Symmetries of Particles and Fields 2018/19. 1 Introduction. 2 Lie Groups. 2.1 Metric spaces. Example. In ordinary three-dimensional (real) space:

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1 Introduction Symmetries of Particles and Fields 8/9 The laws of physics are invariant under many different symmetries: Parity, time reversal, charge conjugation P, T, & C - discrete symmetries Translations and rotations Galilean invariance Lorentz transformations special relativity Internal symmetries e.g. isospin, flavour SU3 Gauge symmetries Electrodynamics Maxwell, the Standard Model: SU3 SU U We shall study each of these in turn. The overall structure of the course is: Introduction to Lie groups and Lie algebras Space-time symmetries; constructing actions with symmetries Compact groups representations Applications in physics: Noether s theorem; isospin; quark model; spontaneous symmetry breaking; chiral symmetry; gauge theories; QCD; Higgs mechanism; electroweak theory Lie Groups Definition A group is a set of elements G = {g} with a composition law G G G i.e., the product of any two group elements is itself a group element with the properties. Associativity: g g g 3 = g g g 3. Unit element or identity element : g = g = g 3. Inverse: gg = g g = Definition A finite group is a group with a finite number of elements. Example. Z, elements {+, }. C, P, T charge conjugation, parity, time reversal are all Z groups. Z N, elements {e πin/n : n =,..., N}. Dihedral groups D N etc. See Symmetries of Quantum Mechanics course. Definition 3 An infinite group is a group with an infinite number of elements, which we will assume to be labelled by a set of continuous parameters {α}. We write G = {gα}. The dimension of the group is the number of independent parameters. Example. In ordinary three-dimensional real space: Spatial translations, x i x i + a i ; dim G = 3. Spatial rotations, x i R ij n, θ xj ; R T R = ; dim G = 3 an axis n parameters and one angle φ. Definition 4 A topological group is a continuous group, for which the parameters {α} constitute a topological space admits a definition of nearness. Very important: you don t need to know topology for this course, but we choose use to use the formal definitions here and only here. A topological space is a set of points, together with a set of neighbourhoods for each point, which satisfy a set of axioms relating points and neighbourhoods. [See books for details.] The group is connected if the topological space is connected, and compact if the topological space is compact. For a metric space compact means closed and bounded Heine Borel theorem. The opposite of a topological group is a discrete group. Example. Translations: connected but not compact. Rotations + reflections: compact but not connected: det R = ± reflections give ve det. Definition 5 A Lie Group is a topological group, for which the group multiplication has an analytic structure: if gα i = gβ i gγ i then α i = φ i β i, γ i where the φ i are analytic functions of the β i and γ i.. Metric spaces Metrics will feature in many parts of this course; let s define or revise their properties here. Consider a real vector space with a basis {x µ }, µ =,..., N. Define a real symmetric matrix called the metric g µν, then define x g µν x µ x ν using the Einstein summation convention. Now choose a basis in which:. g µν is diagonal, g µν = diagλ,..., λ N where λ i are the real eigenvalues of g µν.. The eigenvalues are either zero or ±. Under the rescaling x µ sx µ, g µν s g µν, x stays fixed. We can rescale each λ i separately. Since s > the signs cannot change, so in this basis g µν = diag,,...,,,...,,. We call this the canonical basis. For example, in three dimensional space g ij = δ ij, x = x + x + x 3 Euclidean space. In four dimensional space-time g µν = diag,,,, x = x + x + x 3 x 4. space time

2 We will generally write x 4 x and use a space-time metric with the opposite signature: g µν = diag,,, whereupon x = x x + x + x 3. time space A metric with a mix of + and elements is called an indefinite metric. Metrics with zero eigenvalues occur in the theory of Lie algebras - see later. Raising and lowering indices: if det g µν, i.e., the metric has no zero eigenvalues, then g µν is invertible. We can then use the metric to raise and lower indices: x µ g µν x ν and so forth. Then x = x µ x µ = x µ x µ = g µν x µ x ν = δ µ ν x µ x ν. where we used x µ = δ µ ν x ν relabelling of indices, with δ µ ν = δ ν µ = { µ = ν µ ν We can also raise indices on the metric: g µν = g µα g νβ g αβ g νβ g αβ = δ ν α, i.e., g µν is the inverse of g µν. N indices {}}{ Similarly for the totally antisymmetric Levi-Civita symbol ɛµν αβ and derivatives with respect to x µ µ x µ = g µν = g µν ν x ν ɛ µν αβ = g µµ g νν g αα g ββ ɛ µ ν α β, exercise: check that xα x β = δα β, etc. If the metric is definite all the elements have the same sign, with no zeros we can choose a basis in which g µν = δ µν. In this case we can ignore the distinction between upper and lower indices.. Matrix Groups A useful way of defining groups is through the linear transformation x µ D µ ν α x ν µ, ν =,..., N. D µ ν α is the µν th element of the invertible N N matrix Dα. This gives: The general linear groups GLN, R or GLN, C for real and complex vector spaces, respectively, and the The special linear groups SLN, R or SLN, C if det D =. If D GLN, R and D T D = DD T =, we get the orthogonal groups, ON. If D SLN, R and D T D = DD T =, we get the special orthogonal groups, SON. If D GLN, C and D D = DD =, we get the unitary groups, UN. If D SLN, C and D D = DD =, we get the special unitary groups, SUN. You can think of x µ as the µ th component of a N-component column vector x. We also find the pseudo orthogonal groups e.g., SO3,, the Lorentz group: the metric on the space {x µ } is indefinite, as it has 3 plus signs and minus sign. The pseudo unitary groups. These preserve an indefinite sesquilinear metric involving complex conjugation. We will not mention them again. The symplectic groups SpN. See tutorial question.5. The exceptional groups G, F 4, E6, E7, and E8. The orthogonal and unitary groups leave the metric δ µν invariant: δ µν G Dµ α D ν β δ αβ = δ µν. Similarly the special groups leave the totally antisymmetric Levi-Civita tensor ɛ µν α invariant, ɛ µν α G Dµ µ D ν ν D α α ɛ µ ν...α = det D ɛ µν...α = ɛ µν...α since det D =.3 One Dimensional Groups Example. Translations on the real line, x x + a. Clearly if gagb = ga + b then the group is Abelian: gagb = gbga. There is a simple unitary representation ga = e iaλ with λ a fixed parameter because gagb = e iaλ e ibλ = ga+b. Theorem All one dimensional Lie groups are such that a can be chosen such that gagb = ga + b. Proof. The proof, originally due to Aczel, is a little tricky. Using associativity and analyticity of the composition law, we may write: gx gy gz = gx gy gz associativity gx gφy, z = gφx, y gz analyticity φx, φy, z = φφx, y, z analyticity again Differentiate the last expression with respect to z: φx, y φy, z y = y=φy,z z φφx, y, z z Now set z =, noting that since g =, then φy, = y, to obtain φx, y y ψy = ψφx, y where ψy = We can solve this differential equation by rewriting it as and integrating with respect to y, to obtain φx, y = ψφx, y y ψy We shall see later that g µν = δ µν for compact groups. ρφx, y = ρy + cx φy, z z z= 3 4

3 where x dt ρx = ψt, ρ = and cx is an arbitrary function of x, an integration constant. To verify this is the correct solution, differentiate it with respect to y, which gives equation * directly. We can determine cx by setting y =, which gives ρx = + cx, and hence ρφx, y = ρx + ρy Finally, if we change the parameterisation of the group element gx ḡρx, we have ḡρx ḡρy = gx gy = gφx, y = ḡρφx, y = ḡρx + ρy Thus the group composition law is additive when written in terms of the new barred parameters. Corollary All one dimensional Lie groups are Abelian. Example. The rotation group SO x cos φ sin φ y sin φ cos φ x. y Let z = x + iy, then z e iφ z. But e iφ U. So U and SO are isomorphic. These groups are distinct from the translation group: φ [, π so U, SO are compact, while a R, so the translation group is not compact as a,. Note: a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic. From the standpoint of group theory, isomorphic groups have the same properties and need not be distinguished. Source: Wikipedia! Theorem All compact Abelian Lie groups are isomorphic to U U U. It follows that in this course most of the groups will be non-abelian: g g g g. Corollary The finite cyclic groups C n or Z n = {e mπi/n, m =,..., n } are subgroups of U..4 Representations Any set of matrices D µ ν g satisfying the group composition law, D µ ν g D ν ρ g = D µ ρ g g constitutes a representation of G. Representations may be reducible or irreducible just as for finite groups. Note that if D is a representation, so is D. A representation for which D = D is a real representation. Anything which transforms under G may be classified according to the representation under which it transforms: 5 Scalars: s s i.e., transforms trivially. Vectors: v µ D µ ν g v ν where Dg is the defining or fundamental representation. Tensors: t µν... α D µ µ D ν ν t µ ν... β or t α D α β g t β, where D α β is some larger possibly reducible representation of G. Invariant Tensors: t µν... t µν... under G. The components of an invariant tensor are unchanged by the action of the group. Theorem 3 If G is compact, then for every representation there is an equivalent unitary representation just as for finite groups: Maschke s Theorem. Proof. We limit ourselves to a sketch of the proof. Starting from the usual inner product x, y, we define a new inner product x y g G Dg x, Dg y where Dg is a representation of the group G acting on x, and the summation integration for continuous groups is over all elements of G. Let h G be an arbitrary group element. Using the rearrangement theorem 3 we can write the new inner product as x y = Dgh x, Dgh y = Dg Dh x, Dg Dh y g G g G so D D = with this inner product. = Dh x Dh y = x D hdh y The proof breaks down if G is not compact because the sum/integral on the RHS is not well defined. The action of a group on a Hilbert space must be through a unitary representation, to conserve probability: = ψ i ψi = ψ i ψ lδ il = D i j ψ j D l k ψ k δ il = ψ j ψ kd i j D l k δ il = ψ j ψ kd D jk D i j D i k = δ j k, i.e., D D =, where as usual we have denoted the unit matrix by. 3 Lie Algebras 3. Generators Consider a Lie group G = {gα}, and choose α a, where a [, Dim G], such that g =. When the α a are small, gα is close to the unit element/matrix, and we can use Taylor s theorem to write 4 gα = + iα a T a + Oα this uses analyticity, where T a i gα α a. α= 3 Roughly speaking, the rearrangement theorem states that if we multiply all the elements g i in the group by a fixed element h, we recover all the elements g i once and once only, but in a different order, i.e., a rearrangement. 4 NB We re using the summation convention here. Take care to contract only upper indices with lower indices. 6

4 The matrices {T a } are the generators of the group. The factor of i is a useful convention: if Dg is a unitary representation, the corresponding representation of the generators is hermitian when α a are real: = D D = iα a T a T a + Oα T a = T a. The number of generators = dim G. It can be argued that antihermitian generators are more natural as they avoid the factors of i. Example. SUN: g g = gg =, det g =, so T a = T a, tr T a = the latter follows from the constraint det g = and det A = exp tr ln A. Thus the generators are hermitian and traceless. For SU, we generally choose the Pauli matrices: i σ = σ = i σ 3 =. For SU3, the generators are the Gell-Mann matrices {λ a : a =,..., 8} see later. Since UN doesn t have the constraint det D =, the generators of UN are {, T a } where T a are the SUN generators. Example. SON: g T g = gg T =, det g =. Since g is real, we generally omit the i s in the expansion of gα, and in the definition of the generators, whereupon Ta T = T a, tr T a =, i.e., the generators are real antisymmetric. For SO. For SO3, we may choose. The additional condition tr T a = is automatic as the generators are antisymmetric, so ON and SON share the same generators. Note: if we choose hermitian generators for SON, their elements are purely imaginary. 3. Commutators Consider f = ghg h with f, g, h G. If either g = or h = then f =. Take g, h infinitesimal: 5 where γ = Oαβ and α T α a T a, etc. g = + iα T + iα T + Oα 3 h = + iβ T + iβ T + Oβ 3 then f = + iγ T + iγ T + Oγ 3 The quadratic terms follow from Taylor-expanding g to second order in α, and using g α = gα or gα = gα to express the terms with second derivatives in the form shown. This is a useful exercise do it! 5 Strictly, we take the parameters α a, β a. 7 In fact, by substitution exercise [α T, β T ] = i γ T + Oα3, β 3 All terms of Oα, Oβ, Oα, and Oβ cancel exercise. Since this is true for arbitrary α a, β b, we must have a relation γ c = c c abα a β b, for some set of numbers c c ab, and hence [T a, T b ] = it c c c ab where there is an implicit sum over the index c [, Dim G]. The quantities c c ab are called the structure constants of the group: they define the Lie algebra L G of the Lie group G. We use lower case letters to denote the Lie algebras of SUN, SON, etc, by sun, son, etc. 3.3 Exponentiation Consider a one dimensional subgroup G G, {gt} = {gαt} labelled by a single parameter t. We can choose t R such that so α a =. Differentiate with respect to s: gs gt = gs + t; g = by Theorem g s gt = g s + t and set s =, which gives a differential equation for gt that we can solve easily: d dt gt = g gt gt = exptg. But, we can also write g = g dα a α a α= dt = iγ a T a where γ a dαa t= dt, t= so if g G G, gt = expitγ a T a, therefore every element of G that can be connected to the identity has this form. But the set {e iβa T a } has dimension equal to dim G and coincides with G for β a infinitesimal. So if we know the generators we know all parts of the group connected to the identity. Furthermore gh = exp{iα T } exp{iβ T } = exp{iγ T } for some γ, because gh G. Now the Baker Campbell Hausdorff formula tells us that for any two matrices A and B, expa expb = expa + B + [A, B] + 3! [A, [A, B]] + [[A, B], B] + so if we know L G we can evaluate all the commutators and thus γα, β at least in principle. To summarise: Lie group Lie algebra by linearisation. Lie algebra connected part of a Lie group by exponentiation. 8

5 Note that G L G uniquely, but not vice versa: several groups may have the same Lie algebra. These groups are identical locally in the neighbourhood of, but not globally. Fortunately, for a given L G there is only one connected group G which is simply connected no holes: this is called the universal covering group. Example. ON, SON have the same generators, so on = son the Lie algebras are isomorphic. However, for ON, since DD T =, det D =, so det D = ±, while for SON we have det D = + by definition. We can t get from det D = to det = + by a continuous transformation, so ON is not connected, whereas SON is connected but not simply see later. We write SON = ON/Z. For rotations, N =, 3, think of Z as the parity. For example, for SO, choosing the generator to be real antisymmetric T =, T =, and hence e φt = cos φ + T sin φ exercise. The matrices and O, but not SO. The simply connected covering group of SON is called SpinN. The simply connected covering group of SO3 is SU see tutorial question Properties of Structure Constants Recall the Lie algebra L G of a Lie group G: [T a, T b ] = it c c c ab then the Jacobi identity can be written A a d e A b e c A b d e A a e c = ia e d c ce ab, where we used the anti-symmetry of the structure constants, i.e. c e ca = c e ac in the second term on the left-hand side LHS, and similarly for the first term on the right-hand side RHS. In matrix notation [A a, A b ] = ia e c e ab. 3.5 The Adjoint Representation If a vector v transforms under G : v gv, then hv ghv = ghg gv. So if v gv, h ghg gives the action of the group on itself. Thus consider h = ghg with h infinitesimal, then h is also infinitesimal: h = + iα a T a + h = + iβ a T a + where β a = D a bg α b for some D a bg by linearity = + it a D a bg α b + Relabelling indices in the last expression a b, it follows from the first and third lines above that gt a g = T b D b ag. Pictorially: v h hv g gv ad g = ghg g The matrices D b ag form a representation the adjoint representation of the group G: ghv. Clearly c c ab = c c ba by anti-symmetry of the commutator: [T a, T b ] T a T b T b T a = [T b, T a ]. D b a = δ b a, T b D b cg D c dh = g T c g D c dh = h g T d g h = h g T d hg = T b D b dhg. Under rescaling T a λt a, c c ab λc c ab. 3. The c c ab are real. Proof. We know already that if Dg is a representation of G, then D g is also a representation. So if T a are a representation of L G, Ta are also a representation. But [ Ta, Tb ] = i T c c c ab, so c c ab = c c ab. 4. The Jacobi identity: [T a, [T b, T c ]] + [T b, [T c, T a ]] + [T c, [T a, T b ]] = Exercise: verify that this identity holds for any three square matrices T a, T b, T c. Substituting from the Lie algebra, we find exercise c d ae c e bc + c d be c e ca + c d ce c e ab =. so DgDh = Dhg as required. Note that with the definitions used here, D acts on the left if g acts on the right. The generators in this representation are just the structure constants described above. To verify this, set g = + iα a T a, so g = iα a T a, and D c a = δ c a + iα b A b c a then so A b c a = icc ba as required. gt a g = T a + iα b T b T a T a T b + Oα = T a iα b T c ic c ab = T c δ c a + iα b ic c ba, It follows that the adjoint representation is a real representation recall that the structure constants are real, so the elements of A b are i times a real number, which we call a real representation. Note that the elements of A b would actually be real we used anti-hermitian generators! 5. The structure constants form a representation of L G, known as the adjoint representation. If we define Dim-G Dim-G Dim-G matrices A a with elements A a b c icb ac 9

6 Example. SU and SO3. Take the generators of SU as σ i: then as σ σ = iσ 3 etc., [ σ i, σ j] = iɛ ijk σ k, so the structure constants are just ɛ ijk. Consider g = + i ω σ + Oω. Then gσ k g = + i ω σσ k i ω σ = σ k + i ω i[σ i, σ k ] = σ k + ω i ɛ kim σ m, i.e. g σ g = σ + ω σ + Oω This is clearly an infinitesimal rotation SO3, i.e. an infinitesimal rotation through angle ω about the axis ω in R 3. It follows that the adjoint representation of su is the defining representation of so3, thus that su = so3. In fact SO3 = SU/Z : SU is the covering group of SO3 see tutorial question Invariant Tensors An invariant tensor d aa...a n is any tensor that remains invariant under the action of the group, i.e., d a...an = d a...a n Da a g D a a g D a n an g. The simplest way to construct invariant tensors is to consider traces of products of generators d a...a n trt a T a T an G trt a T a T a n D a a D a n an = tr gt a g gt a g gt an g = tr T a g g T a g g T an g g = d a a n i.e., d a a n is an invariant tensor for any representation T a, including the adjoint representation A a. Example. c d bc is invariant use the Jacobi identity. 3.7 Killing Form An important invariant tensor is the Killing form. This is a metric on the group space defined by g ab tra a A b = A a c d A b d c = cc ad c d bc = g ba cyclic property of the trace Since the structure constants are real, g ab is a real symmetric matrix - this is the advantage of using the adjoint representation. Thus we may choose a basis in which it is diagonal g ab = µ a δ ab no sum on a. The eigenvalues µ a of g ab are therefore real, but may be positive, negative, or zero. Indeed, for Abelian groups g ab = because the structure constants are all zero. Convince yourself of this. Under the rescaling T a λt a, with λ R, then g ab λ g ab with λ >. By rescaling each T a separately we may reduce g ab to the canonical form diag+,..., +,,...,,,...,, i.e., µ a =, ±, a. Example. SU: g ij = ɛ ikl ɛ jlk = δ ij. If the generators were rescaled to ɛ ijk we would have g ij = δ ij. We can use the Killing form to lower indices: for example, we may define structure constants with all lower indices: c abc g ad c d bc. This is also an invariant tensor because we can use the Lie algebra to write it in terms of the trace of products of generators: tra a [A b, A c ] = i tra a A d c d bc = ig ad c d bc = ic abc. Since 6 tra a [A b, A c ] = tr[a a, A b ]A c = tr[a c, A a ]A b. it follows that c abc is totally antisymmetric on all three indices. A second invariant tensor with three indices is d abc = tra a {A b, A c } which is totally symmetric. 7 We need consider only one tensor d a a n for each n > 3, the one which is totally symmetric because antisymmetric parts can always be expressed in terms of c abc and tensors of lower degree, by using the commutation relations. Definition 6 The number of independent invariant tensors excluding c abc is called the rank of the group. 4 Simplicity We may consider a Lie algebra L = {T a : a =,..., l} as a vector space spanned by the generators. Then Definition 7 The subspace S L is a subalgebra of L if the commutator [s, s ] S for all s, s S. In words, the commutator of any two elements in the subspace also lies in the subspace. S will be spanned by a new set of generators {S i : i =,..., l }. S is a proper subalgebra provided < l < l. Definition 8 The subalgebra S L is an invariant subalgebra or ideal if [s, t] S for all s S, t L, i.e., if S absorbs L or S knocks L into S. This is analogous to the definition of a normal subgroup: ghg H, h H, g G, H G. Definition 9 The Lie algebra L is simple if it has no proper invariant subalgebras, whereas L is semi-simple if it has no proper Abelian invariant subalgebras. Simple and semi-simple Lie groups are those with simple or semi-simple algebras. Clearly all simple groups/algebras are also semi-simple. 6 tr X[Y, Z] = trxy Z XZY = tr XY Z tr XZY = tr XY Z tr Y XZ = trxy Z Y XZ = tr[x, Y ]Z, etc. 7 The anti-commutator {A, B} AB + BA.

7 4.. Non semi-simple algebras Consider an algebra which is not semi-simple. Then it has a proper Abelian invariant subalgebra S L. Let s take a basis {S i } for S, and extend it to a basis {S i, P p } = {T a } for L; where the indices a, b, c T, i, j, k S, and p, q, r P, i.e. the set {P p } contains all the generators that are in L, but not in S. Then. Since S is Abelian: [S i, S j ] = it a c a ij = by definition, therefore c a ij = ;. Since S is invariant: [S i, T a ] = it b c b ia S implies c p ia = and, by anti-symmetry of the commutator, c p ai =. Then, using. twice and. once, g ia = tra i A a = A i c d A a d c = cc id c d ac = c k id c d ak = c k ij c j ak =, so g ab has the form g pq 4. Cartan s Criterion. It follows that g ab has zero eigenvalues, and det g ab =. A Lie algebra is semi-simple iff 8 the Killing form is invertible, i.e., det g ab. We denote the inverse by g ab, so that g ac g cb = δ a b. g ab is now a pseudo-riemannian metric. For semi-simple groups we may choose a basis in which g ab and g ab are diagonal and both take the form diag,...,,...,, i.e., all eigenvalues are ±. 4. Casimir Operators When L is semi-simple, we may define the quadratic Casimir operator and similarly the higher Casimir operators C g ab T a T b C n d a an T a T an where d a an = g aa g a na n d a a n, n 3. There are as many independent Casimir operators as there are independent symmetric invariant tensors, i.e., the rank of the algebra. The Casimir operators are useful because they commute with the generators: [C n, T a ] =. To show this, note that 8 iff means if and only if. gc n g = d a an gt a g gt an g = d a an T a D a a T a n D a n an = d a an T a T an = C n where the first expression in the last line follows because d a a n is an invariant tensor. Taking g = + iα a T a with infinitesimal α a gives the result: + iα a T a C n iα a T a = C n C n = C n + iα a [ T a, C n] Since Casimir operators are constructed from products of generators, it follows that they also commute with each other: in fact they constitute a maximally commuting set. They are thus useful for labelling representations. This is analogous to using a complete set of mutually commuting operators to label states in quantum mechanics. For semi-simple Lie algebras it is possible to show using Schur s lemma that g ab tr T a T b = CR g ab for some real number CR which depends only on the representation R of the generators T a. Taking the trace of the expression for the definition of the quadratic Casimir operator above, and using this result gives So CR = tr C / dim G. tr C = g ab CR g ab = δ a a CR = Dim G CR For the adjoint Ad representation, CAd = with our normalisation. One often sees instead the normalisation CFun = or or for the fundamental Fun representation. 4.3 Cartan Decomposition Consider a semi-simple Lie algebra L that is not simple: it has a proper invariant subalgebra S L. Use the metric g ab to define an othogonal subspace P, again using indices i, j, k S, and p, q, r P : g ip = tr A i A p = for all A i S, A p P. Now tr A i [A j, A p ] = tr[a i, A j ]A p = because [A i, A j ] S Thus [A j, A p ] P, i.e., [S, P ] P. Because S is invariant, we have [S, P ] S. But S P =, so [S, P ] = all elements of S commute with all elements of P, and therefore L decomposes into a direct sum L = S P. It is easy to show that P is also a subalgebra of L: so [P, P ] P as required. tr[a p, A q ]A i = tr A p [A q, A i ] = [A q, A i ] = because L = S P It is now easy to prove the following theorem: Theorem 4 A semi-simple Lie algebra L is a direct sum of mutually commuting simple Lie algebras: L = L L L n, where the L i are simple. Proof. If L is not simple, use the above result to decompose it as L = S P. 3 4

8 Both S and P are semi-simple otherwise L would not be. If either is not simple, repeat the procedure. Since L is finite dimensional, this procedure obviously stops. Example. so4 = su su see tutorial question 3. sl, C = su su see later. The decomposition of the algebra induces a similar decomposition of a semi-simple group G = G G G n /Z where the G i are simple, and Z is the centre of G defined as Z = {g : gg = g g, g G}; i.e. the set of all group elements g that commute with themselves and with all other group elements. Clearly Z is discrete, as otherwise G would not be semi-simple. Example: SO4 = SU SU/Z. 5 Compact Groups I If the Lie group G is compact, the generators of L G may be taken to be hermitian because there are unitary representations. This has far-reaching consequences. Taking the adjoint representation hermitian, A a = A a, so thus A a b c = icb ac = ic c ab, g ab = tr A a A b = A a d c A b c d = i cc ad ic c bd = c c ad c c bd. Theorem 5 If G is compact L G may be decomposed as L G = L L L n where L is Abelian so L = u u and the Li are simple. Proof. If L G is not semisimple, it has a proper Abelian invariant subalgebra S L, so g ia = for S i S, T a L derived on page 3. But we have just shown that for hermitian A a, g ia = c b ic c b ac, so g ii = c b ic = not summed over i; i.e., c b ic = i, b, c, whence [S, L] =. It follows that L G = L L where L is Abelian and L is semisimple, and thus can be decomposed into simple algebras as previously shown. Theorem 7 If G is compact and semisimple, tr T a T b = CRδ ab for all irreducible representations. The positive real number CR is the Casimir of the representation R. Proof. Omitted uses Schur s lemma. 5. The su algebra This is the angular momentum algebra familiar from quantum mechanics. For su = so3 [T i, T j ] = iɛ ijk T k, g ij = δ ij, so it is compact and semisimple. Define the step operators T ± = T ± it, then [T 3, T ± ] = ±T ± [T +, T ] = T 3 [T +, T + ] = [T, T ] =. The quadratic Casimir is T = T + T + T 3 = T T + + T 3 + T 3, and [T, T i ] =. There are no other Casimir operators so su has rank : T and T 3 form a complete commuting set. T has eigenvalues jj +, j N: use to label representations. T 3 has eigenvalues m = j, j,..., j, m Z: use to label states within representation. If T 3 jm = m jm, T 3 T ± jm = T ± T 3 ± jm = m ± T ± jm, while T jj = T T + + T 3 + T 3 jj = jj + jj because T + jj =. jj is the state of greatest weight. Since we are interested in finite dimensional representations there is also a state of least weight T j m =, for which T j m = T + T + T 3 T 3 j m = m m j m. Since T is a multiple of the unit matrix we must have m m = jj +, so m + j = m j = m + j m j, and m = j since m j. Representations: 3 fundamental 3 adjoint 4 Combining representations: start with the state of largest m and work down : = 3, 3 3 = 5 3, etc. We will show later that all compact semisimple algebras contain su subalgebras which allow for similar constructions of their representations, and composition of representations. 3 So we need consider only semisimple or indeed simple compact groups. NB, this decomposition does not apply to non-compact groups because it relied on the generators being hermitian. Theorem 6 If G is compact, the Killing form g ab is positive definite has non-negative eigenvalues. If L G is also semisimple, we may choose a basis in which g ab = δ ab distinctions between upper and lower indices can thus be ignored. Proof. For any real vector x a g ab x a x b = c c ad c c bd x a x b = x a c c ad all eigenvalues. If L G is semisimple, det g ab, so g ab x a x b > all eigenvalues >. Rescale the generators so that g ab = diag,...,. 5 6

9 6 Space-time Symmetries In this section we shall study the symmetry groups of the space-time of special relativity, namely the Lorentz and Poincaré groups. 6. Kinematics Let us choose a set of space-time coordinates x µ = x, x i, with x = ct. We have a metric tensor 9 η µν which is real, symmetric, invertible, det η µν, with inverse η µν : η µν η νσ = δ µ σ. We can choose an orthonormal frame in which η µν = diag,,,. Space-time is noncompact. Define x µ η µν x ν, so x µ = η µν x ν etc. x µ = x, x i. We use the metric to define length of x µ : x η µν x µ x ν = η µν x µ x ν = x µ x µ = x µ x µ = x x. 6. Lorentz Group Under a change of frame x µ x µ = Λ µ ν x ν so x µ x µ = Λ µ ν x ν. The postulates of Special Relativity are The speed of light c is the same in all frames. Space-time is isotropic the laws of physics are the same in all reference frames. These lead to: x = η µν x µ x ν = η µν Λ µ ρ Λ ν σ x ρ x σ = η ρσ x ρ x σ = x x x Since this expression is true for all x µ then thus the transformation preserves the metric. η µν Λ µ ρ Λ ν σ = η ρσ, In matrix notation, we may write this is as Λ T η Λ = η. Taking the determinant gives: det η = det Λ T det η det Λ det Λ = ±. In an orthonormal frame, choosing ρ = σ = gives Λ Λ i = Since Λ i, we must have Λ, so Λ or Λ. Thus the Lorentz group O3, has 4 disconnected pieces, connected by time reversal T : x x and parity P : x i x i. If we take Λ SO + 3, i.e. det Λ = +, Λ, we have a proper orthochronous Lorentz transformation. We get the other three pieces of the group by taking the products: P Λ, T Λ, P T Λ. 9 NB the metric tensor η µν is not the Killing form of the Lorentz group see later. An orthochronous Lorentz transformation preserves the direction of time. The proper orthochronous transformations are: Rotations: Λ = R Boosts: e.g., Λ =, where R is 3 3 and satisfies R T R =, det R =, so R SO3. cosh η sinh η sinh η cosh η SO +,. We get the usual form of the Lorentz boost along the x axis in units where the speed of light c = if we take cosh η = v, sinh η = v v. The most general orthochronous Lorentz transformation is specified by 3 rotations about orthogonal axes, and 3 Lorentz boosts in orthogonal directions. Therefore SO3, has dimension Lorentz algebra Consider an infinitesimal Lorentz transformation on the four-vector V µ, i.e. in the defining or vector representation of the Lorentz group: V µ = Λ µ νv ν = δ µ ν + ω µ νv ν, i.e., δv µ = ω µ ν V ν = ω µν V ν where ω µν = ω νµ is real and anti-symmetric, so it has 6 real parameters: ω ij = ω ji, i, j =,, 3, give rotations ω ij = ɛ ijk θ n k, and ω i ν i give boosts. In a general representation D, we write an infinitesimal transformation as D + ω = + i ωαβ M αβ where M αβ = M βα with α, β =,,, 3, are a set of six hermitian generators. It may help to think of a = α, β as a single index a =,,... 6, which labels the six generators M αβ = M a of the Lorentz group, i.e. similar to T a in our previous discussions. In the defining vector representation see equation : M αβ µ ν δ = i α µ η βν δ µ β η αν check this explicitly using equations and 3 In words, this is the µ ν matrix element of the α, β th generator M αβ in the defining representation of the Lorentz group. To find the commutation relations [M µν, M αβ ], noting that Λ µν = Λ T µν for an infinitesimal Lorentz transformation, we use the group properties in an arbitrary representation DΛ to write DΛ D + ω DΛ = D + ΛωΛ T DΛ + i ωαβ M αβ DΛ = + i Λ α µ ω µν Λ β ν DΛ ω αβ M αβ DΛ = ω αβ Λ µ α Λ ν β M µν In the last two lines, we raised and lowered some pairs of indices and relabelled dummy indices on the RHS α µ and β ν to give ω αβ on both sides. This holds for all infinitesimal parameters ω αβ, so For example V µ = x µ. DΛ M αβ DΛ = Λ µ α Λ ν β M µν. M αβ 7 8

10 Now take Λ = + ɛ and expand through Oɛ. Substituting DΛ = + i ɛµν M µν into the above expression, and cancelling the leading M αβ from both sides, gives i ɛµν [M µν, M αβ ] = ɛ µ α δ ν β M µν + δ µ α ɛ ν β M µν = ɛ µ α M µβ ɛ β ν M αν = ɛµν M µβ η να M νβ η µα M αν η µβ + M αµ η νβ where the quantity in parentheses is explicitly anti-symmetric under µ ν and α β, and we used the metric η µν to raise and lower indices. We also used the anti-symmetry ɛ µν = ɛ νµ, several times. The result holds for all ɛ µν, which is also anti-symmetric, so which defines the algebra of so3,. [M αβ, M µν ] = i M αµ η βν M αν η βµ M βµ η αν + M βν η αµ 4 Note that the generators M ij, i, j =,, 3, generate a subalgebra: so3 so3,. To see that the subalgebra is indeed so3, we put it into a more familiar form by defining Then J i ɛ ijkm jk. [J i, J j ] = 4 ɛ imn ɛ jpq [M mn, M pq ] = i 4 ɛ imn ɛ jpq M mp η nq M mq η np M np η mq + M nq η mp = i 4 ɛ imn ɛ jpq M mp δ nq M mq δ np M np δ mq + M nq δ mp = i δ ij δ mp δ ip δ mj M mp = i δ ij M mm M ji = i M ij = i ɛ ijk J k where in the last line we inverted the definition of J i to obtain M ij = ɛ ijk J k. So, finally, [J i, J j ] = iɛ ijk J k, which is indeed the familiar algebra of su = so3. [See tutorial question 3..] So J i are the rotation generators. Similarly let K i = M i, then K i are the boost generators. We find and [K i, K j ] = [M i, M j ] = im ij = iɛ ijk J k [J i, K j ] = ɛ imn [M mn, M j ] = ɛ imn i M m η nj M n η mj = iɛ ijk M k = iɛ ijk K k. Thus the algebras satisfied by the generators J i and K i are coupled or entangled. [J i, J j ] = iɛ ijk J k, [K i, K j ] = iɛ ijk J k, [J i, K j ] = iɛ ijk K k. We disentangle them by defining new generators N i, which are linear combinations of J i and K i Note that these are not hermitian, N i = J i ik i N i. We find Similarly [ ] N i, N j and Altogether [N i, N j ] = 4 [J i, J j ] + i [J i, K j ] + i [K i, J j ] [K i, K j ] = 4 i ɛ ijk J k ɛ ijk K k + ɛ jik K k + i ɛ ijk J k = iɛ ijk J k + ik k = iɛ ijk N k = 4 [J i, J j ] i [J i, K j ] + i [K i, J j ] + [K i, K j ] =. [N i, N k ] = iɛ ijk N k [N i, N j ] = iɛ ijk N k [N i, N k ] = iɛ ijk N k [N i, N j ] = ; i.e., N i and N i generate two separate su algebras. These two su algebras are not completely independent, since they are related by hermitian conjugation, or under parity J i J i, K i K i, so N i N i. Casimirs: we can construct two Casimir operators from: the product of two generators; and the product of two generators and the invariant tensor ɛ µνρσ M µν M µν = J K 4 ɛ µνρσm µν M ρσ = J K. The commutation relations show that so3, has rank and is semi-simple, but it s not compact: the Killing form is diag,,,,,. Note that the six Pauli matrices {σ j, iσ j } satisfy the same algebra as {J i, K j }, so they may be chosen as a basis of generators, which generate rotations and boosts respectively.. Now these six traceless complex but not hermitian matrices are the fundamental representation of sl, C, therefore sl, C and so3, are isomorphic: sl, C = so3,. In fact SL, C is the covering group of the Lorentz group SO3, = SL, C/Z, see tutorial question 3.3, just as SU is the covering group of SO Representations of sl, C The generators N 3 and N 3 commute, but they re not hermitian; sl, C is not compact. Since the N i and N i satisfy su algebras, we can construct step operators N ± and N ± as we did for a single su algebra. There are two Casimir operators. N N i N i with eigenvalues mm +. N N i N i with eigenvalues nn + Strictly, { σj, i σj} satisfy the same algebra as {Ji, Kj}. N i J i + ik i. 9

11 where m, n =,,,..., just as for a single su. Since we have two su algebras, we label each representation by m, n, and we label the states within the representation by the eigenvalues of N 3 and N 3, which run from m, m +,..., m, and n,..., n respectively; i.e., there are m + n + states. The spin of the states within the representation m, n varies from m n to m + n, since J i = N i + N i. This is just like adding angular momenta in quantum mechanics. Example.., spin zero: scalar.., and, spin : spinors the fundamental representation of sl, C. 3., spin one: representation of a vector, V µ = V, V, and spin zero see below. 4., and, spin one antisymmetric tensor F µν self-dual and anti-self-dual see later. The adjoint representation is,, F µν has 6 components. 5., spin two: a symmetric traceless tensor T µν. We will explore these in more detail as the course progresses Euclidean Group If we consider imaginary time, i.e. let t it, we recover Euclidean space, η µν δ µν, and the symmetry group is SO4 = SU SU/Z - see tutorial questions 3. and 3.3. Unlike SO3,, SO4 is compact: g ab = diag,,,,,. The two su subalgebras of so4 are completely independent, and each state is characterised by two spins, n, m, both of which are observables correspond to hermitian generators sl, C Spinors The spinor representations, and, of the Lorentz group or more precisely of SL, C are realised by two-component complex Weyl spinors, ψ L and ψ R respectively. The choice of labels L, R is a convention. Spinor indices are usually suppressed although one sometimes sees ψ a,, ψ ȧ,, with a, ȧ =,. Note the dotted index ȧ in the second case. Under an SL, C transformation Under a rotation in SU SL, C ψ L ψ L = Λ L ψ L ψ R ψ R = Λ R ψ R. Λ L,R = exp { i σ ω}, where ω i = ɛ ijkω jk are the rotation parameters. The representation of boosts is not unitary: if ν i = ω i are the boost parameters, Λ L = exp { i σ ω iν} Λ R = exp { i σ ω + iν} Note the change of sign of the boost. Algebraically, this is because the transformations of ψ R and ψ L are generated by N i and N i neither of which are hermitian, but which are related by hermitian conjugation. Physically, we note that under parity ν i ν i, because the direction of the boost is reversed, so Λ L Λ R and ψ L ψ R. Properties of Λ L,R :. Λ L = Λ R NB, in Euclidean space Λ L and Λ R are completely independent.. Since σ σ i σ = σ i, σ Λ L σ = Λ R, σ Λ L σ = Λ T R. 3. Since σ σ i σ = σ T i, σ Λ T L σ = Λ L, and hence ΛT L σ Λ L = σ. Similarly Λ T R σ Λ R = σ. Exercise: check all of these. Consider a left handed LH spinor ψ L,. Under SL, C σ ψ L σ Λ Lψ L = Λ R σ ψ L using result. above and σ =. So σ ψl, is a right handed spinor, which we call the charge conjugate of ψ L. Similarly σ ψr is a left handed spinor. Under charge conjugation C : left-handed. ψ L σ ψl, which is right handed, and ψ R σ ψr, which is From two left handed spinors ψ L and χ L we can construct a scalar, under SLC, since,, =,, just as spin- times spin- gives spin- plus spin- in quantum mechanics. Explicitly: χ T L σ ψ L χ T L Λ T L σ Λ L ψ L = χ T L σ ψ L where we used result 3. above. In particular, if we choose χ L such that ψ R = σ χ L, and hence ψ R = χt L σ, then ψ R ψ L is a scalar. Similarly, ψ L ψ R is a scalar. We can also build a 4-vector from two spinors, since,, =,. Under an infinitesimal Lorentz transformation ψ L ψ L ψ L i σ ω + iν + i σ ω iν ψ L while ψ L σ iψ L ψ L = ψ L ψ L ω i ψ L σ i ψ L since ν i = ω i = ψ L i σ ω + iν σ i + i σ ω iν σ i + i [σ i, σ j ] ω j + {σ i, σ j } ν j ψ L = δ ij ω ij ψ L σ j ψ L + ω i ψ L ψ L. where we recalled that ω i = ɛ ijk ω jk, which we inverted to get ω ij = ɛ ijk ω k. It follows that if we define σ µ, σ i, then ψ L σ µψ L δ µ ν + ω µ ν ψ L σ ν ψ L ψ L Exercise: check this. So ψ L σ µ ψ L is a 4-vector because it transforms in the same way as x µ under Lorentz transformations. Similary, if we define σ µ, σ i, then ψ R σ µ ψ R is also a 4-vector.

12 6.4.3 Dirac Spinors As we saw earlier, the spinors ψ L and ψ R transform into each other under parity, ψ L ψ R. ψl If we want invariance under parity, we need to construct a Dirac spinor ψ with four components, transforming as,, under SL, C ΛL ψl ψ Λ R ψ R Under parity, the Dirac spinor transforms as ψr ψ = ψ L Λ ψ. ψ γ ψ. where γ is a 4 4 matrix, with each representing a unit submatrix We can project ψ onto L and R spinors using the projection operators P L,R = ± γ5, where γ 5 NB Conventions for these projection operators with the opposite signs are perhaps more popular! From Dirac spinors we can construct two scalars under SL, C: ψ L ψ R + ψ R ψ L = ψ γ ψ ψψ even under parity scalar and ψ L ψ R ψ R ψ L = ψ γ γ 5 ψ = ψγ 5 ψ odd under parity pseudoscalar. Note the definition ψ ψ γ. Similarly there are two 4-vectors: where γ µ = ψ L σ µψ L + ψ R σ µψ R = ψγ µ ψ odd under parity vector ψ L σ µψ L ψ R σ µψ R = ψγ µ γ 5 ψ even under parity axial vector σµ σ µ σi, i.e., γ i =. σ i These are the Dirac matrices in the Weyl representation. Remembering that {σ i, σ j } = δ ij, it s easy to check that they form a Clifford algebra {γ µ, γ ν } = η µν, γ 5 = iγ γ γ γ 3, {γ 5, γ µ } =, γ µ = γ γ µ γ, γ 5 = γ 5. These expressions should be familiar if you re taking the Quantum Field Theory course. Note however that we obtained them here using group theory only. We didn t input any quantum field theory or indeed any quantum mechanics. ψ R 6.5 The Poincaré Group Space-time is isotropic and homogeneous: this corresponds to invariance under Lorentz transformations and translations respectively x µ x µ = Λ µ νx ν + a µ These are called inhomogeneous Lorentz transformations. Translations and Lorentz transformations do not commute, because the translation parameters are rotated by a Lorentz transformation. Under two successive transformations, the second having the tilde x µ x µ = Λ µ ν x ν + ã µ = Λ µ ν Λ ν α x α + Λ µ ν a ν + ã µ. If we denote a general Poincaré transformation as Ua, Λ, then Ua, Λ = Ua, U, Λ Ua UΛ and, from the result for two successive transformations above, Uã, Λ Ua, Λ = U Λa + ã, ΛΛ. Let us define the generators of the Poincaré group by Ua = + ip µ a µ a µ infinitesimal UΛ = + i ωµν M µν ω µν infinitesimal. We can deduce the commutation relations by setting:. Λ = Λ = : Uã Ua = Uã + a = Ua Uã, so [P µ, P ν ] =. shorthand notation. a = ã = : M µν satisfy the so3, commutation relations the Lorentz algebra as before. 3. ã =, a infinitesimal, Λ = Λ, so UΛ Ua UΛ = UΛ a,. Then UΛ P µ UΛ = P ν Λ ν µ, i.e., P µ transforms as a 4-vector 3. If Λ is infinitesimal, we find exercise This holds for any 4-vector. [M µν, P σ ] = ip µ η νσ P ν η µσ. In terms of J i = ɛ ijkm jk, and K i = M i, and writing the four momentum as P µ H, P i, the commutations relations become [J i, P j ] = i ɛ ijk P k [J i, H] = [K i, P j ] = i H δ ij [K i, H] = ip i. 5 Together with [P µ, P ν ] =, these commutation relations form the algebra of the Poincaré group. The Poincaré group has dimension, but it s neither compact nor semi-simple, the latter because P µ form an Abelian invariant subalgebra. Despite this, we can still find two Casimir operators:. P = P µ P µ : [P µ, P ] = [M µν, P ] = 3 4

13 . The other Casimir operator is not so obvious, but the square of any 4-vector that commutes with the generators and is linearly independent of P µ will suffice. Define the Pauli Lubanski vector W µ ɛ µνρσm νρ P σ where ɛ µνρσ with ɛ 3 = + is totally antisymmetric. Then W µ P µ = by anti-symmetry, and [W µ, P ν ] = using the commutation relations between M ρσ and P σ. W µ is a 4-vector, so [M µν, W σ ] = iw µ η νσ W ν η µσ Thus W W µ W µ is a scalar, and [P µ, W ] = [M µν, W ] =, so W is a Casimir operator. 6.6 Classification of Irreducible Representations Denote the eigenvalues of P µ by p µ = p, p energy and momentum, and of P by p. Note that p µ are continuous not discrete because we can rescale P µ αp µ for any α without affecting any of the commutation relations. This is always true for generators of invariant Abelian sub-algebras. Let s consider all possible classes of values of the eigenvalues p µ :. p =, p µ =, whence W = in all states. This corresponds to the vacuum the lowest energy state in physics.. p > timelike: set p = m, where m is the mass, so p = m + p. We then have either p > m positive energy or p < m negative energy states the sign of p is an invariant for orthochronous Lorentz transformations. If we go to the rest frame p µ = m,, then W µ = ɛ µνρσm νρ m δ σ so W = and W i = m ɛ ijk M jk = mj i, where J i satisfy the su algebra. Therefore W remember this is a Poincaré invariant has eigenvalues m ss + for s =,,,.... We conclude that all representations with p > are labelled by p i momentum, continuous, spin s, and s 3 = s, s +,..., s, s, a total of s + degrees of freedom. Physically, a state corresponds to a particle of mass m, spin s, 3-momentum p i, and spin projection s 3. Note: this is pure group theory [Weyl & Wigner 939], no quantum mechanics is involved! 3. p = : these are massless states, m =,, so there is no rest frame. Choose instead a frame in which p µ = p,,, p. Only W = seems relevant for physics otherwise we would have continuous spin. Since W µ P µ =, we have W 3 = W and W = W =, so W µ = λp µ. Since W µ and P µ are four vectors, λ is an invariant, called the helicity. Let η µ =,, then λ = W µη µ P µ η µ which is the usual definition of helicity. = ɛ ijkm ij P k P = J P P One can show that: λ = ±s, where s is the integer or half-integer spin of the representation, i.e. λ = or ± or ±,.... topology: SO3, = SL, C/Z. So massless representations are labelled by λ left-handed or right-handed states, states within a representation are labelled by p i. For example, the massless spin- photon has only polarisation states λ = ±, the massless spin- graviton has only polarisation states λ = ±. 4. p < : these are tachyonic states and are unphysical. Since they move faster than the speed of light, they are acausal Fields Casimirs: Semi-simple or compact groups: the Casimirs have a discrete spectrum: states correspond to finite dimensional vectors or spinors or tensors. For example, SU: j, m. Non-semi-simple groups: the Casimirs have a continuous spectrum: states correspond to continuous functions fields. Because the eigenvalues of P µ are continuous, we need to introduce fields φ s x µ with s labelling the spin components. This is sufficient to construct massive and massless particle states by superposition of states x, s φ s x µ, where is the vacuum state. Under a Poincaré transformation a state Ua, Λ, so φx Ua, Λ φx Ua, Λ. For a scalar field U φ x U = φ Λx + a. 6 For an infinitesimal transformation, φx µ φx µ + δx µ, so δφx µ = δx µ x µ φx µ δx µ µ φx with δx µ = a µ + ω ν µ x ν. I.e., µ = x,, µ = x,. Note that under a Lorentz transformation x µ x µ = x ν x µ x ν = Λ µ ν, x ν so if φ is a scalar, µ φ and µ φ are vectors: µ is a vector operator. Similarly µ ν φ is a tensor, φ µ µ φ is a scalar. The operator = = is called the d Alembertian, the x relativistic generalisation of the Laplacian. In terms of generators, from equations 5 & 6, δφ = ia µ [P µ, φ] + i ω µν [M µν, φ] = a µ µ φ + ω µ ν x ν µ φ as δx µ = a µ + ω µ ν x ν. Since this holds for all infinitesimal a µ and ω µν = ω νµ, Writing P µ = H, P i, where J i = ɛ ijkm jk. [P µ, φ] = i µ φ [M µν, φ] = i x ν µ x µ ν φ = x ν P µ x ν P µ φ. [H, φ] = i t φ Energy Heisenberg equation [P, φ] = i φ Momentum [J, φ] = ix φ Angular momentum NB, we are using units where = c =. to restore remember that has dimensions of angular momentum, so let P µ P µ /, M µν M µν /. Since the vacuum has P µ =, it follows that H x, t = i x, t t 6 P x, t = i x, t

14 etc. Schrödinger equation. A Fourier transformation to momentum space is often useful: φp µ d 4 x e ipµxµ φx µ etc., then [P µ, φp µ ] = p µ φp µ etc. For fields which are not scalars under the Lorentz group, there is an extra factor of DΛ = exp i ωµν S µν in the transformation law whence [P µ, φ] = i µ φ while UΛ, a φx UΛ, a = DΛ φλx + a [M µν, φ] = ix µ ν x ν µ φ + S µνφ L µν where L µν is the angular momentum generalised to 4 dimensions we obtained above for the scalar field, while S µν is the intrinsic spin angular momentum in the appropriate representation of the Lorentz group, DΛ = exp i ωµν S µν. For example, for a scalar S µν =, for a Weyl spinor ω µν S µν = i ωµν σ µν, and for a vector i.e. in the defining or fundamental representation ω µν S µν α β = ωα β. Note that:. L µν satisfy the so3, algebra. See tutorial question 4... The Pauli Lubanski vector W µ = ɛ µνρσs νρ P σ because ɛ µνρσ L νσ P σ = ɛ µνρσ x ν P ρ P σ =, since [P ρ, P σ ] =, so only the intrinsic angular momentum i.e the spin contributes to the eigenvalues of W single particle states have no angular momentum. In other words, they can have spin, but not orbital angular momentum. 7 Symmetry in Action We assume that our dynamics either classical or quantum here mainly classical may be derived from an action principle. To construct suitable actions we follow a set of fundamental principles:. Locality: we assume the action has the form S = d 4 x L where d 4 x = dx dx dx dx 3 is the Lorentz invariant integration measure in Minkowski space, and L is the Lagrange density usually informally called the Lagrangian. We assume that L is a function of local fields Φ which could be scalars, spinors, vectors,... and their derivatives at one point x µ only - so there s no action at a distance. Locality is necessary to ensure causality in the quantum theory.. Unitarity: we assume that the action and thus the Lagrangian is real: this is essential to obtain quantum theories in which probability is conserved note in classical physics complex potential absorption Causality: we require that S leads to classical equations of motion that involve no higher than second derivatives: higher order differential equations generally have acausal solutions. This means see below that L contains terms with at most two derivatives µ. 4. Symmetry: we assume that S is Poincaré invariant up to surface terms: since d 4 x is invariant, this means that L must be a Lorentz invariant function of the fields Φx and µ Φx terms involving x µ explicitly will not be translation invariant. We will also consider further symmetries internal symmetries, gauge symmetries, etc. to further restrict the action. 5. Renormalisability: terms in the action with coefficients with dimension M n, n >, generally imply that the quantum theory loses all predictive power at energies of order M. So we either exclude such terms or assume they are small [technical requirement]. 7. Examples Real scalar field: where V φ doesn t contain derivatives. Complex scalar fields: where φ = φ + iφ / with φ and φ real. Weyl spinor: Dirac spinor: Gauge field: 7. Equations of Motion L = µ φ µ φ V φ L = µ φ µ φ V φ φ L = ψ L σµ i µ ψ L or L = ψ r σ µ i µ ψ R L = ψi µ γ µ ψ m ψψ Consider the action for some generic field Φ: S[Φ] = d 4 x LΦ, µ Φ. L = 4 F µνf µν where F µν = µ A ν ν A µ Under an arbitrary change Φ Φ + δφ L δs = d 4 x δl = d 4 x Φ δφ + L µ Φ δ µφ L = d 4 x Φ L L µ δφ + d 4 x µ µ Φ µ Φ δφ where we integrated the second term by parts noting that δ µ Φ = µ δφ by linearity. The last term is a surface term: it vanishes if we insist that δφ = on the boundary of space-time. Then Hamilton s principle δs = for arbitrary δφ implies L µ L µ Φ Φ = 8