Introduction to Supersymmetry

Transcription

1 Introduction to Supersymmetry Christian Sämann Notes by Anton Ilderton August 24,

2 Contents 1 Introduction A SUSY toy model What is it good for? Spinors The Poincaré group Spin and pin groups Summary The SUSY algebra SUSY algebra on R 1, Representations The Wess Zumino model Superspace and superfields Reminder: Graßmann numbers Flat superspace Superfields SUSY invariant actions from superfields Actions from chiral sfields Actions with vector superfields SUSY quantum field theories Abstract considerations Sfield quantisation of chiral sfields Quantisation of Super Yang Mills theory Maximally SUSY Yang Mills theories Spinors in arbitrary dimensions Actions and constraint equations d = 4, N = 4 super Yang Mills theory Seiberg-Witten Theory The moduli space of pure N = 2 SYM theory Duality The exact effective action A Conventions and identities 78 B Solutions to exercises 81 2

3 Preface Supersymmetry, or SUSY for short, is an extension of the classical symmetries of field theories. SUSY was discovered in the early 1970 s and has attracted growing attention ever since, even though there is still no experimental evidence for its existence up to this day. There are essentially two reasons why high-energy physicists keep interested in supersymmetry: From a phenomenological point of view, the supersymmetric extension of the Standard Model provides very reasonable solutions to some of the remaining puzzles in particle physics. On the other hand, supersymmetric field theories have many intriguing features which often can be accessed analytically, making them ideal toy models for theorists. These lecture notes are based on a series of lectures given by myself from April to June 2009 in the School of Mathematics at Trinity College, Dublin. The material covered in these notes was presented during eleven lectures, each lasting 90 mins, which was a little ambitious, retrospectively. The aim of the lectures was to give a reasonable overview of this topic while being thorough enough to provide a graduate student with the necessary tools to do research involving supersymmetric field theories herself. I put some emphasis on motivating spinors, as I still remember struggling with the reasons for their existence when I first encountered them. Besides the standard material presented in almost any course on supersymmetry, I chose superfield quantisation, maximally supersymmetric Yang-Mills theories and Seiberg-Witten theory as additional topics for these lectures. I had just used supergraphs in a research project myself and their usefulness and simplicity was still fresh in my mind. The maximally supersymmetric Yang-Mills theories with their amazing properties were included because of the important role they play in string theory today. Finally, the section on Seiberg-Witten theory demonstrates why supersymmetric field theories are indeed beautiful toy models. If you should find any typos or mistakes in the text, please let us know by sending an to antoni@maths.tcd.ie or saemann@maths.tcd.ie. The most recent version of these lecture notes can be found on the lecture series homepage, On this webpage, there is also a (still preliminary) version of a Mathematica notebook which performs many of the tedious computations necessary for understanding these lectures automatically. I am very grateful to Anton Ilderton for doing such an amazing job with writing up these notes, correcting prefactors and providing typed-up solutions to all the steps left as exercises during the lectures. I would also like to thank all the people attending the lectures at Trinity for making them more interesting and lively by asking questions. Christian Sämann 3

4 General remarks Our metric is (, +, +, +), which differs by a sign from the metric of most quantum field theory textbooks, as e.g. Peskin Schroeder. The remainder of our conventions are introduced in the text, as they are needed. A summary is included in the appendices. Numerous exercises are included, the intent of which is mainly to make the user familiar with the kind of methods, tricks and identities needed to perform SUSY calculations. Full solutions can be found in the appendices. Recommended textbooks and lecture notes (with metric conventions) Recall that the two different choices for Minkowski metric, the mostly plus (, +, +, +) and the mostly minus (+,,, ), are often referred to as the East Coast Metric (ECM) and the West Coast Metric (WCM), respectively. [1] J. Wess and J. Bagger, Supersymmetry and supergravity, Princeton, USA: Univ. Pr. (1992) 259 p. ECM, the essentials. [2] S. Weinberg, The quantum theory of fields. Vol. 3: Supersymmetry, Cambridge, UK: Univ. Pr. (2000) 419 p. ECM, uses Dirac spinors, more physics. [3] I. L. Buchbinder and S. M. Kuzenko, Ideas and methods of supersymmetry and supergravity: Or a walk through superspace, Bristol, UK: IOP (1998) 656 p. ECM, supermathematics, NR theorems, super Feynman rules. [4] S. J. Gates, Marcus T. Grisaru, M. Rocek and W. Siegel, Superspace, or one thousand and one lessons in supersymmetry, Front. Phys. 58 (1983) 1 [hepth/ ]. ECM, many useful things, in particular super Feynman rules. [5] A. Van Proeyen, Tools for supersymmetry, arxiv:hep-th/ ECM, more algebraic. [6] S. P. Martin, A Supersymmetry Primer, arxiv:hep-ph/ ECM, particle physics, MSSM. [7] A. Bilal, Introduction to supersymmetry, arxiv:hep-th/ WCM, for particle physics conventions. [8] J. D. Lykken, Introduction to supersymmetry, arxiv:hep-th/ WCM, useful as a reference. Cover: an otter, and it s supersymmetric partner, the sotter. 4

5 1 Introduction 1.1 A SUSY toy model At the beginning of the 1970 s, people started looking at SUSY toy models. In this section we will discuss a simple model which will illustrate many of the important physical properties of SUSY theories which are commonly discussed in a field theory context. It will also serve to illustrate the fundamentals of many calculations we will later perform in field theory, in a simple and accessible setting. Much of the following discussion can be found in [9, 10] in more detail. 1 Definition. Our model is quantum mechanical. We have a hermitian Hamiltonian H and non hermitian operators Q, Q related through the anti commutator H = 1 2 {Q, Q } 1 2 (QQ + Q Q), (1) where the operators obey the following dimensional SUSY algebra, {Q, Q} = {Q, Q } = 0 [Q, H] = [Q, H] = 0. (2) The Q and Q are called supercharges and generate supersymmetry transformations. We always count real supercharges, so here we have two of them. It is a direct consequence of (2) that Q 2 = Q 2 = 0. 2 Properties. Consider a Hilbert space ( H, ) carrying a representation of H, Q and Q. From the algebra (2), it follows that H is positive definite, for, given any state Ψ H, 2 Ψ H Ψ = Ψ {Q, Q } Ψ = Ψ QQ Ψ + Ψ Q Q Ψ = Q Ψ 2 + Q Ψ 2 0. (3) We see that states in a SUSY theory have non negative energy; note that the minimum may or may not be obtained. To further examine the state space, we diagonalise H and consider the eigenstates n such that H n = E n n. (4) We must treat the two cases E n = 0 and E n > 0 separately. We begin with the case E n = E > 0. We may here introduce the scaled operators a = Q/ 2E and a = Q / 2E which, within the space of states of energy E, obey the algebra {a, a } = 1, {a, a} = {a, a } = 0. (5) This is a simple example of a Clifford algebra, which we will study later on. We now construct all the states with energy E, in analogy to the construction of harmonic 5

6 oscillator states via creation operators acting on the vacuum. Again, we clearly have a 2 = a 2 = 0, from which it follows that the only eigenvalue of a is 0. Call the state with this eigenvalue. We can create no more states by acting on this with a, and we can create only one more by acting with a (since a 2 = 0), which we call + a. Hence we have a subsystem of two states obeying a = +, a + =, a = a + = 0. A simple 2d representation of the algebra (5) is given by the following matrices and vectors, ( ) ( ) ( ) ( ) 0 0 a =, a =, + =, =. (6) We see here a basic example of the existence of two types of states in SUSY theories; + states and states which will later be bosons and fermions. They are transformed into each other by the action of the SUSY generators, and more generally we will see that Q boson = fermion, Q fermion = boson, and similarly for the action of Q. We also have an example of the SUSY property that states of non zero energy are degenerate and appear in pairs; a possible non zero spectrum of a SUSY theory is sketched in Fig. 1. E Figure 1: Schematic of the non zero energy spectrum in SUSY theories: states appear in pairs of equal energy. We now turn to the states of zero energy, H 0 = 0. Directly from equation (3), we must have 0 = Q Ψ 2 + Q Ψ 2, so that a vacuum state 0 exists if and only if Q 0 = Q 0 = 0. (7) 6

7 3 SUSY breaking. The vacuum should be unique in quantum mechanics and therefore invariant under supersymmetry transformations this is just the statement that Q 0 = Q 0 = 0. If, however, there is no such state we say that SUSY is spontaneously broken. Consider introducing a potential V to our system, of one of the forms shown in Fig. 2. Note that the Mexican hat potential possesses a rotational symmetry, but any particular minimum of the potential x 0 with V (x 0 ) = V min breaks the rotational symmetry, but not SUSY. The quadratic potential with a non zero minimum, on the other hand, breaks SUSY but not the rotational symmetry. We will return to these points in a moment, after looking at a representation of the SUSY algebra on a Hilbert space of wavefunctions. x x Figure 2: The quadratic potential on the left breaks SUSY but not the rotational symmetry, whereas a minimum in the Mexican hat potential breaks the rotational symmetries but not SUSY. 4 SUSY quantum mechanics. To have a Hilbert space on which we can act with Q, Q and H we choose a set of functions in L 2 (R) C 2, ( ) ψ + (x) ψ + : bosons, ψ = (8) ψ (x) ψ : fermions, following (6). We now define the SUSY generators, ( ) Q 0 1 (P = + iw ) ( ) 0 0 (P, Q = iw ), (9) where W (x) is a real function, a dash denotes differentiation with respect to x and P = i x. Exercise 1. Show that the Hamiltonian of this system is H 1 2 {Q, Q} = 1 2( P 2 + W 2 ) 1 2 W σ 3, (10) where σ 3 = diag(1, 1) is the usual Pauli matrix. 7

8 The three terms in the above Hamiltonian describe, respectively, kinetic energy, potential energy and a magnetic interaction. Let us look for the vacuum state of this theory. States annihilated by both Q and Q take one of the forms ψ = ( ψ + 0 ) or ( ) 0 ψ = ψ & (P iw )ψ + = 0 = ψ + e W/, & (P + iw )ψ = 0 = ψ e W/. The choice of ground state is dictated by normalisability, since if W (± ) = only ψ + is normalisable, and if W (± ) = only ψ is normalisable. Note that if W ( ) = W ( ) neither of the states are normalisable, and so there can be no ground state wavefunction. The role played by these boundary conditions gives us an explicit realisation of SUSY breaking discussed above. Note that for the cases W (± ) = ±, i.e. a ground state wavefunction exists, the derivative W must be somewhere vanishing, and so the potential W 2 in our theory has a zero, i.e. SUSY is unbroken. Conversely, if W 2 is non vanishing, then the potential is strictly positive. In this case, there is no normalisable ground state wavefunction, and so SUSY is broken. To be more precise, one has to take into account non-perturbative contributions: although W ( ) = W ( ) still allows for points x with W (x) = 0, there has to be an even (counted with multiplicities) number of such points (consider, e.g., the potential W 2 (x) = x 4 ). The true quantum vacuum is unique and will be a superposition of wave functions localised at these points. Here, however, non-perturbative corrections will lift W (x) to be strictly positive and thus break SUSY. This explains why we do not find SUSY ground states for W ( ) = W ( ). 5 Witten index. Introduce the operator ( ) F, which gives +1 on bosonic states and 1 on fermionic states [10]. For our states of non zero energy, we have ( ) ( ) F = 2aa (12) 0 1 More generally, we would have states of arbitrary energy labelled as +; j and ; j where j is some collection of quantum numbers, and ( ) F +; j = +; j, ( ) F ; j = ; j. (11) Consider the operator Tr( ) F n n ( ) F n, (13) where n runs over the energy eigenvalues. This operator counts the number of bosonic states minus the number of fermionic states, or n B n F. However, since states with 8

9 E > 0 always occur in pairs, all this operator really counts is n E=0 B n E=0 F, i.e. the number of bosonic states minus the number of fermionic states in the vacuum sector. This is the Witten index. Varying the parameters of the theory, such as the coefficients in some potential, the volume of the system, etc, and assuming analyticity of the states in these parameters (there are good reasons for this [10]), the index never changes the reason is that only boson fermion pairs of states can move away from the vacuum E = 0, which clearly cannot change the value of n E=0 B n E=0 F. The value of the index therefore tells us something about SUSY breaking: Tr( ) F 0 = SUSY is unbroken, (14) since if the index is non zero then there is at least one state in the vacuum sector. Note that if we find that the index is zero, we cannot infer that SUSY is broken. As the index is analytic, it is easy to calculate. It may be regulated, for example, by [ Tr ( ) F e βh]. 6 The Witten index as an operator index. We now show that the Witten index is, indeed, an index in the sense defined in the mathematics literature, see e.g. [11]. We can do this quite generally. Given a SUSY theory, we split the Hilbert space into bosonic and fermionic sectors, H = H B H F, on which the SUSY generators must take the block off diagonal form Q = ( 0 M 1 M 2 0 ), Q = ( 0 M 2 M 1 0 ), (15) for some operators M 1 and M 2. The above forms follow from the statement that the SUSY generators transform bosons into fermions and vice versa they are the generalisations of the matrices in (6). Now define the operator Q = Q + Q, which is hermitian and annihilates E = 0 states, We then have that n E=0 B Q = ( 0 M 1 + M 2 M 2 + M 1 0 = dim ker(m 2 + M 1 ), and ne=0 F = dim ker(m 1 + M 2 ). Hence, n E=0 B n E=0 F = dim ker(m 2 + M 1 ) dim ker(m 1 + M 2 ) ind M 2 + M 1, (16) as claimed. We also have n E=0 B + n E=0 F = dim ( ker Q /imq ). ) 9

10 α 1 EM α 1 EM α 1 W α 1 S log Λ/GeV α 1 W α 1 S log Λ/GeV Figure 3: The strong, weak and electromagnetic couplings α 1 of the standard model, with and without SUSY. 1.2 What is it good for? We have seen that SUSY theories are constructed from an algebra and, for explicit representations, a function W, the superpotential. Here we collect some motivating reasons for studying SUSY. 1 Coleman Mandula theorem. We may ask the question: can one extend spacetime symmetries non trivially beyond the Poincaré group? The answer goes as follows. Assume G is the symmetry group of a theory with S matrix S such that G contains the Poincaré group, all particles have positive energy, with finitely many particles of mass m < m 0 for all m 0, S matrix elements out S in are analytic and non trivial, then the Coleman Mandula theorem tells us that G = Poincaré group internal symmetries. So the answer to the above question appears to be no. There was a hidden assumption in this theorem, however that the Lie algebra of G was generated by commutators. As we saw above, SUSY algebras, however, include anticommutators and therefore provide a loophole to the Coleman Mandula theorem. 2 Gauge coupling unification. The gauge group of the standard model is SU(3) SU(2) U(1). Various attempts have been made at constructing a grand unified theory, or GUT, which unifies the standard model at some high energy scale, within a single group, be it SU(5), SO(10), E 6, etc. A unified theory would imply a unified coupling; however, the couplings in the standard run as shown in the left panel of Fig. 3, and do not appear to intersect. With SUSY, however, the situation is improved the couplings very nearly unify at the order of GeV. 3 Hierarchy problem. The masses in the standard model are generated by the Higgs particle. Experimentally, we have H 174 GeV, but this is very sensitive to quantum corrections which can be estimated to be of the order GeV. As keeping the experimentally desired value of 174 GeV would require an unnatural amount of fine tuning, it seems that some protection mechanism is at work, and SUSY is a good candidate for 10

11 this. Here, H m 2 H, and we are thus looking at quantum corrections to the mass of the Higgs boson. Many Feynman diagrams contributing to mass corrections in SUSY theories cancel against other Feynman diagrams in which a particle loop is replaced by its superpartner loop, see Fig. 4. H b + H f = 0 Figure 4: In SUSY theories, contributions from bosonic (b) and fermionic (f) superpartners often cancel exactly. 4 Dark matter. The energy content of the universe is roughly 4% ordinary matter, 22% dark matter and 74% dark energy (or cosmological constant). There are most likely a number of different constituents of dark matter. One of the most important candidates besides neutrinos is the neutralino, the lightest particle of the minimal supersymmetric standard model (MSSM) yet to be found. The hope is, of course, that the LHC will find the neutralino and heavier SUSY particles. In fact and add your own pinch of salt here we have already found half of the SUSY spectrum: that of the ordinary standard model. Although this is clearly no evidence for SUSY whatsoever, it is reassuring that there are actually good theoretical reasons for discovering first the particles of the ordinary standard model, if one assumes a MSSM with broken SUSY. 5 Theorist s reasons. SUSY theories are highly constrained by symmetries and are therefore ideal toy models. Local SUSY theories contain gravity, and perhaps even give good theories of quantum gravity. N = 8 supergravity might actually be finite, similarly to N = 4 super Yang-Mills theory [12]. String theories are also much nicer with SUSY included, as the tachyonic states of the bosonic string can be safely removed from the spectrum. In mathematics, it could be that mirror symmetric partners of rigid Calabi Yau manifolds are Calabi Yau supermanifolds, and therefore mirror symmetry might require us to introduce a notion of supersymmetry [13]. 11

12 2 Spinors To understand SUSY QFT, we need a good understanding of spinors. We will cover this topic in some detail in this part of the lectures, and things will be clearer for it later. The aim is to find all irreducible representations ( irreps ) of the Poincaré group, as all fields in physics live in such representations. 2.1 The Poincaré group 1 Definition. The Poincaré group is the group of isometries (maps preserving distance) on Minkowski space R 1,3. It is a non compact Lie group. Its generators are four translations, P µ, and a total of six boosts and rotations, M µν = M νµ. The Lie algebra relations are [P ρ, M µν ] = i ( η µρ P ν η νρ P µ ) iηµρ P ν + symm., (17) [M µν, M ρσ ] = i ( η µρ M νσ + symm. ). (18) In the first line we have introduced the operation symm. which takes account of the symmetries of the generators, i.e. it symmetrises or antisymmetrises as appropriate. For example, the left hand side of (17) is antisymmetric in µ and ν, hence symm. generates the second term on the right hand side of (17). Exercise 2. Check you understand the definition of symm. by generating the remaining three terms on the right hand side of (18). The Poincaré group is R 1,3 O(1, 3), the semi direct product of the abelian group of translations R 1,3 generated by P µ and the Lorentz group O(1, 3) generated by the M µν. It is not a direct product because translations and boosts do not in general commute. We now look in more detail at the Lorentz subgroup. 2 The Lorentz subgroup. Consider the vector representation of O(1, 3). That is, given an element x R 1,3, x = x 0 x 1 x 2 x 3, an element of the Lorentz group will be represented by a 4 4 matrix. There are four special elements of the group we would like to consider. They are: 12

13 = T = P = 1 1 PT = 1 4 The identity element, which takes any vector to itself. The time reversal operator, which takes x 0 x 0, but leaves spatial components unchanged. The parity operator, which takes us to a mirror world x j x j, but leaves the time direction unchanged. The combined parity time transformation which takes x µ x µ. The reason for introducing these special elements is that the Lorentz group splits into four components, each of which is continuously connected to one of the four elements above. These components are written L + 1, L + T, L P, L PT, (19) with denoting those transformations which preserve the sign of the time direction, while + denotes those which preserve the sign of the spatial components. Those transformations in L +, i.e. those continuously connected to the identity element, are called proper orthochronous. Those transformations with determinant +1, i.e. those in L + and L, connected to 1 4 or PT, form SO(1, 3). Note that the four components in (19) are disconnected. Pictorially, the Lorentz group breaks down into components as shown in Fig. 5. L + L L + L SO(1, 3) Figure 5: The disconnected components of the Lorentz group O(1, 3). Exercise 3. Verify the decomposition of the Lorentz group shown in Fig. 5. First show there is no continuous path from SO(1, 3) to any element not in SO(1, 3), which realises the rectangular boundary shown. Next, show there is no continuous path between ele- 13

14 ments with (0, 0) matrix-component +1 and 1, so realising the circular boundaries. 3 Representations of the Poincaré and Lorentz groups. Particles in QFT are described by fields ϕ ρ on R 1,3 which, if we perform a Lorentz transformation x Lx, should transform under some representation of the Lorentz group, i.e. ϕ ρ (x) ρ(l)ϕ(l 1 x), (20) where ρ(l) is some matrix representation of the Lorentz group. Under sequential Lorentz transformations x L 2 L 1 x =: L 3 x we should have ϕ ρ (x) ρ(l 2 )ρ(l 1 )ϕ(l 1 1 L 1 2 x) = ρ(l 3)ϕ(L 1 3 x), (21) and thus the fields ϕ ρ form representations of the Lorentz (and Poincaré) group. To label representations we need the Casimir operators, which commute with every generator of the group. Casimirs for the Poincaré group are P µ P µ and W µ W µ, W µ := 1 2 ϵµνρσ P ν M ρσ. (22) W µ is called the Pauli Lubanski vector. We label representations by the values of P 2 and W 2. There are two cases to consider: 1. Massive representations, m 2 > 0. In this case there exists a rest frame in which P µ = (m, 0, 0, 0) so P 2 = m 2. It can then be shown that W 2 = m 2 s(s + 1) where s Z/2 is the spin of the particle or field. 2. Massless representations, m 2 = 0. In this case there is no rest frame, but we can always choose a frame such that P µ = (E, 0, 0, E), in which case we find P 2 = W 2 = 0. However, it can be shown that W µ = λp µ here, where λ is the helicity of the particle, λ Z/2, and this is used to label the representations. Elementary particles should form ( sit in ) irreducible representations of the Poincaré group. Hence, the problem of writing down all possible fields in QFT is equivalent to finding all possible irreps of the Poincaré group. 4 Remarks. We restrict ourselves in the sequel to finding all irreps of the Lorentz subgroup the extension to the full Poincaré group is trivial. Representations should be linear (i.e. given by linear transformations) and QFT requires unitary representations (up to a phase recall that to obtain physical information from QFT, we calculate mod squared amplitudes, which remove phases). Wigner showed [14] that one can reduce such representations up to a phase, to representations up to a sign, and Bargmann showed later [15] that studying all unitary irreps of the Poincaré group up to a sign corresponds to studying all irreps of the universal covering group. The universal covering group of a group X is a simply connected space Y with a map f : Y X which is locally homeomorphic (locally it looks like the original group) and 14

15 surjective (i.e. it contains every element of the original group). We will also encounter the double cover of a space, where Y is not necessarily simply connected and the map f is 2:1. We turn now to the universal cover of the Lorentz group, beginning with the proper orthochronous component. 5 The universal cover of L +. We begin by showing that the universal cover of L + SO + (1, 3) is SL(2, C). First define the four vector of Pauli matrices σ µ, ( ) ( ) ( ) ( ) i 1 0 σ 0 =, σ 1 = σ 2 = σ 3 =. (23) i Using these matrices, we construct the carrier space on which our representation must act this space contains all of our original vectors x µ. The carrier space is defined by ( ) X(x) := x µ x 0 + x 3 x 1 ix 2 σ µ = x 1 + ix 2 x 0 x 3. (24) The inverse map, which extracts x µ from X, is 1 x µ = 1 2 Tr ( σ µ X ). (25) The vector norm is x µ x µ = det X, where the minus sign is due to our metric conventions. The determinant of this matrix must therefore be preserved under Lorentz transformations L; the relevant representation is ρ(l) SL(2, C). We will give the explicit form of ρ(l) below, but we first note that the action of L on X(x) is L X(x) = ρ(l)x(x)ρ (L), (26) which clearly preserves the determinant of X since det(ρ(l)) = det(ρ (L)) = 1. Now, for a rotation by an angle θ around a unit direction n, and a boost with rapidity tanh α in the direction n, the matrix ρ(l) is [ 1 ( ) ρ(l) = exp α n iθ n σ ] B L R L. (27) 2 (Boosts along an axis commute with rotations about that same axis, hence the exponential factorises.) This representation provides us with a map SL(2, C) SO + (1, 3), i.e. it tells us how to take an element of SL(2, C) and transform it into a rotation and a boost. Note that the map is 2 : 1, since for rotations of angles θ and θ := θ + 2π R θ = R θ, (28) but the presence of both ρ(l) and ρ (L) in (26) means that the sign is killed and these correspond to the same Lorentz transformation. We will see a peculiar consequence of this in the next section. 1 Note the index positions, which appear unnatural. Once we have introduced the σ matrices, the reader will understand that what we should really write is x µ = 1 2 Tr( σµ X). 15

16 Finally, we leave it as an exercise to show that SL(2, C) is simply connected it is therefore exactly the group we need for classifying the irreps of SO + (1, 3). We now look for all irreps of SL(2, C) and for this, we have to study the fundamental representation. Exercise 4. Show that SL(2, C) is simply connected, i.e. two paths connecting elements M, N SL(2,C) can always be continuously deformed into each other ( are homotopically equivalent ). 6 Weyl spinors. In the fundamental representation, the vector (carrier) space is W = C 2, the space of two component complex Weyl spinors ψ, ( ) ψ 1 ψ =, ψ j C. (29) ψ 2 The action of the Lorentz group is L ψ = ρ(l) ψ, (30) where ρ(l) are the matrices (27). Now, since there is only one ρ present, rotations of angle 2π do not leave spinors invariant they pick up a minus sign from (28). Instead, it takes a rotation of angle 4π to bring a spinor back to itself. This shows us that spinors really are something new, and unrelated to vectors which have no such property 2. To begin to make contact with QFT, we need a scalar product which is Lorentz invariant. To do this, we need a dual space 3. We write the dual space as W C 2 which consists of elements χ on which the Lorentz group acts as ( ) χ T χ 1 = χ 2 W, L χ = χ ρ(l 1 ). (31) To construct the scalar product (which is really a bilinear form as it is not positive definite), we need a map M : W W, which takes ψ ψ T M. The pairing (ψ, ψ ) ψ T Mψ (32) must be invariant under Lorentz transformations, i.e. ρ T (L)Mρ(L) = M. The only possible M are ( ) 0 ±1 M =. (33) For entertaining illustrations of this fact, look for Dirac s belt trick, Feynman s plate trick and the game Tangloids. 3 Compare with the example of constructing the scalar product on x µ we construct a map to the dual space using the metric, x µ x µ = η µνx ν, which furnishes us with the inner product x 2 = x µ x µ = x µ η µν x ν. 16

17 Exercise 5. Prove the result (33). We now define W as follows: for ψ α W, ψ α := ϵ αβ ψ β W, where our choice of M is ϵ 12 = 1, ϵ 21 = 1, i.e. ( ) ϵ αβ 0 1 =. (34) 1 0 Then, (ψ, χ) ψ α χ α = ϵ αβ ψ β χ α = ϵ βα ψ β χ α. (35) For the scalar product to be symmetric we find ψ β χ α = χ α ψ β, i.e. spinors must anticommute. We assume this from now on. We will also abbreviate the inner product as follows: (ψ, χ) ψ α χ α or ψχ (= χψ). 7 Complex conjugate Weyl spinors. We also define the complex conjugate representation, W : its elements 4 ψ are labelled by upper dotted indices, ψ α and transform under Lorentz transformations as ρ (L) ψ. The space dual to W is defined by β ψ α = ϵ α β ψ. (36) Here ϵ 12 = 1, ϵ 21 = +1, so ( ) 0 1 ϵ αβ = ϵ α β = 1 0. (37) Note that ϵ αβ ϵ βγ = δ α γ and ϵ α βϵ β γ = δ γ α. 8 Other examples of spinor representations. We have just seen two examples of spinor representations; the Weyl spinors and their conjugates. These representations are labelled by their helicities, We also have the vector representation: W : ψ α, ( 1 2, 0) W : ψ α, (0, 1 2 ) (38) Vector : x µ = σ µ α α xα α, ( 1 2, 1 2 ). (39) The above furnishes us with a map from bispinors, on the left, to vectors on the right. More generally, arbitrary representations of SO + (1, 3) can be constructed from tensor products of the 2d Weyl spinor rep. of SL(2, C), ϕ α1...α m α 1... α n, ( m 2, n 2 ). (40) 4 The reader who is wary of this notation for complex conjugation, or more familiar with Dirac spinors where we write Ψ = Ψ γ 0, might like to think of the Weyl spinor ψ β as ψ α σ 0α β, as this is exactly the same thing. 17

18 For example, the Yang Mills field strength F µν [ µ, ν ] may be written F µν σ [µ α α σν] β β F α αβ β. (41) The antisymmetry in (µ, ν) implies that F α αβ β can be written F α αβ β = ϵ αβ f α β + ϵ α βf αβ. (42) After a Wick rotation to Euclidean space R 4, if f α β = 0 then F is self-dual: F = F, which corresponds to a gauge field configuration known as an instanton, while if f αβ = 0 then F = F ; this is an anti instanton. 9 Other signatures. In Euclidean signature the symmetry group is SO(4), and the corresponding double cover is SU(2) SU(2). For the metric diag(+, +,, ) with Kleinian signature and symmetry group SO(2, 2) the double cover is SL(2, R) SL(2, R). To adapt the above to these metrics, it is a matter of adjusting the sigma matrices by factors of i such that the condition x 2 = det X(x) is maintained. Note that for finding all unitary ray representations for other signatures and other dimensions, the double cover of the group of isometries is sufficient. 2.2 Spin and pin groups So far we have looked at L +, but we are still looking for the full universal cover of O(1, 3). We pursue this below. For a more detailed exposition of the material, see for example [16]. 1 The Clifford algebra. Consider an algebra generated by d objects 5 γ µ, for µ = 0, 1,... d 1, {γ µ, γ ν } = 2η µν 1 d, (43) along with the vector space V = Span(γ µ ). The sign on the right hand side of this equation is pure convention. Most books working with a mostly minus metric (WCM) use the opposite sign. The Clifford algebra C(V ) is C(V ) = n 0 V n = C V V V V V V... V d. (44) Here, C 1 d, V γ µ, V V γ µν γ [µ γ ν], etc, where we antisymmetrise over indices as symmetric combinations of the γ µ can be reduced to expressions containing fewer γ s using the anticommutator (43). The sum in (44) terminates for the same reason. The algebra decomposes as C(V ) = C + (V ) C (V ), (45) 5 We use a heavy script to denote the abstract elements of the Clifford algebra. Later, we will use normal script γ for the usual gamma matrices. 18

19 where C + (V ) contains products of even numbers of γ µ while C (V ) contains products of odd numbers of the γ µ. With the definition of the following operation τ (an antiinvolution) on this space, ( γµ1... γ µn ) τ = ( 1) n ( γ µn... γ µ1 ), (46) we can now define the pin group. 2 The pin group. On the vector space V with isometry group SO(V ), the associated pin group is defined as Pin(V ) := {Λ C(V ) Λ.Λ τ = 1, Λ.V.Λ τ V }. (47) The interesting point is that Pin(V ) carries a natural representation of the group SO(V ), where the action of an element L SO(V ) on C(V ) is given by the following expression with Λ L C(V ): Λ L γ µ Λ τ L = Λ L γ µ Λ 1 L = γ νl ν µ. (48) Note that we could replace τ with the inverse because of the definition of the pin group. 3 Examples. One can now easily find a couple of explicit examples for Λ L. Consider, for example, the parity transformations P. The associated pin group element is given by Λ P = iγ 0, which obeys Λ τ P Λ P = 1 as required using the fundamental anticommutator. We can now check that the action of Λ P reverses the sign of γ µ for µ = 1, 2, 3, but leaves γ 0 alone, hence iγ 0 γ µ ( iγ 0 ) = γ ν P ν µ, (49) and we have correctly identified the parity operator. The time reversal transformation T corresponds to Λ T = γ 1 γ 2 γ 3, and its action reverses the sign of γ µ only for µ = 0. We also want the continuous Lorentz transformations. These are ( ) 1 Λ L = ± exp 8 [γ µ, γ ν ]θ µν, L L +. (50) Writing Σ µν 1 4 [γ µ, γ ν ], one can show that Σ µν satisfy the algebra of the generators of the Lorentz algebra M µν. 4 The spin group. Spin(V ) := { Λ C + (V ) Λ.Λ τ = 1, Λ.V.Λ τ V } = Pin(V ) C + (V ). (51) Here, as above, C + (V ) contains only even numbers of γ s. The relationship between the spin and pin groups is shown in Fig. 6. Spin(V ) and Pin(V ) are connected and form double covers of SO(V ) and O(V ), 1 Z 2 Pin(V ) O(V ) 1, 1 Z 2 Spin(V ) SO(V ) 1, 19

20 ±Λ 1 ±Λ PT ±Λ P ±Λ T Spin(V ) Pin(V ) Figure 6: The pin and spin groups. where V is an arbitrary R p,q. Some explicit examples of spin groups are Spin(4) = SU(2) SU(2), Spin(3, 1) = SL(2,C), Spin(2, 2) = SL(2, R) SL(2, R). We will come back to spinors in arbitrary dimensions in section Representations of the Clifford algebra C(R 1,3 ). We now move on to the representations of the Clifford algebra. This will lead us to all the fields we need to consider in Lorentz invariant theories. The minimal representations are four dimensional, i.e. γ µ = ( 0 σ µ σ µ 0 ) (52) or, explicitly: {( ) ( ) ( ) ( )} γ µ i = 1 0, 0 1, i i, i where σ µ = η µν σ ν, see (23), and σ µ is defined by, (53) σ µ αα := ϵ α βϵ αβ σ µ. (54) β β Note that in (54), both σ and σ are given with their natural index positions. The form of the gamma matrices given here is the so-called Weyl representation. There is also the Dirac representation, commonly employed in QFT textbooks. We always use the Weyl representation in these lectures. The gamma matrices obey {γ µ, γ ν } = 2η µν 1 4. (55) Their conjugates are γ 0 = γ 0 and γ j = γ j. The γ matrices act on a pair Ψ of Weyl spinors χ W and ψ W, ( ) χ α Ψ = ψ α (56) This pair is called a Dirac spinor. We also define γ 5 := iγ 0 γ 1 γ 2 γ 3, with γ 5 = γ 5, which is the matrix ( ) γ 5 =, (57)

21 in our Weyl representation. P ± := 1 2 (1 4 ± γ 5 ). The Weyl spinors are eigenvectors of the projectors Exercise 6. It is very useful to realise that, writing σ µ (σ 0, σ 1, σ 2, σ 3 ), the barred sigma matrices are simply σ µ = (σ 0, σ 1, σ 2, σ 3 ). Prove this result for yourself. This is the origin of the (baffling) choice of seemingly Lorentz ignorant conventions σ µ = σ µ, which we (tacitly used earlier and) shall never speak of again. 6 Parity and time reversal. Recalling 3, the parity and time reversal operators in terms of explicit matrices are given by Λ P = ±iγ 0, Λ T = γ 1 γ 2 γ 3, (58) which can be checked by calculating Λ P γ 0 Λ 1 P = γ 0, and Λ P γ j Λ 1 P = γ j, = Λ P γ µ Λ 1 P = γ νp ν µ, Λ T γ 0 Λ 1 T = γ 0, and Λ T γ j Λ 1 T = γ j, = Λ T γ µ Λ 1 T = γ ν T ν µ. Letting these parity and time reversal operators act on the Dirac spinor, we find ( ) ( ψ i Λ P Ψ = ± i, Λ T Ψ = ψ ). χ iχ (59) Exercise 7. Show that (59) holds for the definitions of Λ P and Λ T given in (58). 7 The Dirac equation and the Lagrangian. We now make contact to physics by giving the equation of motion for Dirac spinors. This is, ( iγ µ µ m ) Ψ = 0, (60) where γ µ µ is C(V )-valued. The factor of γ µ yields the appropriate coupling of vectors ( µ ) to spinors (Ψ). The equations of motion have the following Weyl decomposition in terms of Weyl spinors, iσ µ µ ψ mχ = 0, i σ µ µ χ m ψ = 0. A Lagrangian density which yields the Dirac equation is, (61) L = Ψ ( iγ µ µ m ) Ψ, (62) where Ψ Ψ γ 0. In terms of Weyl spinors it may be written L = i χ σ µ µ χ + iψ σ µ µ ψ mψχ m χ ψ. (63) 21

22 8 Charge conjugation. Along with the discrete symmetries P, T and PT, there is another discrete transformation. Consider the above equations of motion, but including minimal coupling to a hermitian U(1) gauge potential, ( iγ µ ( µ iea µ ) m ) Ψ = 0. Note that if we take the complex conjugate, we obtain ( iγ µ ( µ + iea µ ) m ) Ψ = 0, (64) where the sign of the coupling has changed. We introduce C which acts as follows, The resulting equation of motion for Ψ C is thus Λ C (γ µ ) Λ 1 C = γ µ, Ψ C Λ C Ψ. (65) ( iγ µ ( µ + iea µ ) m ) Ψ C = 0. We want (Ψ C ) C = Ψ, which implies Λ C Λ C = 1. On R1,3 in the Weyl/chiral representation, we can choose Λ C = ±γ 2. Physically, C changes particles into their antiparticles of the same mass but the opposite charge. 9 The CP T theorem. Both P and CP are violated in nature: the first was observed in Cobalt 60 decays in 1956, the second in Kaon decays in Both appear in the weak nuclear force, and it is unknown if CP violation also exists for strong nuclear interactions. This violation is believed to be the reason for the existence of more matter than antimatter. In 1955, Pauli proved that if you have a quantum field theory which is 1. invariant under L +, 2. causal and local, 3. has a Hamiltonian which is bounded below, then the quantum field theory is invariant under the combined transformation CPT. 2.3 Summary Since all our fields will transform under irreps of O(1, 3), we spent some time constructing all possible irreps. Group theory told us that we needed the universal covering group which, for L + for example, is SL(2, C). We discussed the following representations (the examples column contains some objects we will meet later, for reference): 22

23 Carrier space Name Examples in this rep. W = C 2 Weyl spinors ψ α SUSY charges Q α, ϵ α parameters. W = C 2 Dual Weyl spinors ψ α SUSY charges Q α, ϵ α parameters. W = C 2 Conjugate Weyl spinors ψ α SUSY charges Q α, ϵ α parameters. W = C 2 Conjugate, dual Weyl spinors ψ α SUSY charges Q α, ϵ α parameters. Spinors are anticommuting objects. Once we allow for the discrete transformations P, T and PT we need the double cover of SO(1, 3), which is Spin(1, 3), and the double cover of the full group O(1, 3) is the pin group. With the Dirac matrices, ( ) γ µ 0 σ µ = σ µ, (66) 0 the basic representations are given by Dirac spinors Ψ which decompose into two Weyl spinors, ( ) ψ α Ψ = χ α. (67) Note that the natural index structure on σ µ and σ µ is σ µ α α and σµ αα, so that the gamma matrices (66) act naturally on the Dirac spinor (67). 1 Hermitian conjugation. Before continuing it is worth cementing our conventions as regards complex, or hermitian, conjugation of Weyl spinors. The conjugate of a product of objects, be they vector or spinor, is (AB... Z) = Z... B A. (68) There are no minus signs, and all indices remain in their allotted positions. It may be useful in calculation to explicitly put dots over spinor indices which become conjugated spinors. A conjugated spinor is ψ ψ. A consequence of these conventions is that the Pauli matrices behave as just that matrices, rather than bispinors. Any lack of mathematical satisfaction the reader may feel 6 should be more than compensated for by the ease of calculation these conventions provide. For example, we then have (ψχ) = χ ψ χ ψ = ψ χ and (ψσ µ χ) = χσ µ ψ. 6 as, e.g., that the naive Koszul sign rule is not obeyed 23

24 3 The SUSY algebra The SUSY algebra is an extension of the Poincaré algebra we met earlier. A theorem due to Haag et al [17] says that the SUSY algebra we will present below is the only possible extension of the Poincaré group consistent with the axioms of quantum field theory. 3.1 SUSY algebra on R 1,3 1 SUSY algebra. Along with the generators P µ and M µν which obey the algebra (17) (18), we choose N N +, and then for i = 1... N we introduce the supersymmetry charges Q i α and Q i α which obey {Q i α, Q j β } = { Q i α, Q j β} = 0, (69) {Q i α, Q j α } = 2 δ i j σ µ α α P µ, (70) where Q is the complex conjugate of Q. Note the positions of the i indices labelling the different supercharges, which will serve as a bookkeeping device for correct contractions later. Note also that the zero vector component on the right hand side of (70) looks like the Hamiltonian in quantum mechanics, just as we had in our toy model, earlier. The commutators of the SUSY charges with the generators of the Poincaré group are [P µ, Q i = [P µ, Q i α ] = 0, (71) [ Mµν, Q i α] = i(σ µν ) β α Q i β, (72) [ Mµν, Q i α ] where we have introduced (σ µν ) α β 1 ) (σ µα 4 α σν αβ σ να α σ µ αβ, ( σ µν ) α 1 ( ) σ β µ αα σ ν 4 α β σν αα σ µ α β. = i( σ µν ) α β Q β i, (73) If N > 1 we call this the N extended SUSY algebra. The number of real supercharges in the game is 4N (in four dimensions) since each of the Q i is a two component complex Weyl spinor. 2 Theorem of Haag, Sohnius, Lopuszanski. Up to introducing central charges Z [i,j] such that {Q i α, Q j β } = ϵ αβz [i,j], (75) where the Z are just complex numbers, the N extended SUSY algebra is the only extension of the Poincaré group which is consistent with the axioms of relativistic quantum field theory [17]. We will set the central charges to zero in the majority of these lectures for completeness, we retain them only in Sect. 3.2, 6 when discussing the massive representations of the SUSY algebra. (74) 24

25 3.2 Representations 1 Casimir operators. For the massive representations, P 2 remains a Casimir operator. We also label representations by their superspin, C 2, with C 2 = P 2 W (P W )2. (76) The eigenvalues of the superspin operator are m 4 s(s + 1). For the massless representations, we introduce L µ = W µ 1 16 σ αα µ {Q i α, Q i α }, and then label representations by their superhelicity κ, where L µ = ( κ + 4) 1 Pµ, (77) and κ Z/2 (compare with the relation W µ = λp µ defining the helicity for the massless Poincaré representations). 2 Massless representations. As before, to construct the massless representations, we go to the frame P µ = (E, 0, 0, E), and remain there in this paragraph. The key to understanding the massless reps lies in evaluating (70) in our chosen frame. We find ( ) σ µ α α P 0 0 µ =. (78) 0 2E Comparing with our toy model, we find that Q i 1 = Q i 1 = 0 on such states7, since for arbitrary ψ, ψ {Q 1, Q 1 } ψ = Q 1 ψ 2 + Q 1 ψ 2 = 0, while {Q i 2, Q j 2 } = 4Eδi j, i.e. we have N copies of our toy model Clifford algebra. Now, in this frame, the helicity operator is just J 3 M 12. Calculating its commutator with the SUSY charges we find α α [J 3, Q i 2] = 1 2 Qi 2, [J 3, Q i 2 ] = +1 2 Q i 2. (79) These commutators simply say that Q i 2 lowers helicity by 1/2 while Q i 2 raises helicity by 1/2, so that repeated application of Q or Q changes the spin of a state from half integral to integral and back, i.e. it turns bosons into fermions into bosons, etc. Exercise 8. Derive the raising and lowering commutators (79). We now choose a lowest weight state from which to construct all the states in this representation. (This is just what we did in our toy model, starting from the state annihilated by Q.) Our lowest weight state will be h, which is annihilated by all the Q i 2. All we can do to construct other states is act with the Q i 2, but we can act with each at most once, since they anticommute amongst themselves and square to zero. Hence, 7 Note that the term state referes here slightly sloppily to an element of the module which serves as the carrier space of the SUSY representation under consideration. 25

26 there are a total of 2 N states (each of the Q i 2 may be present once or not at all) of the form Q i Q in 2 h. (80) The range of helicities of these states is h up to h + N /2, possessed by the state Q Q Q N 2 h. If it helps, you may like to imagine all such states as being arranged into corners of an N -dimensional cube, where the number of Q i corresponds to the lattice distance to the corner corresponding to the state h. We now give examples of the massless representations for various N. Representations are labelled by their mass and superspin or superhelicity. The states which live inside any such representation are collectively called a (super)multiplet. As described above, these states are related by the action of the SUSY charges, and are of equal mass. A multiplet may be seen as the set of all states which are needed to form a closed representation of the (SUSY) algebra. 3 Example 1: N = 1 multiplets. As in our toy model, we have a 2 1 = 2 state system, comprising the lowest weight state h and h = Q 2 h. For physical reasons we wish to avoid particles with helicities larger than 2 (the theories of which are thought not to be well defined) and with helicities 2 and 3/2, which correspond to the graviton and gravitino (as we want to steer clear of gravity). Hence we have two possibilities, State helicities Name (1, 1 2 ) The vector multiplet (since helicity h = 1 corresponds to the photon, which is a vector). ( 1 2, 0) The chiral multiplet. The 1-dimensional cube is just a line segment with the end points corresponding to the two different states. Taking the CPT conjugates of these multiplets gives us multiplets of helicity (0, 1 2 ) and ( 1 2, 1). 4 Example 2: N = 2 multiplets. Starting from our lowest weight state h, we can now act with either Q or Q to generate two states of helicity h We can also act with both operators to generate a state of helicity h + 1. As before, we wish to steer clear of spin larger than 1, which gives us two possible multiplets, State helicities Name (1, 1 2, 1 2, 0) The N = 2 vector multiplet. ( 1 2, 0, 0, 1 2 ) The hypermultiplet. 26

27 The 2-dimensional cube is a square, and the corners follow the pattern bosonfermion-boson-fermion. The charge conjugate of the vector multiplet has helicities (0, 1 2, 1 2, 1), while the hypermultiplet is its own charge conjugate. 5 Example 3: N = 3 and N = 4 multiplets. The case N = 3 is usually not discussed seperately from the case N = 4 for the following reason: in principle, there are two multiples: (➀ 1, ➂ 1 2, ➂ 0, ➀ 1 2 ) and (➀ 1 2, ➂ 0, ➂ 1 2, ➀ 1), where the plain number in each product is the helicity while the circled number is the number of states with that helicity. In the cubic picture, the eight states correspond to the corners of the cube with lattice distances (0, 1, 2, 3). These two multiplets are each other s CPT conjugate and in a physical theory, we expect both to be present. Then, however, they combine to form the N = 4 supermultiplet, which reads as (➀ 1, ➃ 12 ), ➅ 0, ➃ 12, ➀ 1, (81) We have a total of = 16 states, which correspond to the 16 corners of a 4-dimensional cube with lattice distance (0, 1, 2, 3, 4). The number of states of course matches with our general form of 2 N states, for N = 4. Note that this supermultiplet is its own charge conjugate. It gives the entire field content of N = 4 super Yang Mills theory, which we will meet again later in these lectures. 6 Massive representations. In these lectures we will mostly deal with massless representations (taking mass generation to be due to Higgsing), but for completeness we include this discussion of the massive representations. We consider the most general case by leaving in the central charges (75), {Q i α, Q j β } = ϵ αβz [i,j], { Q i α, Q j β} = ϵ α β Z[i,j]. (82) We go to the rest frame of the particle, in which P µ = (m, 0, 0, 0). In order to understand the representations we will write our commutators, in this frame, in such a way that the technology we applied to the massless representations can also be employed here. First, we use U(N ) rotations to bring the central charges to the following form, Z [i,j] = 0 z 1 z z z (83) Next, introduce the following linear combinations of the supercharges 8, a i α := 1 ( Q 2i 1 α + ϵ αβ Q 2i ) 2 β, b i α := 1 ( Q 2i 1 α ϵ αβ Q 2i ) 2 β, (84) 8 The strange contractions between indices will only appear in this section, in this frame (Lorentz invariance is broken). 27

28 for r = 1... N /2. The only non-vanishing commutators are {a i α, a j β } = (2m zi )δ ij δ αβ, {b i α, b j β } = (2m + zi )δ ij δ αβ, no sum, no sum. Now recall, from our toy model example, that positivity of the Hilbert space requires 2m z i (and so in particular central charges must vanish in the massless case). Assume that k of the z i saturate this bound, i.e. z i = 2m, i = 1... k, for k N /2. Then one of the pair {a i, a i } or {b i, b i } vanishes, which implies either a i or b i must be put to zero. We therefore have a total of 2N 2k non zero fermionic oscillators, and so 22N 2k states. To illustrate, if k = 0 and N = 1, we can construct the multiplets ( ) 1 2, 0, 0, 1, (1, 12 2, 12 ), 0. It follows that representations are constructed just as in the massless case, for k > 0, and those multiplets are called short or BPS. If k takes its maximal value of N /2 then we have an ultrashort multiplet. Note that only ultrashort multiplets can become massless, as it is only then that 2 2N 2k = 2 N and so the number of states matches that in a massless multiplet. 3.3 The Wess Zumino model We are now ready to look at our first SUSY field theory. We begin with a condensing of our notation by combining commutators and anti commutators into a single bracket. 1 Grading and supercommutators. Introduce a parity to every object. This is a Z 2 grading such that everything bosonic (i.e. with integral spin) carries a label 0, while everything fermionic (i.e. with half integral spin) carries the label 1. We denote the parity of an operator by writing a tilde over it, i.e. P µ = 0, Qj α = 1. Noting that our SUSY algebra comprised anticommutators between the odd objects and commutators otherwise, we introduce the supercommutator (85) {[A, B]} := AB ( )Ã BBA. (86) This recovers all our previous commutators (e.g. {[P µ, P ν ]} = [P µ, P ν ]) and anticommutators (e.g. {[Q, Q]} = {Q, Q}). 2 Representation of the SUSY algebra on component fields. We are looking for some set of fields φ, ψ... etc, on which the SUSY algebra closes. As we look for this, we will build a concrete representation of the SUSY generators acting on various fields. On any field, we have P µ = i µ. Given an infinitesimal v µ, we can define the infinitesimal transformation of, say, a scalar φ as δ v φ = v µ P µ φ = iv µ µ φ. (87) 28