BASIC CONCEPTS

3.1.1 Groups and Representations

AgroupGis a set of elementsg1, g2· · · that has

An associative multiplication law, under which g1g2 = g3 for each g1,2 ∈ G, with g3∈G(closure) and (g1g2)g3=g1(g2g3) (associative).

An identity elementI∈GwithIg=gI=g for allg∈G.

A unique inverse elementg⁻¹ for eachg∈G, such thatgg⁻¹=g⁻¹g=I.

The elements may be discrete (with either a finite or countably infinite number) or may depend on a continuous parameter. Anabelian(commutative) group is a special case, defined byg1g2=g2g1for allg1,2∈G. Otherwise,Gis non-abelian. A subset of the elements that itself forms a group under the same multiplication law is asubgroupofG.

(a) The set of integers n with the operation of ordinary addition is an example of an abelian group with a countable number of elements. The identity element is 0 (i.e., 0 +n=n+ 0 =n) and the inverse is−n.

(b) The set of rational numbersrother than 0 under ordinary multiplication forms another countable abelian group. The identity element is 1 and the inverse is 1/r.

89

(c) The cyclic group Z_n consists of the n^th roots of unity, i.e.,G ={1, ωn, ω²_n· · ·ωⁿ⁻¹_n } whereω_n=e^2πi/n.Z_n is abelian and finite.

(d) The quaternion group is finite and non-abelian. It consists of the eight 2×2 matrices {±I,±iσⁱ}, where I is the 2×2 identity andσⁱ, i= 1,2,3, are the Pauli matrices.

Multiplication is defined by the ordinary matrix product in (1.18) on page 4, so, e.g., (iσⁱ)⁻¹=−iσⁱ. The subset{±I,±iσ³}forms an abelian subgroup.

(e) The symmetric groupS_nis the group of permutations ofnobjects. It hasn! elements and is non-abelian for n > 2. The alternating group A_n is the subgroup of even permutations. It is non-abelian forn >3 and hasn!/2 elements.

(f ) The set of non-singularm×mmatricesAforms a non-abelian continuous group under ordinary matrix multiplication. The identity is the m×m identity matrix and the inverse is the matrix inverseA⁻¹.

ALie groupGis a continuous group for which the multiplication law involves differen-tiable functions of the parameters that label the group elements. Most of the Lie groups of interest in particle physics are compact, which means that the parameters form a compact manifold (the Lorentz group, described in Section 10.2.2, is a notable exception). A Lie group and its multiplication law can be defined, at least for elements close to the identity (infinitesimal transformations), in terms of its associated Lie algebra, which consists ofN generators(operators)Tⁱ, i= 1,2· · ·N, and their commutation rules

[Tⁱ, T^j] =ic_ijkT^k, (3.1)

where a summation on k is implied and the c_ijk = −c_jik are the structure constants of G. Without loss of generality, one can choose the Tⁱ to be Hermitian, in which case the structure constants are real. If all of the c_ijk = 0, then G is abelian; otherwise, it is non-abelian. An element of a compactGcan be represented as a formal power series involving the generators, by the unitary operators

U_G(β~) = exp[−i

N

X

i=1

βⁱTⁱ]≡e^−i~β·T~ ≡

∞

X

k=0

(−i~β·T~)^k

k! , (3.2)

whereβ¹· · ·β^N areN continuous real parameters and theTⁱ are Hermitian. In particular, the identity element isU_G(0) =I, and the inverse of U_G(β~) is

U_G(β~)⁻¹=U_G(−β~) =e^i~β·T~ =U_G(β~)^†. (3.3) For small|β~|it is sufficient to keep just the linear term in (3.2),

U_G(β~)'I−i~β·T~+O(βiβ_j), (3.4) i.e., the generators of the Lie algebra describe the group elements close to the identity. The Lie algebra also defines the group multiplication law for arbitraryβ. That is,~

U_G(α~)U_G(β~) =e^−i~α·T~e^−i~β·T~ ≡U_G(~γ). (3.5)

~γ(~α, ~β) can in principle be expressed in terms of the Lie algebra (the Baker-Campbell-Hausdorffconstruction), although there is no closed form expression in general. However, for small|α~|and|β~|

~γ(~α, ~β)·T~ = (~α+β~)·T~− i

2[~α·T , ~~ β·T~] + h.o.t. (3.6)

Now consider a set ofn×ndimensional matricesLⁱ, i= 1,2· · ·N. If theLⁱ satisfy the same algebra as the generators of a Lie algebra,

[Lⁱ, L^j] =ic_ijkL^k, (3.7)

then theLⁱ(sometimes writtenLⁱ_n) are said to form arepresentationof the algebra, and are Hermitian for the choice of HermitianTⁱ. That is, we are considering theTⁱ to be abstract operators, while theLⁱ are a specific matrix realization. Similarly, then×nmatricese^−i~β·L~ form a representation of the group elementsUG(β~) and have the same multiplication law.

For the compact groups one can choose Hermitian generators normalized such that

Tr (LⁱL^j) =T(L)δij. (3.8)

The Dynkin index T(L)>0 depends on which representation is being considered and on an overall normalization convention, but is independent ofiandj. With these conventions, one can show thatc_ijk is totally antisymmetric in all three indices (Problem 3.2).

3.1.2 Examples of Lie Groups

The simplest example of a Lie group is the abelianG=U(1), with a single generatorT, so that

U_G(β) =e^−iβT, U_G(α)UG(β) =e^−i(α+β)T. (3.9) There is an n = 1 dimensional representation, withL = 1 and group elements U_G(β)→ e^−iβ. U(1) is named for this representation, i.e., the 1×1 dimensional unitary matrices (phase factors).

G = SU(2) is a non-abelian group withN = 3 generators and c_ijk = _ijk. SU(2) is named for its defining representation, the 2×2 unitary matrices (U(2)) with the extra constraint that they arespecial, i.e., their determinant is unity (SU(2)). The generators of the defining representation are Lⁱ = ^τ₂ⁱ, where τⁱ ≡ σⁱ are the Pauli matrices in (1.17), so their Dynkin index is ¹₂ by (1.18). The Lⁱ are Hermitian, so the group representation elements

U(β~)≡e^{−i~β·~τ2} = cosβ

2I−isinβ

2βˆ·~τ (3.10)

are unitary 2×2 matrices. Furthermore, Trτⁱ= 0, so they are special, deth

e^{−i~β·~τ2}i

=e^{−iTr(~β·~τ2)}=eⁱ⁰= 1. (3.11) Theadjointrepresentation ofSU(2) is the 3×3 representation constructed from the struc-ture constants, (Lⁱ_adj)jk = −i_ijk. There are additional representations for n = 4,5· · · ∞. SU(2) is useful in nature for describing rotational invariance, the approximate isospin in-variance of the strong interactions, and the weak isospin gauge symmetry of the electroweak interactions.

The group SU(3) plays two major roles in the standard model: as a gauge symmetry associated with color for the strong interactions (QCD), and as an approximate global flavor symmetry of the strong interactions (theeightfold way). SU(3) can be defined in terms of its defining representation, the 3×3 unitary matrices with determinant one. There are N = 8 generators, with (Hermitian) matrices in the defining relation given by Lⁱ₃ =λⁱ/2, where the Gell-Mann matricesλⁱ, i= 1·· ·8, are given inTable 3.1. The structure constants c_ijk=f_ijk, i, j, k= 1· · ·8,are listed inTable 3.2. The Gell-Mann matrices satisfy

Tr λⁱ

2 λ^j

2

=1

2 δ^ij, Trλⁱ= 0 → det e⁻ⁱ

β·~~λ 2

= 1. (3.12)

There are two diagonal matrices, λ³ and λ⁸, i.e., SU(3) has rank two. The λⁱ satisfy the commutation (Lie algebra) and anticommutation rules.¹

λⁱ, λ^j

= 2ifijkλ^k,

λⁱ, λ^j =4

3δ^ijI+ 2dijkλ^k. (3.13) Thedijkare symmetric in all 3 indices, with the nonzero ones listed inTable 3.2.SU(3) has many other representations, such as a second inequivalent 3-dimensional conjugate repre-sentationLⁱ₃∗ =−λ^i∗/2 =−λ^iT/2, and the 8 dimensional adjoint (Lⁱ_adj)jk=−if_ijk. There are severalSU(2) andU(1) subgroups ofSU(3), such as theSU(2) associated withL^1,2,3.

TABLE 3.1 The Gell-Mann matrices.^a λⁱ= τⁱ 0

TABLE 3.2 The nonzero (totally antisymmetric) structure constantsfijk for SU(3), and the nonzero (totally symmetric)dijkdefined by the anticommutators of the Gell-Mann matrices.

3.1.3 More on Representations and Groups

Therankof a Lie group is the number of generators that are simultaneously diagonalizable.

The diagonal generators correspond to conserved quantum numbers if they commute with the Hamiltonian.U(1), SU(2), and SU(3) have rank 1, 1, and 2, respectively.

Twon×nrepresentationsLⁱ and L⁰ⁱ ofGareequivalentif allN of them are simulta-neously related by a similarity transformation, i.e., if there exists an n×nunitary matrix U such that

L⁰ⁱ=U LⁱU^† for i= 1· · ·N. (3.14) Otherwise they are inequivalent.

1It is sometimes convenient to defineλ⁰≡p

2/3I. Then, Tr (λⁱλ^j) = 2δ^ijand{λⁱ, λ^j}= 2dijkλ^k, for i, j, k= 0,1· · ·8,withd0jk=p

2/3δjk.

A representationLⁱ isreducibleif it is equivalent to a representation

L⁰ⁱ =











L⁰ⁱ_A 0 0 0 0 L⁰ⁱ_B 0 0 0 0 L⁰ⁱ_C 0

0 0 0 . ..











(3.15)

in which each element is simultaneously block diagonal (with the same block dimensions).

Otherwise, it isirreducible (an IRREP). States transforming according to a reducible rep-resentation separate into sectors not related by the symmetry, while all of the states in an IRREP are related. Simple Lie groups have an infinite number of IRREPs, and they frequently have inequivalent IRREPs of the same dimension.

A fundamental representation is, roughly speaking, a representation from which the others can be generated by direct products, in analogy to the way that any angular mo-mentumj in quantum mechanics may be generated by combining 2j angular momenta ¹₂. The defining representations (m) of SU(m), such as the 2 of SU(2) in the example, are fundamentals.

Theadjointorregular representationof a Lie group is theN×N dimensional represen-tation constructed from the structure constants,

Lⁱ_adj

jk=−icijk. (3.16)

It is straightforward to show thatLⁱ_adj satisfy (3.7) (Problem 3.3). The adjoint is essential for defining the self-interactions of the gauge fields in a non-abelian gauge theory.

If Lⁱ_n is an n dimensional representation of a Lie algebra, then the conjugate Lⁱ_n∗ ≡

−L^i∗_n = −L^iT_n is also a representation. L_n is real² if it is equivalent to Lⁱ_n∗, i.e., if there exists a unitaryU such that −L^i∗_n =U Lⁱ_nU^† fori= 1· · ·N. Otherwise, it iscomplex. The adjoint representation Lⁱ_adj is always real, with U =I. The 2 ofSU(2) is real, i.e.,

L2^∗ =−τ^i∗

2 =τ²τⁱ

2τ², (3.17)

so thatU =τ². The higher-dimensional SU(2) representations are also real. On the other hand, the mofSU(m) form >2 isnotequivalent to the m^∗, which is also a fundamental represention. For example, L3^∗ = −^λ2^i∗ in SU(3) is not equivalent to L3 = ^λ₂ⁱ. This is important for the Higgs Yukawa couplings in extensions of the electroweakSU(2) group to higher symmetries.

The Simple Lie Groups

Two groups G₁ and G₂ commute if [gi,ˆg_j] = 0 for all g_i ∈G₁, ˆg_j ∈ G₂. Then, one can define thedirect product groupG=G₁×G₂with elementsg_iˆg_j, or direct products of more than two groups, such as the standard model groupSU(3)×SU(2)×U(1). Asimplegroup is (non-rigorously) a non-abelian group such as SU(3) that is not a direct product.³ A semi-simple group is basically a direct product of simple groups, i.e., a Lie group with no U(1) factors, such asSU(3)×SU(2).

2Mathematics books typically work in terms ofiL, motivating the term “real”.

3More precisely, a subgroupHof a Lie groupGis aninvariant subgroupifghg⁻¹∈Hfor allg∈G, h∈H.

Gis simple if it contains no invariant subgroups (other than the identity andGitself), and semi-simple if it contains no abelian invariant subgroups. Compact semi-simple Lie groups are either simple or the direct product of two or more simple groups.

Cartan has given a classification of the simple Lie algebras. The classification as well as the IRREPs and their properties are elegantly derived fromDynkin diagrams(Slansky, 1981), but here we only give the results. There are four countably infinite series ofclassical Lie algebras, and fiveexceptionalalgebras, as listed inTable 3.3. The four series correspond to simple matrix conditions for thedefiningrepresentations of the associated groups:

SU(m), m= 2· · · ∞, correspond to them×mcomplex unitary matricesU_G=e^−i~β·L~ with unit determinant,

U_GU_G^† =I, detU_G= 1, (3.18)

which implies that β~·~L are the traceless Hermitian matrices.⁴ U_G leaves invariant the inner product of two m-dimensional complex vectors, i.e., y^†x = y^0†x⁰, where x⁰=U_G^†xand similarly fory⁰.

SO(m) are the m×m real orthogonal matrices O_G with unit determinant⁵ (i.e., rotations in anm-dimensional real space)

O_GO^T_G=I, detO_G= 1, (3.19)

so thatO_G =e^−i~β·~L with i~β ·~L real and antisymmetric. The inner producty^Txof two real vectors is left invariant under anO_G transformation.

Sp(2m) are the real 2m×2m symplectic matricesM, defined by

M^TSM=S, (3.20)

where S is the skew symmetric matrix S =

0m Im

−Im 0m

, where 0m and I_m are, respectively, them×mzero and identity matrices. They therefore leave invariant the quadratic formy^TSx, wherexand yare 2m-dimensional real vectors.

The defining representations ofSU(m) andSp(2m) are also fundamental and can be used to generate the higher-dimensional IRREPS as direct products. For SO(m) one can derive higher tensor representations (including the adjoint) from the defining orvector(m). How-ever, there are additional double-valued fundamentalspinorrepresentations, similar to the familiar 2 ofSO(3)∼SU(2) (see, e.g., Li, 1974; Slansky, 1981). All of the IRREPS can be generated as direct products of the fundamental spinor. SU(m), SO(m), and some of the exceptional groups have found considerable application in physics. Recently, Sp(2m) has emerged in connection with string theory.

Casimir Invariants

ACasimir invariantis a functionf(T) of the group generatorsTⁱthat commutes with them, [f(T), Tⁱ] = 0. By Schur’s lemmathe corresponding functionf(L) of an n×nIRREP L is a multiple of the identity. The coefficient may depend on the representation and may be used to label it. The simplest example is thequadratic Casimir

L~²=

N

X

i=1

LⁱLⁱ≡C₂(L)I. (3.21)

4TheU(m) group (which is not simple) is related byU(m) =SU(m)×U(1), whereU(1) is defined in (3.9) withT them×midentity matrix.

5The orthogonal groupO(m) consists of the transformationsOGandROG, whereOG∈SO(m) andR, which represents a reflection in an odd number of dimensions, is a diagonal matrix with elements±1 and detR=−1.

TABLE 3.3 The Cartan classification of simple Lie algebras.^a

Cartan label Classical group N Range

A<sub>`</sub> SU(`+ 1) `(`+ 2) `≥1

B<sub>`</sub> SO(2`+ 1) `(2`+ 1) `≥2

C` Sp(2`) `(2`+ 1) `≥3

D<sub>`</sub> SO(2`) `(2`−1) `≥4

G₂ 14

F4 52

E₆ 78

E7 133

E₈ 248

aThe groups SO(6) ∼ SU(4), SO(4) ∼ SU(2)×SU(2), SO(3) ∼ SU(2) ∼ Sp(2), Sp(4) ∼ SO(5), and SO(2)∼U(1) have the same Lie algebras but may differ for non-infinitesimal transformations. The subscript in the first column is the rank.

A familiar example is J~² = j(j+ 1)I for the angular momentum j representation of the rotation group.C₂(L) is related to the Dynkin indexT(L) defined in (3.8) by

T(L)N =C₂(L)n. (3.22)

The quadratic Casimir of the adjoint T(Ladj) =C2(Ladj) is also written as C2(G). From (3.16),

c_iklc_jkl=C2(G)δij. (3.23)

The quadratic Casimirs and Dynkin indices for the defining and adjoint representations of the classical Lie algebras are given inTable 3.4(see also van Ritbergen et al., 1999). Other useful identities (which can be used to construct other invariants) are

LⁱL^jLⁱ=

C2(L)−1 2C2(G)

L^j c_ijkLⁱL^j = i

2C2(G)L^k,

(3.24)

from which TrLⁱ= 0 for the generators of a simple Lie group.

TABLE 3.4 Quadratic Casimirs and Dynkin indices for the defining representationLnand adjoint representation of the classical Lie algebras.

G N C2(G) n T(Ln) C2(Ln)

SU(m) m²−1 m m ¹₂ ^m_2m²⁻¹

SO(m) ^m(m−1)₂ 2(m−2) m 2 m−1

Sp(2m) m(2m+ 1) m+ 1 2m ¹₂ ^2m+1₄

More onSU(m)

Properties of the SU(m) IRREPs and their direct products can be found systematically from theYoung tableaux(e.g., Cheng and Li, 1984; Patrignani, 2016) or the more general

Dynkin methods. However, many aspects of SU(m) are simple enough to “do it yourself.”

For example, the fundamental Lⁱ_m ≡ λⁱ/2 with Tr (λⁱλ^j) = 2δ^ij can be written as an obvious generalization of the 3×3 matrices inTable 3.1, and the structure constants can be calculated from them. An important property ofSU(m) (that does not generalize to the other simple groups) is that theLⁱ_m along with the identity form a complete set that can be used to write anym×mmatrix,

M = 1

mTr (M)I+1 2

m²−1

X

i=1

Tr (M λⁱ)λⁱ, (3.25) where the coefficients are generally complex. In particular, the anticommutator

λⁱ, λ^j is a linear combination

λⁱ, λ^j = 4

mδ^ijI+ 2dijkλ^k. (3.26)

where the totally symmetricdijk generalize those inTable 3.2.This allows one to generalize theSU(2) Fierz identity inProblem 1.1on page 5 toSU(m),

(χ^†₄~λ χ₃)·(χ^†₂~λ χ₁) = 2ηF(χ^†₄χ₁)(χ^†₂χ₃)− 2

m(χ^†₄χ₃)(χ^†₂χ₁), (3.27) where η_F = +1 if the χ_i arem-dimensional complex vectors or complex scalar fields and η_F =−1 for anticommuting fermion fields.

SU(m) tensor methods, discussed inSection 3.2.3, are especially useful for constructing SU(m) singlets from direct products.

3.2 GLOBAL SYMMETRIES IN FIELD THEORY

Im Dokument The Standard Model and Beyond (Seite 104-111)