Lie Groups and Lie Algebras in Particle Physics

Transcription

1 Lie Groups and Lie Algebras in Particle Physics João G. Rosa Department of Physics and CIDMA, University of Aveiro & Department of Physics and Astronomy, University of Porto These notes are part of a 2 hour lecture course given at the University of Porto, Portugal, to Physics Masters students. The course is part of a larger curricular unit on Mathematical Methods in Physics, which includes a previous part on discrete groups and applications to Condensed Matter Physics. I will nevertheless try to make these notes as self-contained as possible, recalling where necessary some of the concepts and results that were taught in the previous part of the course. These notes are not meant as an extensive review of Lie groups and Lie algebras, and for those interested in learning more about the mathematical aspects of group theory and applications to particle physics I recommend the following textbooks: Howard Georgi, Lie algebras in particle physics Westview Press, 999. J. Fuchs and C. Schweigert, Symmetries, Lie Algebras and Representations Cambridge University Press, I am indebted to my PhD student Catarina Cosme for her help in typing part of these notes. Contents General properties of Lie Groups and Lie Algebras 2. Example: the Lie group SU Cartan-Weyl basis Example: the Cartan-Weyl basis for SU Lorentz and Poincaré groups Lorentz group Relation with SL2, C Representations of the Lorentz group Spinor bilinears Parity and charge conjugation Poincaré group Massive representations Massless representations SUN groups Irreducible representations of SUN SU3 and the quark model Branching SU3 representations into SU2 representations Problems 46

2 General properties of Lie Groups and Lie Algebras Let us start by recalling the mathematical properties of a group: DEFINITION: A group G is a set with a map G G G known as group multiplication satisfying the following properties: Associativity: g.h.l = g. h.l, g,h,l G, Identity: e G e.g = g, g G, Inverse: g G g G g.g = g.g = e. In particle physics we are mostly interested in representations of a group, which define the concrete realization of group transformations. A group representation R is a map that associates to each group element a linear transformation acting on a particular real or complex vector space, V : R : G GLV. The dimension of the representation corresponds to the dimension of the associated vector space. A representation is reducible if and only if there is a subspace U V left invariant by group transformations, i.e. Rgu U for all u U and g G. It is otherwise called irreducible and these are the representations that will mostly interest us in particle physics applications. A Lie Group is a continuous group, i.e. in which all elements g G depend continuously on a continuous set of parameters g = g α, α = {α a }, a =,..., N. 2 We can assume, without loss of generality, that the identity element corresponds to the origin in the space of parameters: e = g α α=0, 3 such that for any representation of the group: R α α=0 = I. 4 In a neighbourhood of the identity element, we may then expand R α in a Taylor series: R dα = I + ix a dα a +..., 5 where we consider Einstein s summation rule that repeated indices should be summed over. In the above expression we have defined: X a i α a D α α=0, 6 and used the notation dα a to denote an infinitesimal change in the {α a } parameters, thus obtaining a representation of a group element arbitrarily close to the identity. The {X a } vectors are known as group generators. We include a factor i in their definition such that, for unitary representations, R α R α = I, the group generators are hermitian operators X a = X a as we will see below. Group multiplication then allows us to obtain any other finite element of the group by multiplying 5 by an infinitesimal 2

3 transformation an arbitrary number of times: R α = lim I + i α k ax a = exp iα a X a, 7 k k α where dα a = lim a k k. This defines the exponential map or exponencial parametrization of the group elements. We may thus write the group elements in terms of its generators, at least in a neighbourhood of the identity. For a unitary representation, e iαaxa e iαax a = I, which implies Xa = X a as anticipated. As opposed to the group elements, the generators form a linear vector space and any linear combination of the generators is itself a group generator. Let us consider a one-parameter family of group elements given by: U λ = exp iλα a X a. 8 In this case, group multiplication is quite simple and yields: U λ U λ 2 = exp iλ α a X a exp iλ 2 α a X a = exp i λ + λ 2 α a X a 9 = U λ + λ 2. However, for group elements that are generated by different linear combinations of the generators this is not so simple, since in general: exp iα a X a exp iβ b X b exp i α a + β b X a. 0 Since any group element admits a parametrization in terms of the exponential map, we must have that: exp iα a X a exp iβ b X b = exp iδ a X a, for some set of parameters {δ a }, a =,..., N. Continuity and differentiability of the group elements then allows us to find the {δ a } parameters by expanding both sides of Eq. in a Taylor series. We first note that: iδ a X a = ln [ + exp iα a X a exp iβ b X b ] ln + K, 2 and that, for small K, Thus, expanding up to to quadratic order in the α a and β a parameters: ln + K = K K K = exp iα a X a exp iβ b X b = + iα a X a 2 α ax a iβ b X b 2 β bx b = iα a X a + iβ a X a α a X a β b X b 2 α ax a 2 2 β ax a Hence, we have that: iδ a X a = iα a X a + iβ a X a α a X a β b X b 2 α ax a 2 2 β ax a α ax a + β a X a = i α a + β a X a α a X a β b X b + 2 α ax a β b X b + 2 β bx b α a X a +... = i α a + β a X a 2 [α ax a, β b X b ] +..., 5 3

4 where we have taken into account that the generators are linear operators that do not, in general, commute with each other. We thus find that: [α a X a, β b X b ] = 2i δ c α c β c X c +... iγ c X c Since this must hold for any choice of parameters, we conclude that: γ c = f abc α a β b 7 and that, hence, where the constants satisfy: [X a, X b ] = if abc X c, 8 f abc = f bac, 9 since the commutator is antisymmetric, [A, B] = [B, A]. These are known as the group s structure constants and define the Lie algebra of the the group G, LG, i.e. the set of generators with the closed commutation properties above. The commutator 8 defines the fundamental properties of the Lie algebra and thus plays a similar role to the group multiplication. We thus conclude that: δ a = α a + β a 2 γ a which implies: exp iα a X a exp iβ b X b = exp i α a + β a X a 2 [α ax a, β b X b ] +..., 2 which is known as the Baker-Campbell-Hausdorff BCH relation. The higher-order terms that we have discarded above correspond to commutators of commutators, e.g. [α a X a, [α c X c, β b X b ]] and are thus determined by the structure constants. Hence, the Lie algebra 8 is sufficient to completely define the group multiplication in a finite neighbourhood of the identity. The structure constants are an intrinsic property of the Lie algebra and are independent of its representation, being determined solely by the group multiplication rule and by continuity. Each representation of the group then defines a representation of the associated Lie algebra. A useful property to note is that the structure constants are real if there is a unitary group representation. DEM: Since in a unitary group representation the generators are hermitian operators we have, on the one hand, that: [X a, X b ] = if abcx c = if abcx c, and, on the other hand, that: [X a, X b ] = X a X b X a X b = X b X a X a X b = [X a, X b ] = if abc X c. so that f abc = f abc. Q.E.D. The group generators satisfy the Jacobi identity: [X a, [X b, X c ]] + [X b, [X c, X a ]] + [X c, [X a, X b ]] =

5 This can be shown in a straightforward way by expanding all the commutators, so we leave it as an exercise. The structure constants themselves can be used to define an important representation of the Lie algebra known as the adjoint representation. This can be done by defining the N N matrices [T a ] bc if abc, 23 the commutator of which is given by: [T a, T b ] cd = T a T b T b T a cd = T a ce T b ed T b ce T a ed = f ace f bed + f bce f acd. 24 From the Jacobi identity we have that: [X a, [X b, X c ]] = [X a, if bcd X d ] = if bcd [X a, X d ] = f bcd f ade X e, 25 and so f bcd f ade + f cad f bde + f abd f cde X e = 0 Xe LG f bcd f ade + f cad f bde + f abd f cde = If we now interchange the indices d and e, we obtain: f ace f bed + f bce f aed = f abe f ced, 27 from which we conclude that: [T a, T b ] cd = f abe f ced = f abe f ecd = if abe T e cd, which can be written as: [T a, T b ] = if abc T c, 28 such that the T a matrices satisfy the commutation relation of the Lie algebra in Eq. 8. The dimension of the adjoint representation corresponds to the number of independent generators, i.e. to the number of real parameters required to specify a group element. Note that the generators are pure imaginary matrices in the adjoint representation if the structure constants are real. A more formal way of defining the adjoint representation is given by the map: ad : L G M L G ad X Y = [X, Y ], X, Y LG. 29 5

6 We may write this in components by choosing a basis of generators for the Lie algebra {T a }, a =,..., N: ad T a T b = [T a, T b ] = if abc T c ad T a cb = if abc = [T a ] bc = [T a ] cb, in agreement with the definition above. Note that we have used that f abc = f acb as we will show explicitly below. The adjoint representation of the Lie algebra naturally induces a representation of the group, also known as the adjoint representation of the group: Ad : G GL L G Ad g T a = gt a g, g G, T a LG. 30 We can easily check that these two representations are related through the exponential map. Writing a group element as g = exp iα a T a, we have that: Ad g T a = exp iα b T b T a exp iα b T b = + iα b T b +... T a iα b T b +... = T a + iα b [T b, T a ] +... = T a + iα b ad T b T a +... = exp iα b ad T b T a. 3 Let us now introduce a series of definitions and theorems that can be used to classify different Lie algebras. DEFINITION: A sub-algebra A L G is a linear space such that: X,Y A [X, Y ] A. 32 We can highlight the following special cases of sub-algebras: A sub-algebra is abelian if: X,Y A, [X, Y ] = 0, 33 and the same applies naturally to the full Lie algebra LG. A sub-algebra is an ideal if: X A Z LG [X, Z] A, 34 and a proper ideal if, in addition, A L G, {0}. DEFINITION: A Lie algebra is simple if it does not contain any proper ideals and semi-simple if it does not contain abelian ideals except {0}. 6

7 THEOREM: A Lie algebra is semi-simple if and only if: L = L... L N 35 where L i, i =,..., N, are simple algebras. A rigorous proof of this theorem is out of the scope of these lectures, but let us consider a generic example that illustrates these general definitions and properties. Consider the product of two groups G = G G 2, where the Lie algebras LG and LG 2 are simple, i.e. have no proper ideals as defined above. We can write a generic element of G in the matrix form: g 0 g = G, 0 g 2 for g i G i, i =, 2, such that the elements of the Lie algebra, i.e. the generators, can be written as: T = T 0 0 T 2 L G. Let us then consider the sub-algebra LG with elements of the form: T T 0 =. 0 0 Then, we have for the commutator between a generic element of LG and an element of this sub-algebra: [T, T ] = = = T 0 T 0 T 0 T 0 0 T T 2 T T 0 T T T 0, 0 0 where T = [T, T ] L G. Hence, the elements of the sub-algebra LG constitute a proper ideal and LG cannot be simple. It is easy to see, by a similar reasoning, that the elements of LG 2 also form a proper ideal, and that there are no other ideals since LG and LG 2 are simple algebras. Thus, the full Lie algebra LG = LG LG 2 will be semi-simple if these ideals are not abelian. DEFINITION: The Killing form or Killing metric associated with a Lie algebra L G is a symmetric bilinear map: Γ : L G L G R defined by: Γ X, Y = Tr [ad X ad Y ], X, Y L G. 36 This defines an inner product within the Lie algebra in the adjoint representation. Considering a basis of generators 7

8 {T a } for the Lie algebra we obtain for the Killing metric components: γ ab = Γ T a, T b = Tr [T a T b ] = T a cd T b dc = if acd if bdc = f acd f bdc. 37 THEOREM: A Lie algebra is semi-simple if the Killing form is non-degenerate: Γ X, Y = 0 X LG Y = 0, or equivalently that detγ 0. DEM: Let us suppose that Γ is non-degenerate and that A L G is an abelian ideal. We can then choose a basis {T a, T α } for the Lie algebra where T a generate the elements in A and T α correspond to the remaining generators of L G. Then, for X L G and Y A we have: ad X ad Y T a = [X, [Y, T a ]] = 0, since [Y, T a ] = 0. Also: ad X ad Y T α = [X, [Y, T α ]] = a α a T a, 38 since [Y, T α ] A and so [X, [Y, T α ]] A as well. We thus conclude that the Killing metric has the form: Γ X, Y = Tr [ad X ad Y ] [ ] 0 = Tr 0 0 = 0. Since by assumption Γ is non-degenerate, we must have Y = 0, and so there cannot exist any non-trivial abelian ideals and the algebra is semi-simple. The proof in the opposite direction follow an analogous reasoning. Q.E.D. THEOREM: If L G is a compact Lie algebra, i.e. if the underlying Lie group is compact, being defined in term of a compact manifold of parameters, then the Killing form is positive semi-definite: Γ X, X 0, X LG, 39 and if the algebra is simple, we have: Γ X, X > 0, X LG. 40 Although we will not prove this theorem in this course, we can use it to show an important result that we have anticipated above. If the Killing form is positive definite, then we can choose an orthonormal basis for the Lie algebra such that: Γ T a, T b = δ ab, 4 which just means that we may diagonalize the Killing metric and choose an appropriate normalization for the generators. 8

9 In this basis, the structure constants are completely antisymmetric. DEM: Γ [T c, T a ], T b + Γ T a, [T c, T b ] = Tr [T c T a T a T c T b ] + Tr [T a T c T b T b T c ] = Tr [T c T a T b ] Tr [T a T c T b ] + Tr [T a T c T b ] Tr [T a T b T c ] = 0, using the cyclic property of the trace. Thus, Γ if cad T d, T b + Γ T a, if cbd T d = if cad Tr [T d T b ] + if cbd Tr [T a T d ] = if cad δ db + if cbd δ ad = f cab + f cba = 0, 42 so that f cab = f cba. Since by the definition the structure constants are antisymmetric in the first two indices, this implies that they must be completely antisymmetric. Q.E.D. THEOREM: For a semi-simple Lie algebra the quadratic Casimir operator: C γ ij T i T j, 43 where γ ij = γ ij, commutes with all the generators: [C, T k ] = 0 Tk LG. 44 DEM: [C, T l ] = γ ij [T i T j, T l ] = γ ij T i [T j, T l ] + γ ij [T i, T l ] T j = iγ ij T i f jlm T m + iγ ij f ilm T m T j = iγ ij f jlm T i T m + iγ ij f jlm T m T i = iγ ij f jlm T i T m + T m T i = if jlm γ ij T i T m + T m T i = 0, since f jlm is antisymmetric while γ ij T i T m + T m T i is symmetric under the interchange of the indices j and m, noting that the Killing metric is symmetric. Q.E.D. An important result in the theory of group representations is Schur s Lemma, which can be cast in the following form: SCHUR S LEMMA: For an irreducible representation R of a group G over a complex vector space V, if there exists a linear transformation P GLV that commutes with the action of all group elements: [P, Rg] = 0 g G, 45 then this operator must be proportional to the identity, P = λi for some complex constant λ. 9

10 DEM: Consider the space of eigenvectors of the operator P with eigenvalue λ: Eigλ = {v V : P v = λv}. 46 This must be a non-empty set for some λ since detp λi = 0 has at least one solution in C. Then, for v Eigλ we have that: P Rgv = RgP v = λrgv, 47 which implies that Rgv Eigλ, i.e. that the eigenspace is left invariant by the action of the group. Since, by assumption, the representation is irreducible, than Eigλ = V, which implies that P = λi. Q.E.D. It is easy to see that Schur s Lemma applies to the irreducible representations of both a Lie group and its Lie algebra. This then leads us to the conclusion that, in any irreducible representation r of LG, the Casimir operator is proportional to the identity: C = C r I dimr, 48 where C r is a number that is characteristic of the representation. In the orthonormal basis where γ ij = δ ij for a simple and compact Lie algebra, we have: C = δ ij T r i T r j = i T r i 2 = C r Idimr. 49 We may also consider the quadratic operator: [ M ij = Tr T r i ], T r j, 50 for the Lie algebra generators in a given representation r. The commutator of this operator with a generator in the adjoint representation is then given by: [T i, M] jk = T i jl M lk M jl T i lk [ ] [ = if ijl Tr T r l, T r k + if ilk Tr [[ ] ] [ = Tr T r i, T r j, T r k Tr [ = Tr T r i T r j T r k T r j T r i T r k = 0, T r j T r j ], T r l [ T r i + T r j ]], T r k T r i T r k T r j T r k ] T r i 5 using the cyclic property of the trace. Schur s Lemma than implies that M = C r I dimad. 52 The two Casimir constants C r and C r are thus related via: Tr r C = C r dim r = Tr T r i i = Tr r M 2 = C r dim ad, 53 0

11 such that C r = dim r C r. 54 dim ad We note, as an aside, that the Casimir operators play a very important role in particle physics, since they characterize the irreducible representations in which each type of field transforms under symmetry operations associated with particular Lie groups, as we will discuss in more detail later on.. Example: the Lie group SU 2 To better understand the general concepts and definitions introduced in this section, let us look in detail into a particular example, that of the group of 2 2 unitary matrices with unit determinant, denoted as SU2. This will also serve as a warm up exercise to the more detailed study of SUN representations and applications to particle physics that we will do later on. The group is formally defined as SU2 = { U GL C 2 : U U = I, det U = }. For a generic matrix in this group: U = a c b d U = a c b d. The unitarity condition yields for the matrix components: UU = a 2 + b 2 ac + bd 0 = ca + db c 2 + d 2 0 ac + bd = 0 a c + b d = 0 a = b = d c This implies that a generic 2 2 unitary matrix can be written in the form: U = and the unit determinant condition yields: α β, β α det U = α 2 + β 2 =. The SU2 group can thus be defined by: { α SU2 = β } β : α, β C, α 2 + β 2 = α, such that each element of the group is specified by two complex numbers subject to a single real condition, thus having three independent degrees of freedom. To determine the associated Lie algebra, we consider infinitesimal group transformations arbitrarily close to the identity, U = I + it +... to obtain that: UU = I + it +... I it +... = I T = T.

12 and also that: det U = det I + it +... = + itrt = Tr T = 0. Recall that for a generic matrix with eigenvalues λ i, det A = i λ i and TrA = i λ i. If A I + X +... then λ i = + ɛ i +..., where ɛ i are the arbitrarily small eigenvalues of X, such that det A = i + ɛ i = + i ɛ i +... = + Tr X, which we used above. These results then imply that the Lie algebra of SU2 is given by: L SU 2 = { T M C 2 : T = T, Tr T = 0 }. 55 A generic matrix in the SU2 Lie algebra may then be written in the form: T = a c b = d a which, along with the condition TrT = a + d = 0, leads us to the conclusion that the diagonal entries must be real with a = d, while the non-diagonal entries are complex conjugate, leaving only three real degrees of freedom, in agreement to what we found above for the group elements. b The basis of matrices for the SU2 algebra is conventionally chosen to be given in terms of the three Pauli matrices, T i = σ i /2, i =, 2, 3: It is easy to check that these matrices satisfy: 0 0 i σ =, σ 2 =, σ 3 = 0 i 0 c d, [σ i, σ j ] = 2iɛ ijk σ k, {σ i, σ j } = 2δ ij, σ i σ j = δ ij + iɛ ijk σ k, 57 where the anti-commutator {A, B} = AB + BA. This thus implies that [T i, T j ] = iɛ ijk T k, 58 such that the SU2 structure constants are f ijk = ɛ ijk in terms of the totally antisymmetric Levi-Civita tensor. In this basis we also have that Tr T i T j = 4 Tr σ iσ j = 4 [δ ijtr I + iɛ ijk Tr σ k ] = 2 δ ij. We note that it is possible to choose the normalization of the generators such that Tr T i T j = δ ij but the above is the most conventionally used normalization. The Killing metric for the SU2 algebra is given by: γ ij = ɛ ikl ɛ jlk = 2δ ij, 2

13 as can be checked explicitly for the different components: γ = ɛ kl ɛ lk = ɛ 23 ɛ 32 + ɛ 32 ɛ 23 = 2 γ 22 = ɛ 2kl ɛ 2lk = ɛ 23 ɛ 23 + ɛ 23 ɛ 23 = 2 γ 33 = ɛ 3kl ɛ 3lk = ɛ 32 ɛ 32 + ɛ 32 ɛ 32 = 2 γ 2 = ɛ kl ɛ 2lk = 0 γ 3 = ɛ kl ɛ 3lk = 0 γ 23 = ɛ 2kl ɛ 3lk = 0. This yields det γ = 8 0, so that the algebra is semi-simple. In fact, the SU2 algebra is simple since there are no proper ideals, as can be inferred from the commutation relations satisfied by the Pauli matrices. The Pauli matrices define the fundamental representation of the SU2 algebra, also known as the spin-/2 representation. This representation acts on 2-dimensional vectors known as SU2 doublets. The adjoint representation or spin- representation is given by: T ad i jk = if ijk = iɛ ijk, with generators: T ad = i , 0 0 T ad 2 = i , T ad 3 = i Another important representation is the complex conjugate representation: R U = U = e T, where T = i j α jσ j. We can use that σ i = σ 2σ i σ 2, as can be explicitly checked: σ 2 σ σ 2 = σ 2 iɛ 23 σ 3 = iɛ 23 iɛ 23 σ = σ, σ 2 σ 2 σ 2 = σ 2, σ 2 σ 3 σ 2 = σ 2 iɛ 32 σ = iɛ 32 iɛ 23 σ 3 = σ 3, to show that: σ 2 T σ 2 = i α σ + α 2 σ 2 α 3 σ 3 = i α σ + α 2 σ2 + α 3 σ3 = T. This then implies that: R σ2t σ2 σ2t σ U = e = e 2. 3

14 Now note that, for generic matrices A and B: e ABA = n ABA n n!, and that ABA n = ABA ABA... ABA = AB n A, so that we have: e ABA = A n B n A = Ae B A. n! Using this result, we may write the complex conjugate representation in the form: R U = σ 2 e T σ 2 = σ 2 RUσ 2. Hence, the fundamental and complex conjugate representations are not really independent and, in fact, are said to be equivalent. The fundamental representation is then said to be pseudo-real..2 Cartan-Weyl basis To define the Cartan-Weyl basis, which is extremely useful in determining and classifying representations of Lie groups and associated algebras, we will consider semi-simple Lie algebras with generators {T i }, i =,..., N. Let us take a linear combination of the generators: H = a i T i L, 60 where we use the Killing metric γ ij and its inverse γ ij to lower and raise indices, such that the Einstein summation convention requires repeated upper and lower indices to be summed over. Note that in the previous calculations we worked with lower indices exclusively, which is consistent in the Lie algebra basis where the Killing metric is the identity. However, as we will see in the Cartan-Weyl basis this does not hold and we must use this more correct version of the summation convention. With the linear combination of generators above, we can study the eigenvalue problem: ad H T = [H, T ] = ρt. 6 This can be written in components on the given basis as: [ a i ] T i, T j = ρtj a i f k ij T k = ρt j a i f k ij ρδ k j T k = 0 det a i f k ij ρδ k j = 0, where the structure constants are now defined as: [T i, T j ] = f k ij T k. 4

15 CARTAN S THEOREM Choosing the linear combination H with the largest possible number of eigenvalues ρ:. The eigenvalue ρ = 0 can be degenerate with multiplicity l = rankl and the corresponding eigenspace is generated by {H i }, i =,..., l, such that [H i, H j ] = All the non-vanishing eigenvalues ρ 0 are non-degenerate. The proof of Cartan s Theorem is out of the scope of this course, but we can study its implications. First, we can immediately conclude that: [H, H i ] = 0, [H, E α ] = αe α, α Now, since [H, H] = 0, we must have H = λ i H i. In addition, the Jacobi identity implies that: [H, [H i, E α ]] = [H i, [E α, H]] [E α, [H, H i ]] = α [H i, E α ], which means that [H i, E α ] is an eigenvector of H with eigenvalue α 0 which, by Cartan s Theorem, is non-degenerate. Hence: [H i, E α ] = α i E α. 64 Therefore, we have that: [H, E α ] = λ i [H i, E α ] = λ i α i E α = αe α 65 from which we infer the relation α = λ i α i. We may then write the structure constants in the form: We may further consider the Jacobi identity for the generators H, E α and E β : [H i, E α ] = f β iα E β f β iα = α iδα β. 66 [H, [E α, E β ]] + [E β, [H, E α ]] + [E α, [E β, H]] = 0 [H, [E α, E β ]] + α [E β, E α ] β [E α, E β ] = 0 [H, [E α, E β ]] = α + β [E α, E β ], 67 such that [E α, E β ] is an eigenvector of H with eigenvalue α + β. This leads to two possible commutation relations: [E α, E β ] = N αβ E α+β, α + β i [E α, E α ] = fα α H i, α + β = 0 This yields the structure constants: f γ αβ γ = N αβ δα+β. 69 5

16 Having found the form of the structure constants in the basis {H i, E α }, we may compute the components of the Killing metric in this basis: γ iα = f γ iβ f β αγ = α i δ γ β = α i N αβ δ N β αγδα+γ β α+β = 0, 70 β since α 0, or also since N αβ is anti-symmetric and δα+β is symmetric under the exchange of the indices α and β. Similarly, we obtain: γ αβ = f = N αγ δ µ γ αγ fβµ µ α+γ N βµ δ γ β+µ γ = N αγ N β,α+γ δβ+α+γ. 7 This implies that γ αβ 0 if and only if α + β = 0. We may then choose to normalize the E α generators such γ α α =. Finally, we have: γ ij = f β iα f α jβ = α i δα β α j δ α β = α i α j δ α α = α α i α j. 72 This result allows to derive the remaining structure constants: i fα α = γ ij f α αj = γ ij f jα α = γ ij f = γ ij f β jα γ β α = γ ij α j = α i. 73 Let us summarize the above results. Cartan s Theorem allows us to find a basis for a semi-simple Lie algebra {H i, E α }, i =,..., l = rank L, known as the Cartan-Weyl basis, with the following commutation relations: [H i, H j ] = 0 [H i, E α ] = α i E α [E α, E β ] = N αβ E α+β, α + β 0 [E α, E α ] = α i H i, 74 The abelian sub-algebra spanned by {H i } is known as Cartan s sub-algebra. The vectors α = α i i=,...,l are denoted α jα 6

17 as the root vectors. In this basis the Killing metric takes the form: γ ij = α α i α j γ iα = 0 γ α α = γ αβ = 0, α + β In the Cartan-Weyl basis, we can find an SU2 sub-algebra. To see this, let us start by choosing H = α i H i for a given root vector α. The subset of generators {H, E α, E α } for each root vector α then satisfies the closed algebra: [H, E α ] = α i [H i, E α ] = α i α i E α [H, E α ] = α i [H i, E α ] = α i α i E α [E α, E α ] = α i H i = H. 76 To see that this is an SU2 algebra, note that defining σ ± = σ ± iσ 2 and using the commutation relations for the Pauli matrices, we have: [σ ±, σ 3 ] = [σ, σ 3 ] ± i [σ 2, σ z ] = σ 2 ± iσ ± i σ ± iσ 2 = ±iσ ± [σ +, σ ] = [σ + iσ 2, σ iσ 2 ] i [σ, σ 2 ] + i [σ 2, σ ] = 2iσ 3, 77 which has the same form as Eq. 76 identifying H σ 3 and E ±α σ ±. that: Let us now consider a representation r of the Lie algebra on a vector space V. Let us choose a basis { λ } for V such H r i λ = λ i λ, 78 where we note that since all H i generators in the Cartan sub-algebra commute they have a common eigenspace. The l-dimensional vectors λ = λ, λ 2,..., λ l are then designated as weight vectors of a given representation. Now, note that: H i E α λ = E α H i λ + [H i, E α ] λ = λ i E α λ + α i E α λ = λ i + α i E α λ, 79 such that E α λ is an eigenvector of H i with eigenvalue λ i + α i, i.e. with weight vector λ + α. We thus see that the E ±α generators act as raising and lowering operators in a given representation and can be used to find all the weights in the representation. For the adjoint representation, where the vector space V coincides with the Lie algebra itself, we have: adh i H j = [H i, H j ] = 0, adh i E α = [H i, E α ] = α i E α. 80 This means that the eigenstates corresponding to the generators H i have null weights, while for the eigenstates E α the weights coincide with the roots of the Lie algebra. For tensor representations of the group Rg = R g R 2 g and associated representations of the Lie algebra, we 7

18 have: H i λ = λ i λ, H 2 i µ = µ i µ, 8 such that H i λ µ = H i λ µ + λ H 2 i µ = λ i + µ i λ µ, 82 such that the weights of tensor representations correspond to the sum of the weights of the individual representations..3 Example: the Cartan-Weyl basis for SU2 To illustrate the advantages of using the Cartan-Weyl basis, let us return to our SU2 example, which is a rank Lie algebra, i.e. there is only one generator in the Cartan sub-algebra. Recalling that the basis of generators in the fundamental representation is T i = σ i /2, i =, 2, 3, we can take H = T 3 corresponding to the diagonal generator and define E ± = T ± it 2 /2, such that: [H, E ± ] = ±E ±, [E +, E ] = 2 H. 83 Thus, the SU2 algebra has roots α ± = ±. For a given SU2 representation we label the states in the associated vector space basis as j, m, where m denotes the weights of the representation: H j, m = m j, m 84 and the spin j = maxm is the highest weight in the representation, which is then denoted as a spin-j representation. For the fundamental representation, the eigenvalues of H = σ 3 /2 are ±/2, so that the fundamental representation is the spin-/2 representation as mentioned earlier. The adjoint representation has three weights, m = 0, ± as can be inferred explicitly from the form of the adjoint generators in Eq. 59, thus corresponding to the spin- representation. For a generic spin-j representation, we can obtain all the states in the basis using the raising and lowering operators E ±. It is conventional to use instead the raising and lowering operators J ± = 2E ±, and from our discussion in the previous sub-section we must have: J j, m = N m j, m, J + j, m = N m j, m, 85 with constants N m that we wish to determine. On the one hand, we have that: j, m J + J j, m = J j, m 2 = N m 2 j, m j, m = N m 2, 86 assuming that the states are normalized. On the other hand, we also have that: j, m J + J j, m = j, m [J +, J ] j, m + j, m J J + j, m = j, m 2H j, m + N m+ 2 = 2m + N m This then yields the recurrence relation: N m 2 = 2m + N m+ 2, 88 8

19 with boundary condition N j+ = 0 since by definition j is the highest weight and J + j, j = 0. It is easy to verify that the solution is then: N m = jj + mm, 89 such that J + j, m = jj + mm + j, m +, J j, m = jj + mm j, m. 90 In particular, J j, j = 0, such that the weights in each spin-j representation are m = j, j +,..., j, j, being a 2j + -dimensional representation. For each spin-j representation, the Casimir operator is then given by: C = Ti 2 = 2 2 H2 + 4 J +J + 4 J J +. 9 i As we have seen, Schur s Lemma implies that this is proportional to the identity, and to obtain the proportionality constant we can take the trace of the Casimir operator in a given representation: Tr j C = j, m 2 H2 + 4 J +J + 4 J J + j, m m = m 2 + jj + mm j, m J+ j, m + jj + mm + j, m J j, m m m = m jj + mm + jj + mm m m m = jj + 2 m = jj + 2j +, 2 and since the trace of the 2j + 2j + identity matrix is 2j +, we conclude that the Casimir of a spin-j representation is: m 92 Cj = jj We thus see that the SU2 representations correspond to the familiar spin angular-momentum states, with the Casimir operator C = J 2 /2. 9

20 2 Lorentz and Poincaré groups The first groups that we will study in detail are the most fundamental groups in relativistic particle physics - the Lorentz group and its extension known as the Poincaré group. We will start by looking at the Lorentz group in detail and then explore its extension to include space-time translations. 2. Lorentz group The Lorentz group is the group of space-time transformations that preserve the relativistic space-time distance, corresponding to coordinate changes between two inertial frames moving at constant velocity with respect to each other. We can formally define it as: L = O3, = { Λ GLR 4 : Λ T ηλ = η } 94 where the 4-dimensional Minkowski metric is given by: η = diag, +, +, The Lorentz group transformations are at the heart of the theory of Special Relativity, such that coordinate changes of the form: x µ x µ = Λ µ νx ν 96 preserve the infinitesimal line element: ds 2 = η µν dx µ dx ν, 97 where we recall that µ = 0 corresponds to the time coordinate and µ = i =, 2, 3 correspond to the spatial coordinates. The invariance of the line element follows trivially from the group definition: ds 2 = η µν dx µ dx ν = η µν Λ µ αdx α Λ ν βdx β = Λ T µ α η µν Λ ν βdx α dx β = η αβ dx α dx β = ds In components, the Lorentz group matrices satisfy: η µν Λ µ αλ ν β = η αβ, 99 and include the well-known boosts and spatial rotations. Examples: Boost along the x direction: γ βγ 0 0 Λ = B x = βγ γ , where β = v/c and γ = / β 2, with v denoting the boost velocity and c the speed of light in vacuum. Writing 20

21 β tanh φ, it is easy to see that the boost matrix can be written in the form: cosh φ sinh φ 0 0 B x = sinh φ cosh φ , such that a boost can be seen as a hyperbolic rotation or rotation by an imaginary angle. Rotation about the z axis: Λ = R z = 0 cos θ sin θ 0 0 sin θ cos θ Generic spatial rotations form the O3 matrix group: O3 = { O GLR 4 : O = 0 0 R 3, R 3 R T 3 = I 3 } O3, 03 where R 3 are 3 3 orthogonal matrices. The Lorentz group includes four distinct components. To see this, note that: detλ T ηλ = det Λ 2 detη = detη det Λ = ±. 04 In addition, we have that for α = β = 0 in Eq. 99 η µν Λ µ 0 Λν 0 = Λ i Λ i 0 2 = η 00 =, 05 such that Λ = + i Λ i 0 2 Λ 0 0 Λ We may thus split the Lorentz group into the sub-groups: L + = { Λ L : det Λ = +, Λ 0 0 } L + = { Λ L : det Λ = +, Λ 0 0 } L = { Λ L : det Λ =, Λ 0 0 } L = { Λ L : det Λ =, Λ 0 0 }. 07 These four sub-groups are disconnected from each other, since one cannot continuously change the sign of the determinant or of the Λ 0 0 component. It is also conventional to define the proper Lorentz group as: L + = L + L

22 and the orthocronous Lorentz group as: L = L + L. 09 For this reason, the L + sub-group is also known as the proper orthocronous Lorentz group, which includes the elements continuously related to the identity I 4 as the boosts and rotations mentioned above. The parity and time-reserval transformations are also part of the Lorentz group, such that: P = diag+,,, L, T = diag, +, +, + L, P T = diag,,, L +. 0 Along with the identity matrix, these elements form an abelian sub-group of L, L 0 = {I 4, P, T, P T }. Any element of the Lorentz group can be written as a unique product of an element of L 0 and an element of L +, as illustrated in the diagram below. L + PT L + P T L L 2.. Relation with SL2, C DEFINITION: The group SL2, C of complex 2 2 matrices with unit determinant is defined as: SL2, C = { M MC 2 : detm = }. Let us define σ µ = I 2, σ i, where σ i, i =, 2, 3, are the Pauli matrices. We may then define a map: τ : R 4 { S MC 2 : S = S } τx = x µ x 0 + x 3 x ix 2 σ µ = x + ix 2 x 0 x 3. 2 We can use this map to define a map between SL2, C and the Lorentz group: Λ : SL2, C GLR 4 x = ΛMx = τ M τxm, 3 22

23 i.e. τx = M τxm. In components: σ µ x µ = σ µ ΛM µ νx ν = M x α σ α M. 4 To show that the matrices ΛM defined through this map belong to the proper orthocronous Lorentz group, let us first note that: x T ηx = η µν x µ x ν = x i x i 2 = detx µ σ µ = detτx. 5 Thus, x T ηx = detτx = detσ µ Λ µ νx ν = detm σ µ x µ M = detm 2 detσ µ x µ = x T ηx, 6 so that ΛM preserves space-time distances and hence ΛM L. Noting that the map is continuous and that Λ±I 2 = I 4, since: σ µ x µ = σ µ Λ±I 2 µ νx ν = x α σ α 7 for M = ±I 2 SL2, C, we conclude that only the elements of L continuously connected to the identity can be obtained through this map, i.e. that ΛM L +. We thus say that SL2, C is the double-cover of the proper orthocronous Lorentz group: L + = SL2, C/Z 2, 8 where the Z 2 group factor identifies the elements continuously connected to +I 2 and I 2 in SL2, C, since these yield the same element of L +. Let us now consider the Lie algebra of SL2, C, which can be obtained by considering infinitesimal transformations: detm = deti 2 + T +... = + TrT = TrT = 0, 9 which implies the following definition for the Lie algebra: LSL2, C = { T MC 2 : TrT = 0 } = span {J i, K i } i=,2,3 20 where the six generators of the Lie algebra are defined by: J i = i 2 σ i, K i = 2 σ i 2 and satisfy the following commutation relations: [J i, J j ] = ɛ ijk J k, [K i, K j ] = ɛ ijk J k, [J i, K j ] = ɛ ijk K k, 22 which follow trivially from the commutators of the Pauli matrices. Note that the tracelessness condition corresponds to a complex constraint on the 4 complex components of the Lie algebra matrices, leading to 3 complex degrees of freedom or equivalently 6 real degrees of freedom, which yields the dimension of the SL2, C Lie algebra. We may proceed in a similar fashion to determine the Lie algebra of the proper orthocronous Lorentz group, which via the map defined above should coincide the one of SL2, C. For an infinitesimal Lorentz transformation Λ = I 4 + ˆT 23

24 we then have: Λ T ηλ = I 4 + ˆT T ηi 4 + ˆT = η + ˆT T η + η ˆT = η, 23 such that { LL + = ˆT MR 4 : ˆT = η ˆT } T η. 24 In components, we have: In particular, this yields the conditions: ˆT µ ν = η µα ˆT T β α η βν = η µα η βν ˆT βα. 25 ˆT 0 0 = ˆT 0 0 = 0, ˆT 0 i = ˆT i 0, ˆT i j = ˆT j i, 26 so that the symmetric ˆT 0 i sector has 3 independent components and the anti-symmetric ˆT i j sector has also 3 independent components. The dimension of the Lorentz and SL2, C algebras thus coincides and we can define the Lorentz group generators: Ĵ i = 2 ɛ ijkσ jk, ˆKi = σ 0i, 27 where σ µν ρ σ = η ρ µη νσ η ρ νη µσ 28 can be shown to satisfy the commutation relation: [σ µν, σ αβ ] = η µβ σ να + η µα σ βν + η νβ σ αµ + η να σ µβ. 29 The generators Ĵi are associated with spatial rotations and span the anti-symmetric components ˆT i j, while the ˆK i generators are associated with boosts and span the symmetric components T 0 i. One can also use the above commutator to show that these satisfy the same commutation relations as their unhatted SL2, C counterparts, which we leave as an exercise Representations of the Lorentz group Using the generators of either SL2, C or L +, we may define: J ± i = 2 J i ± ik i, 30 such that it is easy to show that: [J ± i, J ± j ] = ɛ ijkj ± k, [J + i, J j ] =

25 This defines two independent, i.e. commuting, SU2 algebras. Therefore, the irreducible representations of L + and SL2, C correspond to the pairs of representations j +, j, where j ± denotes the spin of each SU2 representation. EXAMPLES Left-handed Weyl spinors /2, 0: In this representation, J + i = iσ i /2 and J i = 0, corresponding to J i = iσ i /2 and K i = σ i /2, which is the fundamental representation of SL2, C as we have seen above. Using the map between SL2, C and the Lorentz group, we can write Lorentz transformations in this representation as: Λ L M = e 2 si +it i σ i, s i, t i R 32 and the vector space in this representation corresponds to 2-component left-handed spinors transforming as: ψ L x Λ L Mψ L x. 33 Right-handed Weyl spinors 0, /2: In this representation, J + i = 0 and J i = iσ i /2, corresponding to J i = iσ i /2 and K i = σ i /2. We can then write Lorentz transformations in this representation as: Λ R M = e 2 si it i σ i, s i, t i R 34 and the vector space in this representation corresponds to 2-component right-handed spinors transforming as: ψ R x Λ R Mψ R x. 35 Note that: Λ L M = e 2 si +it i σ i = e 2 si +it i σ 2σ iσ 2 = σ 2 e 2 si it i σ i σ 2 = σ 2 Λ R Mσ 2 36 such that the complex conjugate left-handed Weyl representation Λ L M is equivalent to the right-handed Weyl representation. We then say that a right-handed spinor transforms in the complex conjugate representation of SL2, C. Similarly, we have that: ΛL M = e 2 si +it i σ i = e 2 si it i σ i = Λ R M, 37 such that Λ R = Λ LM and the right-handed representation gives the contragredient representation of Λ L M. Dirac spinors /2, 0 0, /2: In this composite representation Lorentz transformations correspond to 4 4 matrices: acting on four-component Dirac spinors: Λ D M = Λ L M 0 0 Λ R M, 38 ψ D x Λ D Mψ D x. 39 Dirac spinors thus include both a left-handed and a right-handed component. 25

26 Vectors /2, /2: This coincides with the fundamental representation of the Lorentz group, with: Λ V M = e si ˆKi+t i Ĵ i, s i, t i R 40 denoting 4-dimensional matrices acting on 4-vectors: A µ x Λ V M µ νa ν x, 4 for which a particular case is A µ = x µ. It is common to denote the left- and right-handed spinor indices in the form: ψ La Λ L M b a ψ Lb, ψ Rȧ Λ R M ḃ ȧ ψ R ḃ. 42 The vectorial representation thus carries both a left-handed and a right-handed index, and can be equivalently expressed in spinor form as: A aȧ = σ µ aȧ A µx 43 Note that since Λ L M = M and Λ R M = M up to the equivalency established above, a Lorentz vector in spinor form transforms as: A aȧ M b a M ḃ ȧ A b ḃ = M b a M ḃ ȧ σ µ bḃa µ = Mσ µ M aȧ A µ = σ µ aȧ Λ V M µ νa ν, 44 using the map between the SL2, C and Lorentz groups, so we see that the vector transformation defined above is equivalent to the spinor transformation laws. Tensors: 2, 2 2, 2 The tensor product of two vector representations of the Lorentz group can be obtained from the tensor product of spin-/2 representations of SU2: 2, 2 2, 2 = 2 2, 2 2 = 0, 0 = 0, 0, 0,, 0, 45 where the first two terms correspond to the symmetric part of the tensor product and the last two terms correspond to the anti-symmetric part. The latter, in particular, corresponds to an anti-symmetric Lorentz tensor with two vector indices: F µν = F νµ. 46 We may define the dual tensor: F µν = 2 ɛ µνρσf ρσ, 47 26

27 from which we may define: F ± µν = 2 F µν ± F µν, 48 such that F µν = F + µν + F µν. It is easy to check that F ± µν = ±F ± µν. We thus obtain a decomposition of the antisymmetric tensor in terms of a self-dual and an anti-self dual component. Taking into account that under a Lorentz transformation: F µν Λµ ρ Λν σ F ρσ, Fµν Λµ ρ Λν σ F ρσ, 49 we can see that the self-dual or anti-self dual character of the tensor is preserved under Lorentz transformations, so that F + µν and F µν correspond to the, 0 e 0, irreducible representations of the Lorentz group. We note that the Maxwell tensor F µν = µ A ν ν A µ transforms in this representation, and its components describe the electric and magnetic fields in terms of the electrostatic potential and the vector potential in terms of the 4- vector potential A µ = φ, A. In this case, the duality transformation exchanges the components of the electric and magnetic fields E i B i, which constitutes a symmetry of the electromagnetic interactions. The symmetric part of the tensor product yields a Lorentz scalar in the 0, 0 representation, corresponding to scalar fields φ that are invariant under Lorentz transformations, and a symmetric and traceless tensor h µν in the, representation. In general relativity, perturbations of the metric about flat Minkowski space of this form correspond to gravitational waves and to the putative graviton particles that correspond to the latter in the yet unknown quantum formulation of the theory Spinor bilinears Weyl spinors are used to describe spin-/2 particles and the associated fields that generalize the wavefunction in the relativistic formulation of quantum mechanics. It is thus useful to discuss some of the Lorentz invariant quantities that we may construct from spinor fields and which may thus appear in the Lagrangian function that describes such particles. The most important terms in a Lagrangian are the quadratic terms, which lead to linear terms in the equations of motion via the Euler-Lagrange equations. These then correspond to kinetic and mass terms for the fields, with the former including field derivatives. Majorana mass term: ψ T Lσ 2 ψ L Λ L Mψ L T σ 2 Λ L Mψ L ψlλ T T LMσ 2 Λ L Mψ L ψl T σ2 Λ L Mσ 2 σ2 Λ L Mψ L ψ T Lσ 2 Λ L MΛ LMψ L ψ T Lσ 2 ψ L, 50 where we have used that Λ T L = ΛR M = σ 2 Λ L Mσ 2 as obtained in Eq. 36 and 37. A term of the form ψr T σ 2ψ R is also invariant under Lorentz transformations. Mass terms for spin-/2 left-handed or right-handed fields can be constructed with Majorana terms of this form provided that they are allowed by other symmetries. 27

28 Dirac mass term: χ L ψ R Λ L Mχ L Λ R Mψ R χ L Λ L MΛ RMψ R χ L Λ R MΛ RMψ L χ L ψ R. 5 A term of the form χ R ψ L is also Lorentz invariant for similar reasons. Note that Dirac mass terms combine the left- and right-handed parts of a Dirac spinor, which are given by distinct Weyl spinors, while the Majorana terms combine spinors in the same Weyl representation. In the Standard Model, as we will see later, the masses of the spin-/2 charged leptons and quarks corresponds exclusively to Dirac mass terms, while the neutrino masses may possibly have a contribution from Majorana terms. Weyl currents: j Lµ = χ L σ µψ L Λ L Mχ L σ µ Λ L Mψ L χ L Λ L Mσ µλ L Mψ L χ L M σ µ Mψ L χ L σ νλ ν µψ L Λ ν µj Lν. 52 Thus the left-handed Weyl current j Lµ transforms as a Lorentz vector, the same occurring for the analogous righthanded Weyl current j Rµ involving right-handed spinors. To obtain Lorentz invariant quantities, we may contract these currents with other vectors in a Lorentz invariant way. For example, the fermion kinetic term given by: χ L,R σµ µ ψ L,R 53 is a Lorentz scalar since: µ = x µ xν = x µ x µ x ν = Λ ν µ ν. 54 In a similar way, we can couple the fermionic current to the electromagnetic potential in a Lorentz invariant way, such that j L,Rµ describes the electric current associated with a charged fermion: η µν j µ L,R Aν η µν Λ µ αλ ν βj α L,RA β = η αβ j α L,RA β Parity and charge conjugation The parity operator acts on the Lorentz group generators in the adjoint representation as: P ĴiP = Ĵi, P ˆK i P = ˆK i, 56 as one can check by explicitly constructing the matrices associated to the generators. Hence, the angular momentum operators Ĵi, which generate spatial rotations, are pseudo-vectors that remain invariant under spatial reflections, while the boost generators ˆK i are normal vectors that change direction under parity transformations. From this we may also 28

29 conclude that: P J ± i P = 2 P Ĵi ± i ˆK i P = J i, 57 such that a parity transformation exchanges the left- and right-handed representations. Charge conjugation, which exchanges particles and the corresponding anti-particles, has similar consequences. For example, for a right-handed Weyl spinor ψ R we may define the conjugate spinor as: ψ c R σ 2 ψ R, 58 such that under a Lorentz transformation: ψ c R σ 2 Λ R M ψ R σ 2 σ 2 Λ L Mσ 2 ψ R Λ L Mψ c R. 59 Hence, the conjugate of a right-handed spinor is a left-handed spinor and, analogously, the conjugate of a left-handed spinor χ c L = σ 2χ L transforms in the right-handed representation. 2.2 Poincaré group The transformations of the Poincaré group include both Lorentz transformations and space-time translations, forming the group: P = { Λ, a : Λ L, a R 4}, 60 such that the action of the group elements in a coordinate system is given by: x µ x µ = Λ µ νx ν + a µ. 6 The group multiplication law is given by: Λ, a.λ 2, a 2 x = Λ, a.λ 2 x + a 2 = Λ Λ 2 x + a 2 + a = Λ Λ 2 x + Λ a 2 + a = Λ Λ 2, Λ a 2 + a x. 62 The identity element is naturally e = I 4, 0 and for each group element we have the inverse element: Λ, a = Λ, Λ a, 63 since Λ, a.λ, a = Λ, Λ a.λ, a = Λ Λ, Λ a Λ a = I 4, 0 = e

30 As should be familiar in quantum mechanics, spatial translations are generated by the linear momentum operator, while the Hamiltonian generates time translations: Ψx + ɛ, t + τ = Ψx, t + iɛ i Ψx, t iτ Ψx, t i t = Ψx, t + iɛ ˆPΨx, t iτĥψx, t, 65 for infinitesimal translations ɛ and τ in space and time, respectively. We can assemble these two operators in the 4- momentum relativistic operator generating space-time translations: ˆP µ = i µ. 66 The Lorentz group generators that we have determined earlier also admit a representation in terms of differential operators: ˆM µν = ix µ ν x ν µ, 67 satisfying the commutation relations: [ ˆM µν, ˆM µν ] = i η µβ ˆMνα + η µα ˆMβν + η νβ ˆMαµ + η να ˆMµβ, 68 which, apart from a factor of i, are the same commutators as for the σ µν generators in Eq. 29. We can also easily check that: The Poincaré algebra is thus given by: [ ˆM µν, ˆP ρ ] = i η νρ ˆPµ η µρ ˆPν = i σ µν σ ˆP ρ σ, [ ˆP µ, ˆP ν ] = LP = span ˆP µ, ˆM µν. 70 We note that, as previously defined: Ĵ i = 2 ɛ ijk ˆM jk = ɛ ijk ˆx j ˆPk 7 are the angular momentum operators generating spatial rotations, while ˆK i = ˆM 0i are the boost generators. An important quantity to define is the Pauli-Lubanski vector: Ŵ µ = 2 ɛ µνρσ ˆM νρ ˆP σ, 72 with the following properties: µ Ŵ µ ˆP [Ŵµ, ˆP ] ν [Ŵµ, ˆM ] αβ [Ŵµ Ŵα], = 0, 73 = 0, = i η µβ Ŵ α η µα Ŵ β, = iɛ µαβν Ŵ β ˆP ν, 74 30

31 The Lorentz invariant quadratic operators: ˆP 2 = ˆP µ ˆP µ, Ŵ 2 = ŴµŴ µ, 75 commute with all the generators of the Poincaré group, thus constituting the Casimir operators of the Poincaré group and from which we can label the different irreducible representations. We may write, in particular: Ŵ 2 = 4 ɛ µνρσɛ µ ˆM νρ σ αβ γ αβγ ˆP ˆM ˆP = 4 [ η ναη ρβ η σγ η ργ η σβ η νβ η ργ η σα η ρα η σγ η νγ η ρα η σβ η ρβ η σα ] ˆM νρ σ αβ γ ˆP ˆM ˆP = 2 ˆM αβ αβ ˆPγ ˆM ˆP γ + ˆM αβ γ αγ ˆPβ ˆM ˆP = 2 ˆM αβ αβ ˆM ˆP 2 + ˆM αβ αγ ˆM ˆPβ ˆP γ, 76 where we used the form of the contraction of two Levi-Civita tensors and the commutation relations 69. As we know from the theory of Special Relativity, the Casimir opertor ˆ P 2 = m 2 for a particle of mass m, so that different irreducible representations of the Poincaré group will correspond to particles with different mass. We also have that the state of a particle with mass m and 4-momentum p is an eigenstate of the ˆP µ generator: ˆP µ m, p, σ = p µ m, p, σ 77 where we have denoted by σ any other quantities that characterize the state in a given representation and that we wish to determine. Under a Lorentz transformation in a given representation RΛ, 0 we obtain a state with 4-momentum Λp: RΛ, 0 m, p, σ = m, Λp, σ. 78 Let us check that this is also an eigenstate of ˆP µ. First, let us note that ˆP µ, being a group generator, transforms in the adjoint representation and is a Lorentz vector: Thus, RΛ, 0 ˆP µ R Λ, 0 = Λ ν µ ˆP ν. 79 ˆP µ RΛ, 0 m, p, σ = RΛ, 0R Λ, 0 ˆP µ RΛ, 0 m, p, σ = RΛ, 0Λ ν µ ˆP ν m, p, σ = Λ ν µ p ν RΛ, 0 m, p, σ = Λp µ m, Λp, σ. 80 This means that, in constructing the states in the different representations of the Poincaré group, we may take as reference state a state with a given momentum, such that states with an arbitrary momentum can be obtained from this reference state by performing a Lorentz transformation. The choice of this reference momentum will be different for particles with and without mass, such that we must analyze these two cases separately. 3