡ p |x〉 = |i1 . . . in〉xpi1...in with xpi1...in ≡ xip1 ...ipn

here p only acts on the tensor components xi, but not on the basis tensors |i〉n

• on the other hand, we could also let p act on the basis tensors (but not on the tensorcomponents). Because of

|xp〉 = |i1 . . . in〉xip1 ...ipn = |ip−11. . . ip−1

n〉xi1...in

the action of p on the basis tensors is given by

p |i1 . . . in〉 = |ip−11. . . ip−1

n〉 = |ip−1〉n

Therefore we have

p |i〉n = |j〉nD(p)ji with D(p)ji = δj1ip−11. . . δjnip−1

n= δjp1 i1 . . . δjpn in

• example: m = 2, n = 3, p = (123), and

|x〉 = |111〉+ 3 |112〉+ 4 |122〉+ 2 |211〉+ 5 |212〉+ 4 |221〉

i.e., we have

x111 = 1 , x112 = 3 , x121 = 0 , x122 = 4x211 = 2 , x212 = 5 , x221 = 4 , x222 = 0

– possibility 1, action on tensor components: x(123)i1i2i3

= xi2i3i1 , i.e.,

x(123)111 = 1 , x

(123)112 = 0 , x

(123)121 = 2 , x

(123)122 = 4

x(123)211 = 3 , x

(123)212 = 4 , x

(123)221 = 5 , x

(123)222 = 0

and thus

|x(123)〉 = |111〉+ 2 |121〉+ 4 |122〉+ 3 |211〉+ 4 |212〉+ 5 |221〉

– possibility 2, action on basis tensors: p−1 = (321)→ p |i1i2i3〉 = |i3i1i2〉

p |111〉 = |111〉 , p |112〉 = |211〉 , p |121〉 = |112〉 , p |122〉 = |212〉p |211〉 = |121〉 , p |212〉 = |221〉 , p |221〉 = |122〉 , p |222〉 = |222〉

and thus

|x(123)〉 = |111〉+ 3 |211〉+ 4 |212〉+ 2 |121〉+ 5 |221〉+ 4 |122〉

92 8. The tensor method to construct irreps of Gl(m) and its subgroups

• important point #1: The representation matrices D(g) and D(p) have the followingsymmetry: For p ∈ Sn and ip = ip1 . . . ipn we have

Dji = Djpip

i.e., the matrices are invariant if the same permutation acts simultaneously on the indicesof i and j (only the order of the factors of g in D(g) or of the Kronecker deltas in D(p)changes)

• important point # 2: the matrices D(g) (g ∈ Gl(m)) and D(p) (p ∈ Sn) commute, i.e.,

pg |i〉n = gp |i〉n

(proof in exercises)

• both the representation D(g) of Gl(m) and the representation D(p) of Sn on V nm are in

general reducible– from Sec. 5 we know how to decomposeD(p) in irreducible components: by applying

the irreducible symmetrizers of the algebra Sn– in the following we will see that this process also leads to a reduction of D(g)

8.3. Decomposition of the tensor space into irreduciblesubspaces under Sn and Gl(m)

8.3.1. Symmetry classes in tensor space

• let (as in Sec.5):– Θp

λ be a Young tableau– epλ be the corresponding Young operator– Lλ = reλ | r ∈ Sn be the irreducible left ideal generated by eλ

• in the following we will learn:– a subspace of the type

span(reλ |α〉)for fixed |α〉 ∈ V n

m and arbitraryr ∈ Sn (i.e., reλ ∈ Lλ)

is invariant and irreducible under Sn– a subspace of the type

span(epλ |α〉) for fixed epλ and arbitrary |α〉 ∈ V nm

is invariant and irreducible under Gl(m)– the tensor space V n

m can be decomposed such that the basis tensors have the form|λ, α, a〉 withλ = symmetry class, given by the Young diagramα = index for the different invariant irreducible subspaces under Sna = index for the different invariant irreducible subspaces under Gl(m)

• for a given Young tableau the epλ |α〉 | |α〉 ∈ V nm are called tensors of symmetry Θp

λ

8.3. Decomposition of the tensor space into irreducible subspaces under Sn and Gl(m) 93

• for a given Young diagram the repλ |α〉 | r ∈ Sn, |α〉 ∈ V nm are called tensors of

symmetry class λ

• consider a subspace Tλ(α) = reλ |α〉 | r ∈ Sn for fixed |α〉, then Tλ(α) is either empty,or– Tλ(α) is invariant and irreducible under Sn– the representation of Sn on Tλ(α) is given by the irrep that is generated by eλ onSn

proof:– let |x〉 ∈ Tλ(α), then we have by definition

|x〉 = reλ |α〉 for some r ∈ Sn→ p |x〉 = pr︸︷︷︸

r′∈Sn

eλ |α〉 ∈ Tλ(α)

→ Tλ(α) is invariant under Sn– let rieλ be a basis of Lλ, then rieλ |α〉 is a basis of Tλ(α). The action of Sn onLλ is:

p |rieλ〉 = |prieλ〉 =∣∣∣rjeλ⟩Dλ(p)ji ∀p ∈ Sn

→ the action of Sn on Tλ(α) is

prieλ |α〉 = rjeλ |α〉Dλ(p)ji→ Tλ(α) is irreducible and the reprsentation matrices on Tλ(α) are the same asthose on Sn

8.3.2. Totally symmetric and totally antisymmetric tensors

• let Θλ=s =

n boxes︷︸︸︷· · · , i.e., es = s the total symmetrizer of Sn. Because of pes = es (for

all p ∈ Sn), Ls is 1-dimensional→ for given |α〉 the irreducible subspace Ts(α) = span(es |α〉) is 1-dimensional. Suchtensors are totally symmetric (in all indices):

es |α〉 = 1n!∑p

p |i〉n αi = |i〉n1n!∑p

αip

→ the tensor components are symmetric in all indices→ all p ∈ Sn leave a totally symmetric tensor invariant→ the representation of Sn on Ts(α) is the 1-dimensional trivial representation (p→ 1)

• example: m = 2, n = 3 → es = 16 [e + (12) + (13) + (23) + (123) + (321)]. In this

case there are 4 different totally symmetric tensors:

|α〉 = |111〉 es |α〉 = |111〉 ≡ |s, 1, 1〉

|α〉 = |112〉 es |α〉 = 13(|112〉+ |121〉+ |211〉

)≡ |s, 2, 1〉

|α〉 = |122〉 es |α〉 = 13(|122〉+ |221〉+ |212〉

)≡ |s, 3, 1〉

|α〉 = |222〉 es |α〉 = |222〉 ≡ |s, 4, 1〉


• the space spanned by the tensors of symmetry class s is called T ′s

• totally antisymmetric tensors (λ = a) exist for n ≤ m, i.e., only up to rank n = m (thisis obvious from the fact that for n > m there are at least two of the n indices of thetensor components that are the same and thus all the components vanish)– La and Ta(α) are 1-dimensional– the representation of Sn on Ta(α) is the 1-dimensional irrep p→ (−1)p

• example: tensors of rank 2 (n = 2) in m dimensions

es |ii〉 = |ii〉 i = 1, . . . ,m

es |ij〉 = 12(|ij〉+ |ji〉

)i 6= j

→ m+ 12(m2 −m) = 1

2(m2 +m) totally symmetric tensors

es |ii〉 = |ii〉 i = 1, . . . ,m

es |ij〉 = 12(|ij〉 − |ji〉

)i 6= j

→ 12(m2 −m) totally antisymmetric tensors

8.3.3. Tensors with mixed symmetry

• consider again the example of rank-3 tensors in m = 2 dimensions, and in particulartensors of symmetry Θλ=κ = 1 2

3 with eκ = [e+ (12)][e− (13)].From Sec. 5.4: Lκ = span(eκ, (23)eκ)1) first pick |α〉 = |112〉

eκ |α〉 = [e+ (12)](|112〉 − |211〉

)= 2 |112〉 − |211〉 − |121〉 ≡ |κ, 1, 1〉

(23)eκ |α〉 = (23)(2 |112〉 − |211〉 − |121〉

)= 2 |121〉 − |211〉 − |112〉 ≡ |κ, 1, 2〉

for all r ∈ S3, reκ |α〉 is a linear combination of these two tensors→ these two mixed tensors form a basis for a 2-dimensional subspace Tκ(1) that isinvariant and irreducible under S3 (see Sec. 5.4)

2) now pick |α〉 = |221〉

eκ |α〉 = 2 |221〉 − |212〉 − |122〉 ≡ |κ, 2, 1〉(23)eκ |α〉 = 2 |212〉 − |221〉 − |122〉 ≡ |κ, 2, 2〉

this is a basis for another 2-dimensional invariant irreducible subspace Tκ(2)

• |κ, 1, 1〉 and |κ, 2, 1〉 are tensors of symmetry Θκ and span the 2-dimensional spaceT ′κ(1) ≡ eκ |α〉 | |α〉 ∈ V 3

2 (exercises)– T ′κ(1) is invariant under Gl(2):

geκ |α〉 = eκg |α〉 ∈ T ′κ(1)

– T ′κ(1) is irreducible under Gl(2) (exercises)

8.3. Decomposition of the tensor space into irreducible subspaces under Sn and Gl(m) 95

• |κ, 1, 2〉 and |κ, 2, 2〉 are tensors of symmetry Θ(23)κ and span the 2-dimensional space

T ′κ(2) = e(23)κ |α〉 | |α〉 ∈ V 3

2 . T ′κ(2) is also invariant and irreducible under Gl(m)

• the spaces T ′κ(a) (a = 1, 2) contain all tensors of symmetry class κ

• thus the 8-dimensional tensor space V 32 is fully reduced:

V 32 = T ′s ⊕ Tκ(1)⊕ Tκ(2)

invariant under S3

= T ′s ⊕ T ′κ(1)⊕ T ′κ(2)

invariant under Gl(2)

we can choose as basis tensors of V 32 :

– the 4 totally symmetric tensors |s, α, 1〉 (α = 1, . . . , 4) from Sec. 8.3.3

– the 4 tensors |κ, α, a〉 with α = 1, 2 and a = 1, 2

8.3.4. Complete reduction of the tensor space

from the example of Sec. 8.3.3 we can deduce the following general properties (proofs see TungSec. 5.5)

• two subspaces of V nm that are invariant and irreducible under Sn and belong to the same

symmetry class λ are either identical or disjoint

• two invariant irreducible subspaces belonging to different symmetry classes are alwaysdisjoint

• the tensor space can be fully decomposed into invariant irreducible subspaces under Sn:

V nm =

∑λ⊕

∑α⊕

Tλ(α)

here only Young diagrams with at most m rows occur

• the basis tensors of Tλ(α) are |λ, α, a〉 with a = 1, . . . ,dim(Tλ(α)

)• the basis tensors can be chosen such that the representation matrices of Sn on Tλ(α) areidentical for all α belonging to the same symmetry class λ:

p |λ, α, a〉 = |λ, α, b〉 Dλ(p)ba︸︷︷︸independent of α

∀p ∈ Sn

• the decomposition of V nm w.r.t. the symmetry classes of Sn automatically leads to a

decomposition of V nm into invariant, irreducible subspaces under Gl(m):

– the subspaces T ′λ(a) that are spanned by the |λ, α, a〉 with fixed λ and fixed a areinvariant and irreducible under Gl(m)

– the irrep of Gl(m) on T ′λ(a) that is furnished by the |λ, α, a〉 is independent of a:

g |λ, α, a〉 = |λ, β, a〉 Dλ(g)βα︸︷︷︸independent of a

∀g ∈ Gl(m)


• there are two formulas for the dimensions of the irreps of Gl(m) constructed in this way:

dim(Dλ)

=

m−1∏k=1

det[(λi↑# of boxes in row i of Θλ

−m− i)m−j]i,j=1,...,m

=

m−1∏k=1

m∏i<j

(λi − λj − i+ j)

=∏ij

m+ j − ihij↑

hooklength of box ij (see Sec. 5.6)

product over all boxes of Θλ,i = row index, j = column index

• back to the example of V 32 :

dim(D

)= det

(3 11 1

)= 2 , dim

(D

)= det

(4 10 1

)= 4

= 23 ·

31 ·

11 = 2 , = 2

3 ·32 ·

41 = 4

→ T ′s is a 4-dimensional invariant irreducible subspace under Gl(2) (or: T ′s consists of 41-dimensional invariant irreducible subspaces under S3).Symbolic notation for V 3

2 = T ′s ⊕ T ′κ(1) ⊕ T ′κ(2): = fundamental representation ofGl(m) (here m = 2), then

⊗ ⊗ = ⊕ ⊕

8.4. Irreps of U(m) and SU(m)

• the irreps of Gl(m) from Sec. 8.3.4 are also (subduced) representations of the subgroupsof Gl(m). In general these do not have to be irreps, but for U(m) and SU(m) they areirreps (but not for O(m))

• because of U(m) = SU(m) ⊗ U(1) the irreps of U(m) are given by

DλU(m)(U) = (detU)kDλ

SU(m)(U) for U ∈ U(m) with k ∈ Z

therefore in the following we only need to consider SU(m)

• for SU(m) the two irreps corresponding to the Young diagrams (λ1, . . . , λm) and (λ1 +k, . . . , λm + k) are equivalent, e.g., for m = 5 and k = 1:

∼

Proof: the two diagrams differ by a factor of (detU)k, and detU = 1 for U ∈ SU(m)

• from the proof one also sees that the Young diagram (m boxes) corresponds to thetrivial representation of SU(m), i.e., U → 1 for all U ∈ SU(m). Functions that transformunder SU(m) in this representation are called SU(m) scalars or SU(m) singlets (butsuch functions transform in the totally antisymmetric irrep of Sm)

8.5. Complex conjugate representation 97

• Irrep of SU(2):– fundamental representation: with dimension 2– trivial representation: with dimension 1– m = 2 → the Young diagrams have at most 2 rows– every irrep corresponding to a diagram with 2 rows is equivalent∗ either to , e.g.,

∼ ∼ ∼

∗ or to a diagram with 1 row that is obtained by cutting off all columns with 2boxes, e.g.,

∼ ∼

→ except for , we only need to consider diagrams with 1 row– dimensions of the irrep corresponding to a diagram with 1 row and k boxes:

∏ij

m+ j − ihij

=k∏j=1

2 + j − 1k − j + 1 = (k + 1)!

k! = k + 1

→ for SU(2), every k ∈ N corresponds to one and only one irrep with dimensionk + 1

• irreps of SU(3):– fundamental representation: with dimension 3

– trivial representation: with dimension 1– m = 3 → the Young diagrams have at most 3 rows

– all irreps are equivalent either to or to a diagram with at most 2 rows: Θλ =(λ1, λ2, 0) with

dim(Θλ) = 12 det

(λ1 + 2)2 λ1 + 2 1(λ2 + 1)2 λ2 + 1 1

0 0 1

= 12(λ1 + 2)(λ2 + 1)(λ1 − λ2 + 1)

8.5. Complex conjugate representation

• in the following we only consider SU(m)

• in the fundamental representation of SU(m) we have m objects, ϕ1, . . . , ϕm, that trans-form as

ϕi → ϕ′i = ϕjUji for U ∈ SU(m) (∗)

the fundamental representation has dimension m and is is often denoted by m

• the matrices U∗ also form a representation of SU(m) since the map U → U∗ preserves thegroup multiplication. This representation is called (complex) conjugate representationand is denoted by m


• conjugation of (∗) yields

ϕ∗i → ϕ∗j (Uji)∗ = ϕ∗j (U †)ij

we now define an object ϕi that transforms like ϕ∗i (but is not equal to ϕ∗i )

ϕi → ϕj(U †)ij

i.e., the ϕi transform in the conjugate representation.

• upper (lower) indices are called contravariant (covariant)

• more generally one can define tensors that transform in the product representationm⊗ · · · ⊗m︸︷︷︸

p factors

⊗m⊗ · · · ⊗m︸︷︷︸q factors

:

ϕj1...jqi1...ip

→ ϕ`1...`qk1...kp

Uk1i1. . . U

kpip

(U †)j1`1 . . . (U†)jq`q

and then reduce them (using the Young diagrams of Sp and Sq)

• for SU(2) the representations 2 and 2 are equivalent→ SU(2) has only real and pseudoreal irreps (the latter are equivalent to their conjugaterepresentations)

• SU(3)– for the irreps with Young diagram Θλ = (λ1, λ2, 0) define

p1 = λ1 − λ2 = # of columns with 1 boxp2 = λ2 = # of columns with 2 boxes

→ dim(Dλ) = 12(λ1 + 2)(λ2 + 1)(λ1 − λ2 + 1) = 1

2(p1 + 1)(p2 + 1)(p1 + p2 + 2)

– this irrep is also denoted by (p1, p2)– the irreps (p1, p2) and (p2, p1) have the same dimension and are complex conjugates

of each other, e.g.,

= (0, 1) ∼ (1, 0)∗ = ∗ = 3= (0, 3) ∼ (3, 0)∗ = ∗ = 10

8.6. Reduction of the product of two irreps

• suppose we have two irreps Dλ and Dµ of Gl(m), U(m), or SU(m) with Young diagramsΘλ and Θµ.Problem: Reduction of the rpoduct representation Dλ ⊗Dmu

• there is a simple graphical rule for this:1) For 1 ≤ i ≤ # of rows of Θµ, write the number i in all boxes of row i of Θµ.2) Consider all possibilities to add the boxes of Θµ to the Young diagram Θλ in the

order "first the 1s, then the 2s, etc." , according to the following rules:a) In every step the resulting Young diagram must be allowed and must not have

more than m rows.

8.6. Reduction of the product of two irreps 99

b) The same number must not appear more than once in the same column.

c) If the numbers are read in the order "rows from top to bottom, every row fromright to left", then in this sequence of numbers there must never (meaning atevery step of reading) be more i’s than (i− 1)’s.

3) If two Young diagrams created in this way have the same shape, they are onlycounted as different if the i’s are distributed differently.

4) For SU(m), columns with m boxes can be deleted (except for the trivial irrep)

• always check the dimensions on both sides!

• example 1: SU(2)

5⊗ 4 =(j = 2

)⊗(j = 3

2)

= ⊗ 1 1 1 =(

1 ⊕ 1

)⊗ 1 1

=(

11 ⊕ 11 ⊕ 1 1

)⊗ 1

= 11 1 ⊕ 111 ⊕ 1

1 1 ⊕ 11 1

= ⊕ ⊕ ⊕

= 8⊕ 6⊕ 4⊕ 2 =(j = 7

2

)⊕(j = 5

2

)⊕(j = 3

2

)⊕(j = 1

2

)

• example 2: SU(3)

3⊗ 3 = ⊗ 1 = 1 ⊕ 1 = 8⊕ 1

or 3⊗ 3 = ⊗ 12 =

(1 ⊕ 1

)⊗ 2 = 1

2 ⊕ 21 = 8⊕ 1

3⊗ 3 = ⊗ 1 = 1 ⊕ 1 = 6⊕ 3

3⊗ 3⊗ 3 = (6⊕ 3)⊗ 3 =(

⊕)⊗ 1 = 1 ⊕ 1 ⊕ 1 ⊕ 1

= 10⊕ 8⊕ 8⊕ 1

8⊗ 8 = ⊗ 1 12 =

(1 ⊕ 1 ⊕ 1

)⊗ 1

2

=(

1 1 ⊕ 11 ⊕

1

1⊕ 1

1

)⊗ 2

= 1 12 ⊕

1 1

2⊕ 1

1 2 ⊕1

12

⊕1

21

⊕ 112

= ⊕ ⊕ ⊕ ⊕ ⊕

= 27⊕ 10⊕ 10⊕ 8⊕ 8⊕ 1


8.7. Applications in (particle) physics

8.7.1. The Goldstone theorem

• suppose the Hamiltonian H of a system has a global, continuous symmetry group G oforder nG with generators Si, i.e.,

[H, Si] = 0 for all i = 1, . . . , nG

• normally the groundstate |0〉 of the system is then also symmetric under G, i.e.,

eiεiSi |0〉 = |0〉 → Si |0〉 = 0 for all i

• however, it could happen that the ground state does not have the full symmetry but onlya smaller symmetry group H ⊂ G of the order nH , i.e.,

Si |0〉

= 0 for i ≤ nH6= 0 for i > nH

Terminology: The symmetry is broken spontaneously from G to H.

• Let H |0〉 = 0. What is the energy of the state Si |0〉?

HSi |0〉 = (HSi − SiH) |0〉 = [H, Si] |0〉 = 0

→ the state Si |0〉 has the same energy as the ground state

• there are nG − nH such states. These states are called Nambu-Goldstone (NG) bosons

• because of 0 = E =√~p2 +m2 → m for p→ 0, the NG bosons are massless

• example: Ferromagnet in 3D– H is invariant under G = SO(3)– at low temperature (T < Tc) the ground state exhibits a spontaneous magnetization:〈 ~M〉 6= 0→ the ground state is only symmetric under H = SO(2), i.e., under rotations inthe plane perpendicular to 〈 ~M〉

– the number of NG bosons is nG − nH = 3 − 1 = 2, these are spin waves, i.e., slowvariations in ~M(~x). They propagate in the plane perpendicular to 〈 ~M〉→ 2 degrees of freedom

• another example: Chiral symmetry breaking in QCD (NG bosons = pions)

8.7.2. SU(2) isospin

• experimental observation: among the observed hadrons there are small groups (multi-plets) with roughly the same mass (= eigenvalue of H), e.g.,

proton p and neutron n: mp ≈ mn ≈ 940MeVthe 3 pions, π0, π− and π+: mπ0 ≈ mπ+ ≈ mπ− ≈ 140MeV

• theoretical explanation:– the strong interactions are independent of the electric charge

8.7. Applications in (particle) physics 101

– the small mass differences within a multiplet are due to electroweak interactions

• as usual, degenerate states should transform in irreps of an internal symmetry group→ find a group that explains the observed particle spectrum (i.e., degeneracies = dimen-sions of the irreps)

• an early attempt (before the discovery of quarks) was isospin symmetry: consider p andn, and define an object with two components:

N =(pn

)

– this object exists in a 2-dimensional space (isospin space)– consider SU(2) transformations in this space, with generators I1, I2, I3

– by definition we take I3 |p〉 = 12 |p〉 and I3 |n〉 = −1

2 |n〉– the strong interactions should be invariant under SU(2)isospin, i.e., the Hamiltonian

of the strong interactions should commute with all 3 generators:

[H, ~I] = 0

– N transforms in the 2-dimensional fundamental (or doublet) representation (I = 12)

of SU(2)isospin

• similarly other particles transform in other irreps of SU(2)isospin, e.g., the pions form anisospin triplet (I = 1) with

π+ : I3 = 1 , π0 : I3 = 0 , π− : I3 = −1

• the hypercharge Y is defined via

Q = I3 + 12Y with Q = electric charge

e.g., p and n have Y = 1, the 3 pions have Y = 0

• the isospin multiplets are distinguished by other quantum numbers of the strong inter-actions (B = baryon number, Y , I, J = spin, P = parity). For all particles within amultiplet the quantum numbers are equal, only I3 differs

8.7.3. SU(2) flavor

• we now know that hadrons are composed of quarks q with spin 12 :

– baryons (∼ qqq): spin = 12 ,

32 , . . .

– mesons (∼ qq): spin = 0, 1, . . .

• baryon number B:– +1

3 for quarks, −13 for antiquarks

– +1 for baryons, −1 for antibaryons, 0 for mesons and all other particles

• the interactions of quarks are described by QCD


• in nature there are Nf = 6 quark flavors: u, d, s, c, b, t. Two of them are very light (u, d),one is light (s), and 3 are heavy (c, b, t)

• in low-energy experiments we only observe hadrons containing u and d→ for now, consider only Nf = 2, i.e., a 2-dimensional flavor space

• the reason for the isospin invariance of the hadron masses is that for mu ≈ md theQCD Lagrangian is invariant under SU(2)flavor, i.e., the internal symmetry group isSU(2)flavor

• the 2-dimensional fundamental representation of SU(2)flavor is furnished by

q =(ud

)up quark (Q = 2

3 , I3 = 12 ;Y = 1

3)down quark (Q = −1

3 , I3 = −12 , Y = 1

3)

i.e., q transforms as a doublet under SU(2)flavor (I = 12 , Y = 1

3)

• in the quark model the two nucleons have quark content

p ∼ uud (Q = 1, I3 = 12;Y = 1)

n ∼ udd (Q = 0, I3 = −12;Y = 1)

(∼ means that we ignore permutations of the quarks for now), i.e., we have productstates of the form (1

2

)⊗(1

2

)⊗(1

2

)(here, 1

2 denotes the 2-dimensional fundamental represnetation with I = 12 and Y = 1

3)

• since particles transform in irreps of the symmetry group, we have to decompose theproduct representation into irreducible components:(1

2

)⊗(1

2

)⊗(1

2

)=((1)⊕ (0)

)⊗(1

2

)=(3

2

)⊕(1

2

)⊕(1

2

)or expressed in dimensions:

2⊗ 2⊗ 2 = (3⊕ 1)⊗ 2 = 4⊕ 2⊕ 2

or expressed as Young diagrams:

⊗ ⊗ =(

⊕)⊗ = ⊕ ⊕ = ⊕ ⊕

in Sec. 8.7.4 we will learn:

– the doublet(pn

)corresponds to a linear combination of the two 2-dimensional irreps

(I = 12 , Y = 1) in the RHS

– the 4-dimensional irrep (I = 32 , Y = 1) corresponds to the ∆-baryons

• in the quark model, mesons consist of quarks and antiquarks . Antiquarks are obtained byapplying the charge conjugation operator C = −iγ2K (where K = complex conjugationoperator):

Cu = u Cd = d


– consider a SU(2) transformation of the quark doublet:

g =(α −β∗β α∗

)∈ SU(2) with αα∗ + ββ∗ = 1

(u′

d′

)= g

(ud

)→ C

(u′

d′

)= Cg

(ud

)= g∗C

(ud

)

i.e.,(u′

d′

)= g∗

(ud

), thus the anti-doublet

(ud

)transforms in 2

– since SU(2) is pseudo real, 2 is equivalent to 2, i.e., we can also combine u and d ina doublet in such a way that it transforms in 2

– to see this consider h =(

0 −11 0

)∈ SU(2)

g∗ = h†gh ,

(u′

d′

)= h†gh

(ud

)→ h

(u′

d′

)= gh

(ud

)

i.e., h(ud

)=(−du

)transforms in 2 just like

(ud

), i.e., as an isospin doublet with

I3(−d) = 12 , I3(u) = −1

2

• for the mesons we start with product states of the form 2⊗2 ∼ 2⊗2. The decompositioninto irreducible components is:

2(ud

)⊗ 2(−du

)= 3⊕ 1

from this we can read off the triplet and singlet states (see example at the end of Sec.8.3.2: triplet = |11〉 , 1√

2(|12〉+ |21〉), |22〉 and singlet = 1√2(|12〉 − |21〉))

– the isospin triplet (I = 1, Y = 0) corresponds to the pions:

I3 = 1 π+ = −ud

I3 = 0 π0 = 1√2

(uu+ dd)

I3 = 1 π+ = du

all these states are symmetric underu↔ −d, d↔ u

– the singlet state is 1√2(uu− dd). In Sec. 8.7.4 we will see that this is the ω-meson

• the introduction of the doublet(−du

)was not absolutely necessary; we could also have

constructed the generalized projection operators for the decomposition 2⊗ 2 = 3⊕ 1 toobtain the projections on span(uu−dd) and span(π+, π0, π−). The choice of an eigenbasisof I3 in the 3-dimensional subspace then gives the quark content of the pions


8.7.4. SU(3) flavor and the quark model

• at higher energies also the strange quark shows up→ consider Nf = 3, i.e., a 3-dimensional flavor space with internal symmetry groupSU(3)flavor

• additional quantum number: strangeness S, with Y = B + S

B I I3 Y S Q

u 13

12

12

13 0 2

3d 1

312 −1

213 0 −1

3s 1

3 0 0 −23 −1 −1

3

• QCD processes leave S (and thus Y ) invariant

• LQCD is only invariant under SU(3)flavor if mu = md = ms. Because mu ≈ md < ms,this symmetry is not exact but broken explicitly to SU(2)I⊗U(1)Y→ no perfect degeneracy, but small mass differences within an SU(3) multiplet (Gell-Mann–Okubo formula)

• the 3-dimensional fundamental representation of SU(3)flavor is furnished by q = (u, d, s)T

• in the quark model mesons consist of a quark and an antiquark (which transform in 3),i.e., we start with product states of the form 3 ⊗ 3 and decompose them in irreduciblecomponents as in Sec. 8.6, yielding

3⊗ 3 = 8⊕ 1 or ⊗ = ⊕

i.e., we should find multiplets of roughly degenerate mesons containing 8 particles or 1particle, respectively

• experimental observation: The lightest (groundstate) mesons indeed form an octet anda singlet (together also called nonet), with quantum numbers B = 0 and JPC = 0−+

– pseudoscalar meson octet(scalar beacaue of J = 0,pseudo because of P =−1)

– pseudoscalar singlet: ψ1with I = Y = 0

π−(du)

ψ8(uu, dd, ss)

π+(ud)

-1 1 I3

K0(ds)

K+(us)

K−(su)

K0(sd)

π0

(uu, dd)

Y

1

-1

-12

12

••• •

•

•

•

•

I = 12 , m = 496MeV

I = 1, m = 137MeVI = 0, m see below

I = 12 , m = 496MeV

• in reality it is a bit more complicated:– consider the three states with I3 = Y = 0:∗ π0 is the state of the isospin triplet, i.e., π0 = 1√

2(uu− dd)

∗ ψ1 is the SU(3) singlet state, i.e., ψ1 = 1√3(uu+ dd+ ss)

∗ ψ8 is the SU(3) octet, isospin singlet state. This state must be orthogonal toπ0 and ψ1, i.e., ψ8 = 1√

6(uu+ dd− 2ss)


– ψ1 and ψ8 have the same quantum numbers (I = 0 and JPC = 0−+)∗ if SU(3) were exact, ψ1 and ψ8 would be physical states (i.e., particles) sincethey transform in different irreps of SU(3)∗ however, SU(3) is broken explicitly→ states transforming in different irreps, but with the same quantum numbers,can mix

η(548 MeV) = ψ8 cos θ − ψ1 sin θη′(958 MeV) = ψ8 sin θ + ψ1 cos θ

the physical particles are η and η′. θ is called nonet mixing angle (experimentalvalue: θ = −22.6)

• in addition there are excited qq-states (rotations, vibrations, etc.). The first excitedmeson nonet has quantum numbers B = 0 and JPC = 1−−

• in the quark model baryons consist of 3 quarks, i.e., we start with product states of theform 3⊗ 3⊗ 3. Decomposition into irreducible components as in Sec. 8.6 yields

⊗ ⊗ = ⊕ ⊕ ⊕

or 3⊗ 3⊗ 3 = 10↑S

⊕ 8↑MS

⊕ 8↑MA

⊕ 1↑A

with S = totally symmetric tensors under S3, i.e., permutations of quarksMS = tensors of mixed symmetry: symmetric under exchange of the first

two quarksMA = tensors of mixed symmetry: antisymmetric under exchange of the

first two quarksA = totally antisymmetric tensors

i.e., we should find multiplets of (nearly) degenerate baryons consisting of 10, 8, and 1particles, respectively

• Experimental finding: The lightest (i.e., ground state) baryons form an octet and a de-couplet:

– baryon octet (B = 1, JP = 12

+)

Σ−(dds)

Σ0(uds)

Σ+(uus)

I3

n(udd)

p(uud)

Ξ−(dss) Ξ+(uss)

Λ

Y

••• •

•

•

•

•

I = 12 , m = 939MeV

I = 0, m = 1116MeV

I = 1, m = 1193MeV

I = 12 , m = 1318MeV

– baryon decouplet (B = 1, JP = 32

+)

Σ∗−(dds)

Σ∗0(uds)

Σ∗+(uus) I3

∆−(ddd)

∆++(uuu)

∆0(udd)

∆+(uud)

Ξ∗−(dss)

Ξ∗+(uss)

Ω−(sss)

Y

•• •

• • • •

• •

•

I = 32 , m = 1232MeV

I = 1, m = 1385MeV

I = 12 , m = 1530MeV

I = 0, m = 1672MeV

• Where are the singlet and the second octet?– baryons are fermions, whose wave functions must be totally antisymmetric (in space,

spin, flavor, color)


– baryons are color singlets, i.e., they transform under SU(3)color in the irrep→ the spin-flavor part must be totally symmetric

– for the spins of the 3 quarks in the baryon we have (here the Young diagrams arefor SU(2)spin)

⊗ ⊗ =(

⊕)⊗

= ⊕ ⊕ = ⊕ ⊕

or 2⊗ 2⊗ 2 = 4↑S

⊕ 2↑MS

⊕ 2↑MA

i.e., we must combine(10↑S

⊕ 8↑MS

⊕ 8↑MA

⊕ 1↑A

)SU(3)flavor

and(4↑S

⊕ 2↑MS

⊕ 2↑MA

)SU(2)spin

– this gives the following possibilities for (SU(3) , SU(2)) multiplets

S : (10, 4) + (8, 2)MS : (10, 2MS

) + (8, 4) + (8, 2) + (1, 2)MA : (10, 2MA

) + (8, 4) + (8, 2) + (1, 2)A : (1, 4) + (8, 2)

e.g., the totally symmetric octet corresponds to the linear combination

(8, 2)S = 1√2

((8MS

, 2MS) + (8MA

, 2MA))

and similarly for the other combinations– only the totally symmetric spin-flavor multiplets (10,4) and (8,2) give a totally

antisymmetric wave function of the baryon→ in the ground state there is only the decouplet and one octet, but no singlet andno second octet; however, the latter two exist in excited states

• another method to derive this:– consider the 3 quarks as 6 states with 3 flavors and 2 spin states per flavor→ approximate SU(6)spin−flavor symmetry

– the decomposition into irreducible components under SU(6) is

6⊗ 6⊗ 6 = 56S ⊕ 70MS⊕ 70MA

⊕ 20A

– the 56-dimensional irrep of SU(6) subduces a representation of SU(3)flavor; thisrepresentation is reducible, and we obtain:

56S = 103/2

dim.=10·4

⊕ 81/2

dim.=8·2

this corresponds to the baryon decouplet (spin = 32) and the baryon octet (spin

= 12), respectively

• the SU(3)flavor symmetry is seen not only in the mass spectrum, but also in scatteringexperiments:


– e.g., consider scattering of a meson (P ) from the pseudoscalar meson octet and abaryon (B) from the baryon octet

– the scattering amplitudes for, e.g., πp or Kp or πΣ scattering are all related due toSU(3) symmetry

– using the Wigner-Eckart theorem, all scattering amplitudes for PB scattering canbe expressed in 6 complex matrix elements and the CG coefficients in the decom-position (see Sec. 8.6)

8⊗ 8 = 27⊕ 10⊕ 10⊕ 8⊕ 8⊕ 1

or ⊗ = ⊕ ⊕ ⊕ ⊕ ⊕

• more examples: see Halzen & Martin, Quarks and Leptons, Sec. 2

Appendix A

A.1 Proof of Cayley’s theorem

• we consider a group G = g1, . . . , gn of order n.

• if we look at the multiplication table of the group, we notice that the i-th row is givenby

(gig1, . . . , gign) = p(g1, . . . , gn)

where p ∈ Sn, i.e., every row is just a permutation of the group elements.

• from this it is obvious that there must be an isomorphism between G and a subgroup ofSn (i.e., the subgroup includes the permutations that appear in the multiplication table).

A.2 Proofs of the four theorems

Theorem 1: Every representation of a group G (of finite order) can be converted to a unitaryrepresentation.

• suppose that the action of g ∈ G on a vector space V is represented by D(g). Then weneed to show that ∃S such that

SD(g)S−1 = U(g) ∀ g ∈ G , U(g) : unitary

• define a matrix M by

M =∑g

D(g)†D(g)

this matrix is positive definite, and can thus be written as

M =∑g

D(g)†D(g) ≡ S†S

• now we look at the scalar product⟨U(g)x

∣∣ U(g)y⟩

=⟨SD(g)S−1x

∣∣∣ SD(g)S−1y⟩

=⟨S−1x

∣∣∣ D(g)†S†SD(g)S−1y⟩

• now we use that

D(g)†S†SD(g) = D(g)†∑h

D(h)†D(h)D(g) =∑h

D(hg)†D(hg) = S†S

• so we have ⟨U(g)x

∣∣ U(g)y⟩

=⟨S−1x

∣∣∣ S†SS−1y⟩

= 〈x | y〉

and thus U(g) is unitary

108

A.2 Proofs of the four theorems 109

Theorem 2 (Schur’s lemma 1): A matrix A which commutes with all matrices of an irrepU(G) is proportional to the identity matrix.• without loss of generality we can take the matrices U(g) of the irrep to be unitary, and

the matrix A to be Hermitian. Otherwise we can decompose A = (Hermitian + anti-Hermitian). Then we can choose the basis of the carrier space such that A is diagonal:

A |niα〉 = λi |niα〉 α specifies the degeneracy

• then, with [A,U(g)] = 0, we have

AU(g) |niα〉 = U(g)A |niα〉 = λiU(g) |niα〉

⇒ U(g) |niα〉 is also an Eigenvector of A with Eigenvalue λi⇒ therefore the vectors |niα〉 form an invariant subspace (for fixed i); but this can

only be true if this subspace is already the whole carrier space itself, since U(G) isan irrep. Hence the additional label i is infact superfluous, and we can set λi = λ

⇒ A = λ1

Theorem 3 (Schur’s lemma 2): Suppose we have two irreps of a group G: Γ1(G) withdimension λ1 and Γ2(G) with dimension λ2. If there exists a λ2 × λ1 matrix M such that

MΓ1(g) = Γ2(g)M ∀ g ∈ G (∗)

then if λ1 6= λ2 we have M = 0.If λ1 = λ2, then either M = 0 or detM 6= 0. In the case M 6= 0 the two representations areequivalent.• without loss of generality we can assume λ1 ≤ λ2

• also we can take the two irreps to be unitary. Then, taking the hermitian conjugate of(∗), we have

Γ1(g)†M † = M †Γ2(g)†

Γ1(g−1)M † = M †Γ2(g−1)→ MΓ1(g−1)M † = MM †Γ2(g−1)(∗)⇒ Γ2(g−1)MM † = MM †Γ2(g−1)

MM † is a square matrix that commutes with all Γ2(g). Therefore by Schur’s lemma 1we have

MM † = c1 , detMM † = cλ2

• if λ2 = λ1 = λ, then M is a square matrix with detM = cλ/2

– if c 6= 0, M is non-singular and Γ1 and Γ2 are equivalent– if c = 0, we have MM † = 0. This implies M = 0, since

0 =(MM †

)kk

=∑i

Mki(M †)ik =∑i

MkiM∗ki =

∑i

|Mki|2 ⇒Mki = 0

• if λ1 < λ2 M is rectangular. Define a square matrix N = (M0), i.e., the missing spaceis filled up with zeros. Then,

NN † = MM † , detNN † = |detN |2 = 0 = detMM †

since MM † = c1 it follows that MM † = 0 and therefore M = 0 (see above)

110 Appendix A Appendix

Theorem 4 (orthogonality relations): Let Γi(G) and Γk(G) be two non-equivalent, unitaryirreps of G with order(G) = n. Then we have

n∑j=1

(Γi(Aj)µν

)∗Γk(Aj)µ′ν′ = n


• first we define a matrix M by

M =∑j

Γi(A−1j )XΓk(Aj) (∗∗)

where X is an arbitrary λi × λk matrix.

• multiplying this by Γk(A`) from the right gives

MΓk(A`) =∑j

Γi(A−1j )XΓk(Aj)Γk(A`) =

= Γi(A`)∑j

Γi((AjA`)−1

)XΓk(AjA`) =

= Γi(A`)M

where we have used that AjA` also runs over all group elements.

• now we look at the case i 6= k. Then Γi and Γk are non-equivalent and it follows thatM = 0 (froms Schur’s lemma 2).

• writing (∗∗) in component form yields

Mµµ′ =∑j

∑αα′

Γi(A−1j )µαXαα′Γk(Aj)α′µ′

until now the matrix X was arbitrary; we now choose all the Xαα′ = 0 except for oneparticular element Xνν′ which we set to 1

• then,

0 =∑j

Γi(A−1j )µνΓk(Aj)ν′µ′ =

∑j

Γi(Aj)†µνΓk(Aj)ν′µ′ =∑j

(Γi(Aj)νµ

)∗Γk(Aj)ν′µ′

• for the case i = k we have

MΓi(Aj) = Γi(Aj)M

i.e., M commutes with all Γi(Aj), and therefore M = c1

• again, writing (∗∗) in component form gives

cδµµ′ =∑j

∑αα′

Γi(A−1j )µαXαα′Γi(Aj)α′µ′

here we can also choose all Xαα′ = 0 except for one Xνν′ = 1, so that (c = cνν′)

cνν′δµµ′ =∑j

Γi(A−1j )µνΓi(Aj)ν′µ′

A.3 The Kronecker product 111

• to evaluate cνν′ we set µ = µ′ and sum over µ:

cνν′λi∑µ=1

δµµ︸︷︷︸λi

=n∑j=1

λi∑µ=1

Γi(A−1j )µνΓi(Aj)ν′µ =

n∑j=1

(Γi(Aj)Γi(A−1

j ))ν′ν

=n∑j=1︸︷︷︸n

(1)νν′

⇒ cνν′ = n

λiδνν′

inserting cνν′ into the equation above and using the fact, that Γi(A−1j )νµ = Γi(Aj)†νµ =

(Γi(Aj)µν)∗, gives the equation from the theorem

A.3 The Kronecker product

• the Kronecker product of a n ×m matrix A and a k × ` matrix B is defined in thefollowing way:

A⊗B =

A11B · · · A1mB

... . . . ...An1B · · · AnmB

=

=

A11B11 · · · A11B1` · · · A1mB11 · · · A1mB1`... . . . ... . . . ... . . . ...

A11Bk1 · · · A11Bk` · · · A1mBk1 · · · A1mBk`... . . . ... . . . ... . . . ...

An1B11 · · · An1B1` · · · AnmB11 · · · AnmB1`... . . . ... . . . ... . . . ...

An1Bk1 · · · An1Bk` · · · AnmBk1 · · · AnmBk`

• with this in mind one can see that the formula,

(A⊗B)(i−1)k+s,(j−1)`+t = AijBst,

given in Sec. 2.6 is indeed correct, since to get a matrix element with Aij one has firstto “jump” down (i − 1) B-blocks of (row) size k and “jump” over (j − 1) B-blocks of(column) size ` to the right. From there one can move to Bst in the usual manner.

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