MATH 742 ADVANCED TOPICS IN MATHEMATICAL PHYSICS REPRESENTATION THEORY OF THE POINCARÉ GROUP FINAL PROJECT

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1 MATH 742 ADVANCED TOPICS IN MATHEMATICAL PHYSICS REPRESENTATION THEORY OF THE POINCARÉ GROUP FINAL PROJECT 1 Introduction The Poincaré Group The Poincaré Algebra Casimir elements of the Poincaré algebra Representations of the Poincaré Group Hilbert spaces Space of classical fields Acknowledgements Appendix Rotation References Introduction As far as common sense is concerned, the world surrounding humans has three infinite space dimensions and it is evolving as time increases (whatever these quantities are precisely). When developing his Special Theory of Relativity (SR), Albert Einstein considered time, which somehow stands apart based on intuitive knowledge, as merely pèp an additional dimension of reality (or spacetime more precisely). 1 Apart from this preliminary assumption that the world has the topology of R 4, the theory stems from two easy to grasp (although maybe hard to believe) physical axioms [1]: Axioms of Special Relativity: The speed of light is constant. It has the same value, c = 1 in some system of units, no matter in which condition whoever measures it. The principle of relativity, which heuristically says that the result of an experiment should not depend on the observer (or more properly on the system of coordinates 2 used by this observer). When learning about SR, I have been surprised and seduced by the amount of interesting theoretical facts all of which being confirmed by experiments that can be deduced from these simple starting points using only a generous amount of mathematical cleverness. This exemplifies what I find particularly appealing and exciting in mathematical physics. It is a symbiotic association of ideas from physicists and mathematicians, people with different objectives and background, that yields otherwise unachievable discoveries. 1 This is something I personally still find hard to accept. Time is so much different. Why does it appear to be going in one direction while we can move back and forth in space? The typical argument of entropy is not really satisfying to me 2 A system of coordinates (or a frame of reference) is a set of four real numbers x = (x 0, x 1, x 2, x 3 ), where x 0 t is a measure of the time elapsed since a certain event and where x 1, x 2, x 3 are usual distances from a chosen origin along three orthogonal axes. 1

2 In this document, we push a step forward the mathematical deductions stemming out of these axioms using the tools of representation theory of Lie groups. As we will see, the mathematical objects that transform under representations of the isometry group of the SR spacetime are classical fields and elements of quantum Hilbert spaces. They are used by physicists in Quantum Field Theories to model virtually every interactions in the universe. The discussion starts with a definition of the Poincaré group from first principles and some fundamental remarks about it. We then study the Poincaré algebra and its Casimir elements because of their importance in the representation theory. In the mathematical process, many links with physics are being made. The representation theory is finally tackled, culminating essentially in the Wigner s classification and in the identification of types of particles in Nature. An appendix also discuss questions related to the subject of rotation in the context of Lie groups. While reading about it, I sincerely found the subject of the Poincaré group very rich and intellectually enlightening. Because of its numerous ramifications, a choice has to be made between a long and detailed presentation and a shorter but less explicit discussion. For concision needs, I choose the latter option, so our goal is rather to weave a web between Poincaré-related ideas than to prove very precise statements. 2 The Poincaré Group Of course, several different systems of four real numbers x μ, μ {0,1,2,3}, can be used to describe whatever happens in spacetime. Physicists conscientiously define a special class of systems of coordinates that are called inertial. It can be shown [1] using the axioms of SR that, no matter which system of inertial coordinates is chosen, the interval 3 Δs 2 (Δx 0 ) 2 + (Δx i ) 2 between two points x and x + Δx in spacetime is invariant. (This property is sometimes taken as a definition of inertial systems of coordinates, but I find this practise somewhat unsatisfying on physical grounds.) If this is regarded as a generalized 3 distance between x and y, it suggests to identify our spacetime with a generalized metric space. The latter is properly called Minkowski spacetime M. It is a space that has the topology of R 4 and its vector space structure along with the Minkowski metric (or quadratic form), a map such that i=1 η M M (x, y) x 0 y 0 + x i y i R. The output of this map is the inner product of x and y. It is often denoted η μν x μ y ν, where a sum on μ, ν is understood (as always in the present text) and where 3 i=1 η μν diag( ). Note that the interval defined above is just the inner product of Δx with itself. Let us finally note that an isometry (or a symmetry) on M is defined just as in the case of conventional metric spaces: it is a map from M to itself that preserves the distances. These preliminaries now allow us to define the Poincaré group (see table below). We have two definitions that are equivalent because, as seen above, inertial frames of references are characterized by the fact that the interval (the generalized length ) has the same value no matter in which frame it is measured. 3 I use generalized because distances are usually assumed to be positive-definite in mathematics. 2

3 Poincaré group P Mathematical definition: Group of isometries of Minkowski spacetime. Active point of view: Actual transformation, same coordinates. Physical definition: Group of conversions between inertial frames of reference in Minkowski spacetime. Passive point of view: Change of coordinates, no actual transformation. It turns out that the mathematical definition hides an active point of view of the transformations while the physical definition is more passive in essence. Proposition 1 P is a non-compact Lie group. [2] Without loss of generality 4 [3], an element g of the Poincaré group is always assumed to be a linear transformation of the form x μ g x μ Λ μ ν x ν + c μ, (1) where Λ μ ν are the entries of a 4-matrix representing 5 a so-called Lorentz transformation and where c μ are the entries a constant 4-vector representing a translation in spacetime. Since the interval must be conserved, we have a constraint on Λ μ ν (but not on c μ ): η ρσ Δx ρ Δx σ = η μν Δx μ Δx ν = η μν Λ μ ρλ ν σδx ρ Δx σ Δx η μν Λ μ ρλ ν σ = η ρσ. (2) This mimics the fact that the Poincaré group is the direct product [4] of the Lorentz group L = O(1, 3) (the subgroup of P leaving the origin fixed) and the group of translations R 3,1 : Proposition 2 P = O(1,3) R 1,3. 6 As seen in problem set 2, O(1,3) can be separated into four connected components according to the determinant of Λ μ ν, a matrix representing an element of O(1,3), and to the sign of Λ 0 0. Given the simple relation with P, the same is true in the case of P: P { P + = L + R 1,3 = ISO (1,3), Proper (det Λ = 1) orthochronous (Λ 0 0 > 0) transf. P = L R 1,3, Improper (det Λ = 1) orthochronous (Λ 0 0 > 0) transf. P + = L + R 1,3, Proper (det Λ = 1) non-orthochronous (Λ 0 0 < 0) transf. P = L R 1,3, Improper (det Λ = 1) non-orthochronous (Λ 0 0 < 0) transf. 4 Suppose otherwise there were an order-two term Λ μ abx a x b in the Taylor expansion of the transformation. Plugging in (2) gives that Λ μ ab = 0. We can believe the rest to be true for higher order terms. 5 By using 4-matrices and 4-vectors, we are already working in a particular 4-dimensional representation of P. This is uncomfortable since we want to treat representation theory latter, but it is inevitable (as far as I know) in order to understand the group we are working on. We could call it the fundamental representation. 6 I could not find a satisfying proof of this decomposition. 3

4 Of these subspaces, only P + R 1,3 = ISO (1,3) is a subgroup because it contains the identity of P. It can be shown that any element of the other subspaces are obtainable from an element of SO (1,3) R 1,3 via a timeinversion and/or a space-inversion or (parity transformation). P L L L + SO(3) R 3 As a concluding remark, note that the general form of (1) and our understanding of the Poincaré group allow us to guess that some Lorentz transformations should be rotations in 3-dimensional space. We will see this more clearly latter, but let us record here this additional subgroup of P. A rotation is a proper transformation that leaves the time fixed, so it is orthochronous. Hence, SO(3) L +. Similarly, translations in R 3 form a subgroup of R 1,3. L + R 1,3 L Sketch of the most important subspaces and subgroups (in bold) of P 3 The Poincaré Algebra The Poincaré algebra p is the Lie algebra associated with P. It is a vector space along with a bracket [, ] p p p, which can be specified by its action on a basis of vectors called the generators of P. We will obtain three common and useful sets of generators. The commutation relations (specifying the bracket) for these sets will be the following: First set Second set Third set 7 Generators J μν, P μ J i, K i, P i, H L i, R i, P i, H [L i, L j ] = iε ijk L k [R i, R j ] = iε ijk R k [L i, R j ] = 0 Commutation relations Notation [J μν, J ρσ ] = i(η νσ J μρ + η μρ J νσ η μσ J νρ η νρ J μσ ) [J μν, P ρ ] = i(η μρ P ν η νρ P μ ) [P μ, P ν ] = 0 η μν η μν [J i, J j ] = iε ijk J k [J i, K j ] = iε ijk K k [K i, K j ] = iε ijk J k [J i, P j ] = iε ijk P k [K i, P j ] = ihδ ij [P i, P j ] = [P i, H] = 0 [J i, H] = 0 [K i, H] = ip i J i 1 2 ε ijkj jk K i J 0i ε ijk is the sign of the permutation of ijk i, j, k {1,2,3} [L i, P j ] = 1 2 (iε ijkp k Hδ ij ) [R i, P j ] = 1 2 (iε ijkp k + Hδ ij ) [L i, H] = Pi 2 [R i, H] = Pi 2 L i 1 2 (Ji + ik i ) R i 1 2 (Ji ik i ) Table 1 Commutation relations of p expressed in three important sets of generators 7 This set is actually valid for the complexification so(1,3) C so(1,3) C of the Lorentz algebra because L i and R i will be defined as complexified Lorentz generators. 4

5 As done during the semester, we now use a group element infinitesimally close to the identity to obtain information about the Lie algebra. In the fundamental representation, the Lorentz transformation is 8 Λ μ ν = δ μ ν + λ μ ν (λ μ ν δ μ ν) and (2) forces on it the following constraint: η μν (δ μ ρ + λ μ ρ)(δ μ σ + λ μ σ) = η ρσ η μν (δ μ ρλ ν σ + λ μ ρδ ν σ) = 0 to first order λ ρσ = λ σρ where η μρ λ ρ ν λ μν The elements of the Lie algebra of L being 4 by 4 antisymmetric matrices in this representation (as just shown) and because such a matrix has 6 degrees of freedom, the Lorentz algebra has dimension 6. Since there is no constraint on the infinitesimal translations, the Poincaré algebra has dimension 10. A generic element of so(1,3) and r 1,3 (Lie algebras for the Lorentz and translations groups) is typically written as follows. i 2 ω μνj μν so(1,3) p ic μ P μ r 1,3 p (ω μν, c μ R, J μν = J νμ and ω μν = ω νμ ) (3) Note that these are abstract vectors which do not depend on the defining representation the latter was only useful to obtain the antisymmetry of λ. The factor of a half is for later convenience and the i will allow us eventually to identify the generators with hermitian operators. Note that there is some freedom regarding the sign of these expressions. This choice is irrelevant for most applications as we can simply redefine ω μν or c μ to absorb the minus sign. Some complications can however occur when considering the representations on the space of functions (see section 4.2). A derivation of one of the commutation relations of the first set in table 1 is detailed in [2]. It uses only material that we developed so far. The other relations can be worked out similarly using the same technique. ~ ~ ~ The Poincaré algebra is well defined at this point but physicists prefer using the set of generators 9 J i 1 2 ε ijkj jk, K i J 0i, P i, H P 0, (4) as they can be given a physical interpretation. The J i have the commutation relations of rotation-related Lie algebras (see table 1 and especially appendix 1), so the interpretation is immediate. Similarly the P i are quite obviously generating translations in R 3 (because of how they arise). Detailing convincingly the interpretation for the generators and the group elements associated with K i and H would take us away from our main concern, but let us nevertheless record in table 2 the important elements on the Poincaré group along with their generator and their physical interpretation. The following change of variables occurred: 8 This remark requires material to be introduced latter. λ μ ν is usually denoted ω μ ν even though the profound signification is not the same. Here is why: λ μ ν = i 2 ω ρσ(j ρσ ) μ ν = i 2 ω ρσi(η σμ δ ρ ν η ρμ δ σ ν) = 1 2 (ω ν μ ω μ ν) = ω μ ν. 9 The definition of J i is equivalent to J 1 = J 23, J 2 = J 31, J 3 = J 12 because of the factor of a half. 5

6 Group element Generator 11 0 ζ 1 ζ 2 ζ 3 t ζ ω μν = ( 1 0 θ 3 θ 2 ) c ζ 2 θ 3 0 θ μ = η μν c c ν = ( 1 1 c ) 2 ζ 3 θ 2 θ 1 0 c 3 Rotation e iθ J, θ R 3 Angular momentum J Boost e iζ K, ζ R 3 Translation e ic P, c R 3 Inverse 10 time-evolution e ith, t R Momentum P Energy H Table 2 Generators, corresponding group element and their physical interpretation The commutation relations from the first set of generators allow very straightforwardly to find the commutation relations of this set (see table 1 again). ~ ~ ~ A third set of generators is defined by L i 1 2 (Ji + ik i ), R i 1 2 (Ji ik i ), P i, H. Let us focus on the Lorentz algebra. Since we are using complexified vectors, the set is actually generating the complexified version of so(1,3), which is so(1,3) C so(1,3) C. We see from table 1 that it contains two commuting sub Lie algebras obeying the commutation relations associated with rotation (see the appendix). This gives the next decomposition. Proposition 3 so(1,3) C su(2) C su(2) C 3.1 Casimir elements of the Poincaré algebra The Poincaré algebra p has two 12 Casimir element, vectors commuting with the generators of the algebra. The table below gives them as well as a physical interpretation that will be explained in section 4. Casimir element P 2 η μν P μ P ν W 2 = η μν W μ W ν where W μ 1 ε 2 μνρσp ν J ρσ is the Pauli-Ljubanski pseudo-vector Interpretation Mass Spin (and mass) Table 3 Casimir elements of the Poincaré algebra 10 The time evolution operator e ith familiar from Quantum Mechanics corresponds to a passive time translation, which explains that from our (active) perspective we get an inverse time evolution. When letting time evolve, we indeed actively push everything backwards. 11 J (J 1 J 2 J 3 ); K (K 1 K 2 K 3 ); P (P 1 P 2 P 3 ). Additional note: I would be able to justify the first line of my table 2, but I could not find any reason why we should give these interpretation to the generators. Maybe we define the angular momentum, the momentum and the energy as being represented by these operators 12 I did not find any proof that the Poincaré algebra had only two Casimir elements. 6

7 Proof [P μ, P 2 ] = 0 is trivial [J μν, P 2 ] = J μν η ρσ P ρ P σ η ρσ P ρ P σ J μν = η ρσ (P ρ J μν + [J μν, P ρ ])P σ η ρσ P ρ P σ J μν = η ρσ (P ρ J μν + i(η μρ P ν η νρ P μ ))P σ η ρσ P ρ P σ J μν = η ρσ P ρ J μν P σ + iη ρσ η μρ P ν P σ iη ρσ η νρ P μ P σ η ρσ P ρ P σ J μν = η ρσ P ρ (P σ J μν + [J μν, P σ ]) + iδ μσ P ν P σ iδ νσ P μ P σ η ρσ P ρ P σ J μν = η ρσ P ρ (P σ J μν + i(η μσ P ν η νσ P μ )) η ρσ P ρ P σ J μν = η ρσ P ρ P σ J μν + iη ρσ P ρ η μσ P ν iη ρσ P ρ η νσ P μ η ρσ P ρ P σ J μν = iδ μρ P ρ P ν iδ νρ P ρ P μ = 0 The proof that W 2 is a Casimir is more tedious. It is detailed in [5]. Proposition 4 Proof W μ P μ = 1 2 ε μνρσp ν J ρσ P μ W μ P μ = 0 = 1 2 ε μνρσp ν (P μ J ρσ + [J ρσ, P μ ]) = 1 2 ε μνρσp ν (P μ J ρσ + iη ρμ P σ iη μσ P ρ ) = 1 4 ε μνρσp ν P μ J ρσ ε νμρσp μ P ν J ρσ + i 2 ε μνρση ρμ P ν P σ i 2 ε μνρση μσ P ν P ρ = 1 4 ε μνρσp ν P μ J ρσ 1 4 ε μνρσp ν P μ J ρσ + i 2 ε μνρση ρμ P ν P σ i 2 ε ρνσμη ρμ P ν P σ = 0 4 Representations of the Poincaré Group I was disappointed to find no reference trying to be absolutely exhaustive in describing the representations of P. People usually focus instead on some interesting cases. I distinguished two important classes of representations motivated by the needs of theoretical physics. Both implement via ρ P GL(V) a Poincaré transformation g P on some vector space V. In one case, the space is a Hilbert space arising in Quantum Field Theory ( see below for clarifications ). In the other case, V is a space of classical fields (tensor valued functions of spacetime). We look at the two cases separately. 4.1 Hilbert spaces In Quantum Mechanics, the physical state of a system is described (up to a factor of e iθ, θ R) by an element ψ of a Hilbert space H (a certain complex vector space). The Hilbert space is endowed with an inner product H H C with these properties [6] φ ψ = ψ φ φ aψ 1 + bψ 2 = a φ ψ 1 + b φ ψ 2 aφ 1 + bφ 2 ψ = a φ 1 ψ + b φ 2 ψ 7

8 ψ ψ 0. (There are other technical conditions for H to be a Hilbert space.) The probability for the system in state ψ to be measured in a state ψ i is P(ψ ψ i ) = ψ i ψ 2. Just as in our previous discussion in section 2, the main physical features of a system should not depend on the inertial frame of reference. In particular, the above probability should not depend on the inertial system of coordinates used to measure it. Here is a theorem from Eugene Wigner. Proposition 5 (Wigner s theorem) A Poincaré transformations g is represented on the Hilbert space by an operator U = ρ(g) = exp(ia) (a p) that is either (i) Unitary, φ ψ = Uφ Uψ, and linear U(aφ + bψ) = au(φ) + bu(ψ) or (ii) Antiunitary, φ ψ = Uφ Uψ, and antilinear U(aφ + bψ) = a U(φ) + b U(ψ) According to Weinberg [6], any Poincaré transformation that can be made trivial by a continuous change of a parameter (all of them, then) is represented by a unitary transformation. We also have the following [4]. Proposition 6 Non-compact groups do not have finite-dimensional unitary representations Therefore (see proposition 1), we are looking here for irreducible infinite-dimensional unitary representations of p, which will give, via the exponential map, representations of P. Since we want ρ(g) to be a unitary operator on a Hilbert space, the corresponding generator a has to be Hermitian: 1 = ρ(g) ρ(g) = exp(ia) exp(ia) = exp(ia ia ) a = a if [a, a ] = 0 ~ ~ ~ The rest of this subsection describes Wigner s classification [6, 7] of irreducible infinite-dimensional unitary representations of P. We first use the eigenstates ψ p of P μ (the generators defined in section 3) as a basis of the Hilbert space 13. We assume here that P μ ψ p = p μ ψ p, where p μ R 4 is regarded as a 4-vector. A trick of the classification consists of writing p μ as a standard momentum k μ via a Lorentz transformation: p μ = Λ μ νk ν. This simplifies the problem as, in a sense, we record only the essential features of p μ by writing identicaly every similar enough eigenvalues. It turns out that the only functions of p μ independent of Lorentz transformations are p 2 = η μν p μ p ν (obviously) and the sign of p 0 in the case of η μν p μ p ν 0 (this is well-known to physicists but I did not reproduce an argument here for mathematicians). There are thus 6 classes of standard momentum (table 4). Standard k μ Little group Interpretation (a) p 2 < 0 p 0 > 0 (m, 0,0,0) SO(3) Particle of mass m and spin s = 0, 1 2, 1, (b) p 2 < 0 p 0 < 0 ( m, 0,0,0) SO(3) 13 This makes sense on a physical basis, but I confess that I can provide no compelling mathematical reasons for this choice 8

9 (c) p 2 = 0 p 0 > 0 (κ, κ, 0,0) ISO(2) Massless particle with unconstrained helicity (d) p 2 = 0 p 0 < 0 ( κ, κ, 0,0) ISO(2) (e) p 2 > 0 (0, n, 0,0) SO(1,2) Tachyon (f) p μ = 0 (0,0,0,0) SO(1,3) Vacuum Table 4 Standard momentum and little groups [6] Subgroups of the Lorentz group leaving k μ unchanged are called little groups. They are given in table 4. [6] gives a few comments about how they are obtained and argues that the representations of P can be found from representations of the little groups via the method of induced representations. In our case, let us just mention that only the cases (a), (c) and (f) have a physical interpretation. To see it, we use the Casimir elements (or operator in the current context) found previously. Schur s lemma implies that the Casimir operators are proportional to the identity operator in irreducible representations. They just act as multiplicative constants, so they are typically used as labels for the irreducible representations. We have P 2 ψ p = η μν P μ P ν ψ p = η μν p μ p ν = m 2, which has the interpretation of a mass squared because of the interpretation given to P μ and because of the relativistic equation of energy (again well-known to physicists). Let us continue our investigation by studying the cases (a), (c) and (f) separately. Case of (a) This is an irreducible representation corresponding to something (we usually say a particle) with positive mass. Let us work out what the other invariant gives. Since W μ P μ = 0, so W 0 = 0. The other components of W μ are W μ P μ ψ p = k μ W μ ψ p = mw 0 ψ p = 0, such that W i = 1 2 ε i0jkp 0 J jk W 2 = η ii W i W i = 1 4 ηii ε i0jk ε i 0j k P0 J jk P 0 J j k = m2 4 ηii ε 0ijk ε 0i j k Jjk J j k = m 2 ( ε ijkj jk ) i = m 2 J 2, using (4). If the Hilbert space H is written in the basis of eigenvectors ψ j of J 2, then the representation theory of so(3) su(2) (seen in class) yields W 2 ψ j = m 2 J 2 ψ j = m 2 s(s + 1) ψ j where s = 0, 1 2, 1, 3 2, is the spin 14 of the particle. All massive particles in nature correspond to this paradigm. This is probably one of the most important and powerful result in the present text. 14 I am not convinced that this should be identified as the spin. Why is it not an angular momentum for example? 9

10 Note that it is not entirely surprising that so(3) came over the scene since it is the Lie algebra associated with the little group of (a). Case of (c) Here the particle is massless but similarly the irreducible representations of the little group can be used to complete the derivation. The little group ISO(2) is the group of rotations and translations in a plane. It is not compact as SO(3) was, so we do not expect unitary representation because of proposition 6. The only way to get unitarity is to set to zero the non-compact generators, which leaves only the rotation generator orthogonal to the plane. (It happens to be J 1 here as can be deduced from [8].) It coincides here with the definition of the helicity operator of the particle h = J P P (because p 1 = κ is the only non-zero component). The helicity is quantized in the real world but, unlike the spin, this cannot be obtained from the representation theory of the Poincaré group. It only becomes apparent upon quantization of the fields in Quantum Field Theory [8]. Case of (f) It describes the vacuum. 4.2 Space of classical fields [9] A field is a function of Minkowski spacetime M: φ M x φ(x), where φ(x) is in general a finite-dimensional tensor even though we focus here on vector fields φ a (x). Under a Poincaré transformation g (active transformation), a vector field is expected to transform as φ a (x) g m a bφ b (m 1 (x)), m a b Mat 4, m 1 M M i.e. the transformed field at x μ depends linearly on the initial field evaluated at the untransformed point 15. The matrix m a b quite obviously represents a Poincaré transformation (m a b = ρ(g) for a certain representation map ρ), so we just described a physical reason for finding the irreducible finite-dimensional representations of p, especially on the space of vectors. What about the function m 1? In a certain system of coordinates, x is actually a 4-vector x μ, so m 1 can be thought of as a map from R 4 to itself. It is also a finite-dimensional representation of the same Poincaré transformation g. However, x is only the argument of a function and we would prefer to have a representation ρ(g) acting on the function itself, i.e. φ a (x) g ρ(g)ρ(g)[φ b (x)] ρ(g)φ b (m 1 (x)). (5) 15 To understand the appearance of the inverse Poincaré transformation in the argument, it is useful to consider the example of a rotation by θ of a scalar field (like temperature for example). Suppose there is a hotspot at some place to ease visualization. The coordinate stay the same but the field changes: φ(x) φ (x). However, the new field is really just the old field at the untransformed coordinate: φ(x) φ (x) = φ(m 1 (x)) 10

11 This motivates the search for infinite-dimensional representations of p, especially on the space of functions. Finite-dimensional representations, space of vectors In this context, the generators are matrices. For a reason that I have not seen explained anywhere, the finitedimensional representations are always discussed relatively to the Lorentz group; the translations are left aside. We already know in some details the 4-dimensional fundamental representation as we used it to understand better the Lorentz group in section 3. In particular, we found that the elements of so(1,3) are 4 by 4 antisymmetric matrices in this representation. This gives a hint on what the actual matrices should look like. The commutation relations would help us finding pretty straightforwardly that (J ρσ ) μ ν = i(ησμ δ ρ ν η ρμ δ σ ν). (6) This is interesting and very important, but conceptually slightly oversimplified. A more systematic way to obtain this representation and other significant ones would be to use the decomposition from proposition 3: so(1,3) C su(2) C su(2) C. From the representation theory of su(2) C, this tells us that the finite-dimensional irreducible representations of so(1,3) C are labeled by a pair (s 1, s 2 ), s 1, s 2 = 0, 1 2, 1, 3 2, of numbers that again have the interpretation of a spin (of the field here). They are representations of dimension 2s s = 2(s 1 + s 2 + 1). Table 5 gives the most important special cases. The generators of su(2) C and the relations between the different basis of generators introduced in section 3 can be used to get the generators in any basis. This would be a straightforward way to obtain (6) for example. Name of the repr. Label Dim. Generators Comments Trivial (0,0) 1 (J ρσ ) μ ν = = φ is called a scalar field or a Lorentz scalar if it is constant over spacetime. Spinorial ( 1 2, 0) 2 L i = σi = σi, 2 R i = = Spinorial (0, 1 2 ) 2 L i = = 0 2 2, R i = σi = σi 2 2 L i = σi , Fundamental ( 1 2, 1 2 ) 4 R i = σi 2 or (J ρσ ) μ ν = i(ησμ δ ρ ν η ρμ δ σ ν) φ a are called left-handed Weyl spinors φ a are called righthanded Weyl spinors Table 5 Important finite-dimensional representations of the Lorentz group. σ i are the Pauli matrices (see appendix). Other important representations (Majorana, Dirac) can be obtained from the spinorial representations. Infinite-dimensional representations, space of functions Here, the generators need to be operators (we will use script letters to distinguish from the generators of finite dimensional representations). A trick can be used to obtain the expression of the generators. When the latter are 11

12 found, the group elements are obtainable via exponentiation as usual. We use (1) to express m 1 in the fundamental representation, and (3) gives φ(m 1 (x)) = φ ((Λ 1 ) μ ν (xν c ν )), φ(m 1 (x)) = φ ((exp + i μ 2 ω ρσj ρσ ) (x ν c ν )) ν = φ ((δ ν μ + i 2 ω ρσ(j ρσ ) μ ν + ) (xν c ν )) = φ (x μ c μ + i 2 ω ρσ(j ρσ ) μ ν xν + ). A Taylor expansion of φ about x μ yields φ(m 1 (x)) φ(x μ ) c μ μ φ(x μ ) + i 2 ω ρσ(j ρσ ) μ ν xν μ φ(x μ ) (cμ μ ) 2 φ(x μ ) + = exp ( c μ μ + i 2 ω ρσ(j ρσ ) μ ν xν μ ) φ(x μ ) so, using (5) and (6), we can readily identify the operator representation of P μ and J μν : ic μ P μ c μ μ P μ iη μρ ρ. i 2 ω μνj μν i 2 ω ρσ(j ρσ ) μ ν xν μ J ρσ (J ρσ ) μ ν xν μ = i(x ρ η σμ μ x σ η ρμ μ ) Note that the c μ introduced with P μ corresponds exactly to the c μ that we had before; hence the same notation. This is because they both correspond to the Poincaré transformation g and because of the 4-vector interpretation given to the coefficient of the generators (see just before table 2). We could verify that the commutation relations (table 1) are respected. Table 6 collects our new results. Translations P μ = iη μν ν = i μ Lorentz transformations J μν = x μ P ν x ν P μ Table 6 Generators of an important infinite-dimensional representations of the Poincaré group ~ ~ ~ Before concluding, let us notice that representations of the Lorentz group appeared in both the finite and infinite cases. Renaming J μν S μν and J μν L μν and reusing our notations from the beginning of this section, we have, for a Lorentz transformation g, so where ρ(g) = e i 2 ω μνs μν and ρ(g) = e i 2 ω μνl μν, φ a (x) g ρ(g)ρ(g)[φ b (x)] = e i 2 ω μνj μν [φ b (x)], J μν S μν + L μν. This is mostly a symbolic expression, but it links beautifully to the notion of total angular momentum from Quantum Mechanics, which is the sum of the spin and orbital angular momentums. 12

13 5 Acknowledgements As a concluding remark, I want to thank Johannes Walcher for encouraging me to work on the Poincaré group. This project allowed me to organize much more neatly many Poincaré-related ideas and gave rise to some new questions (many of which being disseminated in this text) that will need to be answered in the upcoming months. I also thank him for helping me understanding the rotation-related Lie groups (in the appendix). 6 Appendix Rotation We review here how the notion of rotation in R 3 connects to Lie groups and Lie algebra. This is a key discussion for understanding the representation theory of the Poincaré group. SO(3) and SU(2) The most natural Lie group representing rotations in R 3 is the group of proper operations preserving the usual length of R 3 ; SO(3). In the canonical basis of R 3, SO(3) {g Mat 3 (R) g T g = 1, det g = 1}. Using a g = 1 + εa infinitesimally close to the identity ( ε R, ε 1 ), we obtain constraints on a matrix representation of so(3) a: 1 = g T g = (1 + εa T )(1 + εa) = 1 + ε(a + a T ) a T = a 1 = det a = det(1 + εa) = 1 + ε tr(a) tr(a) = 0 so(3) {a Mat 3 (R) a T = a, tr a = 0} A choice of generators so that G g = exp( iθ i J i ) is thus which obey i 0 i 0 J 1 = ( 0 0 i) J 2 = ( 0 0 0) J 3 = ( i 0 0) 0 i 0 i [J i, J j ] = iε ijk J k. An analogous reasoning on SU(2) {g Mat 2 (C) g g = 1, det g = 1} gives su(2) {a Mat 2 (C) a θ = a, tr a = 0} = {( 3 θ 1 iθ 2 ) θ θ 1 + iθ 2 θ i R} 3 A set of generator is (the Pauli matrices) which obey σ 1 = ( ) σ2 = ( 0 i i 0 ) σ3 = ( ) [σ i, σ j ] = 2iε ijk σ k 13

14 The identification J i σi shows that 2 Proposition 7 su(2) so(3) SU(2) and SL(2,C) The group SL(2, C) {g Mat 2 (C) det g = 1} has Lie algebra A + ib C + id sl(2, C) {g Mat 2 (C) tr a = 0} = {( ) A, B, C, D, E, F R}. E + if A ib Meanwhile, su(2) C a + iα b ic + i(β iγ) su(2) C {( ) a, b, c, α, β, γ R} b + ic + i(β + iγ) a iα a + iα (b + γ) + i(β c) {( ) a, b, c, d, e, d R} (b γ) + i(β + c) a iα So there is an obvious equality Proposition 8 sl(2, C) = su(2) C 7 References [1] Schutz, B., A First Course in General Relativity, 2 nd edition, Cambridge University Press, Cambridge, [2] Moore, G. and Burgess Cliff, The Standard Model: A primer, Cambridge University Press, Cambridge, [3] Peskin, M. and Schroeder, D., An introduction to Quantum Field Theory, Westview, [4] Drake, K. et al., Representations of the Symmetry Group of Spacetime, URL: [5] Müller-Kirsten, H. and Wiedemann, A., Supersymmetry, World Scientific, Singapore, [6] Weinberg, S., Quantum Theory of Fields, Vol. 1, Cambridge University Press, Cambridge, [7] Wigner, E., On unitary representations of the inhomogeneous Lorentz group, Ann. of Math., Vol. 40, No. 1, [8] Murayama, H., 232A Lecture Notes: Representation Theory of Lorentz Group, URL: [9] Maciejko, J., Representations of Lorentz and Poincaré groups, URL: [10] Pal, P., Dirac, Majorana and Weyl fermions, arxiv: v2 [hep-ph],