Group Theory and the Quark Model

Transcription

1 Version 1 Group Theory and the Quark Model Milind V Purohit (U of South Carolina) Abstract

2 Contents 1 Introduction Symmetries and Conservation Laws Introduction Finite Groups 4 1 Subgroups, Cosets, Classes and Direct Products 4 Representations of Groups 5 The Symmetric Group S n 6 1 S 6 S 7 4 The regular representation 8 4 Reducible and Irreducible representations 8 41 Reduction of Representations 9 4 Reduction of S n 10 4 Introduction to Young Tableaux 1 5 Infinite groups and Lie groups Introduction 15 6 The Special Unitary Groups: SU(N) Generators and algebra of SU(N) 16 6 Roots and Rank 18 6 Clebsch-Gordan coefficients Young tableaux in the context of SU(N) Dimensionality of Representations 1 64 Reduction of direct products of representations 1 64 Basis Vectors of Representations 7 The Quark Model 8 Conclusion and References 9 References 1

3 1 Introduction Our interest in group theory stems from its applications to particle physics, which are many Fundamentally, when a group of transformation operators commutes with the Hamiltonian, the theory of the group they form can be brought to bear on the energy eigenstates Group theory can then be used to classify these states, to specify their wavefunctions, to tell us how to combine such states (generalization of the addition of angular momenta), to work out selection rules for matrix elements and so on Perhaps even more significant than this practical, calculational aspect is the central role that group theory plays in formulating the fundamental theories of electromagnetic, weak, strong and possible new interactions This is the saga of gauge theories which has truly been the big leap in physics after relativity, quantum mechanics and field theory For most of our needs the symmetry groups we shall require will be the infinite groups SU(N), which also serve as a model in other cases The groups SU(N) will be applied to angular momentum, isospin, and mainly to the quark model in this chapter However, we will spend enough time on group theory just so the reader is prepared to appreciate applications in the context of the standard model in particular, and to theory-building in general Since symmetry operations typically satisfy the requirements of a group the vast mathematical apparatus of group theory can be brought to bear on physical problems exhibiting some kind of symmetry For instance, we can use the properties of SU() to label states, to determine the multiplicity of representations (see below), to obtain wavefunctions and their symmetry properties etc Lagrangians for theories are often built with a specific group at their core describing the desired invariances of the physical theory under certain transformations We will study the bare minimum of group theory necessary for our purposes In particular, we will focus on the finite group S n and the infinite groups SU(N) Symmetries and Conservation Laws Symmetry considerations help us to think about and to classify properties of nature Often, we have to construct laws of nature and an observed or expected symmetry guides us to the correct form of the law An example is finding the force on a charged particle due to a charged sphere In solving such problems, we find that symmetries lead to conservation laws Noether s theorem (1917) shows how symmetries of nature lead to conservation laws and vice versa For instance, translational symmetry leads to conservation of linear momentum, rotational symmetry leads to conservation of angular momentum, time evolution invariance leads to conservation of energy and gauge invariance leads to conservation of various charges Classically, of course, the Euler-Lagrange equations lead directly to conservation laws: d dt ( ) L = L q q = 0 if L does not depend on q In quantum mechanics, we use the Schrödingerequation of motion i h Ψ t = HΨ

4 to show that if an operator A does not explicitly depend on time then it is a constant of motion if it commutes with the Hamiltonian operator H, ie, that d dt A = ī Ψ [H, A] Ψ h Even if we did know the detailed laws of nature, the underlying symmetries help us to think about the resulting theory We may get partial insight into problems without knowing the full dynamics Understanding atomic spectra is an example, as is the Wigner-Eckart theorem Recall that the Wigner-Eckart theorem allows us to factorize a matrix element of a tensor operator into a Clebsch-Gordan coefficient and a reduced matrix element The reduced matrix element does not depend on the magnetic quantum numbers which specify the orientation of a system in space, but only on the total angular momentum and other, non-geometric quantum numbers Classically, an example of symmetry operations is provided by rotations If a classical object is located at some position x, then a rotation leaves the object in a new position x where x = Rx and R is the rotation matrix, an orthogonal x matrix In quantum mechanics, all information about a quantum is incorporated in its state vector Ψ A rotation or other symmetry operation is then carried out by operating on Ψ with an operator, let s call it O, as shown: Ψ Ψ = O Ψ If we require that the scalar product of two state vectors remain invariant under these operations it immediately follows that the operators are unitary, ie, Ψ Φ = Ψ Φ implies that O O = I where I is the identity operator However, if we require only that the physically observable squares of matrix elements be invariant, ie, only that Ψ Φ = Ψ Φ then we find that the operator can also be antiunitary, in the sense that it is antilinear Recall that an antilinear operator T has the property that T a Ψ = a T Ψ where a is any complex number Further, Wigner showed that any such mapping Ψ Ψ which preserves probabilities is rephase-invariant, ie, that the phases of state vectors can be arbitrarily changed and retain the linear addition of states Thus we shall mainly be concerned with unitary or anti-unitary operators, and most often with elements of the Special Unitary group where the overall phase is ignored

5 Introduction Finite Groups A group is defined as a set of quantities (or operations) a, b, c, for which multiplication is defined and which have the following four properties: Closure under multiplication: the multiplication operation ab must result in another group element c, ie, c must be a member of the group This means that the multiplication table consists entirely of group elements Associativity under group multiplication: As in the usual sense, ie, the product a(bc) must equal the product (ab)c Identity: A unique element I exists with the property for any element a in the group Ia = ai = a Inverse: Every element a has an inverse a 1 which is also an element of the group such that aa 1 = a 1 a = I For instance, the natural numbers 1,,, do not form a group under addition because there is no identity element (zero is not a natural number) and there is no inverse However, when we consider the set of integers (which include zero and negative numbers) they do form a group under addition Another really simple example of a group is provided by the set of two operations Identity (or no-op) and Mirror Reflection If the group has a finite number of elements it is called a finite group, otherwise an infinite group The number of elements of a finite group is called the order of the group The laws of commutativity may or may not hold, ie, it may or may not be true that group multiplication is commutative If the commutation relation holds, the group is called Abelian For instance, the group R() which describes rotations in two-dimensional space is Abelian (convince yourself of this) On the other hand, R(), the group describing rotations in three-dimensional space, is non-abelian Sometimes a group is extended to form an algebra, which means that the elements form a linear vector space Thus, we can also add elements to each other, not just multiply them The addition operation leads to a zero for the algebra, but the zero of the algebra does not have an inverse 1 Subgroups, Cosets, Classes and Direct Products It may happen that a subset of the elements of a group G themselves form a group In that case the subset is called a subgroup If g is an element of a group and S is a subset of the group of order h S, then the set obtained by multiplying each element of S by g on the left is called the left coset gs of the group Similarly, we can define the right coset Sg Clearly, the cosets have the same number of elements h S as the subgroup itself Note that 4

6 (i) Two cosets of a subgroup are either identical or have no element in common We can see this as follows Let the subgroup S have elements s i If two elements of left cosets are equal eg, g 1 s 1 = g s Then we see that g = g 1 s 1 s 1 = g 1 s where s is an element of the subgroup S Thus, the left coset g S = g 1 s S = g 1 S because of the closure property of the subgroup S (ii) Each element g of the group G appears in at least one coset This is because eg, the left coset gs 1 i contains g and because gs 1 i is a group element (iii) Therefore, the whole group splits up into cosets of S, each of order h S Therefore, h S is a factor of h Two elements g 1 and g of G are said to be equivalent if there exists some group element g such that g 1 g 1 g = g A subset of elements in which each element is equivalent to another is called a class Classes typically contain elements which are similar to one another in some physical way: thus, an element in a class can be obtained from another by first executing a third group element, followed by the other element in the class, followed by the inverse of the first operation An example is given in the discussion of the group S described below, for instance The identity element is in a class by itself If a subgroup S has the property g 1 Sg = S for every element g in G, then S is called an invariant subgroup Invariant subgroups are thus made up of a whole class or whole classes If there is a mapping from a group G 1 onto a group G which preserves group multiplication, then the two groups are called homomorphic If the correspondence between the group elements is one-to-one then the groups are called isomorphic The direct product of two groups G 1 and G exists if each element of G 1 commutes with every element of G These elements act on different spaces Thus, the elements of the direct product group G can be written as g 1 g or as an ordered pair (g 1, g ) Group multiplication is then defined by g = gg = (g 1, g )(g 1, g ) = (g 1g 1, g g ) Representations of Groups In general, elements of a group may be represented by linear operators acting on a linear vector space (a homomorphism), but almost always we will be interested only in matrix representations The elements of a group can be represented by square matrices, which act on the vectors in the vector space, which in turn are linear combinations of the basis vectors A representaion D of a group G, is such that there is a matrix D(g) which represents each element g and group multiplication of elements is preserved by multiplication of the corresponding matrices In other words, for any two elements a and b of the group if c is the element which satisfies the condition ab = c 5

7 the corresponding matrices M(a), M(b) and M(c) of the representation must satisfy the same condition in matrix form: M(a)M(b) = M(c) If the matrices are n n in size then we call it an n-dimensional representation Of course, we can always represent all the elements of a group by a unit matrix in one or more dimensions; such a representation would be called a trivial representation Mostly, we will be interested in a faithful representation in which the matrices corresponding to different group elements are themselves distinct Two representations are equivalent if their matrices are related by an equivalence transformation: M (a) = U 1 M(a)U That is, a single unitary matrix U must simultaneously transform all matrices in one representation into the corresponding matrices in the other We will be interested in unitary matrices U which transform the basis states upon which M(a) perform symmetry operations into a new set The equivalence transformation is analogous to the transformation of operators and one can easily verify that the multiplication table for an equivalent representation is the same as the original All the matrices representing elements belonging to a single class have the same character (group-theory speak for trace) in any given representation In fact, we often think of elements belonging to a class as being of a given type, for instance elements could form a class of rotations, or a class of reflections The fact that all elements in a class have the same character makes it easy to create character tables and to reduce a representation into a sum of irreducible representations (the Clebsch-Gordan decomposition) The Symmetric Group S n This is the group of permutations of n distinct objects Clearly, S n has n! elements You may want to prove that all elements of S n which are related to each other by a cyclic permutation form a class The group S n is centrally important in quantum physics Symmetry under permutations is, of course, crucial in systems with identical particles The symmetric groups are also important because they are connected with the groups U(N) and SU(N) and also because all finite groups are isomorphic either to S n or to one of its subgroups (see also the discussion of regular representations later) 1 S For example, the objects 1 and can be arranged as 1 or 1 We say that the symmetric group S has the elements e (identity element) and P 1 where P 1 exchanges the positions of 1 and Thus, a wavefunction Ψ(1, ) (which may or may not be expressed as a product of two one-particle functions Ψ(1, ) = Ψ a (1)Ψ b ()) is transformed by P 1 as follows: P 1 Ψ(1, ) = Ψ(, 1) Using P 1 we can create the symmetrizer S 1 and anti-symmetrizer A 1 as follows: S 1 = (e + P 1 ) and A 1 = (e P 1 ) 6

8 [Note: Not all functions can be anti-symmetrized For instance, A 1 Ψ a (1)Ψ a () = 0 Also, note that after symmetrizing or antisymmetrizing you may need to normalize the wavefunctions] S If we have three objects, for instance 1, and, then the element [1 ] means leave the objects in their place, but [ 1 ] stands for interchanging 1 and while leaving in place If we have objects, say u, d and s, then But [1]uds = uds [1]uds = dus [1][1]uds = dsu [1]uds = dsu therefore, [ 1] = [1 ] [ 1 ] There are a total of 6 elements of the group S : e = [1], σ 1 = [1], σ = [1], C + = [1], C = [1], σ = [1] Note that we may use a different notation In our notation [ 1] means that element is put into the first position etc One can imagine writing the reverse For instance, we may choose to write this as ( 1 ) meaning that element 1 is put into location, goes to location 1 etc in which case again uds dsu Yet another notation, perhaps even more explicit, for [ 1] or ( 1 ) is ( 1 ) 1 In this notation, which is very explicit, the order of the columns does not matter Regardless of notation, it should be clear that S has order 6 and the identity element is e A pictorial way to view S is shown in Fig 1 Here we can view S as the group of symmetry operations on an equilateral triangle: we label the vertices 1, and and consider the elements to be no change (identity), reflections σ around the medians ( operations) and rotations C by 10 and 40 These operations all leave the triangle invariant Note that here there are classes: the identity element, the three reflections, and the two rotations The reflections, for example, are equivalent to each other and can be obtained from each other by preceding the operation with a rotation and following the reflection with the inverse rotation The elements that result from group multiplcation of two group elements are shown in the multiplication table below Since we discussed symmetry properties in the context of S, it worth noting that the symmetrizer and anti-symmetrizer for S can be written as S 1 = 1 + P 1 + P + P 1 + P 1 P 1 + P 1 P 1 A 1 = 1 P 1 P P 1 + P 1 P 1 + P 1 P 1 7

9 1 1 e C + C σ 1 σ 1 σ Figure 1: Elements of S as operations on an equilateral triangle The identity is the element called e and shows the triangle undisturbed The C ± operations represent rotations of the triangle, while the σ operations involve reflections around the three medians of the triangle 4 The regular representation To motivate the connection between finite groups and the symmetric group in particular, we should recognize a special representation called the regular representation, where we use the group elements themselves as the basis That is, the basis vectors are n-dimensional where n is the order of the group, and the basis vector representing the i th element is zero except for the i th component which is equal to 1 The matrices representing each element are simply permutation matrices, which reproduce the group multiplication table We see here a connection between an arbitrary group and the symmetric group 4 Reducible and Irreducible representations If there is an equivalence transformation which simultaneously turns all the matrices in a given representation into submatrices along the diagonal (block diagonal form) it is called a reducible representation If no such transformation can be found we have an irreducible representation (The block diagonal form is one like the matrix below in which only the submatrices along the diagonal contain non-zero elements) 8

10 Table 1: Group Multiplication table for S e σ 1 σ σ C + C e e σ 1 σ σ C + C σ 1 σ 1 e C + C σ σ σ σ C e C + σ σ 1 σ σ C + C e σ 1 σ C + C + σ σ 1 σ C e C C σ σ σ 1 e C + We will mostly be interested in unitary representations, ie, representations by unitary matrices Since all irreducible representations can be shown to be equivalent to unitary irreducible representations, in all that follows we will implictly assume that we are dealing with unitary irreducible representations The number of distinct (not equivalent) irreducible representations of a group are limited by the following relation between their dimensionalities d i and the order n of the group: d i = n i How do we know whether a given representation is reducible or not? It turns out to be sufficient to find the sum of squares of the characters Denoting the character of element g k in the i th representation as χ (i) (g k ), the sum over character squares determines the reducibility of a representation as follows: χ (i) (g k ) = n (Irreducible Representation) k χ (i) (g k ) > n (Reducible Representation) k 41 Reduction of Representations A powerful theorem called the orthogonality theorem helps in the process of reducing or decomposing a reducible matrix representation into irredicuble ones along the diagonal The 9

11 theorem states that given two representations Γ (i) (g k ) and Γ (j) (g k ) of group elements g k, the following relation holds: k Γ (i) αβ (g k)γ (j) γδ (g k) = n d i δ ij δ αβ δ γδ where α, β, γ, and δ are matrix indices, n is the order of the group, and d i is the dimensionality of the i th representation This orthogonality theorem implies that the n-component vector of matrix elements in a specified row and column of a given representation is orthogonal to all other such vectors! An interesting corollary follows Since every group has a trivial one-dimensional irreducible representation where every element is represented by unity, we see that in all other representations all the matrices must add up to zero There are many other important relations that follow from this theorem, but here we will mention only one: the character orthogonality theorem, which follows from the group orthogonality theorem and which states that χ (i) (g k )χ (j) (g k ) = nδ ij k This theorem can be used to reduce a representation into irreducible representations, since the character of a group element in any representation is simply the sum of the characters of all contained irreducible representations and characters do not change under similarity transformations If the i th irreducible representation is contained a i times, ie, if Γ = a 1 Γ (1) a Γ () a N Γ (N) then the characters of elements g k in representation Γ will be given by χ(g k ) = i a i χ (i) (g k ) Together with the character orthogonality theorem this relation immediately tells us that a i = χ (i) (g k )χ(g k ) where denotes an average over all group elements, and we have decomposed the representation Γ into irreducible representations 4 Reduction of S n We can now try to construct matrix representations of S n Let us start with S The simplest basis states we can imagine are ( ) ( ) 1 0, 0 1 The group elements are the identity e and the permutation P 1 which transforms each state into the other: ( ) ( ) e =, P =

12 Now we try to reduce these two matrices directly, ie, by finding a suitable basis We see that the similarity transformation using the matrix S = 1 ( ) will transform the basis vectors to s = 1 ( 1 1 ) and a = 1 ( ) 1 1 These are eigenvectors of P 1 with eigenvalues ±1 Under the permutation P 1 the new basis vectors thus transform into themselves (up to a phase factor) and we have reduced the representation Indeed, in the new representation the elements e and P 1 are transformed into block diagonal form: g SgS T transforms e and P 1 into e = ( ) ( ) , P 1 = We see that the new matrices are reducible into two 1x1 matrices Indeed, if we have any arbitrary two-particle state ψ(1) then we can define two basis functions as either ψ(1) and ψ(1) or as a symmetric and antisymmetric combination S 1 = 1 + P 1 and A 1 = 1 P 1 The latter pair form the basis states of the symmetric and antisymmetric representations of the S group What if we tried this for S? Now, of course, there are! = 6 basis states: ψ(1), ψ(1), ψ(1), ψ(1), ψ(1), ψ(1) The totally symmetric and antisymmetric combinations and ψ S = ψ(1) + ψ(1) + ψ(1) + ψ(1) + ψ(1) + ψ(1) ψ A = ψ(1) + ψ(1) + ψ(1) ψ(1) ψ(1) ψ(1) form the basis states of one-dimensional representations of S Two other representations of S exist, both of which are two-dimensional We could form six column vectors, each with 6 rows, with a single 1 and five 0 s to represent the six different arrangements of three objects Then we would have to create matrix representations of the 6 group operations and reduce the space to subspaces where the group operations do not mix the subspaces This work has been done by mathematicians and here we will quote the results It turns out that the dimensionality, number and basis states of such reduced subspaces can be neatly summarized by the use of Young tableaux 11

13 4 Introduction to Young Tableaux In order to enumerate irreducible representations and their dimensionalities in general it is better to turn our attention to the use of Young tableaux, a powerful tool for the study of groups Young tableaux consist of rows of boxes stacked on top of each other To be precise, the stacks of empty boxes are sometimes called Young frames to distinguish them from filled boxes which are called tableaux, but we will just use the term tableaux for both The properties of groups are translated into rules for constructing the tableaux For instance, one rule states that all rows must be left-justified, ie, all rows are aligned at their left edge An example of an allowed tableau is We may place numbers into the boxes, for instance to denote the number of the object (or particle) in each state (box) Thus, we can write 1 In the case of the symmetric group, we must enforce three rules for Young tableaux: The number of boxes in any row must not exceed the number in the row above The numbers in the boxes may not decrease as we go across a row The numbers in the boxes must increase as we go down a column Examples of legitimate tableaux include: 1 1 However, the following are NOT allowed: 1 and and We are now ready to see how to get the dimensionalities of irreducible representations For S n, create all possible tableaux that satisfy the rules given above with the numbers 1 through n entered into the tableau Each configuration represents one or more irreducible representations The dimensionality of each irreducible representation is given by the number of ways in which the numbers can legitimately be entered in the boxes for the corresponding configuration Thus, in the case of S we have two possible tableaux: and 1

14 For each of these, there is only one possible way in which we may enter the numbers 1 and : 1 and 1 Thus, S has only two irreducible representations each of dimensionality 1, as expected from our discussion of S above 1

15 A more complicated example is provided by enumerating the irreducible representations of S 4 : It turns out that in S n there are exactly as many (inequivalent) irreducible representations as the dimensionality of a tableau Thus, there are exactly irreducible representations in S 4 of the kind 1 4 Thus we reduce the 4 element group S 4 as follows: 4! = 1 1 = 4 The representations with the number of rows and columns interchanged are called conjugate representations: and are conjugate and are conjugate is self-conjugate In general, the rows represent symmetry and columns represent antisymmetry For instance, the tableau 1 represents a symmetrized version of a two-particle state while 1 14

16 represents the anti-symmetrized version Thus, one sees that the rule for not repeating a number as one goes down a column makes sense: antisymmetrizing a state which is already symmetric will give zero Of course, tableaux such as are of mixed symmetry: symmetric across the top row, but antisymmetric with respect to exchange of any two particles in a column 5 Infinite groups and Lie groups 51 Introduction As mentioned earlier, in quantum mechanics almost all transformation operators are unitary Of course, these operators operate on states defined in some basis, and two operations (such as two rotations) are equivalent to a single, combined operation Although infinite in number, the unitary operations typically are quantified using a finite number of parameters such as angles, and together they form a group (satisfy closure and the other group properties) Therefore, we can benefit from the theory of infinite groups, and specifically the theory of unitary groups A Lie group is an infinite group of elements which depend in a continuous, differentiable manner on some parameters The groups of rotations in two and three dimensions, R() and R(), are examples of Lie groups: their elements are continuous, differentiable matrix functions of the angles of rotation A continuous group that has a finite number, n, of parameters is called a n-parameter finite continuous group Thus, we may write the element of a Lie group with parameters θ 1, θ θ n as A(θ 1, θ θ n ) or A( θ) For instance, the group of all n n unitary matrices known as U(n) can be generated from parameters θ and generators λ (which are Hermitian matrices, constant for each representation): U( θ) exp( i θ λ) If, in addition, we do not care about an overall phase and are interested in only those unitary matrices which have determinant +1, then U(n) becomes the special unitary group SU(n) and the generators must be Hermitian as well as traceless In the late 19 th century Sophus Lie showed that essentially we can understand an entire continuous group just by studying the properties of elements lying close to the identity ie, those generated by a infinitesimal changes to the identity element using the generators An example of such a continuous group is SO(n), the group of special orthogonal matrices which describes rotations in n-dimensions In an odd number of dimensions this excludes the parity transformation x i x i A continuous group is called compact if the parameters are all bounded and if their domain of variation is closed, ie, convergent sequences converge to a point within the domain We will not concern ourselves with further qualifications, such as simple Lie groups and semisimple Lie groups; suffice it to say that U(1) is not simple while the SU(n) are simple Lie groups 15

17 6 The Special Unitary Groups: SU(N) In quantum mechanics, all transformations that we study are unitary, because they preserve the norms of state vectors: then to preserve the norm we require that If α α = U α α α = α U U α = α α = 1 This implies that U should be a unitary matrix Now, all unitary matrices U can be written as exp(ih) where H is a Hermitian matrix [This is quite easy to prove - try it as an exercise] Let us consider the case of unitary matrices Each matrix has four complex elements, but the condition of unitarity reduces the number of real parameters back to four Similarly, the Hermitian matrices in the exponential have four real parameters A general Hermitian matrix can be written as H = θ 0 I + θ σ where I is the identity matrix, σ is the triplet of Pauli spin matrices, and θ is a vector of parameters Since I commutes with all matrices, we can write U = exp(θ 0 I) exp( θ σ) The first term simply gives an overall phase to the states, which is typically not interesting Hence, we focus on the remaining unitary matrix U = exp( θ σ) Consider next the group of all unitary matrices which have determinant +1 Recall that unitary matrices interest us because they leave scalar products invariant All unitary matrices of a given order which have determinant +1 form a Lie group called SU(N) These can be obtained continuously from the identity matrix, while matrices with determinant -1 cannot Thus SU(N) will turn out to be very useful in describing transformations (symmetry operations) in particle physics Interest in SU(N) stems from the fact that we remove an overall phase factor from unitary matrices to form SU(N) matrices (in quantum mechanics the overall phase is often irrelevant) (The eigenvalues of a unitary matrix must all have modulus unity and thus their product, which is the matrix determinant, must also have modulus unity) 61 Generators and algebra of SU(N) We define the generators of SU(N) by ( ) U(0,, dθi,, 0) I G i i lim i U dθ i 0 dθ i θ i θ=0 16

18 where the identity element I is the element for which θ = 0 Thus, each element U( θ) = exp( i θ G) Clearly, SU(N) has N 1 parameters because the matrices must be unitary (N conditions) and have determinant +1 (one condition) Writing the elements of matrices infinitesimally different from the identity matrix as U ij = δ ij + du ij we see, using the properties of unitarity and +1 determinant, that and du ik = du ki du ii = 0 i Thus we see that the generators of SU(N) are Hermitian and traceless Clearly, again there are N 1 independent Hermitian and traceless generators How many of these commute? If we were to simultaneously diagonalize the maximum number of them, we would have at most N 1 such generators because of the tracelessness condition Using the fact that the G i are Hermitian, we see that their commutators must be anti- Hermitian Since the G i span the space of traceless Hermitian matrices, we must be able to write down each commutator as the following linear combination: [G i, G j ] = ig ijk G k where the g ijk are real coefficients These coefficients are called the structure constants and serve to define the Lie group in a representation of any dimensionality The above relation is called the Lie algebra of the generators - the generators form a Lie algebra while the original group matrices form the Lie group Taking the Hermitian adjoint of this equation immediately leads to the fact that the g ijk are antisymmetric with respect to interchange of the first two indices In fact, it can be shown that they are antisymmetric with respect to interchange of any two indices An important point to keep in mind about Lie Algebras is this: the Lie algebra of a group is very important from a physics point of view, since the commutator encapsulates the physics being described by the group theory An example of a Lie algebra is provided by the generators of the group SU() - the components of angular momentum These are known to satisfy the relation [J i, J j ] = i hɛ ijk J k and therefore the ɛ ijk are the structure constants of SU() (in representations of all dimensionalities) Using the Jacobi identity for matrices A, B and C [[A, B], C] + [[B, C], A] + [[C, A], B] = 0 it is possible to show that the structure constants themselves satisfy the Lie algebra In other words a representation of the group (called the adjoint representation) can be created 17

19 by taking the generators to be the structure constants g ijk written as the matrices (g k ) ij Clearly, these generators are n n matrices where n = N 1 Finally, one can show that if G i satisfy the commutator algebra, so do the G i This new representation may or may not be equivalent to the original Conventionally, the following prescription for writing down the generators of SU(N) (up to a constant) is often used The commuting, diagonal, generators are written as , , The matrices with offdiagonal terms can be written as the combinations (T ji + T ij ) and i(t ji T ij ) where T ij has a 1 in the i, j location (i j) and zeros elsewhere Convince yourself that you can generate the Pauli spin matrices in this fashion The result is that we can write the generators of the fundamental representation of SU() as G 1 = 1 ( ) 0 1, G 1 0 = 1 ( ) 0 i, G i 0 = 1 ( ) Similarly, the generators of the fundamental representation of SU() are eight in number and are given by the Gell-Mann matrices λ 1 = , λ = 1 0 i 0 i 0 0, λ = , λ 4 = , λ 5 = i 0 0 0, λ 6 = , i λ 7 = i, λ 8 = i We see from the form of the SU() generators that the first seven include the Pauli spin matrices as components, and thus include SU() transformations SU() for flavor contains the three SU() transformations u d, d s, and s u, where the necessary diagonal SU() generator is a linear combination of the SU() diagonal generators λ and λ 8 These two diagonal generators are used to label states within a multiplet and can be identified, for flavor SU(), with I and 1 + S 6 Roots and Rank For any Lie algebra we can attempt to find the maximum number of commuting generators G D These are generators which either commute with all the other generators or can be etc 18

20 made to do so by taking appropriate linear combinations of the other generators Thus, they solve the eigenvalue equation [G D, G X ] = ρg X where G X is a generator or a linear combination of generators The eigenvalues ρ for a given diagonal generator are called roots The maximum number of diagonal generators for a group is called the rank (r) of the group For instance, in SU() there is only one diagonal generator: J z The roots are 0, ± h: [J z, J z ] = 0, [J z, J ± ] = ± hj ± A root diagram in SU() would simply be a straight line with the roots indicated by dots There are some important properties of the rank of a group worth mentioning We begin by defining a Casimir operator A Casimir operator is a non-linear combination of generators which commutes with all the generators of a group Thus, for instance, J is the Casimir operator for the SU() of angular momentum: it commutes with each component of J It turns out that the number of Casimir operators is equal to the rank of a group The eigenvalues of a Casimir operator are the same for all members of a representation or multiplet Thus, both the spin up and spin down state of an electron are characterized by s = 1/ while each state is characterized by a different value of m In general, we can label multiplets by the eigenvalue of the Casimir operators and their members by the eigenvalues of the diagonal generators, called the weights (or weight vectors, in general) SU() thus has two Casimir operators: one which is the sum of the squares of the eight generators, and another cubic in the generators There are other interesting properties of the rank r of a group For instance, in the case of SU(N) where r = N 1, there are r fundamental representations of the group A fundamental representation of SU(N) is a representation with dimensionality N Thus, in SU() of color or flavor, the quarks and anti-quarks both have their own, distinct, fundamental representations A more important property of the rank r lies in the connection to physics There are only a finite number of Lie algebras with a given rank - and these were all classified by Cartan Thus, if a physical problem appears to involve multiplets characterized by a certain number r of observables, we know immediately to look at only those Lie algebras of rank r already classified by Cartan Further, if we are talking about quantum-mechanical unitary transformations, we almost certainly want to investigate SU(r+1) 6 Clebsch-Gordan coefficients Clebsch-Gordan coefficients are used in the addition of angular momenta and, in general, in the decomposition of the direct product of two representations of a group into irreducible representations To see what this means, we first return to the familiar problem of addition of angular momenta Given two systems of angular momenta j 1 and j and quantum numbers m 1 and m, the complete set of states which describe the combined object are most easily written as j 1 m 1 j m These states span a complete set in the so-called direct product representation We simply took the direct product of single-particle states; in other words if we were talking 19

21 of wavefunctions we simply multiply the (j 1 + 1) eigenfunctions of one particle with the (j + 1) eigenfunctions of the other to get a total of (j 1 + 1)(j + 1) direct product wavefunctions The combined system can have angular momenta j which vary from j 1 j to j i + j and for each of which m varies, as usual, from j to +j It is easy to show that there are still (j 1 + 1)(j + 1) states labeled with the quantum numbers j and m (Try this as an exercise!) How do we transform between these two bases? Inserting the completeness relation I = m 1 m m 1 m (1) m 1,m we can write the expansion jm = m 1,m m 1 m m 1 m jm () where the scalar products m 1 m jm describe the expansion of jm in terms of the direct product states m 1 m and are called the Clebsch-Gordan coefficients Similarly, one can define Clebsch-Gordan coefficients for the decomposition of the direct product of two SU() representations into irreducible representations These and other Clebsch-Gordan coefficients are tabulated extensively in the literature 64 Young tableaux in the context of SU(N) Young tableaux for SU(N) are similar to those for S n with only a few important differences First, we now use the boxes to denote the particles (the first box represents the first particle, the second box the second particle and so on) and the states of particle are written into the boxes Thus, the triplet and singlet configurations of a spin-1/ particle may be written as Triplet: and Singlet: + Note that + denotes the symmetric state ( )/ On the other hand + denotes the antisymmetric state ( + + )/ There are two new rules that are used in constructing Young tableaux for SU(N), the first of which is: SU(N) tableaux can have a maximum of N columns The reason for this is that we put the states of the fundamental representation into each box and any column with more than N boxes will necessarily repeat a state It will then be impossible to antisymmetrize with respect to interchange of those two boxes, as required by the antisymmetry of columns The rules for Young tableaux stated earlier in section 4 continue to apply 0

22 641 Dimensionality of Representations What is the dimensionality of the representation pictured by a Young tableau? If we start with the top row and write the number of boxes in each row as λ i, we can denote a tableau by the set of N numbers (λ 1, λ, λ N ) Thus the tableau is represented by λ = (4,,, 1) Alternatively we can denote the tableau by the set of N 1 numbers (p 1, p, p N 1 ) which are the differences between the λ i : p 1 = λ 1 λ, p i = λ i λ i+1 etc In our case here p = (1, 0, ) It can be shown that the dimensionality d of SU() tableaux is given by d = (p 1 + 1) Similarly, the dimensionalities of representations of SU(), SU(4) etc are given by d = 1! (p 1 + 1)(p + 1)(p 1 + p + ) d 4 = 1!! (p 1 + 1)(p 1 + p + )(p 1 + p + p + )(p + 1)(p + p + )(p + 1) etc These formulas give us the dimensionality of irreducible representations; we will need them in what follows 64 Reduction of direct products of representations We will be interested in the direct product of representations and the dimensionality of irreducible representations formed after the direct product is reduced For instance, when we combine the fundamental representations of quarks and anti-quarks to form mesons, what kinds of multiplets (irreducible representations) should we expect the mesons to fall into? For this task, we will need a final rule, the second rule specific to SU(N) Young tableaux This rule applies to the reduction of the direct product of two irreducible representations When combining two irreducible representations, put a row number in each row for the second representation For the resulting representations, put the boxes of the second representation to the right of or below the boxes of the first representation Draw a path starting from the rightmost box of the top row, going to the leftmost box and continuing in a similar fashion with the next row At every point in this path the number of boxes encountered with the number m must be greater than or equal to the number of boxes encountered with the number m + 1 Also, continue to apply the earlier rules regarding numbers in tableaux boxes to these numbers, ie, they must increase going down in columns and they may not decrease from left to right in a row The distinct tableaux thus formed are the irreducible representations in the decomposition 1

23 Thus, we can reduce the direct product of the two fundamental representations of SU() as follows: 1 = Basis Vectors of Representations We expect states to give us 6 symmetric and antisymmetric states Think of two quarks with flavors u, d and s combining to give symmetric versions of uu, dd, ss, ud, ds and su, while yielding antisymmetric versions of ud, ds and su 7 The Quark Model 8 Conclusion and References In conclusion, group theory is a vast subject and we have examined only a very few features of greatest relevance to us in our study of quantum mechanics and particle physics Group theory can be extensively employed in understanding various aspects of physics The mathematics itself is quite fascinating and books on group theory tend to be rather enjoyable There is an element of magic to it: we gain understanding in leaps by discovering hidden connections and find regularities we didn t know even existed! For a physicist the trick is to constantly ask oneself how the mathematics describes the physics Students of particle physics will find the books by Jones, Lichtenberg and Georgi to be particularly good references

24 9 References References [1] Group Theory by M Hammermesh, Addison-Wesley, (196) [] Lie Algebras in Particle Physics (Frontiers in Physics) by H Georgi, Perseus Press (1999) [] Unitary Symmetry and Elementary Particles by D B Lichtenberg, Academic Press (1970) [4] Groups, representations, and physics by H F Jones, A Hilger, (1990)