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An Introduction to Supersymmetry

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An Introduction to Supersymmetry ETH Zurich¨ Proseminar in Theoretical Physics Prof. Gaberdiel An Introduction to Supersymmetry Author Supervisor Thomas Hahn Dr. Ida Zadeh May 14, 2018 Contents 1 Introduction 1 2 Lorentz and Poincare Groups and their Algebras 2 2.1 Lorentz Group . 2 2.2 Lie Algebra . 4 2.3 Lorentz Algebra . 5 2.4 Poincare Group . 6 2.5 Poincare Algebra . 6 3 Spinors 7 3.1 Introduction to the Spinor Representation . 7 3.2 Spinors . 8 3.3 Constructing an Action . 9 3.4 The Dirac Equation . 10 3.5 Chiral Spinors . 11 3.6 Weyl Equation . 11 3.7 Majorana Fermions . 12 3.8 Weyl Spinors and Invariant Tensors . 12 4 N=1 super-Poincare Algebra and its Representations 13 4.1 General Irreducible Representations of the Homogenous Lorentz Group . 13 4.2 Graded Lie Algebra . 14 4.3 N=1 Super-Poincare Algebra . 15 4.4 Commutators with Internal Symmetries . 16 4.5 Representations of the Poincare Group . 17 4.6 N=1 Supersymmetry Representations . 18 4.7 Massless Supermultiplet . 19 4.8 Massive Supermultiplet . 20 5 Wess-Zumino Model 21 5 5.1 γ and γ5 ........................................................ 21 5.2 On-Shell Non-Interacting Model . 22 5.3 Off-Shell Interacting Model . 22 5.4 On-Shell Interacting Model . 22 5.5 Conserved Supercurrent . 23 6 Extended Supersymmetry 24 6.1 Extended Supersymmetry Algebra . 24 6.2 Massless Representations for N > 1.......................................... 25 6.3 Massive Representations for N > 1.......................................... 25 1 Introduction During the course of this paper I will try to give an overview of what supersymmetry (SUSY) is and how to work with the mathematical structures that it contains. The next two chapters will introduce the necessary mathematical tools, which we require for chapters four through six. Some sections, such as 3.3, 3.4, and 3.6 are not necessary to understand later chapters, but give further insight into the physical interpretation of these mathematical constructs and introduce some notation that will be used afterwards. This chapter is based on [1] and [2]. Without going too much into detail, I will give a short introduction as to what SUSY is. Up until the 1960s, attempts were made to combine all known symmetries into a single group. It wasn't until Coleman and Mandula released their famous "no-go" theorem, that it was realized all such efforts would be in vain. This theorem states that the most general bosonic symmetry can be written as a direct product of the Poincare group and internal symmetries [1], i.e. G = GPoincare × GInternal : (1) However, the Coleman-Mandula theorem has a loophole. In 1975, Haag, Lopuskanski, and Sohnius managed to extend the theorem to include fermionic symmetry generators, which are given by the letter Q, that map bosons to fermions and vice-versa. As a result, it turns out that the most general symmetry group can be written as the direct product of the 1 super-Poincare group, which is given by the Poincare generators and the fermionic generators Q, and internal symmetries [1]. We therefore get that G = Gsuper−Poincare × GInternal : (2) Let's look at transformation properties of Q. If we apply a unitary operator U that represents a full rotation of 360◦, we get that UQjbosoni = UQU−1Ujbosoni = Ujfermioni (3) UQjfermioni = UQU−1Ujfermioni = Ujbosoni : (4) We also know that under a 360◦ rotation, the two states transform as Ujfermioni = −|fermioni (5) Ujbosoni = jbosoni : (6) Since the sum of all fermionic and bosonic states form the basis of our Hilbert space, we get that UQU −1 = −Q; (7) which transforms exactly like a fermionic state [2]. During the course of chapter 4 we will calculate the anticommutator fQ; Qyg = QQy + QyQ. It should be noted that while the Qs will not be unitary in general, fQ; Qyg is. In addition, the anticommutator will have positive eigenvalues, since h:::jQQyj:::i + h:::jQyQj:::i = jQyj:::ij2 + jQj:::ij2 ≥ 0 : (8) A further analysis will show that this anticommutator can be written as a linear combination of the energy and momentum operators #»#» fQ; Qyg = αE + β P: (9) In addition, the sum over all the Qs is proportional to the energy operator, as all other terms will cancel out X fQ; Qyg / E: (10) all Q Since the left-hand-side is positive, the energy spectrum can either be only positive or negative. If, for physical reasons, we require energies bounded from below, the proportionality factor is positive. This would then require our energy values to be non-negative. In chapters 4 and 6, we will analyze the representations of the super-Poincare group, which correspond to families of particles. Since any supersymmetry transformation will turn a particle into something else, these families will always contain more than one particle [2]. This tells us that all families, which are also called supermulitplets, must contain at least one boson and one fermion. Before we start with section 2, there is one more thing I would like to mention that will not be discussed in later chapters. As we will see, our Qs will commute with the energy and momentum operators. This in turn, means that Q changes neither the energy nor the momentum of the particle and therefore all states in the supermultiplet must have the same mass. However, we do not see this in nature, which tells us that either supersymmetry isn't fundamental to nature or there is spontaneously broken symmetry, which means that our ground state will not be invariant under SUSY transformations. Spontaneously broken symmetry would lift this mass degeneracy [2]. 2 Lorentz and Poincare Groups and their Algebras This chapter is based on [3] and [4]. 2.1 Lorentz Group The Lorentz group O(1; 3) is given by all linear transformations in R4 that conserve the Minkowski metric. In other words, 4 O(1; 3) = fA 2 GL(4; R)j(Ax; Ax) = (x; x); 8x 2 R g ; (11) where (x; y) = x0y0 −x1y1 −x2y2 −x3y3 and x; y 2 R4. If we write A, where A 2 O(1; 3), as a matrix, this can be interpreted as follows: 0 1 0 0 0 1 T B 0 −1 0 0 C A gA = g; where g = B C : (12) @ 0 0 −1 0 A 0 0 0 −1 2 While these matrices have a great many number of properties, we are only interested in a handful of them. In the following, I will make certain claims and prove them as well [3]. Claim 1: det(A) = ±1 Proof: det(AT gA) = det(AT )det(g)det(A) = det(g) ; (13) which implies that det(A) = ±1. 2 Claim 2: (A00) ≥ 1 Proof: AT gA = g ; (14) where A = (Aij) implies that X AikgklAjl = gij : (15) k;l For i = j = 0, we now have X A0kgklA0l = g00 : (16) k;l We also know that gkl = 0 for k 6= l. This relationship therefore allows us to remove one of the indices, which gives us 2 2 2 2 A00 − A01 − A02 − A03 = 1 : (17) 2 Since all entries of A are real, their squares must be greater than or equal to zero. This, in turn, proves that A00 ≥ 1. We will use these two claims to split O(1; 3) into four classes. But before that, we will examine some elements of the Lorentz group [3]. Orthogonal Transformations in R3: 0 1 0 0 0 1 B 0 C B C (18) @ 0 R A 0 where R 2 O(3) is an orthogonal transformation. These matrices form a subroup. Lorentz Boosts: 0 cosh(χ) 0 0 sinh(χ) 1 B 0 1 0 0 C L(χ) = B C (19) @ 0 0 1 0 A sinh(χ) 0 0 cosh(χ) This matrix represents a Lorentz boost in the z-direction with rapidity χ 2 R. Lorentz boosts in the x- or y-direction can be constructed analogously. These matrices are used in special relativity to change from one inertial system to another. One of their more important properties is that L(χ1)L(χ2) = L(χ1 + χ2). Space Inversion: 0 1 1 B −1 C P = B C (20) @ −1 A −1 Time Inversion: 0 −1 1 B 1 C T = B C (21) @ 1 A 1 These two discrete Lorentz transformations, along with 1 and PT , generate an Abelian group of order 4. We will now define two additional subgroups of O(1,3), of which the latter will help us create the four classes. O+(1; 3) = fA 2 O(1; 3)jA00 ≥ 1g (22) SO+(1; 3) = fA 2 O+(1; 3)jdet(A) = 1g (23) 3 In the last paragraph I have ever so slightly made a proposition, but have not proven it. I have namely stated that both of these sets are subgroups, which beforehand does not necessarily have to be true. The following two proofs will do just that[3]. Claim 3: O+(1; 3) is a subgroup of O(1; 3). More specifically, it is the set of Lorentz transformations that maps the vectors inside the future light cone(Z+) to vectors inside the future light cone(Z+). Proof: We will prove the second statement first. Let A 2 O+(1; 3). If x 2 Z+ then Ax must lie within the light cone, since 0 ≤ (x; x) = (Ax; Ax). We# » must now show that Ax indeed lies in the future light cone, i.e. (Ax)0 > 0. We will now introduce #» 1 2 3 the following notation: A0 = (A01;A02;A03) and x = (x ; x ; x ). The 0-component of Ax will then be: # » 00 0 #» x = A00x + A0 · x (24) Now since (a; a) = 1, where a is the first row of A(this will not be proven, but can easily be shown by checking that AT is# » a Lorentz transformation and then using definition (12)), a must lie within the light cone.
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