sign in sign up
<<
Home , Cartan subalgebra, Class of groups, Complex Lie algebra, Component (group theory), Group theory, Real representation

DRAFT 2019 February 26

Mathematical Methods in - 231B – –

Eric D’Hoker Mani L. Bhaumik Institute for Department of Physics and Astronomy University of California, Los Angeles, CA 90095, USA [email protected]

The purpose of this course is to present an introduction to standard and widely used methods of in Physics, including Lie groups and Lie , , tensors, , structure theory of solvable and simple Lie algebras, homogeneous and symmetric spaces. Contents

1 Definition and 5 1.1 A little history ...... 5 1.2 Groups ...... 6 1.3 Topological groups ...... 9 1.4 Lie Groups ...... 12 1.5 Vector spaces ...... 13 1.6 Lie Algebras ...... 13 1.7 Relating Lie groups and Lie algebras ...... 15 1.8 Tangent vectors, , and cotangent space ...... 16

2 Lie groups and Lie algebras 18 2.1 The general GL(n)...... 18 2.2 Closed of the ...... 21 2.3 The orthogonal groups SO(n) and Lie algebras so(n)...... 21 2.4 The Sp(2n) and Lie sp(2n)...... 23 2.5 The SU(n) and su(n)...... 25 2.6 Cartan subalgebras and subgroups and the ...... 26 2.7 Summary of matrix Lie groups and Lie algebras ...... 26 2.8 Non-semi-simple matrix groups ...... 30 2.9 differential forms, , and measure ...... 31 2.10 Spontaneous breaking, parameters ...... 37 2.11 Lie supergroups and Lie superalgebras ...... 38

3 Representations 41 3.1 Representations of groups ...... 41 3.2 Representations of Lie algebras ...... 43 3.3 Transformation of a under a representation ...... 44 3.4 Direct sum of representations ...... 45 3.5 Schur’s Lemma ...... 46 3.6 Unitary representations ...... 47 3.7 of representations ...... 49 3.8 Characters of Representations ...... 50

4 Representations of SL(2; C), SU(2), and SO(2, 1) 53 4.1 Irreducible representations of the SU(2) ...... 53 4.2 Finite-dimensional representations of sl(2, C)...... 54 4.3 Infinite-dimensional representations of sl(2; C)...... 57 4.4 Harmonic oscillator representation of so(2, 1; R)...... 57 4.5 Unitary representations of so(2, 1; R)...... 58 4.6 An example of the continuous series ...... 60

2 5 Tensor representations 62 5.1 Tensor product representations ...... 62 5.2 Symmetrization and anti-symmetrization ...... 63 5.3 Representations of SU(3)...... 64 5.4 Representations of SU(n)...... 66

6 representations 68 6.1 Spinor representations of SO(3; R)...... 68 6.2 The Clifford Algebra ...... 69 6.3 Representations of the Clifford algebra ...... 70 6.4 Spinor representations of so(d)...... 72 6.5 Reducibility and Weyl spinor representations ...... 73 6.6 Charge and complex conjugation ...... 74 6.7 Spinor representations of so(d − 1, 1; R)...... 75 7 Roots, weights, and representations 78 7.1 The Cartan- ...... 78 7.2 Weights ...... 79 7.3 Roots ...... 80 7.4 Raising and Lowering operators ...... 82 7.5 Finite-dimensional representations ...... 82 7.6 The example of SU(3) ...... 84 7.7 The root of An = sl(n +1)...... 86 7.8 The root lattice of Dn = so(2n)...... 87 7.9 The root lattice of Bn = so(2n +1)...... 89 7.10 The root lattice of Cn = sp(2n +1)...... 90

8 Representations of the classical Lie algebras 92 8.1 Weyl reflections and the ...... 92 8.2 Finite-dimensional irreducible representations of sl(n + 1; C)...... 93 8.3 Spinor representation of so(2n + 1; C)...... 94 8.4 Spinor representations of so(2n; C)...... 94 9 The structure of semi-simple Lie algebras 96 9.1 Some properties of simple roots ...... 96 9.2 Classification of finite-dimensional simple Lie algebras ...... 97

10 Weyl’s formulas 102 10.1 Characters on the maximal ...... 103 10.2 Weyl’s first formula ...... 104

3 Bibliography

Standard texts for physicists • H. Georgi, Lie Algebras and Particle Physics, Benjamin/Cummings, 1982; • B.G. Wybourne, Classical Groups for Physicists, Wiley, 1974; • W. K. Tung, Group Theory in Physics, World Scientific; • P. Di Francesco, P. Mathieu, D. S´enechal, Conformal Theory, Springer 1997, Chapters 13 and 14.

Classics and more mathematical • H. Weyl, Classical Groups, Princeton, 1946. • C. Chevalley, Theory of Lie Groups, Princeton, 1948; • E. Cartan, The Theory of Spinors, Hermann, Paris 1966, Dover; • S. Helgason, Differential , Lie Groups, Symmetric Spaces, Acad. Press 1978; • M. Nakahara, Geometry, and Physics, Institute of Physics Publishing (2005) (no group theory per se, but great for topology and differential geometry);

Very useful but more mathematical • B.C. Hall, Lie Groups, Lie Algebras, and Representations, Springer Verlag Graduate Texts in Mathematics 222 (2015); • W. Fulton and J. Harris, Representation Theory, Springer Verlag, 1991; • D.P. Zelobenko, Compact Lie Groups and Their Representations, Tansl. Math. Monographs, Vol. 40, Am. Math. Soc., 1973; • N.J. Vilenkin, Special Functions and the Theory of Group Representations, Translations of Mathematical Monographs, Vol 22, Am. Math. Soc., 1968; • F.D. Murnaghan, The Theory of Group Representations, Johns Hopkins, 1938;

Others • A.W. Knapp, Lie Groups beyond an Introduction, Birkh¨auser2004; • J. Humphreys, Introduction to Lie Algbras & Representation Theory, Springer, 1980; • N. Jacobson, Lie Algebras, Wiley, 1962; Dover, 1979; • I.R. Porteous, Clifford Algebras and , Cambridge, 1995;

4 1 Definition and examples of groups

In this section, we shall begin with a little history, and then present the basic definitions and some simple examples of groups, subgroups, topological groups, discrete groups, parametric groups, Lie groups, and Lie algebras.

1.1 A little history

The mathematical concept of a group goes back to Joseph-Louis Lagrange (1736 - 1813) and Niels Abel (1802 - 1829), and was articulated by Evariste Galois (1811 - 1832) in the context of subgroups of the group of of the roots of a . Lie groups and Lie algebras were introduced by (1842 - 1899) around 1870. In physics the concept of a group arises almost invariably in the context of a symmetry or invariance of a physical quantity or of an equation which described a physical quantity. Greek antiquity appreciated the beauty of regular polygons and polyhedra. Galileo and Newton undoubtedly understood the notions of translation and invariance, such as in Galilean relativity, and used them without invoking the notion of a group. Poincar´e(1854 - 1912) observed the invariance of Maxwell’s equations (without sources) under Lorentz and Poincar´etransformation. It is fair to state that Einstein (1879 - 1955) pioneered elevating the symmetry of the equations to a physical principle in his 1905 theory of special relativity (Poincar´einvariance), but even more clearly in his 1915 theory of general relativity (invariance under general coordinate transformations). Since then, group theory has become a dominant organizing principle of modern physics: Herman Weyl (1885 - 1955) applied group theory to , Lev Landau (1908 - 1968) based his theory of second order phase transitions on the group-theoretic symmetry properties of an order parameter; Murray Gell-Mann (1929 - present) used SU(3) group theory to predict the Ω particle; and the of Particle Physics is built on the Yang-Mills theory for the gauge group SU(3)c ×SU(2)L ×U(1)Y , though some of this symmetry is spontaneously broken and not manifest at low energies. One of the biggest theoretical questions in modern theoretical and experimental physics is to find out what lies beyond the Standard Model. The most popular theoretical specu- lations are grand unification, which is based on extending SU(3)c × SU(2)L × U(1)Y to a larger unifying group, and which is based on extending the Poincar´egroup to a , with fermionic parameters. One of the most active areas of research over the past decade has been the conformal bootstrap program, based on exploiting the prop- erties of the to study quantum field that do not necessarily admit a Lagrangian or Hamiltonian description.

5 1.2 Groups

A G equipped with a ? is a group G? provided G and ? satisfy,

1. : For any pair of elements g1, ∈ G the operation obeys g1 ? g2 ∈ G;

2. Associativity: For any triplet of elements g1, g2, g3 ∈ G the operation obeys (g1 ? g2) ? g3 = g1 ? (g2 ? g3) = g1 ? g2 ? g3; 3. Identity: There exists an element e ∈ G such that e ? g = g ? e = g for all g ∈ G; 4. Inverse: For every g ∈ G, there exists a g−1 ∈ G such that g ? g−1 = g−1 ? g = e. The group imply that the identity element e is unique, and that for every element −1 g, the inverse g is unique. The direct product G1 × G2 of two groups G1 and G2 with respective operations ∗1 and ∗2 is a group with elements (g1, g2) under the operation ∗ defined by (g1, g2) ∗ (h1, h2) = (g1 ∗1 h1, g2 ∗2 h2) with g1, h1 ∈ G1 and g2, h2 ∈ G2. A group G? is commutative or Abelian if the product satisfies g1 ? g2 = g2 ? g1 for all g1, g2 ∈ G. Otherwise, the group is said to be non-commutative or non-Abelian. When no confusion is expected to arise, one often drops reference to the operation in denoting a group, so that

G∗ is simply denoted G.

1.2.1 Subgroups, invariant subgroups, and simple groups

A H of a group G is a subset H ⊂ G such that h1 ? h2 ∈ H for all h1, h2 ∈ H and h−1 ∈ H for all h ∈ H. The group G itself and the group consisting of the identity element {e} only both trivially form subgroups of the group G. Every subgroup of an is Abelian, but a non-Abelian group may have Abelian and/or non-Abelian subgroups. A subgroup H of a group G is referred to as an invariant subgroup, or , iff it is invariant under all the inner of G by conjugation,

γ ∗ H ∗ γ−1 ∈ H for all γ ∈ G (1.1)

An inner of a group G is any transformation T : G → G which preserves the multiplication structure of the group, so that T (g1)∗T (g2) = T (g1∗g2) for all g1, g2 ∈ G with T (e) = e and T (g−1) = T (g)−1. The key example of an inner automorphism is conjugation, −1 Tγ(g) = γ ∗ g ∗ γ for γ ∈ G. Inner automorphisms, viewed as maps from G to G, form a group under the operation of composition of maps, Tγ ◦ Tγ0 (g) = Tγ∗γ0 (g). Every subgroup of an Abelian group G is an invariant subgroup. The quotient space G/H of a group G and one of its subgroups H is the set of equivalence classes in G under the of right-multiplication by elements in H. Two elements g1, g2 ∈ Γ are equivalent to one another iff there exists an element h ∈ H such that −1 g2 = g1 ? h, or equivalently g1 = g2 ∗ h . The containing an element g ∈ G is denoted [g], and all the elements of G in the class [g] are obtained from g by right multiplication by H, and are thus given by g ∗ H.

6 If H is a normal subgroup of G, then the quotient space G/H is itself a group, and thus a subgroup of G. To prove this assertion we will show that the operation ∗ of G induces a multiplication on the classes of the quotient space G/H. Let [g1] and [g2] be the classes of G/H which contain respectively the elements g1 and g2. The general elements of G in those classes are respectively g1 ∗ h1 and g2 ∗ h2 where e h1, h2 ∈ H. Now consider the product of two arbitrary representatives of the classes [g1] and [g2], which is given by g1 ∗ h1 ∗ g2 ∗ h2. When H is a normal subgroup of G, we may use the conjugation relation −1 g2 ∗ h1 ∗ g2 = h ∈ H so that

g1 ∗ h1 ∗ g2 ∗ h2 = g1 ∗ g2 ∗ h ∗ h2 (1.2)

By definition the element g1 ∗ g2 ∗ h ∗ h2 belongs to the class [g1 ∗ g2], so that the operation ∗ on G induces an operation ? on the classes. When H is not a normal subgroup of G, −1 we cannot use the relation g2 ∗ h1 ∗ g2 = h ∈ H and then it is generally not possible to consistently the product of arbitrary members of two classes to a single class. A group G is referred to as a iff its only invariant subgroups are the group G itself and the trivial subgroup of the identity {e}. Simple groups constitute the building blocks of general groups which, under certain conditions, may be decomposed into a direct product of simple groups and certain Abelian groups.

1.2.2 Fields

A field F is defined to be a set F equipped with two operations, which are usually referred to as addition +, and multiplication ×, and which are such that F+ is an Abelian group 0 with identity element 0, F× is an Abelian group with identity 1, and the operations are related by the property of distributivity for all a, b, c ∈ F,

a × (b + c) = a × b + a × c (1.3)

Fields form the basis of virtually all the algebra needed in physics.

1.2.3 Examples Some of the simplest groups under the operations of addition and multiplication are,

Z+ 0 rationals Q+ Q× 0 reals R+ R× 0 complex C+ C× 0 H+ H× (1.4)

7 0 0 0 0 with the subgroup inclusions Z+ ⊂ Q+ ⊂ R+ ⊂ C+ ⊂ H+ and Q× ⊂ R× ⊂ C× ⊂ H× where the subscript 0 stands for the removal of 0, the identity element of addition. The 0 above-mentioned groups are all Abelian, except for H× which is non-Abelian. Combining addition and multiplication, Q, R and C are fields, but H is not a field with this definition since its multiplication is non-Abelian. In physics the fields of interest are R and C. The group of integers has truncations to finite groups, defined as follows,

Zn = Z/nZ = {0, 1, ··· n − 1}+ addition mod n 0 0 Zp = Z/pZ = {1, ··· p − 1}× multiplication mod p for p a prime where the mod acts as follows. For N ∈ Z and n ∈ N, we define q ≡ N (mod n) such that N = pn + q with 0 ≤ q < n. Note that for multiplication it is crucial to have p a , as otherwise the product of two elements may vanish mod p, as for example in (2 × 3)(mod 6) = 0. When p is prime, Zp forms a field which is denoted Fp. Another example is the group of permutations Sn of a set An with n elements. More generally, a group G acting on a set A by permuting its elements is a transformation group acting on A. When the number of elements of A is n < ∞ one always has G ⊂ Sn, but G need not be the full group. Examples of transformation groups include many of the transformations we use in physics, including translations on the real line by discrete translations such as in a or by continuous translations, and similarly for rotations. A very important is matrix groups, which are sets of square matrices with the operation of . They will be studied in great detail in this course. Here we shall just quote the most general of the matrix groups GL(n) which is the general linear group of n × n matrices. The entries of the matrices may be in Z, Q, R, C or even quaternions H and are then denoted respectively by, GL(n, Z) ⊂ GL(n, Q) ⊂ GL(n, R) ⊂ GL(n, C) ⊂ GL(n, H) (1.5) In each case the form a normal subgroup under multiplication, respectively given by Z2, Q, R, C, H (one often omits the zero superscript and the multiplication sub- script). The quotients of each group GL(n) by its invariant subgroup is again a group referred to as the special linear groups, and denoted by,

SL(n, Z) ⊂ SL(n, Q) ⊂ SL(n, R) ⊂ SL(n, C) ⊂ SL(n, H) (1.6) The other matrix groups will defined as closed subgroups of one of the above. Finally, rectangular m × n matrices form a group under addition of matrices. This group is equivalent to a subgroup of GL(m + n) under multiplication. Denoting two m × n matrices by M1 and M2, the correspondence is exhibited by the following matrix identity, I M  I M  I M + M  m 1 m 2 = m 1 2 (1.7) 0 In 0 In 0 In where Im and In respectively denote the identity m × m and n × n matrices.

8 1.3 Topological groups The notions of open and closed sets of a set G, of a G, a metric topology, a Cauchy , and continuity were introduced in section 2 of the notes for 231A, and we shall not repeat the definitions here. A G is a group G equipped with a topology so that G is also a topological space and the group operation is continuous. For later convenience we shall recall here a few properties of a topological group G which will be of fundamental importance in group theory.

1.3.1 Connectedness A subset S of a topological group G is path-connected (which we shall simply refer to as connected when no confusion is expected to arise), if any two points x, y ∈ S can be joined by a f : [0, 1] → S given by f(s) for s ∈ [0.1], such that f(0) = x and f(1) = y. Note the criterion of continuity and the requirement that f([0, 1]) must be entirely contained in S. A set S which is not connected is disconnected. The set of all disconnected components of S forms a group, denoted by π0(S) and referred to as the zero-th of S. A topological group G where every point is disconnected from every other point is called discrete and we have π0(g) = g for every g ∈ G.

Let G be a topological group and Gid the connected of G which contains the identity. Then Gid is an invariant subgroup and every connected component of G is congruent to G0 with π0(G) = G/Gid.

The set of connected components of the direct product of two topological groups G1,G2 (or more generally of two topological spaces) is the direct product of the sets of components,

π0(G1 × G2) = π0(G1) × π0(G2) (1.8) so that the number of disconnected components is multiplicative.

1.3.2 Simply-connectedness Let S be a connected subset of a topological group G.A closed curve C ⊂ S is given by a continuous function C : [0, 1] → S such that C(0) = C(1). A subset S is simply-connected if every closed curve in S can be continuously deformed to a point, while remaining in S. To make this more explicit, a closed curve C can be deformed to a point p provided there exists a continuous function Cˆ from [0, 1] × [0, 1] → S such that,

Cˆ(s, 1) = C(s) Cˆ(s, 0) = p s ∈ [0, 1] (1.9)

Subsets which are not simply-connected are also very important, and one can provide a measure for how much they fail to be simply connected. The key ingredient is the notion of

9 homotopy. Two curves C0 and C1 in S which have the same end points p, q are homotopic to one another provided there exists a continuous function Cˆ : [0, 1] × [0, 1] → S such that,

ˆ ˆ C(s, 1) = C1(s) C(1, t) = C1(1) = C0(1) = p ˆ ˆ C(s, 0) = C0(s) C(0, t) = C1(0) = C0(0) = q (1.10)

Homotopy induces an equivalence relation between curves, either with the same end-points or between closed curves in S. Thus we may state the condition of simply-connectedness as the fact that all closed curves in S are homotopic to points. The equivalence classes of closed curves in S under the homotopy equivalence relation are referred to as the elements of the first homotopy group or π1(S). The classes form a group under composition of curves by choosing two representatives C1 and C2 which have a point p in common and then composing the curves at the point p.

Under direct product of two connected topological groups G1,G2, the fundamental group behaves as follows,

π1(G1 × G2) = π1(G1) × π1(G2) (1.11)

For example, the fundamental group of U(1) = S1 = SO(2) is isomorphic to Z (see the corresponding discussions in the 231A notes). Since S1 is connected, we apply the above 1 1 2 result for the torus and we have π1(S × S ) = Z . Since R is simply connected and has 1 π1(R) = 0, the fundamental group of the cylinder is π1(S × R) = Z.

1.3.3 Compactness

A cover of a subset S of a topological group G is defined to be a class of open sets On S with n ∈ N such that S ⊂ On and thus covers S. A subspace S of a topological space is compact if for every cover {On}n∈N one can extract a finite sub-cover. The Heine-Borel states that a subset of Rn, with the topology induced by the Euclidean metric on Rn, is compact if and only if it is closed and bounded in the sense of the Euclidean metric topology. If G1,G2 are two compact spaces, then their direct product G1 × G2 is compact. On the other hand, if G1 is not compact then the direct product of G1 with any group is not compact. It is important to point out that a given set may be compact under one topology and non-compact under another. For example, consider the complex plane C = R2. Under the topology induced by the Euclidean metric, R2 is non-compact since it is unbounded. However, we could equip C with a different metric, such as for example, |dz2 ds2 = (1.12) (1 + |z|2)2

10 With this metric, infinity z = ∞ is a finite distance away from any other point in C, and infinity reduces to just a single point. Strictly speaking the point z = ∞ does not belong to C, but we can add it to C to produce the set, C¯ = C ∪ {∞} (1.13) ¯ 2 ¯ The set C with the metric ds can be covered by just two open sets C = O1 ∪ O2, where we can take for example O1 = {z ∈ C, |z| < 2} and O2 = {z ∈ C, |z| > 1}. Thus, the set C¯ with the metric topology induced by ds2 is a compact set, and C¯ is referred to as a (one-point) compactification of C.

1.3.4 A topological space M is a topological provided every point p ∈ M has an open neighborhood which is homeomorphic to an open set in Rn. A map is a provided it is bijective, continuous, and its inverse map is also continuous. By continuity, the value of n must be constant throughout the manifold, and n is referred to as the dimension of M. Some important aspects of manifolds are as follows. n 1. The homeomorphism ϕα from an open neighborhood Uα of a point p ∈ M into R provides‘local coordinates for Uα. The pair (Uα, ϕα) is called a chart (think of a geographical map), and an atlas (Uα, ϕα)α∈S is a set S of charts such that, [ Uα = M (1.14) α∈S so that all points in M may be described by at least one local coordinate system.

2. The only way the open sets Uα can cover M is by having non-trivial intersections with one another. The intersection Uα ∩ Uβ of two open sets Uα and Uβ is an open set. If the intersection is non-empty we have two different (i.e. coordinate

sets) for the same point p, namely ϕα(p) and ϕβ(p). Their compositions, −1 n n ψα,β = ϕα ◦ ϕβ : R → R (1.15)

are referred to as transition functions. Since ϕα and ϕβ and their inverses are con- tinuous, the functions ψα,β are automatically continuous. While it follows from the definition of a that the transition func-

tions ψα,β are continuous, we may impose conditions on ψα,β which are stronger than mere continuity. Here are some of the most frequently used extra conditions. • Differentiable manifold or more precisely Ck differentiable manifold provided the order k derivatives of the transition functions exist and are continuous; • Real analytic manifold provided the transition functions are real analytic; • Complex manifold exists provided n is even, and the transition functions are holo- morphic functions of several complex variables.

11 1.3.5 Group manifolds and parametric groups Requiring a topological group G to be a manifold is yet a further restriction that will turn out to be of great importance. When the group G is a manifold, its dimension dim G is defined to be the dimension of G as a manifold. A group G is a parametric group if it can be parametrized (at least locally in the

neighborhood of any point) by a system of real coordinates or parameters t = (t1, ··· , tn),

g(t) = g(t1, . . . , tn) (1.16) so that the group multiplication structure in G is given by,

g(ti) ∗ g(si) = g(fi(t, s)) (1.17) where the real-valued functions fi(t, s) are continuous. Since the parameters locally describe an Euclidean Rn neighborhood, a parametric group G is automatically a manifold endowed with a group structure.

1.4 Lie Groups

A group G is a Lie group if G is a parametric group and the function fi(s, t) expressing the composition of two group elements are real analytic in s, t.

Theorem 1 If the functions fi(s, t) admit first derivatives in s, t and those derivatives are continuous, then the functions fi are analytic in s, t. (Sophus Lie, 1888) Theorem 2 Every parametric group is a Lie group. (This gives a solution to Hilbert’s Vth problem; see D. Montgomery and L. Zippin , 1955 ) These show that the combination of the group structure and the continuity

of fi or first order differentiability automatically implies real analyticity. Continuity in parameters is pervasive in physics, whenever we have a continuous sym- metry, such as rotations or Lorentz transformations, or group transformations. Therefore, our emphasis in these lectures will be on Lie groups, and we shall be brief on discrete groups, such as permutations and crystallographic groups.

1.4.1 Examples of Lie groups

The groups R and C, under addition and multiplication, are Lie groups since the operations on two elements x, y of addition x+y and multiplication xy are real analytic. Similarly, the groups Rn, Cn, and the of m × n matrices are all Lie groups under addition. The groups Z and Q under addition are not Lie groups, as they are not even manifolds.

12 Under multiplication, the matrix groups GL(n, R), GL(n, C),SL(n, R),SL(n, C) are all Lie groups. For example, to see this for the case of GL(n, R) we exhibit the matrix in terms of its entries xij for i, j = 1 ··· , n,   x11 x12 ··· x1n x21 x22 ··· x2n g(x) =   (1.18) ············  xn1 xn2 ··· xnn

The entries xij may be used as real parameters to promote GL(n, R) into a parametric group. Note that the parameters are labelled here by a composite index, but this is equiv- alent to the labelling by a single parameter used in the definition of a parametric group. The composition functions f(x, y) are given by the rules of matrix multiplication, n X g(x)g(y) = g(f(x, y)) fij(x, y) = xik ykj (1.19) k=1 They are continuous functions of x, y, so that GL(n, R) satisfies to axioms of a Lie group, which are in fact real analytic as should be the case for a Lie group. Similarly, GL(n, C), SL(n, R), and SL(n, C) are Lie groups.

1.5 Vector spaces Next, we will introduce Lie algebras, but to do so we shall first recall the definition of a vector space. A vector space V over a field F is a set V, equipped with two operation: 1. under the operation of addition, denoted +, V forms an Abelian group; 2. there is also an operation of scalar multiplication of elements of V by elements of F which satisfies associativity and distributivity axioms, for all α, β ∈ F and all u, v ∈ V, (α + β)u = αu + βu α(u + v) = αu + αv (αβ)u = α(βu) (1.20)

The zero element 0 of F is related to the zero element 0 of V by 0u = 0, and the unit 1 of F acting trivially on V by 1u = u. The fields of interest in physics will be R or C giving respectively a real or a complex vector space. The vector space V will then be isomorphic to Rn, Cn or be an infinite-dimensional generalization, such as a complex function space.

1.6 Lie Algebras

A Lie algebra (defined over a field F which will be either R or C), is a vector space G endowed with a binary operation [x, y] which is referred to as the Lie and which satisfies the following axioms,

13 1. closure: [x, y] ∈ G for all x, y ∈ G; 2. anti-symmetry: [y, x] = −[x, y] for all x, y ∈ G; 3. linearity: [αx + βy, z] = α[x, z] + β[y, z] for all α, β ∈ F and all x, y, z ∈ G; 4. Jacobi identity: [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 for all x, y, z ∈ G.

When the vector space G has dimension n over F, we denote a basis of G by Xa for a = 1, ··· , n and Xa are referred to as generators of the Lie algebra. Closure implies that the Lie bracket of two basis vectors must be a linear combination of the Xa, so that we have the structure relations of the Lie algebra G,

n X [Xa,Xb] = fabcXc (1.21) c=1

The coefficients fabc are referred to as the of the Lie algebra G. By anti-symmetry of the Lie bracket, they are anti-symmetric in the first two indices,

fbac = −fabc (1.22) for all a, b, c = 1, ··· n. By linearity, the product of any two vectors may be decomposed in terms of the structure constants, while the Jacobi identity on the basis vectors,

[Xa, [Xb,Xc]] + [Xb, [Xc,Xa]] + [Xc, [Xa,Xb]] = 0 (1.23) imposes the Jacobi relations on the structure constants,

n X   fadefbcd + fbdefcad + fcdefabd = 0 (1.24) d=1

Conversely, a set of coefficients fabc will correspond to structure constants of a Lie algebra provided fabc is anti-symmetric in the indices a, b and satisfies the Jacobi identity (1.24). We note that, while the Lie bracket [x, y] in the definition of a Lie algebra has been denoted by the symbol customarily used for the xy−yx, a Lie algebra structure does not assume that products such as xy can be defined for x, y ∈ G: these products may or may not exist. For the case of matrix algebras, we will in fact have [x, y] = xy − yx because the product of square matrices exists independently of their commutator.

1.6.1 Examples of Lie algebras

Any vector space V, over R or C, may be endowed with a vanishing Lie bracket for all pairs of elements of V. The corresponding Lie algebra is then referred to as an Abelian Lie algebra. The vector spaces Rn and Cn where the elements are interpreted as translations fall in this .

14 Examples of non-Abelian Lie algebras include the vector space V of n×n matrices where the Lie bracket of two elements x, y ∈ V is the commutator [x, y] = xy − yx and operation which is defined in terms of the product xy of n × n matrices, is always well-defined, and automatically satisfies the Jacobi identity. An example of a non-Abelian Lie algebra is provided by gl(n, R) which is the vector space of all n × n matrices with real entries and the Lie bracket is the commutator. This vector 2 2 space has dimension n and a basis may be given by n matrices Xij with 1 ≤ i, j ≤ n whose entries are all zero, except for entry (i, j) which is 1. Note that in our general discussion

of Lie algebras we used a single index a to label the generators Xa, but here is it much more convenient to use the composite label (i, j). To work out the structure constants for gl(n, R) using this notation, we exhibit the entries of the generator Xij as follows,

(Xij)αβ = δiαδjβ 1 ≤ α, β ≤ n (1.25)

The components of the commutator are then obtained by the rules of matrix multiplication,

n X   [Xij,Xk`]αβ = (Xij)αγ(Xk`)γβ − (Xk`)αγ(Xij)γβ (1.26) γ=1

Expressing the components in terms of Kronecker δ, we find,

n X   [Xij,Xk`]αβ = δiαδjγδkγδ`β − δkαδ`γδiγδjβ = δjkδiαδ`β − δi`δkαδjβ (1.27) γ=1

Recasting the Kronecker δ combinations in terms of components of generators, we arrive at the Lie algebra structure relations of gl(n, R),

[Xij,Xk`] = δjkXi` − δi`Xkj (1.28)

Note that the same basis of generators Xij may be used to span the vector space of n × n matrices with complex entries. They form the Lie algebra gl(n, C) whose structure relations on the generators Xij coincide with those of gl(n, R).

1.7 Relating Lie groups and Lie algebras In the subsections above, Lie groups and Lie algebras have been defined independently of one another. In this subsection, we shall explain how the two are intimately related to one another. We begin by stating the results, and will then partially prove them.

Theorem 3 To every Lie group G there corresponds a unique Lie algebra G generated by the tangent vector fields to G at the identity.

15 The definition and construction of tangent vectors and the tangent space to a point on a manifold will be given in the subsequent subsection. For the special case of a Lie group G, the tangent space at every point on the Lie group manifold may be obtained from the

tangent space at the identity by using left or right translations Lγ, Rγ on G, which are defined by left and right multiplication,

Lγ(g) = γ g −1 Rγ(g) = g γ (1.29)

Left translations form a group γ ∈ GL which is isomorphic to G, and right translations form an independent group γ ∈ GR which is also isomorphic to G. Left and right translations 0 commute with one another, LγRγ0 = Rγ0 Lγ for all γ ∈ GL and γ ∈ GR and, upon setting γ = γ0 reduces to the inner automorphism discussed earlier. Under right translation by an element γ = g, the element g is mapped to the identity. Every point in G may be mapped

to the identity by such a translation. The tangent space Tg at the point g is therefore isomorphic to the tangent space Te at the identity in G. Theorem 4 To every Lie algebra G there corresponds a unique connected and simply connected Lie group G obtained by the exponential map applied to the Lie algebra. The exponential map will be constructed in the next section for matrix groups. The knowledge of the Lie algebra determines the Lie group global properties which cannot be reconstructed from a local treatment, such as connectedness and simply connectedness.

1.8 Tangent vectors, tangent space, and cotangent space

Let M be a manifold of dimension n. Let Uα be a local open neighborhood of a point n n p ∈ M and ϕα : M → R the corresponding homeomorphism to R . We shall assume that the manifold is differentiable at least once, so that all transition functions are C1. n A differentiable curve γ is a map γ :[−1, 1] → M such that ϕα ◦ γ :[−1, 1] → R is differentiable. We shall assume that the point p lies on γ and set γ(0) = p. The tangent

vector to the curve γ at the point p, in the chart Uα is defined by, d d ϕα ◦ γ(t) = ϕα(γ(t)) (1.30) dt t=0 dt t=0 n Since the of ϕα is in R , the tangent vector defined above is an element of the vector space Rn, and we may use the linearity properties of the vector space Rn to add vectors and multiply them by a scalar. The corresponding vector space is defined to be the tangent

space Tp(M), and has dimension n.

The tangent vectors defined in this manner depend on the chart (Uα, ϕα), on the curve γ, and on the choice of the parameter t. It will be convenient to isolate the dependence on γ and t by recasting the tangent vectors in terms of vector fields. To do so, consider a

16 differentiable function f : M → R. In the chart (Uα, ϕα) f may be expressed by the local i i ˜ n coordinates x = ϕα for i = 1, ··· , n in terms of a function f : R → R by, ˜ ˜ 1 n f(p) = f ◦ ϕα(p) = f(x , ··· , x ) (1.31) The derivative of f along the curve γ at the point p is obtained by chain rule, m   ˜ d X d i ˜ ˜ ∂f f(γ(t)) = ϕα ◦ γ(t) ∂if ∂if ≡ (1.32) dt t=0 dt t=0 ∂xi i=1 and defines a on the space of differentiable functions f. The coefficients of the ˜ derivatives ∂if are the components of the tangent vector to γ at p, but might be replaced with arbitrary coefficients which no longer refer to the specific curve used, m X i ˜ Dvf = v (p)∂if (1.33) i=1 It is a standard , when no confusion is expected to arise, to drop the tilde on f, and to identify the points p ∈ M with their local coordinates (x1, ··· , , xn). The map

Dv is linear in f, obeys Leibnitz’s rule,

Dv(fg) = fDv(g) + gDv(f) (1.34) and is referred to as a derivation or vector field. The tangent space Tp(M) at a point p ∈ M is isomorphic to the space of all vector fields. We may view the vector of derivatives ∂i as the basis vectors of Tp(M). Finally, it makes sense to act on a function f by different vector fields. The commutator of two vector fields is again a vector field, n i X  j i j i [Dv,Dw]f = Duf u = v ∂jw − w ∂jv (1.35) j=1

The commutator satisfies the Jacobi identity thanks to the Schwarz identity ∂i∂jf = ∂j∂if.

The vector space dual to Tp(M), in the sense of duality in normed vector spaces, is the ∗ i cotangent space Tp (M) at p, and its basis vectors are the differential forms dx . A general differential 1-form ω may be decomposed in this basis by, m X i ω = ωi(p)dx (1.36) i=1 The total differential d, m X ∂ d = dxi (1.37) ∂xi i=1 is independent of the coordinates chosen in the chart.

17 2 Matrix Lie groups and Lie algebras

In this section, we shall discuss Lie groups and Lie algebras defined in terms of matrices whose entries are real or complex. The group operation is matrix multiplication of n × n matrices which gives again an n × n matrix, and is always associative. The most general such groups were introduced earlier as GL(n, R) and GL(n, C). Fur- ther matrix groups that will be of interest in Physics are subgroups of GL(n) defined by certain equalities between the group elements. Such subgroups are referred to as closed subgroups of GL(n) since generally spaces defined by an equality are closed sets. Matrix groups whose entries are rational or integers, such as GL(n, Q) and GL(n, Z), are occur in Physics as duality groups, but they are not Lie groups. We shall exhibit the relations between Lie groups and Lie algebras, introduce Car- tan subgroups and subalgebras, evaluate the dimension and the rank, establish low-rank coincidences and general inclusions of the standard matrix groups and algebras.

2.1 The general linear group GL(n) The set of n × n matrices with real or complex entries and non-vanishing determinant form groups under matrix multiplication denoted by GL(n, R) or GL(n, C). The is the unit element, and the condition of non-vanishing determinant guarantees that every matrix in GL(n) has an inverse given by the inverse matrix. Some basic topological and group-theoretic properties are as follows.

• GL(1, R) ≈ R0 is Abelian, non-compact, not connected since the positive and negative components are disconnected, and simply-connected. • GL(1, C) ≈ C0 is Abelian, non-compact, connected, but not simply connected since the complex plane has one point removed. • For n ≥ 2, the groups SL(n, R) and SL(n, C) are non-Abelian, non-compact since 2 2 they are unbounded in Rn or Cn and connected. While SL(n; C) is simply-connected, its real counterpart SL(n; R) is not simply connected. • For n ≥ 2, the groups GL(n, R) = SL(n, R) × R0 and GL(n, C) = SL(n, C) × C0 inherit the properties of their factors, and are non-Abelian, non-compact, the former being not connected and the latter not simply connected.

As was already explained earlier, these matrix multiplication groups are all Lie groups.

2.1.1 The general linear Lie algebra gl(n)

The general linear Lie algebra gl(n, R) is the vector space of all n × n real matrices. The Lie bracket is given by the matrix commutator [x, y] = xy − yx, which gives a real n × n

18 matrix for any x, y ∈ gl(n, R), is clearly anti-symmetric and linear in each argument, and automatically satisfies the Jacobi identity, since we have,

[[x, y], z] + [[y, z], x] + [[z, x], y] = [xy − yx, z] + [yz − zy, x] + [zx − xz, y] = 0 (2.1)

for all x, y, z ∈ gl(n, R). Thus, gl(n, R) forms a Lie algebra over R. Similarly, the space of all n × n matrices with complex entries gl(n, C) forms a Lie algebra over C. The special linear Lie algebra sl(n, R) and sl(n, C) are the subspaces of respectively gl(n, R) and gl(n, C) with vanishing , and close under the commutator since tr[x, y] = 0 for all x, y ∈ gl(n).

2.1.2 From GL(n) to gl(n)

From the Lie group GL(n, R) we may construct the corresponding Lie algebra gl(n, R) by expanding the group elements in a neighborhood of the identity element I ∈ GL(n, R) which corresponds to the parameter values xij = δij for i, j = 1, ··· , n. To organize this expansion, we introduce the expansion of the group elements,

2 2 gX (α) = I + αX + O(α ) xij = δij + αXij + O(α ) (2.2)

where X is an arbitrary n × n real matrix. Performing this expansion in all directions

around the identity defines the tangent space Te at the identity, and X ∈ Te. For given X, the group elements gX (α) are labelled by one parameter α and generate a one-parameter subgroup of G. We can now go through the conditions under which GL(n, R) is a group, expand those conditions to linear order in α and deduce their implications on the structure

of the tangent space Te.

Closure of the Lie group requires that the product gX (α)gY (β) be in GL(n, R). To first order in α and β this requires that αX + βY should be in Te, which means that Te is a vector space. The structure of an algebra on Te is obtained by expanding the product gX (α)gY (β) to terms which are first order in both α and β. To obtain these terms it is easiest to study the commutator of group elements,

−1 −1 2 2 gX (α)gY (β)gX (α) gY (β) = (I + αX)(1 + βY )(1 − αX)(1 − βY ) + O(α , β ) = I + αβ(XY − YX) + O(α2, β2) 2 2 = g[X,Y ](αβ) + O(α , β ) (2.3)

The bracket is defined to be the commutator of the matrices,

[X,Y ] = XY − YX (2.4)

which automatically satisfies the Jacobi identity. Thus closure in the Lie group requires

that the commutator [X,Y ] ∈ Te, so that Te with the commutator operation satisfies the axioms of a Lie algebra, and may be identified with the Lie algebra gl(n, R).

19 2.1.3 The exponential map: from gl(n) to GL(n)

Lie’s theorem states that, given a Lie algebra G, there exists a unique connected and simply- connected Lie group G whose Lie algebra is G. Thus, we can ask in the case of matrix groups such as gl(n) and sl(n) (with real or complex field), how one passes from the Lie algebra to the Lie group. This is done with the help of the exponential map, which for matrix algebras is literally given by the exponential function of matrices. The Taylor series of an entire function f(z) has infinite radius of convergence near any point in C. Thus we may identify the function with its Taylor series at the origin,

∞ X zk f(z) = f (k)(0) (2.5) k! k=0

The function f is defined on n × n matrices X by substituting X for z in its Taylor series,

∞ X Xk f(X) = f (k)(0) (2.6) k! k=0 where Xk is the k-th power of the matrix X. The result f(X) is an n×n matrix. Generally, matrix sums of this type are difficult to compute, because one generally does not have a good formula for successive powers of X. However, if X is diagonalizable by a matrix g, −1 namely if one has X = gΛg for some Λ with components Λij = λiδij, then the function may be easily worked out,

−1 f(X) = gf(Λ)g f(Λ)ij = f(λi)δij (2.7)

The calculation may be extended to the case where X cannot be diagonalized, but can only be reduced to Jordan normal form. The exponential map of a matrix X is the exponential function of X, given by eX . The exponential of 0 ∈ G maps to the identity matrix, while the inverse of eX is given by e−X . A very useful relation between the trace and the determinant is given by,

det (eX ) = etr(X) (2.8)

Given an element X of the Lie algebra gl(n) one defines a one-parameter group GX by,

tX gX (t) = e (2.9)

For all X and all values of t, gX (t) ∈ GL(n), as the exponential always produces an . If trX = 0 then det gX (t) = 1 and thus gX (t) ∈ SL(n).

20 2.2 Closed subgroups of the general linear group

Subgroups of the general linear groups, either over R or over C, often arise in mathematics n n and in physics as the elements of GL(n) which leave a on Vn = R or C , or a Hermitian form on Cn invariant. A general bilinear form is specified by an n × n matrix Q, which is real or complex according to the nature of Vn, t Q(x, y) = x Qy x, y ∈ Vn (2.10) If Q is degenerate, namely if the rank of Q is strictly less than n, then the form can be restricted to a subspace Vp with p < n on which the restricted form is non-degenerate. Therefore we may assume that Q is non-degenerate without loss of generality. The subset of linear transformations M ∈ GL(n) which leave Q invariant satisfy, Q(Mx, My) = Q(x, y) M tQM = Q (2.11) for all x, y ∈ Vn. They form a group GQ under multiplication since when Mi satisfies t Mi QMi = Q for i, 1, 2 then their product also satisfies this relation, as may be seen from the following manipulations t t t t (M1M2) Q(M1M2) = M2M1QM1M2 = M2QM2 = Q (2.12)

Multiplication is associative since Mi are matrices; the identity matrix is the unit element of the group; and the inverse of an element M is given by M −1 = Q−1M tQ which makes sense because we have assumed that Q is non-degenerate. The quadratic relation implies that det M = ±1, so that the group GQ is disconnected and has (at least) two connected components, of which the satisfies det M = 1. The bilinear form Q need not be symmetric or anti-symmetric, but may clearly be decomposed into its symmetric and its anti-symmetric parts. This decomposition will give rise to bilinear forms with definite symmetry but each of which is generically degenerate. The non-degenerate forms with definite symmetry will be studied next.

2.3 The orthogonal groups SO(n) and Lie algebras so(n) The orthogonal groups, either over the reals or over the complex numbers, are defined to leave a symmetric bilinear form invariant. The real and complex cases behave differently and it will be convenient to treat them separately from one another.

2.3.1 Classification over R

We define a real-valued symmetric non- on the vector space Vn = Rn in terms of an n × n real Q, t n Q(x, y) = x Qy x, y ∈ R (2.13)

21 Equivalently, one may consider the associated Q(x, x). Since Q is real symmetric it may be diagonalized by an orthogonal matrix M, which satisfies M tM = I t such that Q = M DM and D is a diagonal matrix with entries diδij. In terms of the new coordinates x0, y0, the bilinear form thus reads,

n X Q(x, y) = dixiyi (2.14) i=1

Since we have assumed that the quadratic form is non-degenerate, none of the eigenvalues di vanish, and we shall denote the number of positive and negative eigenvalues respectively by p and q with p + q = n. The pair (p, q) is referred to as the signature of the quadratic form.

The signature is unchanged under real linear transformations on the coordinates xi, and is thus an intrinsic of the quadratic form. By also rescaling the coordinates, one arrives at the canonical bilinear form of signature (p, q),   t Ip 0 Q(x, y) = x Ip,qy Ip,q = (2.15) 0 −Iq where Ip is the p × p identity matrix. One usually considers the bilinear forms with signa- tures (p, q) and (q, p) to be equivalent to one another. Important special cases are,

1. Euclidean signature (n, 0), corresponding to the Euclidean metric on Rn, given by ds2 = Q(dx, dx) = dxtdx; 2. Minkowski signature (n − 1, 1), corresponding to the Minkowski metric on Rn, given 2 t by ds = Q(dx, dx) = dx In−1,1dx; 3. Conformal signature (n − 2, 2), do be discussed later.

The linear group which leaves a real symmetric non-degenerate bilinear form of signature (p, q) invariant is denoted by O(p, q; R) and is defined by,

t M Ip,qM = Ip,q M ∈ O(p, q; R) (2.16)

For n = 2, the groups are Abelian, while for n = p + q ≥ 3, the groups are non-Abelian. The defining relation implies that det M = ±1, so that O(p, q; R) is not connected as a manifold, but rather has two connected components. The component connected to the identity may be defined as follows,

t M Ip,qM = Ip,q det M = 1 M ∈ SO(p, q) (2.17)

For Euclidean signature (n, 0), the groups O(n; R) and SO(n; R) are compact since they 2 are closed and bounded in Rn , while for pq 6= 0 the groups SO(p, q) are non-compact, since

22 2 they are unbounded in Rn . None of these groups are simply-connected, a property which is both delicate and important and will be discussed in detail when we deal with spinors. The associated Lie algebra is so(n, R), the algebraic pairing is the commutator of ma- trices, and the elements m ∈ so(n, R) satisfy,

 SU mt = −I mI m = (2.18) p,q p,q U t T

where S and T are anti-symmetric p × p and q × q matrices respectively, and U is an arbitrary p × q matrix.

2.3.2 Classification over C We define a complex-valued symmetric non-degenerate bilinear form Q on the vector space n Vn = C in terms of an n × n complex symmetric matrix Q,

t n Q(x, y) = x Qy x, y ∈ C (2.19)

Unlike in the real case, there is no need in the complex case to introduce signatures, since a factor of −1 may always be absorbed in the quadratic form by multiplication by a factor of i. Making a general linear transformation on the coordinates, one arrives at the canonical symmetric bilinear form over the complex numbers,

Q(x, y) = xty (2.20)

Its invariance group is denoted by O(n, C) and consists of the elements g ∈ O(n, C) such that gtg = I. It follows that det g = ±1 so that the group has two connected components; the component connected to the identity is denoted SO(n, C). The corresponding Lie alge- bra so(n, C) is generated by the same elements as the real algebra so(n, R) but considered now with complex coefficients.

2.4 The symplectic group Sp(2n) and Lie algebra sp(2n) The symplectic groups, either over the reals or over the complex numbers, are defined to leave a anti-symmetric bilinear form invariant. The real and complex cases behave very similarly and will be treated together. A non-degenerate anti-symmetric bilinear form can only exist in even dimensions, and may be put in canonical form by general linear transformations on the entries,

 0 I  Q(x, y) = xtJ y J = n (2.21) −In 0

23 where x, y ∈ R2n for the real case and x, y ∈ C2n in the complex case. The closed subgroup Sp(2n) of GL(2n) which leaves Q invariant is defined by, Q(Mx, My) = Q(x, y) M tJM = JM ∈ Sp(2n) (2.22) for all x, y ∈ R2n for Sp(2n; R) or x, y ∈ C2n for Sp(2n; C). The associated Lie algebras are sp(2n; R) and sp(2n; C) and we have for either the real or the complex case that m ∈ sp(2n) satisfies mtJ + Jm = 0.

2.4.1 Symplectic structure of Hamiltonian mechanics is based on the following three ingredients:

1. S2n which is a manifold of even dimension 2n, locally parametrized by generalized position qi and momentum pi variables for i = 1, ··· , n; 2. The Hamiltonian H(p, q) is a function on phase space; 3. The Poisson bracket {x, y} is an anti-symmetric bilinear form on phase space. When

this form is non-degenerate, S2n is a . Evaluating the Poisson bracket on the generalized coordinates,

x = (p1, ··· , pn; q1, ··· , qn) {xα, xβ} = Jαβ (2.23) for α, β = 1, ··· , 2n. Equivalently, one defines the symplectic 2-form by,

2n n 1 X X ω = J dx ∧ dx = dp ∧ dq (2.24) 2 αβ α β i i α,β=1 i=1

and this form is closed dω = 0. The linear transformations on xα which leave the Poisson bracket or, equivalently, the symplectic form ω invariant form the real sym- plectic group Sp(2n; R). In this way, the symplectic group is the subgroup of all canonical transformations which act linearly on the canonical variables. The existence of a transformation which acts on the canonical vari-

ables by infinitesimal transformations δaqi and δapi implies, by Noether’s theorem, the existence of a conserved charge Qa. The transformations are reproduced by the action of the Poisson bracket,

δaqi = {Qa, qi} δapi = {Qa, pi} (2.25)

The Poisson bracket of two conserved charges Qa,Qb produces a new conserved charge, so that the vector space of conserved charges produces a Lie algebra under composition by the Poisson bracket,

d X {Qa,Qb} = fabcQc (2.26) c=1

24 and fabc are the structure constants of the symmetry algebra. The Poisson bracket auto- matically satisfies the Jacobi identity when acting on smooth functions, and therefore the symmetry algebra is a Lie algbera. The qualification is analogous to the curl of a gradient vanishing, provided both act on smooth functions.

2.5 The unitary group SU(n) and Lie algebra su(n)

A Hermitian form (sometimes referred to as a ) over Cn is given in terms of a Hermitian matrix Q by,

† n Q(x, y) = x Qy x, y ∈ C (2.27)

with Q† = Q. As with bilinear forms, we assume without loss of generality that the form is non-degenerate. Since Q is Hermitian, we may diagonalize it by a unitary transformation, and its eigenvalues di will be real and non-zero, so that the form may be expressed as,

n X Q(x, y) = di x¯i yi (2.28) i=1

One may further rescale the variables xi, yi by a , which rescales di by an arbitrary but positive number. Thus, we encounter again the possibility of having forms of different signatures (p, q) with p + q = n where p and q count the number of positive and negative eigenvalues respectively. Thus a Hermitian form of signature (p, q) may be reduced to the following canonical form,

† n Q(x, y) = x Ip,q y x, y ∈ C (2.29)

The forms of signature (p, q) and (q, p) are usually assumed to be equivalent to one another. The group which leaves this form invariant is defined by,

† Q(Mx, My) = Q(x, y) M Ip,qM = Ip,q (2.30) and denoted by U(p, q). There is no need to indicate that this is over the complex, since Hermitian assumes complex. The group U(p, q) has a U(1) invariant subgroup, and for this reason one defines the more frequently used group,

SU(p, q) = {M ∈ U(p, q), det M = 1} (2.31)

The most familiar case is when q = 0, in which case the group is denoted by SU(n). The Lie algebra associated with the Lie group SU(n) is denoted by su(n) and is the space of traceless anti-Hermitian n×n matrices m ∈ su(n). More generally, the Lie algebra

25 associated with SU(p, q) is denoted by su(p, q) and consists of n × n complex matrices m which satisfy the relation,   † m1 m3 m Ip,q + Ip,qm = 0 m = † (2.32) m3 m2 where m1 and m2 are anti-hermitian matrices of dimension p × p and q × q respectively, and m3 is an arbitrary p × q matrix.

2.6 Cartan subalgebras and subgroups and the center A crucial role is played by the Cartan subgroup of a Lie group, and the of a Lie algebra. We shall limit the definition here to semi-simple Lie groups (the product of simple Lie groups), and semi-simple LIe algebras (the direct sum of simple LIe algebras), which all the matrix groups are aside from some Abelian cases. The Cartan subalgebra of a semi- G with Lie group G is defined to be its maximal Abelian subalgebra, and is unique up to conjugation by an arbitrary element of the Lie group G. The dimension of the Cartan subalgebra is defined to be the rank of G and G. The Cartan subgroup of G may similarly be defined as the maximal Abelian subgroup of G, or , which is unique up to conjugation. The center Z(G) of a group G is the subgroup of elements in G which commute with all elements g ∈ G. The center Z(G) is a normal subgroup of G so that G/Z(G) is a group.

2.7 Summary of matrix Lie groups and Lie algebras In the table below we give a summary of simple Lie groups, their group relation, Cartan type of their Lie algebra, rank, and dimension. The unit determinant condition is understood for all groups but has been exhibited only for SL groups for brevity. The above Lie groups are all connected. The groups SU(n), SO(n; R), and USp(2n) are compact; all others are non-compact. The groups SL(n; C) and SU(n) are simply connected, while π1(SO(n)) = Z2 for n ≥ 3 and π1(Sp(2n)) = Z2 for all n ≥ 1, so these groups are not simply connected. The group USp(2n) = Sp(2n; C) ∩ U(2n) is the maximal compact subgroup of Sp(2n; C), and is sometimes denoted Sp(n).

2.7.1 between matrix groups of low rank For complex groups we have the following identifications,

• SO(3; C) = SL(2; C)/Z2 • SO(4; C) = SL(2; C) × SL(2; C)/Z2 • SO(5, C) = Sp(4; C)

26 type group group relation rank dimension center 2 An−1 SL(n; C) det M = 1 n − 1 2n − 2 Zn 2 SL(n; R) det M = 1 n − 1 n − 1 1 or Z2 † 2 SU(n) M M = I n − 1 n − 1 Zn † 2 SU(p, q) M Ip,qM = Ip,q, p + q = n n − 1 n − 1 Zn t Bn SO(2n + 1; C) M M = I n 2n(2n + 1) 1 SO(2n + 1; R) M tM = I n n(2n + 1) 1 t SO(p, q; R) M Ip,qM = Ip,q , p + q = 2n + 1 n n(2n + 1) 1 t Cn Sp(2n; C) M JM = J n 2n(2n + 1) Z2 t Sp(2n; R) M JM = J n n(2n + 1) Z2 t † USp(2n) M JM = J, M M = I n n(2n + 1) Z2 t Dn SO(2n; C) M M = I n 2n(2n − 1) Z2 t SO(2n; R) M M = I n n(2n − 1) Z2 t SO(p, q; R) M Ip,qM = Ip,q, p + q = 2n n n(2n − 1) Z2

• SO(6; C) = SL(4; C) For compact groups we have the following identifications at low ranks,

• SO(3; R) = SU(2)/Z2 = USp(2)/Z2 • SO(4; R) = SU(2) × SU(2)/Z2 • SO(5; R) = USp(4)/Z2 • SO(6; R) = SU(4)/Z2 For real non-compact groups we have the following identifications at low ranks,

• SO(1, 2; R) = SL(2; R) = Sp(2; R) • SO(1, 3; R) = SL(2, C)/Z2 • SO(2, 2; R) = SO(2, 1; R) × SO(2, 1; R) • SO(2, 3; R) = Sp(4; R)/Z2 • SO(2, 4; R) = SU(2, 2)/Z2 All these identifications at low rank involve the orthogonal groups and are somehow related to the existence of the Clifford algebra and spinors. Many if not all play one role or another in physics. Let us establish the most frequently used relations.

2.7.2 Pauli matrices

We begin by introducing the Pauli matrices σi for i = 1, 2, 3 which are 2 × 2 Hermitian matrices satisfying the following relations,

tr(σi) = 0 {σi, σj} = 2δijI2 (2.33)

27 It is customary to use a specific representation for these matrices, given by,

0 1 0 −i 1 0  σ = σ = σ = (2.34) 1 1 0 2 i 0 1 0 −1

2.7.3 Proving SO(3; C) = SL(2; C)/Z2

There is a simple one-to-one map between complex 3-vectors z = (z1, z2, z3) with zi ∈ C and traceless complex 2 × 2 matrices,

  3 z3 z1 − iz2 X M(z) = = ziσi (2.35) z1 + iz2 −z3 i=1

The trace of M(x) vanishes while its determinant is the complex quadratic form −det M(z) = 2 2 2 −1 z1 + z2 + z3. Under conjugation gM(z)g by an arbitrary invertible matrix g, the trace continues to vanish so that the result must be a matrix M(z0) of the same type,

M(z0) = gM(z)g−1 (2.36)

Since multiplying g by a complex scalar, including the central element −I ∈ SL(2, C), does 0 0 not affect M(z ) we restrict g ∈ SL(2, C)/Z2. Since det M(z ) = det M(z), the action of g on x must be a complex orthogonal transformation, so that SO(3; C) = SL(2; C)/Z2.

2.7.4 Proving SO(3; R) = SU(2)/Z2

To prove the identification SO(3; R) = SU(2)/Z2, we define the group U(2) by the fact that g ∈ U(2) leaves the following quadratic form on z ∈ C2 invariant under z → gz,   † z1 z z = 1 z = z1, z2 ∈ C (2.37) z2

Now consider the real vector xi for i = 1, 2, 3 given by the following Hermitian forms,

† 1 2 3 xi = z σiz x =z ¯1z2 +z ¯2z1 x = iz¯2z1 − iz¯1z2 x =z ¯1z1 − z¯2z2 (2.38)

† i P3 i i For z z = 1, the vector v has unit length, i=1 v v = 1. The invariant subgroup U(1) of U(2) merely multiplies z by a phase and does not act on vi. This leaves only SU(2), but its center element −I ∈ SU(2) also does not act on vi. Hence the action is only by i SU(2)/Z2. Now this action leaves the length of the three-vector v invariant, and thus acts i by an orthogonal transformation on v , so that SU(2)/Z2 = SO(3; R).

28 2.7.5 Proving SO(4; R) = (SU(2) × SU(2)) /Z2

There is a one-to-one map between real 4-vectors x = (x1, x2, x3, x4) with xi ∈ R and complex matrices of the form,

  3 x4 + ix3 ix1 + x2 X M(x) = = x4I2 + i xiσi (2.39) ix1 − x2 x4 − ix3 i=1

† which span the space defined by the conjugation relation M + M = I2 tr(M). One may also think of these matrices as being isomorphic to quaternions. The following relation gives the Euclidean quadratic form in R4,

† 2 2 2 2 M(x) M(x) = I2 x1 + x2 + x3 + x4 (2.40)

Under independent left and right multiplications by elements (gL, gR) ∈ SU(2)L × SU(2)R,

0 −1 M(x ) = gLM(x)gR gL ∈ SU(2)L, gR ∈ SU(2)R (2.41) we have M(x0)†M(x0) = M(x)†M(x) and hence this transformation performs a rotation on the vector x. However, the transformation (gL, gR) = (−I2, −I2) does not act on M(x), and hence we have the identification SO(4; R) = (SU(2) × SU(2)) /Z2.

2.7.6 Proving SO(1, 3; R) = SL(2; C)/Z2

There is a one-to-one map between real 4-vectors x = (x0, x1, x2, x3) with xi ∈ R and Hermitian 2 × 2 matrices,

  3 x0 + x3 x1 − ix2 X M(x) = = x0I2 + xiσi (2.42) x1 + ix2 x0 − x3 i=1

4 2 2 2 2 The determinant gives the Minkowski quadratic form on R det M(x) = x0 − x1 − x2 − x3 which is invariant under SO(1, 3; R). Under the following action of g ∈ SL(2; C), M(x0) = g†M(x)g (2.43) the determinant, and hence the Euclidean quadratic form is preserved. The center element −I2 ∈ SL(2; C) does not act, and hence we can identify SO(1, 3; R) = SL(2; C)/Z2.

2.7.7 Maximal compact subgroups Compact Lie groups play a central role in Mathematics and in Physics, and the compact subgroups of a non- also play a central role. The maximal compact subgroup of a non-compact group is the largest such subgroup, in the sense of largest dimension and

29 group max compact subgroup SL(n; C) SU(n) SL(n; R) SO(n; R) SU(p, q) SU(p) × SU(q) × U(1) SO(n; C) SO(n; R) SO(p, q; R) SO(p; R) × SO(q; R) Sp(2n; C) USp(2n) Sp(2n; R) U(n) SO(p, q; R) SO(p; R) × SO(q; R)

rank. These groups may generally be obtained as intersection groups with the largest compact group in a given dimension, which is U(n). Thus the maximal compact subgroups are given in the Table below. For example, the maximal compact subgroup of Sp(2n; R) is obtained by computing its intersection with U(2n). Let M ∈ Sp(2n; R) ∩ U(2n), so that the matrix is real and unitary, which means that it belongs also to SO(2n; R) and we have t t M M = I2n as well as M JM = J, which upon combining both shows that MJ = JM. In n × n block decomposition, we have,  0 I  AB J = n M = (2.44) −In 0 CD t The commutation of M with J implies C = −B and D = −A, while M M = I2n implies t t t t A A + B B = In and A B = B A. Identifying J with the complex number i we introduce † the complex n×n matrix m = A+iB, and we see that m m = In using the above relations between A and B, so that m ∈ U(n).

2.8 Non-semi-simple matrix groups The Lie groups considered so far are, 1. Abelian groups , such as for example Rn, Cn, or U(1) = SO(2, R); 2. Simple groups, such as for example SL(n),SU(n) for n ≥ 2, SO(n) for n ≥ 5, and Sp(2n) for n ≥ 1, along with their various different real forms; 3. Semi-simple groups, such as direct products of the above simple groups; 4. Reductive groups, which direct products of semi-simple groups and Abelian groups,

such as for example the gauge group SU(3)c ×SU(2)L ×U(1)Y of the Standard Model. But there are plenty of other matrix groups that play an important role in Mathematics and in Physics. Here, we shall not attempt to give a full classification, but illustrate the ideas with the examples of the group of Euclidean motions and the Poincar´egroup, neither of which belongs in any of the above four categories.

30 2.8.1 The group of Euclidean motions and the Poincar´egroup The Euclidean metric or distance function in arbitrary dimension n is given by, n n X X ds2 = (dxi)2 s2 = (xi − yi)2 (2.45) i=1 i=1 In either formulation, the Euclidean metric is invariant under translations xi → xi+ai, yi → yi + ai by an arbitrary translation vector a = (a1, ··· , an)t which form an Abelian group Rn. The metric is also invariant under rotations x → Rx and y → Ry for an arbitrary rotation R ∈ SO(n, R) represented by an n × n orthogonal matrix acting on the column vector x = (x1, ··· , xn)t, which forms a non-Abelian group. We can assemble these two transformations into a single package, M(R, a)x = Rx + a (2.46) The composition of two transformations is given as follows,

M(R1, a1)M(R2, a2)x = R1R2x + R1a2 + a1

M(R1, a1)M(R2, a2) = M(R1R2,R1a2 + a1) (2.47) From this formula, we see that the transformations close, are associative, have unit element M(I, 0) and inverse M(R, a)−1 = M(R−1, −R−1a), so they form a Lie group which is referred to the group of Euclidean motions, and denoted ISO(n; R). The composition law reveals that translations and rotations act on x in an intertwined manner so that the group is not the direct product of SO(n; R) by Rn, but rather the product is referred to as semi-direct product and denoted by SO(n; R) n Rn. A matrix representation of the group may be obtained by considering (n + 1) × (n + 1) matrices M˜ acting on a vectorx ˜ = (x1, ··· , xn, 1)t ∈ Rn+1. Using a block decomposition of M˜ into blocks of size n × n, n × 1, 1 × n and 1 × 1, we have the following representation, R a M˜ (R, a) = (2.48) 0 1

One verifies that M˜ (R, a)˜x = (Rx + a, 1) so that M˜ (R, a) acts on x as M(R, a) does. The Poincar´egroup is obtained by flipping the sign of one term in the metric, and the Poincar´egroup ISO(1, n − 1; R) = SO(1, n; R) n Rn. Further generalizations are manifest.

2.9 Invariant differential forms, metric, and measure A very important notion is that of invariant differential forms on a Lie group G, and their relation with the Lie algebra G of G. To develop these objects for arbitrary Lie groups requires a bit of differential geometry, so we shall content ourselves here by defining invariant differential forms for matrix Lie groups.

31 2.9.1 Invariant differential forms on a Lie group For a matrix Lie group of dimension d, the group elements g(t1, . . . , td) are represented by square matrices n × n for some value of n, and are parametrized by local coordinates (t1, ··· , td). The partial derivatives of g with respect to ti are again n × n matrices, and the total differential is defined by,

d X ∂g dg = dti (2.49) ∂ti i=1 Generally, the matrix dg is not an element of the Lie algebra, but we can define differential forms that do take value in the Lie algebra by introducing the following combinations,

−1 −1 ωL(g) = g dg ωR(g) = dg g (2.50)

It is manifest that ωL is invariant under left multiplication of g by an arbitrary constant (i.e. t-independent) element h ∈ G, since d(hg) = hdg so that ω(hg) = ωL(g). Similarly, ωR(gh) = ωR(g) is invariant under right multiplication by a constant element h ∈ G.

Now the claim is that both ωL and ωR take values in the Lie algebra G which was defined to be the tangent space to G at the identity element of G. Ig we consider ωL and ωR at the identity element, then the statement is trivial by the definition of the tangent space to a manifold given earlier. Next, consider ωL (an analogous argument holds for ωR) at a point g0 ∈ G which is not the identity. But now we can use the invariance of ωL under left multiplication by g0 to pull the form back to the identity,

−1 ωL(g) = ωL(g0 g) (2.51) but from the right side it is clear that at the point g = g0, the form is evaluated at the identity element and thus again belongs to the Lie algebra. The dimension of the space of left-invariant forms (and of right-invariant forms) is n. Choosing a basis Xa with a = 1, ··· , d for G, we may decompose the forms as follows,

d d X a X a ωL = ωLXa ωR = ωRXa (2.52) a=1 a=1

We shall show later on that ωK and ωR satisfy the Maurer-Cartan equations which will prove that every Lie group admits a flat connection, which has torsion.

2.9.2 Invariant metric on a Lie group

For a matrix group G, one may use the invariant differential forms ωL or ωR to construct the following metric ds2 on G,

2 −1 −1  ds = −tr g dg g dg = −tr (ωLωL) = −tr (ωRωR) (2.53)

32 Expressing the metric ds2 in terms of local coordinates got G, we find,

d X  ∂g(t) ∂g(t) ds2 = γ (t)dtidtj γ (t) = −tr g−1(t) g−1(t) (2.54) ij ij ∂ti ∂tj i,j=1 The metric ds2 is invariant under both left and right multiplication of g, and is thus invariant under the group G × G. The signature of the metric will depend on the form of G, and is dictated by the signature of the quadratic form tr(XaXb). One shows that for a

compact Lie group G, the signature of −tr(XaXb) is Euclidean (all positive signs), so that the metric is then a positive Riemannian metric. For example, in the case of the SO(n, R) the Lie algebra so(n, R) consists of real n × n anti-symmetric matrices. We take a basis Xαβ with 1 ≤ α < β ≤ n, whose components are given by (Xαβ)ij = δαiδβj − δαjδβi and evaluate the traces,

−tr (XαβXγδ) = 2δαγδβδ − 2δβγδαδ (2.55) The trace is non-zero only if (α, β) = (γ, δ) or (α, β) = (δ, γ), but the latter is excluded since it would require β = γ < α = δ, which is excluded by the ordering α < β. Thus, the basis vectors Xαβ with 1 ≤ α < β ≤ d are all mutually orthogonal under the trace pairing, 2 and −tr(Xαβ) = 2. Therefore the invariant metric is positive definite and Riemannian. The Lie algebra of the unitary group U(n) consists of all n×n anti-Hermitian matrices.

We take a basis consisting of the real anti-symmetric matrices Xαβ introduced for so(n) together with the imaginary symmetric matrices iYαβ with 1 ≤ α ≤ β ≤ n for which the components are given by (Yαβ)ij = δαiδβj + δβiδαj. It is immediate to show that tr(XαβYγδ) = 0 and that,

tr (YαβYγδ) = 2δαγδβδ + 2δβγδαδ (2.56) The trace is zero unless (α, β) = (γ, δ) or/and (α, β) = (δ, γ) in which cases it is positive.

Thus, the full basis consisting of the union of {Xαβ} and {Yαβ} is orthogonal and non- degenerate, so that the the invariant metric is positive definite and Riemannian. Now let’s consider the non-compact group SO(p, q; R) with pq 6= 0. For the compact sub-algebra SO(p) we use the basis Xαβ with 1 ≤ α < β ≤ p while for SO(q) we use Xα0β0 0 0 with p + 1 ≤ α < β ≤ p + q, while for the off-diagonal block we use the basis Yαβ0 but this time without the factor of i since those matrices must be real symmetric. The signature of 1 1 this metric is now 2 p(p − 1) + 2 q(q − 1) plus signs and pq minus signs, adding up to the 1 total dimension 2 (p + q)(p + q − 1) of SO(p, q; R).

2.9.3 on a Lie group Having the G × G invariant metric ds2 on a Lie group G, we can investigate its geodesics, namely the curves of extremal distance between two given points. Differential geometry

33 tell us to evaluate the Christoffel connection for the metric and then solve the equation. Here we shall be able to take a short cut, and show that for every element X of the Lie algebra G of G, the one-parameter subgroup,

 tX GX = gX (t) = e , t ∈ R (2.57)

is the unique geodesic through the identity whose tangent vector at the identity is propor-

tional to X. Clearly the curve GX contains the identity for t = 0 and has tangent vector X. To prove that GX is a geodesic, we fix two points, the identity for t = 0, and the point X e for t = 1, and show that the length of GX is extremal under all infinitesimal variations δg(t) of the curve gX which leave the end points unchanged, given by,

g−1(t)δg(t) = ε(t) ∈ G ε(0) = ε(1) = 0 (2.58)

The length of an arbitrary curve g(t) is given by,

1 1 Z   2 `[g] = dt − trg−1g˙(t)g−1g˙(t) (2.59) 0

whereg ˙ is the derivative of g(t) with respect to t. For example the length of the curve GX X 2 1 between the identity and e is given by `[gX ] = (−tr(X )) 2 . To compute the variation in length due to the deformation ε, we first compute the following variation,

δ g−1g˙ = −g−1δgg−1g˙ + g−1δg˙ (2.60)

Using the fact that we parametrize the variations by the Lie algebra valued function ε(t), we have δg˙ =gε ˙ + gε˙, which allows us to simplify the above expression, and obtain,

δ g−1g˙ =ε ˙ + [g−1g,˙ ε] =ε ˙ + [X, ε] (2.61)

The length of the ε-deformed curve is given by,

1 1 Z   2 `[g] = dt − trX +ε ˙ + [X, ε]2 + O(ε3) (2.62) 0 Expanding the trace, we have,

trX +ε ˙ + [X, ε]2 = trX2 + 2trXε˙ + tr(ε ˙ + [X, ε])2 (2.63) where we have used the relation trX[X, ε] = tr(X2ε) − tr(XεX) = 0 due to the cyclic symmetry of the trace. Expanding the length in powers of ε to second order we find,

Z 1  2  1 2 1 (trXε˙) 3 `[g] = `[gX ] + dt −trXε˙ − tr(ε ˙ + [X, ε]) + 2 + O(ε ) (2.64) 0 2 2 trX

34 The first order term vanishes since X is independent of t and ε(0) = ε(1) = 0. This immediately shows that the curve GX has extremal length, and is a geodesic. We may learn more about these geodesics by investigating the contributions to second order in ε. The last term in (2.64) receives contributions only from deformations ε which are parallel to X, but these amount to a reparametrization in t of the curve and are geo- metrically immaterial. Thus, we may assume, without loss of generality, that tr(Xε(t)) = 0 for all t, and we are left with,

Z 1 1 2 3 `[g] = `[gX ] − dt tr(ε ˙ + [X, ε]) + O(ε ) (2.65) 2 0 For a compact Lie group G, the trace of gives a negative definite quadratic form on G. Since the combinationε ˙ + [X, ε] ∈ G, we conclude that for compact G, the geodesics are actually (local) minima of the length functional. For non-compact G, we can draw no such conclusion. We shall return to this issue when we deal with homogeneous spaces.

2.9.4 Invariant measure on a Lie group From the metric, as always, we can construct a volume form since from distances we can compute angles, and from distances and angles we can construct the volume. Since the metric is invariant under G × G, the resulting measure or volume form,

d p Y i dµG = det (γ(t)) dt (2.66) i=1

is also invariant under G × G and is referred to as the Hurwitz- or often just the Haar measure. The Haar measure allows one to integrate functions over the Lie group with strong symmetry properties, which will be of fundamental importance when we deal with representation theory and characters. For compact Lie groups, the volume of G with respect to the Haar measure is finite, Z Vol(G) = dµG (2.67) G These volumes may actually computed and carry interesting topological information.

2.9.5 Invariant metric on a space G/H Many interesting manifolds are obtained by taking the coset space G/H of a matrix Lie group G by one of its Lie subgroups H. For example, the sphere Sn, the hyperbolic upper half space Hn, and complex projective space CP n of arbitrary dimension n ≥ 2 may be

35 constructed as coset,

n S = SO(n + 1; R)/SO(n; R) n H = SO(n, 1; R)/SO(n; R) CP n = SU(n + 1)/SU(n) × U(1) (2.68)

The examples of the sphere S2 = CP 1 and the upper half plane H2 are familiar already. We shall now construct an invariant metric for such spaces. Recall that at the level of the group, the coset space is the quotient of G by the equivalence relation whereby to elements g, g˜ ∈ G are equivalent to one another, g1 ≈ g2, iff there exists an element h ∈ H such thatg ˜ = g h. Now consider the left-invariant differential forms g−1dg, which we have constructed on G, and observe its transformation under right multiplication by an element h of G, so thatg ˜ = g h,   g˜−1dg˜ = h−1g−1 dg h + g dh = h−1(g−1dg)h + h−1dh (2.69)

Clearly, the left-invariant differential form g−1dg depends on the element chosen in G to represent the coset, and is thus not well-defined on G/H. However, we see that the inhomogeneous term h−1dh is always an element of the Lie subalgebra H. To construct an invariant metric on the coset G/H, we will assume that the Lie algebra G of G admits a decomposition into a direct sum of the Lie algebra H of H plus a complement M,

G = H ⊕ M (2.70) such that the mixed trace vanishes, tr(XY ) = 0 for all X ∈ H and Y ∈ M. If the Lie group is compact, then this decomposition is automatically orthogonal under the trace inner product. But the decomposition extends to non-compact cases as well. In particular, this is the case when G/H is a iff H and M satisfy,

[H, H] ⊂ H [H, M] ⊂ M [M, M] ⊂ H (2.71)

Assuming the existence of the decomposition, we now take the projection of the differential form g−1dg onto the space M,

ω(g) = g−1dg (2.72) M the relation between the one-form ω(g) at two points g, g˜ = gh ∈ G which belong to the same equivalence class, and thus correspond to the same point in G/H, is given by,

ω(gh) = h−1ω(g)h (2.73)

36 Therefore, we may now construct a metric ds2 on G/H out of these new differential forms by taking the trace,

ds2(g) = trω(g)ω(g) (2.74)

The trace is invariant under the conjugation by h and thus we have ds2(gh) = ds2(g), and this metric is well-defined on the coset G/H. Furthermore, since it is constructed out of the left-invariant forms g−1dg, the metric ds2(g) is automatically invariant under left translations in G. We leave it up to the reader to show that the round metric on Sn indeed corresponds to this coset metric, and similarly for the other spaces. It is straightforward to check that geodesics on G/H are generated by one-parameter subgroups on G with a tangent vector at the identity given by Y ∈ M.

2.10 Spontaneous symmetry breaking, order parameters Consider a statistical mechanics system with a very large number (strictly speaking an infinite number) of degrees of freedom or a quantum field theory. Suppose its microscopic dynamics is governed by a Hamiltonian or by a Lagrangian which is invariant under a symmetry represented by the action of a group G on the dynamical degrees of freedom. A familiar example is given by translations or rotations in space, which are certainly symme- tries of the microscopic Hamiltonian or Lagrangian of Physics. In quantum mechanics, the ground state of the system is the state of lowest energy, and we shall often adopt the same terminology for a classical statistical mechanical system at very low temperatures (think of it as the ~ → 0 limit of a quantum system). Now the ground state may or may not be invariant under G. Taking again the example of rotation invariance, the ground state of a gas is statistically rotation invariant, and the same is true for a liquid. But the magnetic field of a magnet is not invariant under all rotation, but only under the rotations that leave the magnetic field invariant. This is an example of spontaneous symmetry breaking, a phenomenon which occurs whenever the ground state fails to be invariant under all the of the microscopic system, which are the symmetries of the Hamiltonian or Lagrangian. Thermal fluctuations may alter the symmetry and in the case of a magnet produce a second order phase transition to an unmagnetized (approximately) rotation symmetric state above the Curie temperature. Quantum fluctuations also contribute and may induce a quantum phase transition at very low temperatures. Lev Landau (1908 - 1968) proposed a theory of second order phase transitions based on an order parameter which reflects the change in symmetry of the system. The order parameter is a space-time dependent field (the scalar analogue of electric or magnetic fields), governed by an effective Lagrangian which summarizes the dynamics near the ground state.

37 For every continuous symmetry that is spontaneously broken, Goldstone’s theorem assures us that the spectrum will contain a massless scalar particle, the Nambu-Goldstone boson. Near the ground state energy, the only excitations in the spectrum will be the Goldstone bosons, for which we can write down an effective Lagrangian which must be invariant under Poincar´esymmetry, the symmetry G which and whose ground states must be invariant under a subgroup H of G. The Goldstone field g(x) takes values in G for every point x of space-time which we will just take to be flat Rn. The Lagrangian must be invariant under G, but take values only in G/H, since a rotation of the ground state by H is immaterial. It is immediate to deduce such a candidate Lagrangian in terms of the invariant differential form constructed on G/H,

1   −1 L = tr ωM ωM ωM = g ∂µg (2.75) 2 M These Lagrangians are special cases of what are referred to as non-linear sigma models, and they play a crucial role in many areas of theoretical physics, including particle physics and condensed matter and play a role in differential geometry as well.

2.11 Lie supergroups and Lie superalgebras Fermions obey Fermi-Dirac which requires Fermi fields to obey anti-commutation relations. In the functional integral, Fermi fields are represented by anti-commuting vari- ables referred to as Grassmann variables. (Perturbative Yang-Mills theory in a covariant gauge requires the ghost fields which are also represented by Grassmann variables.) By the Coleman-Mandula theorem (1967), there are no Lie group transformations that exchange bosons and fermions consistently with the physical principle of quantum mechanics and rel- ativity. It was discovered by Golfand and Lichtmann (1969), Ramond (1970), Gervais and Sakita (1971), that algebras using anti-commuting variables exist that can exchange bosons and fermions consistently. A full implementation in field theory was pioneered by Wess and Zumino in 1974, and goes under the name of supersymmetry. The mathematical structures realizing supersymmetry are a generalization of Lie groups in which the parameters live in a Grassmann algebra, and a generalization of Lie algebras with both as well as anti-commutators.

2.11.1 Grassmann algebra We postulate an arbitrary number of anti-commuting variables ξi, i = 1, ··· N with the following properties, {ξi, ξj} = ξiξj + ξjξi = 0 for all i, j = 1, ··· ,N (2.76) In particular this means that the square of each variable vanishes. These variables generate a vector space either over the real or over the complex numbers in which linear combinations

38 of ξi may be formed. We may supplement this vector space by a copy of RM or CM with commuting variables xI with I = 1, ··· M, and the combined vector space is denoted either RM|N or CM|N . One introduces a binary grading by which xI has even grading and each ξi has odd grading, and one only allows linear combinations that preserve the grading. Thus, it is illegal to form xI + ξi. The grading is preserved under multiplication. This restriction is required by the physical super-selection rule that one cannot form a linear combination of a fermion and a boson state. The vector spaces RM|N and CM|N generate a Grassmann algebra which is obtained by taking linear combinations of products of an even number of ξi and linear combinations of products of odd numbers of ξi subject to the grading. Thus, we can take linear combinations such as 2 + ξ1ξ3 + 6ξ1ξ2ξ5ξ7 with even grading, and πξ1 + 19ξ7ξ8ξ9 with odd grading. The Grassmann algebra may appear a bit abstract at first and it may not be immediately clear that such objects actually exist. A convenient construction in terms of matrices is obtained by starting from γµ which satisfy the Clifford-Dirac algebra, {γiγj} = 2δijI i, j = 1, ··· , d (2.77) For d = 3, one realizes the γi by the Pauli matrices γi = σi, and we shall see later how to construct the matrices γi explicitly for arbitrary d. Now take d = 2N even and form the following combinations, ξj = γ2j−1 + iγ2j j = 1, ··· ,N (2.78) Using (2.77), one readily verifies that {ξi, ξj} = 0 for all i, j = 1, ··· ,N.

2.11.2 Super groups

We shall denote the elements of the vector spaces Rm|n and Cm|n by graded columns, x X = (2.79) ξ where xI are even with I = 1, ··· , m and ξi are odd with i = 1, ··· , n. A general linear transformation M takes X to a vector Y = MX in the same vector space, and thus with the same grading. Using block matrix notation, y AB y = Ax + Bξ Y = M = (2.80) υ CD υ = Cx + Dξ where A, B, C, D are matrices of dimension m × m, m × n, n × m and n × n respectively. To preserve the grading, A, D must be even, and B,C must be odd. The product of two matrices M1,M2 is defined as the block-wise product,     Ak Bk A1A2 + B1C2 A1B2 + B1D2 Mk = M1M2 = (2.81) Ck Dk C1A2 + D1C2 C1B2 + D1D2

39 The product closes and is associative. The identity matrix has A = D = I and B = C = 0. The inverse of M exists when the matrices A and D are invertible, and is then given by,

A0 B0 A0 = (A − BD−1C)−1 B0 = −A−1BD0 M −1 = (2.82) C0 D0 D0 = (D − CA−1B)−1 C0 = −D−1CA0

The space of invertible matrices M forms a group GL(m|n; R) or GL(m|n; C) either over R or over C. However, GL(m|n) is not a Lie group in the sense defined earlier, since some of its parameters are not real or complex numbers, but rather Grassmann numbers. Note that the even matrices are not just real or complex numbers, but rather even elements of the Grassmann algebra. For this reason it is referred to as a Lie super algebra.

2.11.3 The Lie super groups SU(m|n) and OSp(m|2n) Closed subalgebras of GL(m|n) are defined by requiring invariance of a non-degenerate even quadratic form on Cm|n or Rm|n. The quadratic form is the sum of quadratic forms in the even variables xi and the odd variables ξi, the latter being very restrictive. In the complex case, it may be diagonalized to the following canonical form,

† † QC = x x + ξ ξ (2.83)

The corresponding Lie supergroup is U(m|n) with maximal even subgroup SU(m) × SU(n) × U(1). In the real case, non-degeneracy requires their number to be even 2n and the canonical form is given by,

t t QR = x x + ξ Jξ (2.84)

and the Lie supergroup is OSp(m|2n) with maximal even subgroup SO(m) × Sp(2n).

2.11.4 Lie super algebras

A Lie super algebra G is a graded algebra with even subspace G0 and odd subspace G1, on which one defines a graded Lie bracket, [·, ·] or {·, ·} depending on the arguments,

1. closure: [G0, G0] ∈ G0,[G0, G1], [G1, G0] ∈ G1, and {G1, G1} ∈ G0; 2. (anti-)symmetry: [x, y] = −[y, x] and {x, y} = {y, x}; 3. linearity: [αx + βy, z] = α[x, z] + β[y, z], {αx + βy, z} = α{x, z} + β{y, z};

4. Jacobi identity: if at most one of x, y, z ∈ G1 we have the usual Jacobi identity, with x ∈ G0 and y, z ∈ G1 we have [x, {y, z}] + {y, [z, x]} − {z, [x, y]} = 0, and with x, y, z ∈ G1 we have [x, {y, z}] + [y, {z, x}] + [z, {x, y}] = 0.

40 3 Representations

A group is an abstract mathematical concept. The transformation of physical quantities is realized through a representation of the group on physical variables. When the transforma- tions are linear, we are dealing with a linear representation (or representation for short), in terms of square matrices or in terms of linear operators. The concept of a representa- tion is of fundamental importance in group theory, and provides a tool to classify groups. Representations of Lie groups will be closely related with representations of Lie algebras, which we shall study in detail. Non-linear representations or non-linear realizations of a group also enter physics and will be discussed in a later section.

3.1 Representations of groups A (linear) representation R of a group G with operation ∗ is a map R : G → GL(N) in terms of which the group operation ∗ of G maps to matrix multiplication in GL(N),

1. R(g ∗ g0) = R(g)R(g0) for all g, g0 ∈ G;

2. R(e) = IN ; 3. R(g−1) = R(g)−1 for every g ∈ G.

If the map is R : G → GL(N, R) one refers to R as a real representation, while if the map is R : G → GL(N, C) one refers to R as a . A special case which is important in physics and in mathematics is when the map R is to the unitary group R : G → U(N) ⊂ GL(N; C), in which case R is referred to as unitary representation. The dimension of the representation R is defined to be N in either case, though it must be stressed that this definition refers to the dimension over R when R is real, and over C when R is complex. For N = ∞, the representation is said to be infinite-dimensional. Two representations R,R0 of the same group G are referred to as being equivalent representations if their dimensions are equal N = N 0 and there exists a matrix A ∈ GL(N) (which is thus invertible) such that

R0(g) = AR(g)A−1 (3.1) for all g ∈ G. Clearly, the matrix A may be viewed as effecting a change of basis in the vector space on which G acts so that we shall not be interested in distinguishing between equivalent representations, and consider only the equivalence classes of representations.

3.1.1 Examples of representations We shall spend a lot of time constructing and examining representations of various groups later on. Here we wish to present just a few examples to clarify the definition.

41 1. The trivial representation assigns to every group element g ∈ G the value R(g) = 1. 0 2. The C× has a two-dimensional representation R in terms of real matrices, mapping a complex number z = a + ib with a, b ∈ R to the matrix, a −b 0 −1 R(z) = R(i) = (3.2) b a 1 0

3. The additive group G+ of M × N matrices admits a representation R in GL(M + N) by (M + N) × (M + N) matrices of the following form for x, y ∈ G+,

1 x R(x) = R(x)R(y) = R(x + y) (3.3) 0 1

4. The group of permutations SN has a representation by N × N matrices. To a permutation σ ∈ SN , we assign a matrix R(σ) whose entries as given by,

R(σ)ij = δi,σ(j) i, j = 1, ··· ,N (3.4)

This matrix has exactly one entry on each row which equals 1, all others being zero. 5. Another representation of the is given by the signature of the permutation, denoted by R(σ) = (−)σ, and takes only the values ±1. In quantum mechanics, a system of N identical bosons is invariant under permutations, while a system of N identical fermions transforms with the signature representation. 6. Matrix groups, introduced in the preceding section, are defined in terms of subgroups of the general linear groups GL(N; R) or GL(N, C) and therefore naturally exhibited in terms of a linear representation, which is referred to as the defining representation. For GL(N), SL(N), SO(N), and SU(N) the defining representation has dimension N while the dimension of the defining representation of Sp(2N) is 2N.

A non-linear representation of a group G arises when G acts not on a vector space, but rather on a manifold such as a coset space G/H of G by one of its subgroups H. A simple example of a non-linear representation of SL(2, R) acting on the real line x ∈ R is given by the M¨obiustransformation,

ax + b a b g : x → g = ∈ SL(2, ) (3.5) cx + d c d R

Non-linear representations are ubiquitous in problems where the is on variables and fields which satisfy non-linear differential equations, such as Einstein’s equations of general relativity or Yang-Mills theory.

42 3.2 Representations of Lie algebras

c Consider a Lie algebra G of dimension n with generators Xa, structure constants fab , for a, b, c = 1, ··· , n, and structure relations,

n X c [Xa,Xb] = fab Xc (3.6) c=1 A representation ρ of a Lie algebra G is a linear map ρ : G → gl(N) where gl(N) is the space of N × N real or complex matrices (see the previous section), such that ρ preserves the Lie algebra bilinear product,

ρ([x, y]) = [ρ(x), ρ(y)] for all x, y ∈ G (3.7)

Note that, by an abuse of notation, [x, y] on the left side stands for the Lie bracket of the Lie algebra G, while [ρ(x), ρ(y)] stands for the commutator of matrices in gl(N). The dimension of the representation ρ is defined to be N (over R or C as in the case of representations of groups, see the preceding subsection). Taking Xa to be a basis of the Lie algebra G with a = 1, ··· , n = dim G, we see that the linearity of ρ implies that ρ(Xa) satisfy the same structure relations as Xa itself,

n X c [ρ(Xa), ρ(Xb)] = fab ρ(Xc) (3.8) c=1 As introduced above, the dimension n of the Lie algebra G is independent of the dimension N of the representation ρ.

3.2.1 Examples of representations of Lie algebras Here we just give some immediate examples of Lie algebras to illustrate their definition and reserve for later a more systematic discussion. 1. The trivial representation assigns to every x ∈ G the zero element, ρ(x) = 0. 2. The defining representation for each matrix Lie group SL(N), SU(N), SO(N), Sp(2N) induces the defining representation with the same dimension of the corre- sponding Lie algebras sl(N), su(N), so(N) and sp(2N). 3. Every Lie algebra G possesses the whose representation matri-

ces are given by the structure constants fabc with a, b, c = 1, ··· , n = dim G,

c c (Fa)b = −fab (3.9)

The Jacobi identity of the Lie algebra G given in (1.24) may be recast in terms of

matrix multiplication for the matrices Fa, effected by the summation over the index

43 d in (1.24). To see this, we recast the Jacobi identity in the following form by using anti-symmetry of the structure constants in their first two indices,

n n X  d e d e X d e fac fbd − fbc fad = − fab fdc (3.10) d=1 d=1 Expressing the four f on the left side and the second f on the right side in terms of F , but leaving the first f on the right side, we find,

n e X d e [Fa,Fb]c = fab (Fd)c (3.11) d=1

Thus the matrices Fa form a representation of the Lie algebra G. This representation has exactly the same dimension as the Lie algebra, namely n, and is referred to as the adjoint representation of G. For an Abelian Lie algebra, the adjoint representation is always equivalent to the trivial representation. In a basis-independent formulation, the adjoint representation of an element x ∈ G

is denoted Adx and acts on an arbitrary element z of G (which makes sense since its dimension is exactly the same as that of the Lie algebra) by,

Adx(z) = [x, z] (3.12)

The fact that this map is a representation of G again follows from the Jacobi identity,

[Adx, Ady](z) = [x, [y, z]] − [y, [x, z]] = [x, [y, z]] + [y, [z, x]]

= [[x, y], z] = Ad[x,y](z) (3.13)

which was used in going from the first to the last line.

3.3 Transformation of a vector space under a representation

The general linear group GL(N; C) naturally transforms the vector space V = CN into V by linear transformations, and analogously for GL(N; R) on V = RN . Therefore, an arbitrary group G may also transform the vector space V through an N-dimensional representation R : G → GL(N) of G. By the same token, the vector space V transforms under an N-dimensional representation ρ of a Lie algebra G, and we have,

R(g): V → V v ∈ V → R(g)v ∈ V g ∈ G ρ(x): V → V v ∈ V → ρ(x)v ∈ V x ∈ G (3.14) where we have represented the elements v of the vector space V as column vectors in a given basis of V , and R(g)v and ρ(x)v are the products of N × N square matrices by a

44 column vector on the right which gives again a column vector. The vector space V and its elements are then said to transform under the representation R of the group G and under the representation ρ(x) of the Lie algebra G.1 In physics, the vector spaces V describe N physical quantities, such as position in space and time transformed by translations, rotations, and Lorentz transformation, or a set of electromagnetic fields, or states in the of a quantum mechanical system with N degrees of freedom.

3.4 Direct sum of representations In this subsection, we shall define the direct sum of representations, and the reducibility of a general representation into a direct sum of representations. We shall write things out for representations of a group G, the transcription to representations of a Lie algebra being immediate. First, we need to review the direct sum of vector spaces.

The direct sum of two vector spaces V1 and V2 of respective dimensions N1,N2 (either over R or C) is a vector space V1 ⊕ V2 of dimension N1 + N2 consisting of ordered pairs (v1, v2) with v1 ∈ V1 and v2 ∈ V2 such that addition on V1 ⊕ V2 is defined by,

0 0 0 0 (v1, v2) + (v1, v2) = (v1 + v1, v2 + v2) (3.15)

1 N1 1 N2 In terms of bases e1, ··· , e1 for V1 and e2, ··· , e2 for V2 a basis of V1 ⊕ V2 may be chosen 1 N1 1 N2 to be e1, ··· , e1 , e2, ··· , e2 thereby clearly giving a vector space of dimension N1 + N2. In this basis, we may represent the vectors in V1 and V2 by column matrices,  1  v1  1   1  v1 v2 ··· N N   1 2 2 2  N1  X i i  v1  X i i  v2  v1  v1 = v e =   v2 = v e =   v1 ⊕ v2 =   (3.16) 1 1 ··· 2 2 ···  v1  i=1   i=1   2 N1 N2   v1 v2 ···  N2 v2 This matrix notation, in a given basis, will be used throughout.

The direct sum of two representations R1 and R2 of the group G, of respective dimen- sions N1,N2, is a representation R1 ⊕R2 of dimension N1 +N2 obtained by taking the direct sum of the separate transformations of G on each vector space, and may be represented in the above basis in terms of the following block-matrix decomposition,   R1(g) 0 (R1 ⊕ R2)(g) = (3.17) 0 R2(g)

1In the mathematics literature, V is referred to as the representation space of R (or ) while in the physics literature, by a confusing abuse of notation, often refers to V as the representation as well. Here, we shall try to stick to the terminology that the vector space and its elements transform under the representation R.

45 The operation of direct sum on the space of representations is commutative. Given a representation R : G → GL(N) acting on a vector space V of dimension N, a vector v ∈ V is said to be invariant under a subgroup H ⊂ G if R(g)v = v for all g ∈ H. Similarly, a vector space W ⊂ V is said to be invariant under a subgroup H ⊂ G if R(g)w ∈ W for all w ∈ W and all g ∈ H. Clearly, V and {0} are trivial invariant

subspaces. The direct sum R1 ⊕ R2 of two representations of G, as constructed above, has two non-trivial invariant subspaces, namely V1 and V2. A representation R : G → GL(N) is said to be reducible if it admits at least one non-trivial invariant subspace under G. A representation R : G → GL(N) is said to be

completely reducible if every invariant subspace V1 has an invariant complement V2, so 0 that V = V1 ⊕ V2. The difference between a reducible representation R and a completely reducible representation R, both with invariant subspace V2, may be illustrated by using the matrix notation, in which case they respectively take the following form,

R (g) 0  R (g) S (g) R(g) = 1 R0(g) = 1 3 (3.18) 0 R2(g) 0 S2(g)

0 The vector space V1 in the upper block is invariant under both R(g) and R (g) for all g ∈ G. 0 Its complement V2 is also invariant under R(g) but not under R (g), since the action of 0 R (g) on V2 will have a component along V1 generated by the block matrix S3(g). A representation R : G → GL(N) acting on the vector space V of dimension N is irreducible if its only invariant subspaces are V and {0}. Irreducible representations form the building blocks for general representations, so that the knowledge of all irreducible representations will suffice to classify all representations.

3.5 Schur’s Lemma

Let R : G → GL(M; C) and S : G → GL(N; C) be two irreducible complex representations of the group G, such that,

AR(g) = S(g)A for all g ∈ G (3.19)

where A is an N × M matrix which is independent of g. If R and S are inequivalent then A = 0. If R and S are equivalent, then A is unique up to a multiplicative constant. In particular, if R = S, then A = λI with λ ∈ C. The last assertion means that any N × N matrix which commutes for all g ∈ G with an irreducible representation R(g) of dimension N is proportional to the identity matrix. To prove Shur’s Lemma, denote by X and Y the vector spaces, of respective dimensions M and N, which are transformed under R and S, so that R(g): X → X and S(g): Y → Y ,

46 and the matrix A is a map from X → Y . Now define the following subspaces,

X0 = Ker (A) = {x ∈ X such that Ax = 0}

Y0 = Im (A) = {y ∈ Y such that y = Ax for some x ∈ X}

It follows from the conjugation relation that X0 is a subspace of X invariant under R since we have AR(g)x = S(g)Ax = 0 for every x ∈ X0 and thus R(g)X0 ⊂ X0. Similarly, Y0 is a subspace of Y invariant under S. To see this, take an element y ∈ Y0 which may be expressed as y = Ax, apply the conjugacy relation to x so that AR(g)x = S(g)Ax = S(g)y, and observe that this implies S(g)y ∈ Y0. Since the representations R and S have both been assumed to be irreducible, the subspaces X0 and Y0 must both be trivial subspaces, namely either {0} or respectively X or Y . We now examine these different cases.

The cases X0 = Y0 = {0} and X0 = X,Y0 = Y are ruled out . The case X0 = X and Y0 = {0} corresponds to A = 0. Finally, the case X0 = {0} and Y0 = Y corresponds to the matrix A being invertible and the representations R and S are then equivalent. The uniqueness up to multiplication by a constant may be seen as follows. Suppose there were a second A0 satisfying A0R(g) = S(g)A0, then take the combination (A0 − λA)R(g) = S(g)(A0 − λA) for λ ∈ C. Now let λ be a root of det (A0 − λA) = 0 (note that it is here that we need to be in C). For that value of λ, the matrix A0 − λA is not invertible, but then by the first part of Shur’s lemma, it must vanish. In the case where S = R, we can clearly take A = I but since A must be unique up to multiplication by a constant, this is in fact the only possibility, which gives rise to the last assertion. An important application is that all irreducible representations R : G → GL(N; C) of an Abelian group G are one-dimensional. Note that there is no analogous result for real representations R : G → GL(N; R) since for example the Abelian group of multiplication given by elements of the form,  a b R(a, b) = (3.20) −b a for a, b ∈ R is reducible over C to a ± ib but not reducible over R.

3.6 Unitary representations Unitary representations will be important because they preserve the Hermitian inner prod- uct in complex vector spaces, and Hilbert spaces of quantum states or of functions. There is also a very important immediate result, valid for representations of finite or infinite dimension, which simplifies the study of unitary representations considerably. Consider a complex vector space V .A Hermitian inner product is a map V × V → C, denoted by (u, v) such that for all u, v, w ∈ V and all α, β ∈ C we have, (v, u) = (u, v)∗ (u, αv + βw) = α (u, v) + β (u, w) (3.21)

47 One defines kuk2 = (u, u), which is a norm on V provided it is positive kuk2 ≥ 0 for all u ∈ V ; definite so that kuk = 0 implies u = 0; and satisfies the triangle inequality ku + vk ≤ kuk + kvk for all u, v ∈ V . The resulting normed vector space is automatically a metric topological space. If V is complete then V is a Hilbert space.

Theorem 5 Every unitary representation is completely reducible. To prove this result, let (x, y) be the inner product on V with respect to which the representation R(g) is unitary, so that we have,

(R(g)x, R(g)y) = (x, y) for all x, y ∈ V (3.22)

Let V1 be an invariant subspace of the representation R(g) so that R(g)V1 = V1. The ⊥ orthogonal complement V2 = (V1) with respect to the inner product (x, y) is defined by,

V2 = {v ∈ V such that (v, x) = 0 for all x ∈ V1} (3.23)

Now let y ∈ V2 then its transform R(g)y satisfies,

(R(g)y, x) = (y, R(g)−1x) (3.24)

for all x ∈ V and thus all x ∈ V1, and all g ∈ G. Since V1 is an invariant subspace of R, −1 the image R(g) is in V1, so that the second inner product vanishes by virtue of y ∈ V2. Therefore we have (R(g)y, x) = 0 for all x ∈ V1 so that R(g)y ∈ V2. Therefore, V2 is an invariant subspace under R(g) and the representation R(g) is completely reducible into the representations on the subspaces.

Theorem 6 Every finite-dimensional representation of a finite or compact group G is equivalent to a unitary representation. To prove this theorem for the case of a finite group G, we consider a representation R : G → GL(N; C) on a vector space V of dimension N < ∞ with a non-degenerate Hermitian inner product (x, y). An arbitrary representation R(g) may not be unitary with respect to this inner product. We form a new inner product by summing over the group, X hx, yi = (R(h)x, R(h)y) (3.25) h∈G

The sum is over a finite number of elements and thus convergent. Now transform both x, y by the representation R(g), X hR(g)x, R(g)yi = (R(h)R(g)x, R(h)R(g)y) (3.26) h∈G

48 and use the fact that R is a representation, so that R(h)R(g) = R(h ∗ g). But the sum over h is the same as the sum over h ∗ g for any h ∈ G, so that we find,

hR(g)x, R(g)yi = hx, yi (3.27) for all x, y ∈ V and all g ∈ G. This means that R is unitary with respect to the new Hermitian inner product hx, yi. The only change required for a compact group G is to replace the summation over G by an integral over the compact Lie group G, using the Hurwitz-Haar measure dµ = dµG, Z hx, yi = dµ(h)(R(h)x, R(h)y) (3.28) G Since G is compact and the integrand continuous in h, the integral is convergent for all x, y. Transforming x, y under R(g), we find, Z hR(g)x, R(g)yi = dµ(h)(R(h ∗ g)x, R(h ∗ g)y) (3.29) G where we have again used the fact that R is a representation so that R(h)R(g) = R(h ∗ g). Changing from integration variable h to h0 = h ∗ g, we use the translation invariance of the measure dµ(h0∗g−1) = dµ(h0) and recover hR(g)x, R(g)yi = hx, yi so that the representation R(g) is unitary with respect to the inner product hx, yi.

Corollary 1 It follows from combining Theorems 3.6 and 3.6 that every finite-dimensional representation of a finite or of a compact group is completely reducible. This result is particularly helpful as it means that for these cases it will suffice to study just irreducible representations, all others being then given by taking direct sums of irreducible representations.

3.7 Tensor product of representations To define the tensor product of representations, we need to first define the tensor product of vector spaces V1 and V2 of respective dimensions N1 and N2. The tensor product of two vectors v1 ∈ V1, v2 ∈ V2 is given by a bilinear map, denoted by,

(v1, v2) → v1 ⊗ v2 (3.30) and all linear combinations of such tensor products define a vector space V1⊗V2 of dimension 1 N1 1 N2 N1N2. In terms of bases e1, ··· , e1 for V1 and e2, ··· , e2 for V2 a basis of V1 ⊗ V2 may be i j chosen to be e1 ⊗ e2 where i = 1, ··· N1 and j = 1, ··· ,N2, thereby clearly giving a vector

49 space of dimension N1 + N2. Using the decomposition of the two vectors onto these basis vectors as given in (3.16), the tensor product v1 ⊗ v2 takes the form,

N1 N2 X X i j i j v1 ⊗ v2 = v1v2 e1 ⊗ e2 (3.31) i=1 j=1

A general element v of the tensor product space V1 ⊗ V2 is given by a general linear i j combination of the basis vectors e1 ⊗ e2 given as follows,

N1 N2 X X ij i j v = v e1 ⊗ e2 (3.32) i=1 j=1 where vij are arbitrary coefficients in the field underlying the vector space V .

The tensor product R1 ⊗ R2 of two representations Ri : G → GL(Ni; C) is a represen- tation of dimension N1N2, namely,

R1 ⊗ R2 : G → GL(N1N2; C) (3.33) whose transformation on V = V1 ⊗ V2 is defined on the tensor product of two vectors by,

(R1 ⊗ R2)(g)(v1 ⊗ v2) = (R1(g)v1) ⊗ (R2(g)v2) (3.34) and in particular on basis vectors of V1 ⊗ V2 by,

N1 N2 i j X X ik j` k ` (R1 ⊗ R2)(e1 ⊗ e2) = (R1) (R2) e1 ⊗ e2 (3.35) k=1 `=1 where we have suppressed the dependence on g. Note that the tensor product is associative,

(R1 ⊗ R2) ⊗ R3 = R1 ⊗ (R2 ⊗ R3) for all R1,R2,R3 (3.36)

3.8 Characters of Representations A group character of a representation R of a group G is a complex-valued function on G which depends on the representation R and on the of each group element. For a finite-dimensional representation R(g), we define,

χR(g) = tr R(g) (3.37)

The trace is taken of an N × N matrix where N = dim R. Invariance under conjugation follows from the fact that R is a representation and the cyclic property of the trace,

−1 −1 χR(h ∗ g ∗ h ) = tr R(h)R(g)R(h) = trR(g) = χR(g) (3.38)

50 One of the most important properties of group characters is that their behavior under the operations of direct sum and tensor product of representations. For representations R1 : G → GL(M; C) and R2 : G → GL(N; C) we have,

χR1⊕R2 (g) = χR1 (g) + χR2 (g)

χR1⊗R2 (g) = χR1 (g) χR2 (g) (3.39) To prove the first line is immediate from inspection of (3.17). To prove the second line, we ik j` recall from (3.35) that the matrix elements of R1 ⊗ R2 are given by (R1) (R2) . To take ii jj the trace, we must sum the diagonal elements (R1) (R2) obtained by setting (k, `) = (i, j) over the composite index ij, which clearly produces the product of the traces.

For the special case of g = e, the identity element of G, we have R(e) = IN so that,

χR(e) = dim(R) (3.40) The direct sum and tensor product formulas for characters then reduce to the dimension formulas derived earlier,

dim(R1 ⊕ R2) = dim(R1) + dim(R2)

dim(R1 ⊗ R2) = dim(R1) dim(R2) (3.41) For infinite dimensional representations, the definition does not make sense as it stands, since in particular the value of the character would diverge at the identity element. Some- times, the character may still be defined in a useful way.

Theorem 7 The following orthogonality relations hold for finite-dimensional irreducible representations R1 and R2 of a finite group G, X χR1 (g) χR2 (g) = NR1 δR1,R2 (3.42) g∈G while for a compact group G with Hurwitz-Haar measure dµ we have, Z

dµ(g)χR1 (g) χR2 (g) = NR1 δR1,R2 (3.43) G where the overline indicates complex conjugation, δR1,R2 equals 1 when R1 and R2 are equivalent to one another and vanishes otherwise, and NR1 are normalizations. The proof proceeds from Shur’s lemma. In fact we shall prove a Lemma first.

Lemma 1 Two irreducible representations R1 and R2 of a compact Lie group G (or a finite group upon replacing the integral by a sum over G), satisfy the orthogonality relation, Z j −1 ` ` j dµ(g) R1(g)i R2(g )k = C(R1) δR1,R2 δi δk (3.44) G

51 where δR1,R2 vanishes when R1 and R2 are inequivalent representations, and equals 1 when R1 and R2 are equivalent in which case the result has been recorded for when the represen- tations are actually equal. To prove the lemma, we introduce the the matrix M defined by, Z j −1 ` j` dµ(g) R1(g)i R2(g )k = Mik (3.45) G

i Multiplying to the left by R1(h)a , summing over i to produce the product R1(h)R1(g) = R1(h ∗ g), changing variables from g to h ∗ g, and using the invariance of the measure gives dµ(g) = dµ(h ∗ g), we find the following relation, Z X a j` j −1 ` X jb ` R1(h)i Mak = dµ(g) R1(g)i R2(g ∗ h)k = Mik R2(h)b (3.46) a G b

j` ` For fixed values of the indices j, k, denote the matrix Mik = Mi . The above relation then implies that M satisfies R1(h)M = MR2(h) for all h ∈ G. By Shur’s lemma, either R1 and R2 are inequivalent, in which case M = 0 for all values of j, k. Or R1 and R2 are equivalent in which case M is unique up to a multiplicative factor by Shur’s lemma. Repeating the same procedure on the indices j, k we conclude the proof of the lemma. The Theorem follows by taking the trace in the indices i, j and separately in k, `.

Theorem 8 A unitary irreducible representation of a compact group is finite dimensional. The proof proceeds by using the orthogonality theorem, and we leave it up to the reader.

Representations of Lie algebras So far we have discussed representations of groups, including Lie groups. The relation between a representation ρ of a Lie algebra G and a representation R of its associated connected and simply connected Lie group G is given explicitly by the exponential map,

R(g) = eρ(x) g = ex (3.47) for all x ∈ G. Thus, having a representation of a Lie algebra will easily produce a repre- sentation of the corresponding Lie group.

52 4 Representations of SL(2; C), SU(2), and SO(2, 1) The group SU(2) is familiar to any physicist. Its elements are 2×2 complex matrices which may be parametrized in a variety of ways. As a matrix we have,

a b g = g†g = I det g = 1 (4.1) c d 2

Equivalently, it may also be parametrized in terms of Euler angles α, β, γ,

σa g = eiαJ3 eiβJ2 eiγJ3 J = (4.2) a 2 where σa are the Pauli matrices.

4.1 Irreducible representations of the Lie group SU(2) We begin by constructing the finite-dimensional irreducible representations of the Lie group

SU(2). The trivial representation is denoted D0. The formulation of SU(2) in terms of 2 × 2 matrices provides the defining representation which is often denoted by D 1 (g) = g, 2 1 and which is manifestly irreducible. The direct sum of n defining representations D 2 is a representation of dimension 2n which is obviously reducible. To construct new irreducible representations of SU(2) we take tensor products of the defining representation with itself and then completely reduce those into irreducible repre- 2 sentations. To do this, we define the vector space V = on which D 1 acts, and denote C 2 i its vectors by vα with components v for i = 1, 2. The representation D 1 acts as follows, α 2

2 2 i X i i0 X i i0 vα → D 1 (g)vα v → D 1 (g) i0 v = g i0 v (4.3) 2 α 2 α α i0=1 i0=1

The tensor product of two vectors transforms under the tensor product representation,

vα ⊗ vβ → (D 1 ⊗ D 1 )(g)(vα ⊗ vβ) (4.4) 2 2

Writing out the tensor products in components we have,

2 2 i j X i j i0 j0 X i j i0 j0 v v → D 1 (g) i0 D 1 (g) j0 v v = g i0 g j0 v v (4.5) α β 2 2 α β α β i0,j0=1 i0,j0=1

Clearly, the tensor product representation D 1 ⊗ D 1 has dimension 4. 2 2

53 Since we are dealing with a finite-dimensional representation of a compact group, we know by theorem 3.6 that the representation D 1 ⊗D 1 is completely reducible. Symmetriza- 2 2 tion and anti-symmetrization of the tensor product provides a general procedure to reduce tensor product representations. In the case of D 1 ⊗ D 1 , we have, 2 2

2 1 i j i j X i j 1 i0 j0 i0 j0 (v v ± v v ) → g 0 g 0 (v v ± v v ) (4.6) 2 α β β α i j 2 α β β α i0,j0=1

Now, the anti-symmetric combination involves,

i j i j ij 1 2 1 2 vαvβ ± vβvα = ε (vαvβ ± vβvα) (4.7) where εij is the anti-symmetric symbol with ε11 = ε22 = 0 and ε12 = −ε21 = 1, and i0 j0 i0 j0 similarly for vα vβ ± vβ vα . But now consider the combination,

2 X i j i0j0 g i0 g j0 ε (4.8) i0,j0=1 which is anti-symmetric in i, j and thus has a single independent component, obtained by 1 2 2 1 setting i = 1 and j = 2, and given by g 1g 2 − g 1g 2 = det g = 1. Thus the anti-symmetric combination is invariant under SU(2) and transforms under the trivial representation D0. The symmetric combination has dimension 3, is irreducible, and is denoted D1.

4.2 Finite-dimensional representations of sl(2, C)

The 3-dimensional Lie algebra with generators Ja for a = 1, 2, 3 and structure relations,

[J1,J2] = iJ3 [J2,J3] = iJ1 [J3,J1] = iJ2 (4.9) is the Lie algebra sl(2; C) or equivalently so(3; C) provided the vector space generated by the generators Ja is over C. When the generators −iJa are real we have the algebra so(3; R), and when the generators Ja are Hermitian it is the Lie algebra su(2) familiar from angular momentum theory. Finally, we may also set J1 = X1,J2 = X2,J3 = iX3,

[X1,X2] = −X3 [X2,X3] = X1 [X3,X1] = X2 (4.10) and now require X1,X2,X3 to be real matrices, which gives the Lie algebra so(2, 1; R). The Lie algebras su(2), so(3; R) and so(2, 1; R) are referred to as different real forms of the sl(2; C), and their representation theory is closely related to that of sl(2; C), with which we will begin.

54 The rank of sl(2; C) is one. We choose the generator of the Cartan subalgebra to be 2 J3 and assume that J3 is diagonalizable. We organize the remaining generators in linear combinations chosen so that the adjoint map AdJ3 is diagonal, 1 AdJ J± = ±J± J± = √ (J1 ± iJ2) (4.11) 3 2 In this basis, the structure relations become,

[J3,J±] = ±J± [J+,J−] = J3 (4.12)

This organization will generalize to all simple Lie algebras in which case the J± generators will correspond to roots. To construct the most general finite-dimensional irreducible representation ρ of sl(2; C), we shall assume that the representation matrices ρ(Ja) are N × N, for as yet undetermined values of N < ∞, and satisfy the structure relations of the Lie algebra,

[ρ(J3), ρ(J±)] = ±ρ(J±)[ρ(J+), ρ(J−)] = ρ(J3) (4.13)

We denote the N-dimensional vector space on which they act by V and the eigenvalues of ρ(J 3) by m. Since there is no assumption of Hermiticity of the generators for sl(2; C), we generally may have m ∈ C. The eigenspace of ρ(J3) may have dimension one or higher, and we label the different vectors with the same eigenvalue m by α. The spectrum of ρ(J3) is the set of quantum numbers S(ρ) = {(m, α)} and has cardinality N. We shall often use the terminology and notations of quantum mechanics, refer to vectors as states, and denote by |ρ; m, αi the basis vectors of a basis of V in which ρ(J 3) is diagonal,

ρ(J 3)|ρ; m, αi = m|ρ; m, αi (m, α) ∈ S(ρ) (4.14)

The generators ρ(J±) map V to V by the definition of a representation, and therefore 0 0 0 0 ρ(J±)|m, α; ρi must be a linear combination of vectors |ρ; m , α i with (m , α ) ∈ S(ρ). Actually, we may restrict the form of this linear combination by evaluating,   ρ(J3)ρ(J±)|ρ; m, αi = [ρ(J3), ρ(J±)] + ρ(J±)ρ(J3) |ρ; m, αi

= (m ± 1)ρ(J±)|ρ; m, αi (4.15)

obtained using the structure relations. Thus we have the following decompositions,

X ± ρ(J±)|ρ; m, αi = Cαβ(m, ρ)|ρ; m ± 1, βi (4.16) β

2 When the Lie algebra is su(2), the generator J3 is Hermitian and thus automatically diagonalizable.

55 On the right side the sum is over all vectors such that (m ± 1, β) ∈ S(ρ). The matrix ρ(J+) raises the eigenvalue of ρ(J3) by 1 while ρ(J−) lowers the eigenvalue by 1.

Next we apply ρ(J±) repeatedly and make use of the assumption that the dimension of the representation ρ is finite. This implies that the eigenvalue of ρ(J3) cannot get raised or lowered indefinitely since this would produce a representation space of infinite dimension.

Thus, there must exist j± with non-zero vectors |j±, α±; ρi and (j±, α±) ∈ S(ρ) such that,

ρ(J+)|ρ; j+, α+i = 0

ρ(J−)|ρ; j−, α−i = 0 (4.17)

This may occur for one or more values of j± and one or more values of α± for each value of j±. Repeatedly applying ρ(J−) to any one of the vectors |ρ; j+, α+i will produce all the vectors of an irreducible representation of sl(2; C). If several α+ occur then applying ρ(J−) will produce the direct sum of as many irreducible representations as there are α+.

We shall now focus on a single irreducible representation produced from a single j+ = j and a single α. In general, this representation is no longer ρ, but rather one of the irreducible

components of ρ. We shall denote this representation by ρj and drop the label α, labelling the states simply |ρj; mi. In mathematics terminology, the quantum number m is referred to as the weight and the corresponding vector |ρj; ji as the highest weight vector. Applying lowering and raising operators we have,

ρj(J±)|ρj; mi = N±(ρj; m)|ρj; m − 1i N+(ρj, j) = 0 (4.18)

Enforcing the structure relation [ρj(J+), ρj(J−)] = ρj(J3) gives,

N+(ρj, m − 1)N−(ρj, m) − N−(ρj, m + 1)N+(ρj, m) = m (4.19)

Considering the sum of this relation over m up to m = j,

j j X   X N+(ρj, n − 1)N−(ρj, n) − N−(ρj, n + 1)N+(ρj, n) = n (4.20) n=m n=m

The second term in the parentheses is the opposite of the first shifted by n → n + 1. Thus

the sum is given by its endpoints and, using the fact that N+(ρj, j) = 0, we obtain,

2N+(ρj, m − 1)N−(ρj, m) = (j + m)(j − m + 1) (4.21)

The formula reproduces N+(ρj, j) = 0 and shows that the only value of m < j for which N− can vanish is m = −j, so that N−(ρj, −j) = 0. Given the highest weight j of the + + representation ρj, the allowed values of m are such that j − m ∈ Z , where Z stands for

56 the zero or positive integers. But −j must be such a value, so that 2j ∈ Z+. Therefore the + states in the representation ρj are given for j − m ∈ Z and,

+ |ρj; mi 0 ≤ j − m ≤ 2j ∈ Z (4.22) and the dimension of the representation is dim(ρj) = 2j + 1. Note that we did not use the assumption that m or j had to be real from the outset: for sl(2; C) this result arises solely from requiring the representations to be finite-dimensional. Imposing the restrictions of Hermiticity for su(2), or reality for so(3; R) changes nothing to the structure and dimensions of the representations derived above. In the case of com- pact groups we know that all finite-dimensional representations are equivalent to unitary representations.

4.3 Infinite-dimensional representations of sl(2; C) Now let us remove the assumption that the representation ρ is finite-dimensional, but still assume that ρ(J3) is diagonalizable, so that we still have the relations,

N+(ρ; m − 1)N−(ρ; m) − N−(ρ; m + 1)N+(ρ; m) = m (4.23)

Consider now a representation ρ for which N−(ρ; j) = 0, so that ρ(J−)|ρ; ji = 0 for a non-zero state |ρ; ji. This time applying m raising operators to |ρ; ji, we find,

−2N+(ρ; j + m)N−(ρ; j + m + 1) = (m + 1)(2j + m − 2) (4.24)

If 2j were a negative , then we recover the finite-dimensional case already discussed. But if we assume that 2j is not a negative integer, then we can apply an infinite number of raising operators for which N+(ρ; j + m) 6= 0 and thus |ρ; j + mi= 6 0. This produces an infinite-dimensional representation of sl(2; C).

4.4 Harmonic oscillator representation of so(2, 1; R) Consider the one-dimensional harmonic oscillator with degrees of freedom a, a† which satisfy the Heisenberg algebra [a, a†] = 1 upon setting ~ = 1, and have the following Hamiltonian and additional generators, 1 H = a†a + J = (a†)2 J = (a)2 (4.25) 2 + −

They satisfy the structure relations of so(2, 1; R) given by,

[H,J±] = ±2J± [J+,J−] = −4H (4.26)

57 Clearly, J+ is a raising operator for H while J− is a lowering operator. The quadratic Casimir take the form,

1 C = H2 − {J ,J } = H2 − 2H − J J (4.27) 2 + − + −

3 and on the harmonic oscillator representation evaluates to C = − 4 . Clearly H is positive, and there must thus be a lowest energy state, which we shall denote |0i such that,

H|0i = E0|0i J−|0i = 0 (4.28)

for an as yet undetermined ground state energy E0. Applying C to |0i gives,

3 1 − = E2 − 2E or E± = 1 ± (4.29) 4 0 0 0 2

n Applying raising operators gives the excited states |ni = J+|0i with energy, 1 E± = 1 ± + 2n n ≥ n (4.30) n 2

The interpretation is as follows: while the operator a† raises the energy by one unit, the raising operator J+ which is in the algebra so(2, 1; R) raises the energy by two units since it is quadratic in a†. The two different “ground states” correspond to the even and odd wave functions of the harmonic oscillator, and are the lowest weight states of different representations of so(2, 1; R). Thus the entire spectrum of the harmonic oscillator is the direct sum of two irreducible representations.

4.5 Unitary representations of so(2, 1; R) We consider a quantum system in one dimension which generalizes the harmonic oscillator, governed by the Hamiltonian H, and the following additional generators,

p2 g x2 H = + + 2 2x2 2 p2 g x2 K = + − 2 2x2 2 i D = (xp + px) (4.31) 2 where x, p satisfy the Heisenberg algebra, [x, p] = i. The operator D generates dilations, and has been normalized so that [D, x] = x and [D, p] = −p. From a quantum point of view, the operators H,K and iD are self-adjoint for g ∈ R, assuming some restrictions

58 on the values of g to be spelled out later. Together with the operators K and D, the Hamiltonian forms the so(2, 1; R) algebra, with the following structure relations, [D,H] = −2K [D,K] = −2H [H,K] = 2D (4.32)

For g ≥ 0, the spectrum of H is discrete, real, and positive, and we choose to work in a basis where H is diagonal. Next, we organize the operators K and D in combinations that raise and lower the eigenvalues of H, i.e. that diagonalize AdH ,

J± = K ± D [H,J±] = ±2J± [J+,J−] = −4H (4.33)

† Since K is self-adjoint and D is anti-self-ajoint, we see that (J±) −J∓, as in the case of the harmonic oscillator. One verifies using the structure relations that the quadratic Casimir operator is given by

2 2 2 2 C = H − K + D = H − 2H − J+J− (4.34) The calculation of C for the particular quantum system considered here may be simplified by using the canonical commutation relations [x, p] = i, and we find that C is proportional 3 to the identity operator, and given by C = g − 4 . Clearly, the spectrum of H is real and bounded from below, so there must be a state of lowest eigenvalue of E0,

H|0i = E0|0i J−|0i = 0 (4.35)

The value of E0 is determined by using the value of the Casimir, and we get the entire spectrum from applying the raising operator, 1 1 E± = 1 ± p1 + 4g E± = 1 ± p1 + 4g + 2n n ∈ + (4.36) 0 2 n 2 Z 1 The spectrum makes sense even when g is negative as long as g ≥ − 4 . The entire spectrum again would appear to be the direct sum of two irreducible representations, as was the case for the harmonic oscillator. However, it is clear from the Hamiltonian that just its potential √ ± part is bounded from below by g, so that the energy E0 must be bounded from below √ + − by g. This is always the case for the E0 branch, but holds for the E0 branch only when 1 9 − 4 ≤ g ≤ 64 . The energy also needs to be bounded from below by the harmonic oscillator − 1 value, and the combined bound gives that the E0 branch can exist only for − 4 ≤ g ≤ 0.

Theorem 9 The irreducible unitary representations of SO(2, 1; R) are as follows. 1. discrete series: C ≥ −1 and there is a lowest (or highest) energy state with energy 1 E0 ≥ 0 (or ≥ − 4 in the above example);

2. continuous series: C < −1 and 0 ≤ E0 < 2;

3. complementary series: C = −1 + λ2 and λ ∈ R.

59 4.6 An example of the continuous series Let me now give an example of a physically relevant representation that produces the continuous series. We note that SO(2, 1) acts transitively on the hyperboloids,

2 2 2 2 H± = {(x, y, z) ∈ R x + y − z = ±1} (4.37)

While H+ is connected by not simply connected, H− has two connected components, each of which is simply connected. Note that the compact subgroup SO(2) of SO(2, 1) rotates

(x, y). Both surfaces H± have constant negative curvature. To find the representations in the continuous series, we diagonalize the Laplace-Beltrami operator acting on functions on either one of these hyperboloids, 1 √ ∆ = −√ ∂ g gmn∂ (4.38) g g m n and the Laplace operator represents the Casimir operator. We consider the connected component of H− which is characterized by z ≥ 1, and introduce the following coordinates,   x = shϕ cos θ H− : y = shϕ sin θ (4.39)  z = chϕ where 0 ≤ θh2π and 0 ≤ ϕ < ∞. The metric is readily obtained,

ds2 = dϕ2 + sh2ϕ dθ2 (4.40) and so is the Laplace operator, ∂2 chϕ ∂ 1 ∂2 ∆ = − − − (4.41) ∂ϕ2 shϕ ∂ϕ sh2ϕ ∂θ2 The differential operators of SO(2, 1) in this representation are given by, ∂ T = −2i 3 ∂θ  ∂ chϕ ∂  T = 2e±iθ i ∓ (4.42) ± ∂ϕ shϕ ∂θ Extra factors of 2 have been inserted so that the algebra is the same as in the preceding examples, and the structure relations are as follows,

[T3,T±] = ±2T± [T+,T−] = −4T3 (4.43) The Casimir operator is,

2 C = T3 − 2T3 − T+T− = −4∆ (4.44)

60 We now solve for the eigenvalue equation:

1 ∂2ψ chϕ ∂ψ 1 ∂2ψ ∆ψ = − Cψ = − − − (4.45) 4 ∂ϕ2 shϕ ∂ϕ sh2ϕ ∂θ2

We shall use the following parametrization for λ2 ∈ R,

C = −1 − λ2 (4.46) and diagonalize T3 simultaneously with ∆, by introducing the eigenfunctions of ∂/∂θ,

imθ ψλ,m(ϕ, θ) = e ψλ,m(ϕ) (4.47)

so that the remaining eigenvalue equation reduces to,

chϕ m2 1  −ψ00 − ψ0 + ψ = + λ2 ψ (4.48) λ,m shϕ λ,m sh2ϕ λ,m 4 λ,m This equation is seen to be the Legendre equation by changing variables x = chϕ, and relabeling the parameter λ in the equation as follows, 1 1 ν(ν + 1) = − − λ2 ν = − + iλ (4.49) 4 2 which converts the eigenvalue equation into the standard form for the Legendre functions,

 m2  (1 − x2)ψ00 − 2xψ0 + ν(ν + 1) − ψ = 0 (4.50) 1 − x2

m m The solutions are conical functions P 1 (chϕ) and Q 1 (chϕ) for which we have usual − 2 +iλ − 2 +iλ completeness relations.

61 5 Tensor representations

Vector and tensor algebra is a topic in physics that has a somewhat separate life from group theory but is in fact equivalent to the representation theory for the classical groups as long as no spinors are involved. Tensor methods are organized by the pictorial Young tableaux and are often more intuitive and practical when only representations of low dimension are involved, and are

5.1 Tensor product representations We consider one of the classical matrix Lie groups G, such as SL(n),SU(n),SO(n), or Sp(n) (for n even), and start out with their respective defining representations, which are n n all of dimension n. The associated vector space Vn is R or C depending on the group. The defining representation matrix will be denoted R(g) for g ∈ G and U(g) and n × n matrix. The group G acts on Vn by linear transformations on its vectors,

G : Vn → Vn v → R(g)v ∈ Vn v ∈ Vn (5.1) or in component notation,

n i X i i0 v → R(g) i0 v i = 1, ··· , n (5.2) i0=1

The transformation of Vn under the defining representation R of G induces a transformation on the tensor product space Vn ⊗ Vn under the tensor product representation R ⊗ R,   G : Vn ⊗ Vn → Vn ⊗ Vn v1 ⊗ v2 → R(g) ⊗ R(g) (v1 ⊗ v2) (5.3)

Note that the group element g is the same in both factors. Equivalently, in components,

n i j X i j i0 j0 v1 v2 → R(g) i0 R(g) j0 v1 v2 (5.4) i0,j0=1

By taking linear combinations in Vn ⊗ Vn of tensor products of vectors we obtain general ij element w ∈ Vn ⊗ Vn whose components may be denoted by w for i, j = 1, ··· , n and is referred to as a tensor of rank 2 since it has two indices of the defining representation, and dimension n2. It transforms under R ⊗ R by,

G : Vn ⊗ Vn → Vn ⊗ Vn w → (R(g) ⊗ R(g))w (5.5) or in components,

n ij X i j i0j0 w → R(g) i0 R(g) j0 w (5.6) i0,j0=1

62 The process may be repeated by taking the tensor product of r copies of Vn to obtain tensors of rank r which transform under the r-fold tensor product of the defining representation R.

G : Vn ⊗ · · · ⊗ Vn → Vn ⊗ · · · ⊗ Vn w → R(g) ⊗ · · · ⊗ R(g) w (5.7) | {z } | {z } | {z } r r r or in components,

n i1,··· ,ir X i1 ir j1,··· ,jr w → R(g) j1 ··· R(g) jr w (5.8)

i1,··· ,ir=1 When no confusion is expected to arise, we shall henceforth omit exhibiting the g-dependence. The representations thus constructed are tensor product representations of the defining representation. For the groups SL(n), SU(n) and Sp(n) we will show later that all finite- dimensional representations may be constructed from tensor product representations. For the groups SO(n), however, tensor product representations will give only a subset of all representations, as these groups have also spinor representations which cannot be obtained from tensor product representations.

5.2 Symmetrization and anti-symmetrization The tensor product representations constructed above are in general reducible. Sym- metrization and anti-symmetrization always provides operations which will reduce the representations into direct sums of representations of smaller dimension, though the rep- resentations obtained this way may or may not be irreducible. We begin by symmetrizing and anti-symmetrizing the tensor product of the representation spaces Vn ⊗ Vn by defining the operations of symmetrization S and anti-symmetrization A on Vn ⊗ Vn, 1 1 S(v ⊗ v ) = (v ⊗ v + v ⊗ v ) S(vi vj) = (vi vj + vjvi ) 1 2 2 1 2 2 1 1 2 2 1 2 1 2 1 1 A(v ⊗ v ) = (v ⊗ v − v ⊗ v ) A(vi vj) = (vi vj − vjvi ) (5.9) 1 2 2 1 2 2 1 1 2 2 1 2 1 2 The symmetrized and anti-symmetrized tensor product representations then act as follows,

n X 1 R : S(vi vj) → (Ri Rj + Rj Ri )S(vkv`) S 1 2 2 k ` k ` 1 2 k,`=1 n X 1 R : A(vi vj) → (Ri Rj − Rj Ri )A(vkv`) (5.10) A 1 2 2 k ` k ` 1 2 k,`=1

1 The dimensions of the representations RS and RA are respectively given by 2 n(n + 1) and 1 2 2 n(n − 1) whose sum is indeed the dimension of R ⊗ R, which is n .

63 5.3 Representations of SU(3)

Denote the defining representation of SU(3) by R : SU(3) → GL(3; C). Actually, since R is a representation of SU(3), then so is the representation R∗. We shall now show that the complex conjugate representation R∗ is equivalent to the antisymmetric tensor product of R with itself. To see this concretely, define the totally anti-symmetric tensor in three indices, (−)σ if (ijk) = σ(123) σ ∈ S ε = 3 (5.11) ijk 0 otherwise Next, we consider the relation det R = 1, expressed in terms of the ε-symbol,

3 X α β γ εαβγ Ri Rj Rk = εijk det R = εijk (5.12) α,β,γ=1

† k Multiplying by (R )δ on both sides, summing over k and using the unitarity relation P † k γ γ k(R )δ Rk = δδ , we have,

3 3 X α β X † k εαβδRi Rj = εijk(R )δ (5.13) α,β=1 k=1

Multiply by εij` on both sides and sum over i, j using the relation,

3 X εijkεij` = 2δk` (5.14) i,j=1 gives,

3 3 X X α β † ∗ εijkεαβδRi Rj = 2(R )δγ = 2Rγδ (5.15) i,j=1 α,β=1 Therefore, the representation R∗ is equivalent to the anti-symmetrized tensor product of two defining representations. We have learned something interesting: each time we anti-symmetrize the tensor prod- uct of two defining representations, we may replace that representation by its equivalent R∗, while the totally anti-symmetrized tensor product of three R gives a singlet representation. We shall use the convention that the vectors in the vector space under which the defining

representation transforms are denoted with a lower index such as vi, while a vector in the dual vector space which transforms under R∗ will be denoted with a barred upper index, v¯i. Thus, a general tensor of SU(3) is of the form,

¯i1···¯ip¯ Ti1···ip (5.16)

64 To reduce the tensor, we separately symmetrize it in the upper and lower indices. To fully reduce the tensor, we remove all the traces between upper and lower indices. To count the dimension of the irreducible representation with p defining andp ¯ anti- defining representations, we first ignore the trace condition and count the dimension of just the tensor ti1···ip . Since we have total symmetrization, it suffices to count the number of ways we can put the three possible values i = 1, 2, 3 in three boxes with the sum of the lengths of the boxes being n. Denote by p1, p2, p3 the number of values 1, 2, and 3 respectively, so that p1 + p2 + p3 = n. Each pi satisfies 0 ≤ pi ≤ n. thus the dimension is,

n n−p1 n X X X (p + 2)! dim(t) = 1 = (n + 1 − p ) = (5.17) 1 2! p! p1=0 p2=0 p1=0

Thus, the dimension of the tensor T fully symmetrized in upper and lower indices, but ignoring the trace is given by,

(p + 2)! (¯p + 2)! (5.18) 2! p! 2!p ¯!

Removing the traces is equivalent to removing the tensor with any pair of upper and lower indices contracted in view of the symmetry of the tensor. Thus, the dimension of the irreducible tensor T is given by,

(p + 2)! (¯p + 2)! (p + 1)! (¯p + 1)! D(p, p¯) = − 2! p! 2!p ¯! 2! (p − 1)! 2! (¯p − 1)! 1 = (p + 1)(¯p + 1)(p +p ¯ + 2) (5.19) 2 We can check this formula for example on the adjoint representation, which is of the form j Ti so that p =p ¯ = 1, and we find dim(T ) = 8, which is the famous meson multiplet of π0, π±,K±,K0, K¯ 0, η, when only the u, d, s quarks are included. Another is the multiplet in which the proton and neutron live, which is the symmetric tensor product of 3 defining representations, so that p = 3 andp ¯ = 0, and we get dim(T ) = 10. This is the multiplet that contains the Ω− consisting of three s quarks, which was predicted on the basis of SU(3) group theory by Murray Gell-Mann for which he got the Nobel Prize. Young tableaux provide a pictorial image of irreducible representations. One begins by denoting the defining representation by a single square 2, and the trivial representation by a center dot •. Taking the tensor product with another copy of the defining representation is indicated by adjoining another box, either to the right or underneath the first box. In the vertical alignment of boxes, the tensor product is anti-symmetrized, while in the horizontal alignment, they are symmetrized. For SU(3), the anti-symmetric tensor product of three defining representations is the trivial representation, so non-trivial representations can have

65 p¯ p ··· ··· ···

either one or two vertically aligned boxes, and the most general irreducible representation labelled by (p, p¯) constructed above has the following Young tableaux, On may now also give a pictorial representation for tensor multiplication of an arbitrary representation by the defining representation: the result is the sum of three irreducible representations, each one obtained by adding a square in each row to the utmost left. Adding the square in the third row makes the corresponding column a singlet and produces the representation (p, p¯− 1). Adding the square in the second row diminishes p by one and increasesp ¯ by one, so we have the representation (p − 1, p¯+ 1), and finally in the first row we get (p + 1, p¯). One verifies that the dimensions add up: 3D(p, p¯ − 1) = D(p − 1, p¯ + 1) + D(p, p¯) + D(p + 1, p¯) (5.20) There are rules for taking the tensor product not just with the defining representation, but with a general representation. We shall see an easier route using weights.

5.4 Representations of SU(n) For the case of SU(n), we introduce the totally anti-symmetric symbol in n indices, (−)σ if i = σ(k) σ ∈ S ε = k n (5.21) i1···in 0 otherwise The determinant relation is now,

X α1 αn εα1···αn Ri1 ··· Rin = εi1···in det R = εi1···in (5.22) α1,··· ,αn

† in Multiplying by (R )k and summing over in gives,

X α1 αn−1 X † in εα1···αn−1kRi1 ··· Rin−1 = (R )k εi1···in (5.23)

α1,··· ,αn in The remaining step is straightforward and now proves that R∗ is equivalent to the (n − 1)- fold anti-symmetric tensor product of the defining representation. Are the symmetric/antisymmetric reductions now irreducible representations? This depends upon what the group is. In general, for representations of G`(n, C) S,A j SU(n) S,A, δi SO(2n + 1),SO(2n) δij Sp(2n) Jij

66 By isolating the parts proportional to these tensors, they are further reduced. Some illustrative examples. SO(n) n

Ti1...ip (5.24)

Now you can take the trace as well on symmetric indices

ip−1ip T i1...ip−2 = Ti1...ip δ (5.25) {ip−1ip} There is also the fully antisymmetric tensor, but it does not lead to new reductions. Sp(2n) One has the defining representation 2n.

Now there is a trace with respect to Jij: on antisymmetric indices

Ti1...ip Ti1...ip−2 = Jip−1ip [ip−1ip] (5.26)

For exceptional groups (they are all subgroups of classical groups) there is usually yet another invariant tensor, which further reduces the representations.

67 6 Spinor representations

In this section, we discuss spinor representations of the orthogonal Lie algebras so(d; R) and so(d − 1, 1; R), starting from the respective Clifford algebras. We begin with a brief review of the spinor representation of the group of rotations SO(3; R).

6.1 Spinor representations of SO(3; R) The fundamental representation of SO(3; R) has dimension 3. All the higher dimensional representations may be obtained by tensor product decomposition, and give us the irre- ducible representations of integer spin j. So where do the spinor representations enter ? There are two ways of looking at it. First, look at the generators of the Lie algebra so(3; R), 0 0 0   0 0 1 0 −1 0 L1 = 0 0 −1 L2 =  0 0 0 L3 = 1 0 0 (6.1) 0 1 0 −1 0 0 0 0 0

which satisfy the following structure relations,

[L1,L2] = L3 [L2,L3] = L1 [L3,L1] = L2 (6.2)

The group elements g ∈ SO(3; R) may then be parametrized by Euler angles, α, β, γ,

g(α, β, γ) = eαL3 eβL1 eγL3 0 ≤ α, β < π, 0 ≤ γ < 2π (6.3)

It may be verified that the above range of parameters covers SO(3; R) exactly once. From the point of view of the Lie algebra, the spinor representations enter because the structure σa relations are also satisfied by setting La = −i 2 where σa are the Pauli matrices. This gives a 2-dimensional representation R which cannot be obtained from the tensor product of the defining representation of SO(3; R). But actually, it is not a genuine representation of SO(3; R), because the Euler angle parametrization now gives,

R(g) = e−iασ3/2e−iβσ1/2e−iγσ3/2 0 ≤ α, β < π, 0 ≤ γ < 2π (6.4)

While g was periodic with period 2π in γ, this is not the case for R(g) since we have,

R(g(α, β, γ + 2π)) = −R(g(α, β, γ + 2π)) (6.5)

so that R(g) is actually double-valued function of g, and the representation R is said to be double-valued or to be a projective representation of SO(3; R). But R(g) is a single- valued representation of SU(2), and in fact coincides with its defining representation. The sign ambiguity is absent in SO(3; R) because it is isomorphic to SU(2)/Z2, the Z2 exactly accounting for the sign indeterminacy.

68 6.2 The Clifford Algebra The generalization to spinor representations for higher rank orthogonal groups makes use of the Clifford algebra whose representations induce spinor representation in the orthogonal groups. We begin by recalling the parametrization and the structure relations of so(d).

The generators Mµν are labelled by a composite index (µν) with 1 ≤ µ, ν ≤ d and they are anti-symmetric, so that Mνµ = −Mµν. The independent generators may thus be chosen to be Mµν with 1 ≤ µ < ν ≤ d. Their matrix elements are given by,

(Mµν)αβ = δµαδνβ − δµβδνα (6.6) The structure relations are given by,

[Mµν,Mρσ] = δνρMµσ − δνσMµρ − δµρMνσ + δµσMνρ (6.7)

Note that one should think of the Kronecker symbols δµν etc as representing the identity matrix in so(d; R). It is possible to generalize the construction of spinors to so(p, q; R) with pq 6= 0 and δµν will then be replaced by the matrix elements of Ip,q, and indeed we shall do so for the Minkowski metric later. In the special case of d = 3, we may recast the

generators Mµν in terms of the notation La used earlier for so(3) as follows,

L1 = M32 L2 = M13 L3 = M21 (6.8) We shall now introduce the Clifford algebra to produce representations of this Lie algebra. The Clifford algebra in d dimensions, for Euclidean signature, is defined by,

{γµ, γν} = 2δµνI µ, ν = 1, ··· , d (6.9) Now let’s assume that we can construct a representation of the Clifford algebra (we shall

do so explicitly soon). The claim is then that if γµ forms a representation of the Clifford algebra, then we automatically obtain a representation of the Lie algebra so(d) with the following representation matrices, 1 1 M = [γ , γ ] = (γ γ − γ γ ) (6.10) µν 4 µ ν 4 µ ν ν µ This result is so simple and so fundamental that it is worth proving it explicitly by com- puting the commutator, 1 [M ,M ] = [γ γ , γ γ ] (6.11) µν ρσ 4 µ ν ρ σ where in each entry of the commutator, we have used the Clifford algebra relations to

convert γνγµ into −γµγν plus the identity matrix which cancels out of the commutator. Next, we write out the commutator, 1 1 [M ,M ] = γ γ γ γ − γ γ γ γ (6.12) µν ρσ 4 µ ν ρ σ 4 ρ σ µ ν

69 By systematically permuting the γ-matrices to the same position with the help of the Clifford relations, we get,

1 1 1 1 [M ,M ] = γ {γ , γ }γ − γ {γ , γ }γ − {γ , γ }γ γ + γ γ {γ , γ } (6.13) µν ρσ 4 µ ν ρ σ 4 ρ σ µ ν 4 µ ρ ν σ 4 ρ µ σ ν Using now the Clifford algebra relation to convert all anti-commutators, we find,

1 1 1 1 [M ,M ] = δ γ γ − δ γ γ − δ γ γ + δ γ γ (6.14) µν ρσ 2 νρ µ σ 2 µσ ρ ν 2 µρ ν σ 2 νσ ρ µ Using the Clifford algebra relations to anti-symmetrize the products of γ-matrices we pro- duce terms proportional to the identity matrix, which all cancel. We find that the matrices M indeed obey the structure relations of so(d; R). To see why the representation matrices have anything to do with spinors, we need to construct representations of the γ-matrices.

6.3 Representations of the Clifford algebra For d = 3 the representation of the Clifford algebra is in terms of the Pauli matrices,

0 1 0 −i 1 0  σ = σ = σ = (6.15) 1 1 0 2 i 0 3 0 −1

The Pauli matrices may also be given in the raising/lowering basis,

1 0 1 1 0 0 σ = (σ + iσ ) = σ = (σ − iσ ) = (6.16) + 2 1 2 0 0 − 2 1 2 1 0

where they satisfy [σ3, σ±] = ±σ± and [σ+, σ−] = σ3. For d = 4, it was Dirac who, in one of his strokes of genius, wrote down the Dirac representation in terms of 4 × 4 matrices for the case of the Minkowski metric. What he used was, effectively, taking the tensor product of the Pauli matrices, though he did not use that terminology. He used this representation to construct the Dirac equation which led to a revolution in physics. For general d we proceed by transforming the Clifford algebra into an algebra of fermions, by defining the following operators, for j = 1, ··· n,

1 1 b = (γ − iγ ) b† = (γ + iγ ) (6.17) j 2 2j−1 2j j 2 2j−1 2j

† when d = 2n is even. When d = 2n + 1 is odd, we add b0 = b0 = γ2n+1. Expressing the † Clifford algebra relations in terms of the bi and bi , we find the following algebra,

† † † {bj, bk} = δjkI {bj, bk} = {bj, bk} = 0 j, k = 1, ··· , n (6.18)

70 when d is even, and when d is odd we supplement the above relations with the anti-

commutators with b0, 2 † (b0) = I {bj, b0} = {bj, b0} = 0 for d odd (6.19) The algebra for d = 2n even consists of n complex fermion operators which are independent of one another. This means that their representation is a tensor product of the representations for each one of the fermion oscillator pairs. We are interested in finite-dimensional representations, because we wish to construct finite-dimensional representations of so(d). To construct the † representation for a single pair, bi, bi we use the fact that it is finite-dimensional so that † there must be some state |−ii 6= 0 in the representation for which bi|−ii = 0. Applying bi † to |−ii, we obtain a new state |+ii = bi |−ii. This state is different from |−ii because when we apply bi to it, the result is |−ii which is non-zero while applying bi to |−ii gave zero. But now were are finished because no other states are obtained by applying the generators † bi or bi . Thus the irreducible representation is two-dimensional, and in fact it is just the algebra of σ± and σ3. To obtain the representation of the full Clifford algebra for arbitrary d, we take the tensor product ground state,

|0i = |−i1 ⊗ |−i2 ⊗ · · · ⊗ |−in (6.20)

It is annihilated by all the bi for i = 1, ··· , n, and is an eigenstate of b0 if d is odd. The most general state is a linear combination of the states,

† m1 † m2 † mn (b1) (b2) ··· (bn) |0i (6.21) n where each mi can take the values 0 or 1, giving a total of 2 states. Thus the representation n [ n ] has dimension 2 = 2 2 , and it is irreducible for all d. We may equivalently return to the representation in terms of the original γ-matrices. The states map as follows, 1 0 |+i → u = |−i → u = (6.22) i + 0 i − 1 and the ground state is given by,

|0i → u− ⊗ u− ⊗ · · · ⊗ u− (6.23) | {z } n † The individual generators map as follows bi → σ− and bi → σ+, from which we can now construct the representations of the b and b† operators, † b1 = σ− ⊗ I ⊗ I ⊗ · · · ⊗ I ⊗ I b1 = σ+ ⊗ I ⊗ I ⊗ · · · ⊗ I ⊗ I (6.24) † b2 = σ3 ⊗ σ− ⊗ I ⊗ · · · ⊗ I ⊗ I b2 = σ3 ⊗ σ+ ⊗ I ⊗ · · · ⊗ I ⊗ I ······ † bn = σ3 ⊗ σ3 ⊗ σ3 ⊗ · · · ⊗ σ3 ⊗ σ− bn = σ3 ⊗ σ3 ⊗ σ3 ⊗ · · · ⊗ σ3 ⊗ σ+

71 † Finally, converting σ± into σ1 and σ2 and bi and βi into γµ, we obtain, the expressions for the γ matrices,

γ1 = σ1 ⊗ I ⊗ I ⊗ · · · ⊗ I ⊗ I γ2 = σ2 ⊗ I ⊗ I ⊗ · · · ⊗ I ⊗ I (6.25)

γ3 = σ3 ⊗ σ1 ⊗ I ⊗ · · · ⊗ I ⊗ I γ4 = σ3 ⊗ σ2 ⊗ I ⊗ · · · ⊗ I ⊗ I

γ5 = σ3 ⊗ σ3 ⊗ σ1 ⊗ · · · ⊗ I ⊗ I γ6 = σ3 ⊗ σ3 ⊗ σ2 ⊗ · · · ⊗ I ⊗ I ······

γ2n−1 = σ3 ⊗ σ3 ⊗ σ3 ⊗ · · · ⊗ σ3 ⊗ σ1 γ2n = σ3 ⊗ σ3 ⊗ σ3 ⊗ · · · ⊗ σ3 ⊗ σ2

We also define the matrix

γ¯ = σ3 ⊗ σ3 ⊗ σ3 ⊗ · · · ⊗ σ3 ⊗ σ3 (6.26)

which satisfies

2 γ¯ = I {γ,¯ γj} = 0 (6.27)

If d is even, so that d = 2n, then γµ, µ = 1, ··· , 2n defined above, indeed satisfy (6.9). If d is odd, we define γd =γ ¯, and then γµ, i = 1, ··· , 2n + 1 satisfies the Clifford algebra. By † construction, all γµ are Hermitian, γµ = γµ. The γ-matrices form only a small subset of all 2n ×2n matrices. All the other generators n n of 2 × 2 matrices are obtained by anti-symmetrized products of the basic γµ,

γµ1µ2···µp ≡ γ[µ1 γµ2 ··· γµp] (6.28)

where [··· ] denotes anti-symmetrization of the p indices. The identity I together with all

matrices γµ1µ2···µp for 1 ≤ p ≤ n forms a complete basis, with the following properties,

1 † 2 p(p−1) (γµ1µ2···µp ) = (−1) γµ1µ2···µp p γ¯ γµ1µ2···µm = (−1) γµ1µ2···µp γ¯ (6.29)

6.4 Spinor representations of so(d)

1 We have shown earlier that the generators Σµν = 4 [γµ, γν] satisfy the structure relations of so(d). Here we shall provide more details and show why this is the spinor representation. We shall denote the representation matrices of the spinor representation of SO(d) by S(g) where g is an element of SO(d). Near the identity g ≈ I, we have,

ν ν ν 2 gµ = δν + ωµ + O(ω ) d 1 X S(g) = I + ω Σ + O(ω2) (6.30) 2 µν µν µ,ν=1

72 The spinor representation S(g) leaves the γ-matrices invariant,

d X ν −1 gµ S(g)γνS(g) = γµ (6.31) ν=1 a relation which follows from integrating the infinitesimal version,

[Σµν, γρ] = δµργν − δνργµ (6.32) Notice that the generators have been normalized here to be anti-hermitian. The reason this representation is a spinor representation may be seen by performing a rotation in just a single direction, say by, i Σ12 = σ3 ⊗ I ⊗ I ⊗ · · · ⊗ I (6.33) 2 | {z } n−1 But this matrix is proportional to a Pauli matrix with coefficient 1/2, so that a rotation by 2π will again produce a minus sign. This double-valuedness is the characteristic property of a spinor representation. For low rank, it may be understood as it was for the SO(3; R) case in terms of SU(2), as we have,

SO(4; R) = SU(2) × SU(2)/Z2 SO(5; R) = USp(4)/Z2 SO(6; R) = SU(4)/Z2 (6.34)

In each case the Z2 identification takes care of the double-valuedness. But the special identifications end there. For d ≥ 7, one sometimes refers to the simply-connected double cover of SO(d) as Spin(d), so that SO(d) = Spin(d)/Z2.

6.5 Reducibility and Weyl spinor representations For d = 2n + 1 odd, the Dirac spinor representation is irreducible. However, for d = 2n

even, the matrixγ ¯ commutes with the representation Σµν,

[¯γ, Σµν] = 0 (6.35) Sinceγ ¯2 = I but tr¯γ = 0, we see thatγ ¯ is not proportional to the identity matrix. Putting these two observations together, we see that the Dirac spinor representation Σ must reducible into two representations of half the dimension of 2n. The components are referred to as Weyl spinor representations Σ+ and Σ−, and are defined by projection onto the ±1 eigenspaces ofγ ¯ by, 1 Σ± ≡ (I ± γ¯)Σ (6.36) 2 µν each of which is irreducible.

73 6.6 Charge and complex conjugation

If the matrices γµ satisfy the Clifford algebra relations, then so must their complex con- ∗ jugate matrices γµ. The matrices γµ are not all real. Since we have found only a single ∗ irreducible representation of the Clifford algebra, and hence γµ must actually correspond to an equivalent representation. Therefore, there exists an invertible matrix C such that,3

∗ −1 γµ = −CγµC (6.37)

for all 1 ≤ µ ≤ d. The matrix C is referred to as the charge conjugation matrix. We may compute C explicitly, in the basis given above for example for d = 2n even, by observing

that half of the matrices γµ are real symmetric while the other half are imaginary anti- symmetric. To find C, we choose a basis, such as the one above, and we have

−1 CγµC = −γµ µ odd −1 CγµC = +γµ µ even (6.38)

It is easy to find C, up to an overall multiplicative factor, by considering the product of the n symmetric Dirac matrices, together with a power of Γ. Choosing the multiplicative factor as follows, we find,

n C = (¯γ) γ1 γ3 γ5 ··· γ2n−1 (6.39)

The matrix C then satisfies,

C†C = CtC = I t 1 n(n+1) C = (−1) 2 C 2 1 n(n+1) C = (−1) 2 I (6.40)

The complex conjugation properties of the representation Σ of so(d) are then give as follows,

 ± −1 C Σij C n even ±∗  Σij = (6.41)  ∓ −1 C Σij C n odd

For n even, the Weyl spinor representations Σ± are self-conjugates, while for n odd, they are complex conjugates of one another. An analogous result may be derived for odd d.

3The sign prefactor is a matter of convention since, if C satisfies the above equation, then Cγ¯ satisfies ∗ t −1 the equation with the opposite sign, γµ = γµ = (Cγ¯) γµ( Cγ¯) .

74 6.7 Spinor representations of so(d − 1, 1; R) When the metric is changed from Euclidean signature (+ + ··· +) to Minkowski signature

(+ ··· + −) with metric ηµν , the structure equation becomes,

[Mµν,Mρσ] = ηνρMµσ − ηνσMµρ − ηµρMνσ + ηµσMνρ (6.42)

The Clifford algebra is defined with respect to the metric η as well and we have,

{Γµ, Γν} = 2ηµνI (6.43) and the spinor representation matrices take the form, 1 Σ = [Γ , Γ ] (6.44) µν 4 µ ν We use capital letters here to represent the γ-matrices to clearly distinguish them from the case of Euclidean signature. It is straightforward to obtain a representation of the Clifford

algebra matrices Γµ in terms of γµ by setting,

Γd = i γd

Γµ = γµ for µ = 1, ··· , d − 1 (6.45)

For even d, we define the matrix Γ¯ such that Γ¯2 = I. The representation Σ obtained this way is a spinor representation for the same reasons as with Euclidean signature. For even d, the Dirac spinor representation is reducible using the matrixγ ¯ in terms of Weyl spinor representations Σ± with half the dimension of 2n. What changes when the signature is changed is the reality conditions that one can im- pose on the representations. Complex conjugation may still be defined in an SO(d−1, 1; R)- invariant way, but there is now a further distinction one can make since transposition is

not related to complex conjugation by Hermiticity. Thus, if Γµ satisfy the Clifford algebra ∗ t relations, then so must Γµ and Γµ, but these representations must be equivalent to Γµ, so that there must exist invertible matrices C and B such that,4

t −1 Γµ = −C Γµ C ∗ −1 Γµ = +B Γµ B (6.46) To find C, we must pick a basis, such as the one above, and we have

−1 C Γµ C = −Γµ µ odd −1 C Γµ C = +Γµ µ even (6.47)

4The sign prefactors are a matter of convention since, if B and C satisfy the above equations, then ΓB t −1 ∗ −1 and ΓC satisfy the equation with the opposite sign, Γµ = +(ΓC)Γµ(ΓC) , and Γµ = −(ΓB)Γµ(ΓB) .

75 For even d = 2n it is easy to find C by considering the product,

¯ n C = (Γ) Γ1Γ3Γ5 ··· Γ2n−1 (6.48)

The matrix C is then the same matrix as in the Euclidean case and satisfies,

C†C = CtC = I t 1 n(n+1) C = (−1) 2 C 2 1 n(n+1) C = (−1) 2 I (6.49)

Complex conjugation on the other hand behaves differently, and is most easily gotten as the composition of Hermitian conjugation and charge conjugation,

∗ † t t −1 t −1 Γµ = (Γµ) = −(Γ0) ΓµΓ0 = Γ0CΓµ(Γ0C) (6.50)

Using the definition of the matrix B and Shur’s lemma, we find that B must be proportional

to CΓ0. We shall define B as follows,

B = CΓ0 (6.51)

For even d = 2n, the complex conjugation properties of the representation are,

 ± −1 B Σµν B n odd ± ∗  Σµν = (6.52) ∓ −1 B Σµν B n even

Thus, for n odd, the spinor representations ψ+ and ψ− are self-conjugates, while for n even, they are complex conjugates of one another. The spinors ψ∗ and Bψ and ΓBψ transform under the same representation. Thus, we may impose the following reality conditions,

ψ∗ = Bψ (6.53) which requires the consistency condition B∗B = I. In view of B∗ = B we have,

∗ 1 n(n+1) B B = −(−1) 2 I = I (6.54)

has the solutions n ≡ 1, 2 (mod 4). Alternatively, one may impose the reality condition,

ψ∗ = ΓBψ (6.55)

which requires that the consistency condition (ΓB)∗(ΓB) = I be obeyed, or

∗ 1 n(n−1) (ΓB) (ΓB) = +(−1) 2 I = I (6.56)

76 has the solutions n ≡ 0, 1 (mod 4). The Majorana and Weyl conditions may be enforced when we have Majorana spinors, and [B, Γ] = 0. So, this can happen only when n ≡ 0, 1, 2 (mod 4). To fulfill the remaining condition, we require,

n+1 B = Γ B Γ = Γ Γ0 C Γ = (−1) B (6.57) so that only n ≡ 1 (mod 4) i.e. d ≡ 2 (mod 8) survives.

77 7 Roots, weights, and representations

The construction given in the earlier section for the representations of sl(2; C) basically con- tains all the ingredients for the construction of the representations of all finite-dimensional representations of an arbitrary semi-simple Lie algebra G over C. The construction for Lie algebras of rank higher than 1 is considerably more complicated, but may be handled with great ease using the method of roots and weights due to Cartan. The adjoint representation of G is such a representation, whose matrices are the struc- ture constants of G and whose dimension is that of G. The classification of all represen- tations thus contains the classification of all semi-simple Lie algebras. The power of the method of roots and weights is illustrated by Cartan’s classification theorem (1920) of all simple finite-dimensional Lie algebras, and by Dynkin’s complete theory of representations.

7.1 The Cartan-Killing form

We consider a Lie algebra G over C of finite dimension n = dimC G and rank r. In a basis c {X1, ··· ,Xn} of G the structure relations are given by the structure constants fab , n X c [Xa,Xb] = fab Xc (7.1) c=1

The Cartan-Killing form is the bilinear symmetric form, or metric, γab on G defined by, n X d c γab = tr (AdXa AdXb ) = fac fbd (7.2) c,d=1 Cartan’s definition of a semi-simple Lie algebra is that its Cartan-Killing form is invertible, in which case we denote the inverse by γab, n X bc c γabγ = δb (7.3) b=1

We shall denote the inner product of two vectors va, wb by, n X ab (v, w) = γ vawb (7.4) a,b=1 One could, of course, take the convention to choose γ proportional to the identity ma- trix, as is done in Georgi’s book. However, this choice leads to unnecessarily complicated normalizations of the Cartan generators. An alternative and equivalent definition of a complex semi-simple Lie algebra is that it is the complexification of the Lie algebra K of a compact matrix Lie group. This definition, advocated for example in Brian Hall’s book, is especially convenient to use.

78 Theorem 10 Let G be a semi-simple Lie algebra with Cartan-Killing form γab. 1. The Cartan-Killing form is invariant under the action of G; P d 2. The combination fabc = d fab γdc is totally anti-symmetric in its indices; 3. For an arbitrary representation ρ of G, we define a symmetric bilinear form by,  tr ρ(Xa)ρ(Xb) = γ(ρ)ab (7.5)

If G is a simple Lie algebra (i.e. its adjoint representation is irreducible), then for any representation ρ of G the form γ(ρ) is proportional to the Cartan-Killing form,

γ(ρ)ab = K(ρ)γab (7.6)

where K(ρ) depends only on ρ and equals 1 for the adjoint representation.

To prove these results, we start from the following identity, valid in any representation,

n n   X d X d 0 = tr [ρ(Xa), ρ(Xb)ρ(Xc)] = fab γ(ρ)dc + fac γ(ρ)db (7.7) d=1 d=1

This relation proves invariance of γ(ρ)ab in any representation ρ. Taking ρ to be the adjoint representation shows that fabc, defined above, is anti-symmetric in its last two indices and is thus anti-symmetric in all indices. To prove 3, we write (7.7) in matrix form t c c t Faγ(ρ) + (Faγ(ρ)) = 0 using (Fa)b = fab . Eliminating Fa between the relations for an arbitrary representation ρ and the adjoint representation we find,

−1 −1 Faγ(ρ)γ = γ(ρ)γ Fa (7.8)

When G is a simple Lie algebra, its adjoint representation is irreducible, and by Shur’s lemma γ(ρ)γ−1 must be proportional to the identity matrix, thereby proving 3.

7.2 Weights The Cartan subalgebra H of G is defined to be the maximal Abelian subalgebra of G for which the adjoint map is diagonalizable. The dimension r of the vector space H over C is defined to be the rank of G. We denote by H1, ··· ,Hr a choice of basis generators of H which, by definition, mutually commute,

[Hi,Hj] = 0 for all i, j = 1, ··· , r (7.9)

Since G is the complexification of the Lie algebra K of a compact Lie group, by the above definition of a semi-simple Lie algebra, we may choose the Cartan generators to be the Her- mitian generators of K. Furthermore, since every finite-dimensional representation of K is

79 equivalent to a unitary representation (by theorem 3.6), we may choose the representations matrices ρ(Hi) in any finite-dimensional representation of the complexified Lie algebra G to be Hermitian as well. For sl(2; C) the Cartan generator may be chosen to be J3. For sl(3; C) there will be two such generators, which we denoted λ3, λ8 in Problem set 2. Consider an arbitrary representation ρ of G, whose dimension N is finite. The repre- sentation matrices ρ(Hi) mutually commute,

[ρ(Hi), ρ(Hj)] = 0 for all i, j = 1, ··· , n (7.10) and may be chosen to be Hermitian, as argued above. Since they mutually commute and are Hermitian, they may be diagonalized in the same basis and have real eigenvalues. We shall label the states of the representation ρ, in the basis where all ρ(Hi) are diagonal, by the real eigenvalues µi of the matrices ρ(Hi),

ρ(Hi)|ρ;µi = mi|ρ;µi µ = (m1, ··· , mn) (7.11)

The array µ of eigenvalues is referred to as the weight vector of the state |ρ;µi. We shall show later that, when ρ is an irreducible representation, all the states are uniquely labelled by the set of all weight vectors. This result is familiar from quantum mechanics, where one uses a maximal set of commuting observables to generate all the quantum numbers.

7.3 Roots

Since the generators Hi mutually commute and are Hermitian, we may choose the remaining generators of G to diagonalize simultaneously the adjoint maps,

[Hi,Eα] = aiEα α = (a1, ··· , an) (7.12) where each ai is real, and α is referred to as the root vector. Diagonalizing the adjoint map generally requires complexifying the Lie algebra. For the case of sl(2; C) there were two such generators which we denoted J± = J1 ± iJ2 which clearly required introducing i ∈ C. The generators satisfied [J3,J±] = ±J± so in this case the root vectors would be (±1). The root vectors α are identical to the weight vectors of the adjoint representation.

Denoting the subspace of G generated by Eα by Gα, and the space of all root vectors α by R, we have the Cartain decomposition of the Lie algebra G as follows, M G = H ⊕ Gα (7.13) α∈R

The result follows from the Hermiticity of the generators Hi of H. In addition, we prove the following results.

80 Theorem 11 Let G be a semi-simple Lie algebra with Cartan subalgebra H = {H1, ··· Hr}. † 1. If α is a root, then so is −α and we have E−α = (Eα) . 2. Opposite root generators close onto a sl(2; C) subalgebra as follows, r X ij [Eα,E−α] = Hα Hα = (αα, H) = γ aiHj i,j=1

[Hα,E±α] = ±(αα,αα)E±α (7.14) 3. The commutator of two generators associated with roots αα,ββ 6= −α vanishes if α +β

is not a root, and gives a generator Eα+β if α + β is a root,

[Eα,Eβ ] ∼ Eα+β (7.15)

4. The root spaces Gα are one-dimensional and the only root proportional to α is −α.

To prove 1. we take the adjoint of [Hi,Eα] = aiEα and use the fact that Hi is Hermitian, † and ai is real. It follows that (Eα) has eigenvalue −ai. To prove 2, we use the Jacobi identity to evaluate the following commutator,

[Hi, [Eα,E−α]] = [Eα, [Hi,E−α]] − [E−α, [Hi,Eα]]

= −ai[Eα,E−α] − ai[E−α,Eα] = 0 (7.16)

i Hence, [Eα,E−α] must be in the Cartan subalgebra of G, and there exist b such that, n X i [Eα,E−α] = b Hi (7.17) i=1

Taking the inner product with Hi and using the Cartan-Killing form, r   X j tr Hi[Eα,E−α] = γijb (7.18) j=1

But the left hand side may be rearranged as tr([Hi,Eα]E−α) which evaluates to aitr(EαE−α). It will be convenient to normalize the generators Eα so that,

tr(EαE−α) = 1 (7.19) i P ij so that b = j γ aj for all i = 1, ··· , r. To prove 3. we consider the commutator [Eα,Eβ ] for α + β 6= 0, and evaluate its commutator with Hi using the Jacobi identity,

[Hi, [Eα,Eβ ]] = [[Hi,Eα],Eβ ] + [Eα, [Hi,Eβ ]] = (ai + bi)[Eα,Eβ ] (7.20)

If α + β is not a root, then [Eα,Eβ ] has no components along Eγ for any root γ, and it can have no components along Hj since the left side vanishes, but the right side would not. Thus if α + β is not a root, then [Eα,Eβ ] = 0. Finally, if α + β is a root, then we have [Eα,Eβ ] ∼ Eα+β , as stated above. Point 4. will be proven later.

81 7.4 Raising and Lowering operators

A generator Eα of G corresponding to a root α modifies the weights of a representation ρ. To see this, we use the structure relations satisfied by the representation matrices,

[ρ(Hi), ρ(Eα)] = aiρ(Eα) (7.21) and evaluate the weight vector µ = (m1, ··· , mr) of a state ρ(Eα)|ρ;µi as follows,   ρ(Hi)ρ(Eα)|ρ;µi = [ρ(Hi), ρ(Eα)] + ρ(Eα)ρ(Hi) |ρ;µi

= (ai + mi)Eα|ρ;µi (7.22) Since all the states in an irreducible representation ρ were uniquely labelled by their weight vector, we conclude that we must have,

ρ(Eα)|ρ;µi = Nα(ρ;µ)|ρ;µ + αi (7.23) where Nα(ρ;µ) are normalization constants which may vanish. Since we have,

n X ij [ρ(Eα), ρ(E−α)] = ρ(Hα) = γ aiρ(Hj) (7.24) i,j=1 for any representation, we obtain relations between the normalizations Nα(ρ;µ) by applying this relation to an arbitrary state |ρ;µi= 6 0,

n X ij ρ(Eα)ρ(E−α)|ρ;µi − ρ(E−α)ρ(Eα)|ρ;µi = γ aiρ(Hj)|ρ;µi (7.25) i,j=1 we obtain the following fundamental relation,

Nα(ρ;µ − α)N−α(ρ;µ) − Nα(ρ;µ)N−α(ρ;µ + α) = (αα,µµ) (7.26)

7.5 Finite-dimensional representations

For a finite-dimensional representation ρ, the vector space generated by applying ρ(Eα) to any given state transforming under the representation ρ must have finite dimension. Consider the state |ρ;µi associated with weight vector µ, and the vector space generated by applying a single root generator multiple times,

κ κ Y ρ(Eα) |ρ;µi = Nα(ρ;µ + (k − 1)α) |ρ;µ + κααi k=1 κ κ Y ρ(E−α) |ρ;µi = N−α(ρ;µ − (` − 1)α) |ρ;µ − κααi (7.27) `=1

82 with κ ∈ N. We shall denote by p and q the largest values of κ such that the states satisfy |ρ;µ + pααi= 6 0 and |ρ; µ − qααi= 6 0 and,

Eα|ρ;µ + pααi = 0 E−α|ρ;µ − qααi = 0 (7.28)

Since the representation ρ has finite dimension, we must have 1 ≤ p, q < ∞. Now we use the fundamental formula (7.26) for all weights µ + kαα with −q ≤ k ≤ p ,

Nα(ρ;µ + (k − 1)α)N−α(µ + kαα) − Nα(ρ;µ + kαα)N−α(µ + (k + 1)α) = (α,µµ + kαα) (7.29)

Summing the contributions over this range,

p X Nα(ρ;µ + (k − 1)α)N−α(µ + kαα) k=−q p p X X − Nα(ρ;µ + kαα)N−α(µ + (k + 1)α) = (αα,µµ + kαα) (7.30) k=−q k=−q

Changing summation variables k → k − 1 in the second sum, we see that all the terms except at the end points of the summation on the left side cancel one another, and we are left with,

Nα(ρ;µ − (q + 1)α)N−α(µ − qαα) p X −Nα(ρ;µ + pαα)N−α(µ + (p + 1)α) = (αα,µµ + kαα) (7.31) k=−q

In view of the definition of p and q we have Nα(ρ;µ + pαα) = 0 and N−α(ρ;µ − qαα) = 0, so that the left side cancels. The vanishing of the right side then gives the equation,

1 1  (p + q + 1)(αα,µµ) + p(p + 1) − q(q + 1) (αα,αα) = 0 (7.32) 2 2

Dividing through by (p + q + 1) gives the following fundamental formula,

2(αα,µµ) = q − p (7.33) (αα,αα)

Since q, p ∈ N, we see that the weights µ lie in a lattice that is dual to the root lattice. Note that in this ratio of inner products the overall normalization of γab cancels out, and one may use the Cartan Killing form defined in an arbitrary representation.

83 7.6 The example of SU(3) The choice Gell-Mann made for the generators of SU(3) is as follows.  0 1 0   0 −i 0   1 0 0  1 1 1 T = 1 0 0 T = i 0 0 T = 0 −1 0 1 2   2 2   3 2   0 0 0 0 0 0 0 0 0  0 0 1   0 0 −i  1 1 T = 0 0 0 T = 0 0 0 4 2   5 2   1 0 0 i 0 0  0 0 0   0 0 0   1 0 0  1 1 1 T = 0 0 1 T = 0 0 −i T = √ 0 1 0 6 2   7 2   8   0 1 0 0 i 0 2 3 0 0 −2

The matrices Ta form an orthogonal basis under the inner product provided by the trace, and their normalization has been chosen so that they satisfy, 1 tr(T T ) = δ (7.34) a b 2 ab Thus the inner product (u, v) may be simply replaced by the vector inner product u · v.

Ultimately, we shall prefer the basis in which T8 is replaced by the generator, 0 0 0  √ 1 3 1 T 0 = 0 1 0 = T − T (7.35) 8 2   2 8 2 3 0 0 −1

0 1 at the cost of having tr(T8T3) = − 4 , and thus a non-diagonal metric. We observe the following subalgebras isomorphic to su(2) or so(3),

S1 = {T1,T2,T3} 0 S2 = {T4,T5,T3 + T8} 0 S3 = {T6,T7,T8}

S4 = {T2,T5,T7} (7.36)

The sub-algebras S1, S2, S3 are in the two-dimensional representation of su(2), and are equivalent to one another, but S4 is in the three-dimensional representation of su(2) or in the defining representation of so(3) and is inequivalent to the first three. Now we proceed to choosing a Cartan subalgebra, and constructing the roots of su(3) and the weights of arbitrary representations of su(3). The rank is 2, and clearly we will

choose the Cartan subalgebra generators to be H1 = T3 and H2 = T8. The states in any representation ρ will be labelled by the weight vector µ = (m1, m2),

Hi|ρ; µi = mi|ρ; µi i = 1, 2 (7.37)

84 The states of the defining representation 3 in the standard basis of C3 correspond to the following basis in terms of the weight vectors λ of the defining representation, 1 0 0 0 = |3; λ1i 1 = |3; λ2i 0 = |3; λ3i (7.38) 0 0 1 where the weight vectors are given by, 1 1   1 1   1  λ1 = , √ λ2 = − , √ λ3 = 0, −√ (7.39) 2 2 3 2 2 3 3 For the adjoint representation, we choose the following linear combinations which diago-

nalize AdH1 and AdH2 ,

T1 ± i T2 = E±α1

T4 ± i T5 = E±α3

T6 ± i T7 = E±α2 (7.40) where the root vectors αi are given by, √ ! √ ! 1 3 1 3 α = (1, 0) α = − , α = , (7.41) 1 2 2 2 3 2 2

Notice that each root satisfies αi · αi = 1, so that the master formula becomes,

2α · β ∈ Z (7.42)

for any pair of roots. We readily verify that 2α1 · α3 = 2α2 · α3 = −2α1 · α2 = 1. But the master formula tells us more than that. If µ is a weight, and if µ + pα and µ − qα are the largest addition and subtraction of µ by the root α, then p and q satisfy,

2α · µ = q − p (7.43)

Taking µ = α2 and α = α1 we have 2α · µ = q − p = −1. This means that p ≥ 1, so α1 + α2 must be a root, and indeed α1 + α2 = α3. We can choose as a basis of all roots the vectors α1 and α2. If 2α1 · µ ∈ Z and 2α2 · µ ∈ Z, then it follows that 2α3 · µ ∈ Z. Now consider a more general representation ρ of su(3). Its weight vectors must satisfy,

2α1 · µ = p1 ∈ Z 2α2 · µ = p2 ∈ Z (7.44)

Solving for µ in terms of p1, p2, we find, p p + 2p  µ = 1 , 1 √ 2 (7.45) 2 2 3

85 Comparison with the weights of the fundamental representation, we see that µ is a linear combination with integer coefficients of the weights of teh defining representation,

µ = (p1 + p2)λ1 + p2λ2 (7.46)

This is no accident: the weights of any representation lies in the to the root lattice, of which the weight lattice of the defining representation is a basis. Of course, this is not the full story since these states must truncate somewhere.

7.7 The root lattice of An = sl(n + 1) Although the rank of the algebra is n, it will be convenient to consider the root vectors in n+1 R , generated by an orthonormal basis of vectors ei for i = 1, ··· , n + 1. Parametrize the Lie algebra sl(n + 1) by the matrices Mij with components,

(Mi,j)αβ = δi,αδj,β (7.47)

The commutator of two such matrices is given by,

[Mi,j,Mk,`] = δj,kMi,` − δi,`Mk,j (7.48)

We shall chose the Cartan generators to be diagonal and traceless,

Hi = Mi,i − Mi+1,i+1 (7.49)

The commutator of Hi with an arbitrary generator Mk,` is given by,  [Hi,Mk,`] = δi,k − δi,` − δi+1,k + δi+1,` Mk,` (7.50)

Therefore, the root vector associated with the generator Mk,` with k 6= ` is given by,

α = ek − e` Eα = Mk,` (7.51)

The set of all roots is given by,

R = {ek − e` for 1 ≤ k 6= ` ≤ n + 1} #R = n(n + 1) (7.52)

One defines a root α = ek − e` to be a positive root iff k < ` and a root α to be a negative root iff −α is a positive root, which is equivalent to k > `. All positive roots may be expressed as a linear combinations of the simple roots,

αi = ei − ei+1 i = 1, ··· , n (7.53)

86 with positive integer, or zero, coefficients. To see this, we decompose the positive root

ek − e` for k < ` as follows, `−1 X ek − e` = (ei − ei+1) (7.54) i=k All the coefficients are either zero or 1, and hence all positive or zero, as announced. The inner products of the simple roots are given as follows,   2 if i = j (αi,ααj) = −1 if |i − j| = 1 (7.55)  0 if |i − j| ≥ 2 Using the Dynkin diagran representation introduced earlier, in which roots are represented by nodes, two roots with inner product −1 are linked by a single link, and two roots which are orthogonal re not linked, we have the following for An = sl(n + 1; C). ···

α1 α2 α3 αn−1 αn

The is the matrix of inner products of the simple roots, normalized in such a manner that the entries have to be integers by construction, and are independent of the overall scaling of the ,

2(αi,ααj) Cij = (7.56) (αi,ααi) Note that the Cartan matrix is symmetric when all the simple roots have the same length, as is the case for he algebra An = sl(n + 1), but will not be symmetric when simple roots of different lengths occur in the root space. For sl(n + 1), the Cartan matrix is simply

Cij = (αi,ααj), and takes the form,  2 −1 0 0 ··· 0 0  −1 2 −1 0 ··· 0 0     0 −1 2 −1 ··· 0 0  C =   (7.57) ··· ··· ··· ··· ··· −1 0     0 0 0 0 ··· 2 −1 0 0 0 0 · · · −1 2

7.8 The root lattice of Dn = so(2n) The algebra so(2n) is generated by 2n × 2n anti-symmetric matrices which we denote by t Mab = −Mba = −Mab for a, b = 1, ··· , 2n and with components,

(Mab)α,β = δaαδbβ − δaβδbα (7.58)

87 Their structure relations are given as follows,

[Mab,Mcd] = δadMbc + δbcMad − δbdMac − δacMbd (7.59)

These are actually the generators of so(2n; R) which is the Lie algebra of the compact Lie group SO(2n; R). We now complexify the algebra and choose the following generators of the Cartain subalgebra,

Hi = i M2i−1,2i i = 1, 2, . . . , n (7.60) √ where the prefactor i = −1 and not the index. The remaining generators may be or- ganized into four sets, depending on the parity of each index, and we have the following commutation relations,

[Hi,M2j−1,2k−1] = −iδi,jM2j,2k−1 − iδi,kM2j−1,2k

[Hi,M2j,2k−1] = +iδi,jM2j−1,2k−1 − iδi,kM2j,2k

[Hi,M2j−1,2k] = −iδi,jM2j,2k + iδi,kM2j−1,2k−1

[Hi,M2j,2k] = +iδi,jM2j−1,2k + iδi,kM2j,2k−1 (7.61)

The roots are identified as follows,

0 α = η ej + η ek j 6= k (7.62) where η = ±1 and independently η0 = ±1, and the associated generators are,

1  E = M + ηM + η0M − ηη0M (7.63) α 2 2j−1,2k−1 2j,2k−1 2j−1,2k 2j,2k

One verifies that Eα is indeed a simultaneous eigenstate with the following eigenvalues,

0 [Hi,Eα] = (ηδi,j + η δi,k)Eα (7.64)

Counting the number of generators, we have n in the Cartan subalgebra and 2n(n−1) root 1 generators, adding up to 2 2n(2n − 1) which indeed is the dimension of so(2n). The roots are usually denoted,

R = {±ej ± ek, 1 ≤ j 6= k ≤ n} (7.65)

where the ± signs are independent of one another. All roots have equal length and satisfy (αα,αα) = 2. By the same lexicographical rule that we used for (n), the positive roots are,

R+ = {ej − ek, ej + ek, 1 ≤ j < k ≤ n} (7.66)

88 The simple roots system may be chosen as follows,

αi = ei − ei+1, i = 1, ··· , n − 1

αn = en−1 + en (7.67)

It is easy to see that every positive root is a linear combination with positive integer coefficients of simple roots. The inner products of the simple roots are given as follows,   2 if i = j for 1 ≤ i, j ≤ n −1 if |i − j| = 1 for 1 ≤ i, j ≤ n − 1 (αi,ααj) = (7.68)  0 if |i − j| ≥ 2 for 1 ≤ i, j ≤ n  0 if i = n − 1, j = n

We find the following Dynkin diagram. Since all roots and all simple roots have the same

αn

···

α1 α2 αn−3 αn−2 αn−1

length, the Cartan matrix is symmetric and given by Cij = (αi,ααj) and is given by a small modification of the one for sl(n + 1; C),  2 −1 0 0 ··· 0 0 0  −1 2 −1 0 ··· 0 0 0     0 −1 2 −1 ··· 0 0 0    C = ························  (7.69)    0 0 0 0 ··· 2 −1 −1    0 0 0 0 · · · −1 2 0  0 0 0 0 · · · −1 0 2

7.9 The root lattice of Bn = so(2n + 1) Since so(2n) is a subalgebra of so(2n + 1) of the same rank, we shall choose the Cartan generators of so(2n+1) to be those chosen for so(2n). The roots ±ej ±ek with 1 ≤ j, k ≤ n of so(2n) are all roots of so(2n + 1) as well. But there are more roots, which are given by,

[Hi,Eα] = η δi,jEα α = η ej η = ±1 (7.70) with associated generators,

Eα = M2j−1,2n+1 + ηM2j,2n+1 (7.71)

89 Hence the system of all roots is given by,

R = {±ej ± ek, 1 ≤ j 6= k ≤ n and ± ej, 1 ≤ j ≤ n} (7.72) where the ± signs are independent of one another. By the same lexicographical rule that we used for sl(n), the positive roots are,

R+ = {ej − ek, 1 ≤ j < k ≤ n and ej, 1 ≤ j ≤ n} (7.73) The simple roots system may be chosen as follows,

αi = ei − ei+1, i = 1, ··· , n − 1

αn = en (7.74) We encounter here, for the first time, roots of different lengths, and simple roots of different lengths. The Cartan matrix is not now symmetric. For example, in the case of n = 3, namely so(7), we have,  2 −1 0  C = −1 2 −1 (7.75) 0 −2 2 The Dynkin diagram is given by ···

α1 α2 α3 αn−1 αn

7.10 The root lattice of Cn = sp(2n + 1) Without derivation we shall give here the root lattice of sp(2n),

R = {±ej ± ek, 1 ≤ j 6= k ≤ n and ± 2ej, 1 ≤ j ≤ n} (7.76) where the ± signs are independent of one another. By the same lexicographical rule that we used for sl(n + 1; C), the positive roots are,

R+ = {ej − ek, 1 ≤ j < k ≤ n and 2ej, 1 ≤ j ≤ n} (7.77) The simple roots system may be chosen as follows,

αi = ei − ei+1, i = 1, ··· , n − 1

αn = 2en (7.78) The Dynkin diagram is given by Superficially, this is looks like the same Dynkin diagram as we had for so(2n + 1), but the key difference is that the role of the long and short roots has been exchanged. Sometimes this is indicated by an arrow pointing from the short to the long roots.

90 ···

α1 α2 α3 αn−1 αn

Coincidences of Lie algebras of low rank From looking at the Dynkin diagrams of the classical Lie algebras, it is immediately manifest that we have the following coincidences,

so(3; C) = sl(2; C) = sp(2; C) so(4; C) = sl(2; C) ⊕ sl(2; C) so(5; C) = sp(4; C) so(6; C) = sl(4; C) (7.79) and that there are no other coincidences.

91 8 Representations of the classical Lie algebras

The classical Lie algebras are An = sl(n + 1; C), Bn = so(2n + 1; C), Cn = sp(2n; C), and Dn = so(2n; C). Their root systems have been constructed in the preceding section. In the present section, we shall discuss the construction of their finite-dimensional representations, and give details for the case of sl(n + 1; C) and so(2n; C) only. The key tool is the following result obtained earlier. Consider a semi-simple Lie algebra G of rank n and a finite-dimensional irreducible representation ρ of G. If µ is a weight of ρ and α a root of G, and µ + kαα is a weight of ρ for −q ≤ k ≤ p with p, q ≥ 1, but µ + kαα is not a weight of ρ for k = p+1 and k = −q −1 then p, q satisfy the following magic formula,

2(αα,µµ) q − p = (8.1) (αα,αα)

As established earlier, the weights of all finite-dimensional representations lie on the weight lattice which consists of all vectors µ such that 2(αα,µµ)/(αα,αα) ∈ Z for all roots α of G.

A basis of the weight lattice is given by the fundamental weights λi for i = 1, ··· n, defined as duals to the simple roots αj by,

2(αj,λλi) = δij (8.2) (αj,ααj) The weights of every finite-dimensional representation ρ span a finite sub-lattice of the

lattice of linear combinations with integer coefficients of the fundamental weights λi.

8.1 Weyl reflections and the Weyl group The system of root and weight vectors is highly symmetric, a property which is reflected in the fact that it is invariant under the Weyl group W (G) generated by Weyl reflections. To quantify Weyl reflections, we start from a weight µ of the Lie algebra G in a finite- dimensional irreducible representation ρ and use the magic formula in the following way,

2(αα,µµ) 2(αα,µµ) If ≥ 0 then q ≥ (αα,αα) (αα,αα) 2(αα,µµ) 2(αα,µµ) If ≤ 0 then p ≥ − (8.3) (αα,αα) (αα,αα)

When 2(αα,µµ)/(αα,αα) is negative, we can always add −2(αα,µµ)/(αα,αα) times α to the weight µ to obtain a new weight µ0,

2(αα,µµ) µ0 = µ − α (8.4) (αα,αα)

92 When 2(αα,µµ)/(αα,αα) is positive, we can always subtract 2(αα,µµ)/(αα,αα) times α to the weight µ to obtain a new weight which turns out to be given by the same formula as µ0. Defining

the following linear operation Πα(µ) on the weight space, 2(αα,µµ) Π (µ) = µ − α (8.5) α (αα,αα) we have just shown that it will take a weight to another weight of the representation ρ.

The operation Πα reverses the sign of the root vector α,

Πα(α) = −α (8.6)

and leaves invariant any vector which is orthogonal to α. Thus, geometrically, Πα reflects the vector µ in the plane orthogonal to α, and is referred to as a Weyl reflection. The

reflection property is confirmed by showing that the square of Πα is the identity,   2(αα,µµ) Π Π (µ) = Π (µ) − Π (α) = µ (8.7) α α α (αα,αα) α The set of Weyl reflections for all roots generates a group, referred to as the Weyl group W (G) which is a subgroup of the group of all permutations of all the roots. The Weyl group provides an invaluable tool for constructing the full weight diagram of a representation, given its highest weight.

8.2 Finite-dimensional irreducible representations of sl(n + 1; C) Using the simple roots of sl(n + 1; C),

αi = ei − ei+1 i = 1, ··· , n (8.8)

which all have length (αi,αα)i) = 2, the fundamental weights λj are defined to satisfy,

(ei − ei+1) · λj = δij 1 ≤ i, j ≤ n (8.9) The solution is unique and given as follows,

j X λj = e` (8.10) `=1 An arbitrary representation is specified by its highest weight vector, n X λ = qiλi (8.11) i=1

The numbers qi correspond to the lengths of the horizontal arrays of boxes in the . For example, taking the case of sl(5; C) with n = 4, the values of qi parametrize the Young tableau as follows.

93 q1 q2 q3 q4

8.3 Spinor representation of so(2n + 1; C) The simple roots of so(2n + 1; C) are given by,

αi = ei − ei+1 i = 1, ··· , n − 1

αn = en (8.12)

The fundamental weights λj for j = 1, ··· , n − 1 obey identically the same equations as the first n − 1 fundamental weights of sl(n + 1; C) and are thus given by,

j X λj = e` j = 1, ··· , n − 1 (8.13) `=1

But the fundamental weight λn satisfies a different equation than for sl(n + 1; C) in view of the fact that αn is a sorter rot, and we have, 2(αn,λλn) = 1 while (αi,λλn) = 0 for i = 1, ··· , n − 1. The solution is uniquely given by,

n 1 X λ = e (8.14) n 2 ` `=1

All the weights of the representation with highest weight λn may be constructed by Weyl reflections in the roots ei, and we get, 1 (±e ± e ± · · · ± e ) (8.15) 2 1 2 n where the ± assignments are independent of one another. Therefore, the representation has dimension 2n, and is referred to as the fundamental spinor representation of so(2n + 1; C).

8.4 Spinor representations of so(2n; C) The simple roots of so(2n; C) are given by,

αi = ei − ei+1 i = 1, ··· , n − 1

αn = en−1 + en (8.16)

94 The fundamental weights λj for j = 1, ··· , n − 2 obey identically the same equations as the first n − 2 fundamental weights of sl(n + 1; C) and are thus given by, j X λj = e` j = 1, ··· , n − 2 (8.17) `=1

But the fundamental weights λn−1,λλn satisfy,

(λn−1,ααi) = (λn,ααi) = 0 i = 1, ··· , n − 2

(λn−1,ααn−1) = (λn,ααn) = 1

(λn−1,ααn) = (λn,ααn−1) = 0 (8.18) Solving these equations, we find, 1 λ = (e + ··· + e + e − e ) n−1 2 1 n−2 n−1 n 1 λ = (e + ··· + e + e + e ) (8.19) n 2 1 n−2 n−1 n

2(αn,λλn) = 1 while (αi,λλn) = 0 for i = 1, ··· , n − 1. The solution is uniquely given by, n 1 X λ = e (8.20) n 2 ` `=1

All the weights of the representation with highest weight λn may be constructed by Weyl reflections in the roots, and we get, 1 λ : (±e ± e ± · · · ± e ) odd number of minus signs n−1 2 1 2 n 1 λ : (±e ± e ± · · · ± e ) even number of minus signs (8.21) n 2 1 2 n where the ± assignments are independent of one another, except for the overall number of minus signs as indicated above. Therefore, we find two independent fundamental spinor representations, each one having dimension dimension 2n−1. A general finite-dimensional irreducible representation ρ of so(2n; C) is given by its highest weight vector λ, which is a linear combination of the n fundamental weight vectors

λi with integer coefficients, n X λ = qiλi qi ∈ Z, qi ≥ 0 (8.22) i=1

When qn−1 + qn is even, then the representation is single-valued and referred to as a tensor representation, while when qn−1 + qn is odd, the representation is double-valued and is referred to as a spinor representation.

95 9 The structure of semi-simple Lie algebras

A semi-simple Lie algebra is specified by its adjoint representation. Thus, in the magic formula, we take the weight vectors µ to be the roots, and find that the root lattice must obey the following conditions, 2(αα,ββ) 2(αα,ββ) M = ∈ M 0 = ∈ (9.1) (αα,αα) Z (ββ,ββ) Z which implies that we also must have, for all roots α, β, 4(αα,ββ)2 MM 0 = ∈ (9.2) (αα,αα)(ββ,ββ) Z This combination is invariant under overall rescaling of the root lattice, and in fact is given by the angle θ between the root vectors αα,ββ,

0 2 MM = 4 cos θ ∈ Z (9.3) This condition is extremely restrictive, and allows for

MM 0 θ Dynkin 0 90o 1 60o, 120o 2 45o, 135o 3 30o, 150o 4 0o, 180o same root

Clearly these conditions provide very strong restrictions on the structure of semi-simple Lie algebras of finite dimension, and in fact lead to their complete Cartan classification, which we shall now carry out.

9.1 Some properties of simple roots One defines a root vector to be positive as one defines in general a weight vector to be

positive. Let the root and weight vectors be given in an ordered orthonormal basis ei for i = 1, ··· , n. A root or weight µ is a positive root iff its first non-zero component is positive. A root or weight µ is a negative root if −µ is a positive root. The only null root or weight is µ = 0. With this lexicographical partial ordering, every root or weight vector is either positive, negative or null.

96 One defines a simple root as a positive root which cannot be written as the sum of two positive roots. The entire root system is determined by giving the simple roots. Some important properties of simple roots are as follows.

1. A semi-simple Lie algebra of rank n has n linearly independent simple roots, which

we shall denote by αi for i = 1, ··· , n. 2. If αα,ββ are simple roots, then α −β is not a root. Obviously, this is true when α = β, so consider α 6= β. Assume that α − β 6= 0 is a root. Then ether α − β is positive or β − α is positive. Consider the first and write α = (α − β) + β, but then α would be the sum of two positive roots, which contradicts the fact that α is simple. 3. As a result, the magic formula tells us that if αα,ββ are two simple roots, we have,

2(αα,ββ) = q − p (9.4) (αα,αα)

but we must have q = 0 since otherwise β − α would be a root, which contradicts the item above. Hence we must have (α,ββ) ≤ 0. Thus the angle θ between different π simple roots must obey 2 ≤ θ < π. The case θ = π is excluded since it would correspond to opposite roots which cannot both be positive. 4. The n simple roots determine all the roots of the algebra, either by using the magic formula, or by using the Weyl group. 5. Every root α can be uniquely decomposed as a sum of simple roots:

n X α = niαi (9.5) i=1

where either all ni ≥ 0 for a positive root α, or all ni ≤ 0 for a negative root α. 6. All the information on the simple roots is provided by the Cartan matrix or equiva- lently by the Dynkin diagram.

9.2 Classification of finite-dimensional simple Lie algebras Dynkin defines a Π-system to be a set of simple roots which satisfy three conditions.

(A) There are n linearly independent simple root vectors αi for i = 1, ··· , n.

(B) Any two distinct simple roots αi,ααj are such that,

2(αi,ααj) = −pij pij ≥ 0, pij ∈ Z (9.6) (αi,ααi)

97 (C) The simple root system is indecomposable, i.e. it cannot be split into two sets of simple roots that are mutually orthogonal to one another. A decomposable simple root system corresponds to a Lie algebra which is the direct sum of two Lie algebras and hence is not a simple Lie algebra. We shall now enumerate all possible Π-systems. The classification theorem is simply done with the help of a few Lemmas. Lemma 2 The only Π system with 3 vectors are

α1 α2 α3 α1 α2 α3

The only angles allowed between two distinct simple roots are 90o, 120o, 135o, and 150o. The sum of the angles between three linearly independent vectors must be < 360o. This means that one and only one angle must be 90o, since otherwise the system would be o decomposable. Suppose θ13 = 90 as in the diagrams above. The remaining angles must o o satisfy θ12 + θ23 < 270 . This requires one angle to be θ12 = 120 and the remaining angle o o θ23 is allowed to be either 120 or 135 , with the two cases depicted in the above figures. Lemma 3 An indecomposable subsystem of a Π-system is a Π-system. This is obviously true. Corollary 2 The only Π-system with a triple link is given by,

This is an immediate consequence of Lemma 3. Lemma 4 If a Π-system contains two vectors connected by a single line, the diagram obtained by shrinking the line away and merging the two vectors into a single circle is another Π-system. To prove this lemma, let α and β be the two vectors joined by a single line, and let Γ be the set of all other vectors. It follows from Lemma 2 that Γ contains no vector connected to both α and β. The vector α + β has the same length as α and β. This follows from applying the magic formula, 2(αα,ββ) 2(αα,ββ) = = −1 (9.7) (αα,αα) (ββ,ββ) Now if γ ∈ Γ and connected to α, then it cannot be connected to β and hence we have (γγ,αα + β) = (γγ,αα), while if γ 0 ∈ Γ and connected to β, then it cannot be connected to α and hence we have (γγ,αα + β) = (γγ,ββ). Hence the set {α + β} ∪ Γ is a Π-system.

98 Corollary 3 No Π-system contains more than one double link. To prove this, consider a diagram with two double links and shrink all the single lines in between the two double links, to end up with a system with two consecutive double links which contradicts Lemma 2 with the help of Lemma 3.

Corollary 4 No Π-system contains a closed loop. Again, shrink all single lines until you get a closed loop diagram with three vectors that contradicts Lemma 2.

Lemma 5 If the configuration on the left is a Π-system, then the configuration on the right must also be a Π-system.

α

γ γ α + β β

To prove this, we use the fact that (αα,ββ) = 0 and the fact that γ, α, β have the same length. It then follows that also (α + ββ,αα + β) = 2(αα,αα), and that its inner product with γ is given by,

2(γγ,αα + β) 2(γγ,αα + β) = −2 = −1 (9.8) (γγ,γγ) (α + ββ,ααβ )

which indeed gives the Π-systen in the right panel of the figure.

Corollary 5 The only branches in a π-system can be tri-valent with single links emanating from the vertex.

Lemma 6 The following four diagrams do not correspond to Π-systems.

2

1 2 3 4 3 2 1

1 2 3 2 1

99 3

1 2 3 4 5 6 4 2

2 1

1 2 3 2 1

The numbers inside the root circles are the coefficients needed to construct a linear combination that vanishes. For the last diagram, for example, we have,

2 (α1 + 2α2 + 3α3 + 2α4 + 2α5 + α6 + α7) = 0 (9.9) hence the root vectors are linearly-dependent. As a result of all the lemmas we have,

1. A triple link can occur only by itself, as is reflected in the algebra G2 of Theorem 9.2. 2. A double link can occur either at the end of a chain of simple links as in the algebras

Bn and Cn of Theorem 9.2, or with just a single length on each side of the double link as in the algebra of Theorem 9.2. 3. The remaining diagrams which have only single links can have at most one trivalent vertex since otherwise the single links that connected the two vertices could be shrunk

to a quadrivalent vertex. This leaves chains without vertices as in the algebras An of Theorem 9.2, or if there is one vertex then one of the links emanating from the vertex must connected to a single root. With the diagrams that are eliminated by Lemma

9.2, this leaves the algebras Dn,,, and of Theorem 9.2.

100 Theorem 12 All finite-dimensional complex simple Lie algebras are given by

··· 2 An dimC An = n − 1

Bn ··· dimC Bn = n(2n + 1)

Cn ··· dimC Cn = n(2n + 1)

Dn ··· dimC Dn = n(2n − 1)

E8 dimC E8 = 248

E7 dimC E7 = 133

E6 dimC E6 = 78

F4 dimC F4 = 52

G2 dimC G2 = 14

The short roots have been indicated in blue. The root systems for the exceptional groups are as follows. 1 R =  ± e ± e , 1 ≤ i < j ≤ 8; (±e ± e ± e ± e ± e ± e ± e ± e ) E8 i j 2 1 2 3 4 5 6 7 8 1 R =  ± e ± e , 1 ≤ i < j ≤ 6; ±(e − e ); ± (±e ± e ± e ± e ± e ± e − e + e ) E7 i j 7 8 2 1 2 3 4 5 6 7 8 1 R =  ± e ± e , 1 ≤ i < j ≤ 5; ± (±e ± e ± e ± e ± e − e − e + e ) E6 i j 2 1 2 3 4 5 6 7 8 1 R =  ± e ± e , 1 ≤ i < j ≤ 4; ± e ; ± (±e ± e ± e ± e ) F4 i j i 2 1 2 3 4  RG2 = ± (ei − ej, 1 ≤ i < j ≤ 3; (2ei − ej − ek), {i, j, k} = {1, 2, 3} (9.10)

101 10 Weyl’s character formulas

Consider a compact matrix group G of rank n with Lie algebra G. Two of the most important problems in group theory are as follows.

1. Decomposition of the tensor product of two irreducible representations R1,R2 of G, M R1 ⊗ R2 = mR(R1,R2) R (10.1) R

where the sum is over all possible irreducible representations R of G. The positive

integer mR(R1,R2) is the multiplicity with which the representation R occurs in the tensor product R1 ⊗ R2.

2. If R is an irreducible representation of the group G, and we have a subgroup G0 of G, then we want to know how R decomposes into a direct sum of irreducible

representations r of the subgroup G0, M R = mr(R)r (10.2) r

where the sum is over all irreducible representations r of G0 and the positive integers mr(R) is the multiplicity with which the representation r occurs in the decomposition of R with respect to G0. This decomposition is sometimes referred to as the branching rules for the subgroup G0.

Both questions could already have been answered by use of the method of weights or tensors, but the most natural approach is the method of group characters. Recall the definition of

the character χR(g) of a representation R of G,

χR(g) = trR(g) χR(e) = dim R (10.3)

Recall that characters behave well under direct sum and tensor product of representations,

χR1⊕R2 (g) = χR1 (g) + χR2 (g)

χR1⊗R2 (g) = χR1 (g) × χR2 (g) (10.4) for all g ∈ G. Furthermore, characters are functions only of the conjugacy class of g,

−1 χR(h g h ) = χR(g) for all g, h ∈ G (10.5)

The method of characters provides a very convenient solution to both the decomposition and the branching rule problem.

102 10.1 Characters on the maximal torus By conjugation, we may diagonalize g to a diagonal matrix of phases,

n iH(ϕ) X g = e H(ϕ) = ϕi Hi (10.6) i=1 where as earlier, we have chosen the Cartan generators Hi to be Hermitian. Since G is compact, every finite-dimensional represnetation R of G is equivalent to a unitary repre- sentation. Thus, by the same token we may diagonalize R(g) in any representation ρ of

the Lie algebra G of G, and choose the representation matrices ρ(Hi) again Hermitian,

n iρ(H(ϕ)) X R(g) = e ρ(H(ϕ)) = ϕi ρ(Hi) (10.7) i=1 Then in general a character is of the form

X i(λλ,H(ϕ)) χR(g) = nλe (10.8) λ nλ is the multiplicity for weight λ in the weight diagram of representation ρ. Now any unitary matrix may be diagonalized by a unitary transformation. For simplic- ity, we shall assume G = U(n), but the derivation can be easily generalized to SU(n) or any of the other compact Lie groups. Therefore we may decompose g as follows,

eiϕ1 0 0 ··· 0   0 eiϕ2 0 ··· 0  +   g = U DUD =  0 0 eiϕ3 ··· 0  (10.9)   ···············  0 0 0 ··· eiϕn

The character in a representation R is then defined by,   eiϕ1 0  ..  χR(g) = χR  .  (10.10) 0 eiϕn

Since characters are class functions, they only depend on the diagonal matrix D and not on U. This is a tremendous simplification. We have at our disposal the orthogonality relations for characters of irreducible representations (primitive characters), given for G = U(n) by, Z ? dµ(g) χR1 (g)χR2 (g) = N δR1,R2 (10.11) G

103 iϕi Now we would like to express χR(g) in terms of in e . The problem is that the measure dµ(g) depends on D but also on the other variables U. There is a standard change of variables that yields us the Jacobian for the change of variables

g → (U, ϕ1, ··· , ϕn) (10.12)

To do so, regard dµ(g) as coming from the invariant metric on G = U(n),   ds2 = −tr g†dg g†dg (10.13)

To decompose the mesure dµ(g) it suffices to decompose the metric, using,

g†dg = −U †D?(dU) U †DU + U †D?(dD)U + U †(dU) (10.14)

In evaluating the metric, the cross terms of dD and dU cancel, and one is left with,   ds2 = −tr(D?dD)2 − 2 tr(U †dU)2 + 2 tr D?(dUU †)D(dUU †) (10.15)

The metric is the sum of a metric on the variables ϕi which is independent of U, and a metric on U which depends on ϕi. Therefore, the measure factorizes and is of the form,

n ? Y dµ(g) = ∆(ϕ)∆ (ϕ) dϕi dµ(U) (10.16) i=1

Each factor dU U † is accompanied by a linear function in one of the phases eiϕi and the dependence on positive and negative phases factroizes. The measure dµ(U) contains n2 −n iϕ1 iϕn 1 factors, so that ∆ must be a polynomial in e , ··· , e of degree 2 n(n − 1).

∆(ϕ) must vanish whenever ϕi = ϕj for i 6= j. This tells us what ∆ should be up to a ϕ-independent constant,

Y iϕi iϕj ∆(ϕi) = (e − e ) (10.17) i

1 which is of degree 2 n(n − 1) and is the only function with those properties.

10.2 Weyl’s first formula The character of any irreducible representation of U(n) has the following form,

X (γ ) R i iϕi χR(g) = γi = e (10.18) ∆(γi)

104

1 1 ... 1

γ γ γ 1 2 n Y γ2 γ2 . . . γ2 ∆(γi) = (γi − γj) = 1 2 n (10.19) . i

and χR(γi) is an anti- in γi,

`1 `1 `1 γ1 γ2 . . . γn . `2 `2 . γ1 γ2 . XT (γi) = . (10.20) ······ . `n `n `n γ1 γ2 γn Here

`k = mk + (n − k) k = 1, . . . , n (10.21) with k X mk = qi (10.22) i=1 and the q’s are the components as follows: X µ = qiλi (10.23) i

where λi are the weights of the fundamental representations:

2(λj,ααi) = δij (10.24) (αi,ααi)

By letting ϕi → 0, and taking the limit, one deduces a formula for dimensions. Y (`i − `j) 1≤i

105