An Introduction to Supersymmetry Frederick McDonnell Mathematical Sciences Dept., Durham University Supervisors: Prof. Wojtek Zakrzewski & Prof. Richard Ward April 27, 2016 Abstract This report aims to provide an introduction to Supersymmetry (SUSY) for those with limited knowledge of field theory. The Klein-Gordon field is discussed and then quantised, with infra- red and ultra-violet divergences examined. We then discuss the Lorentz group, considering representations of its associated Lie algebra, before extending it to the Poincar´egroup with the inclusion of translations. The Dirac field is then introduced from a historical perspective, and quantised in analogy to the Klein-Gordon field. The concept of Supersymmetry is introduced as an extension of the Poincar´ealgebra, and the relations of the resulting Super-Poincar´ealgebra are examined. We then construct representations of this algebra, called supermultiplets. We discuss the massless free Wess-Zumino model, a supersymmetric theory consisting of a massless chiral supermultiplet (a Weyl fermion and a bosonic complex scalar field). We confirm the action is invariant for this model, and recover the SUSY algebra by inserting quantised fields. Contents 1 Introduction 4 1.1 Preamble . 4 1.1.1 Introduction . 4 1.1.2 Conventions . 5 2 Field Theory 6 2.1 Classical Field Theory . 6 2.1.1 Background . 6 2.1.2 Equations of Motion . 6 2.1.3 Noether's Theorem . 7 2.1.4 An example of Noether's Theorem . 8 2.1.5 On-shell and Off-shell . 9 2.2 Moving to Quantum Field Theory . 9 2.2.1 The Klein-Gordon Equation . 9 2.2.2 Conjugate Momentum and Hamilton's Equations . 10 2.2.3 The Stress-Energy Tensor . 10 2.3 Canonical Quantisation of a Scalar Field . 11 2.3.1 Poisson Brackets . 11 2.3.2 Quantisation . 11 2.3.3 The Hamiltonian for the Klein-Gordon field . 11 2.3.4 The Harmonic Oscillator in Quantum Mechanics . 12 2.3.5 Applying the Quantisation of the Harmonic Oscillator . 14 2.3.6 Infra-red and Ultra-violet Divergences . 15 3 The Lorentz & Poincar´egroups 16 3.1 Lorentz Transformations . 16 3.1.1 Defining a Lorentz Transformation . 16 3.1.2 Lorentz Transformations on Scalar Fields . 16 3.1.3 Introduction to the Lorentz Group . 17 3.2 Representations and Spinors . 19 3.2.1 Structure of the Lorentz Group . 19 3.2.2 Representations of the Lorentz Group . 21 3.2.3 Transformations of Spinors . 23 3.3 The Poincar´eGroup . 24 3.3.1 The Poincar´eGroup . 24 3.3.2 Casimirs of the Poincar´eGroup . 24 2 4 The Dirac Field 26 4.1 The Dirac Equation . 26 4.1.1 An introduction to Fermions . 26 4.1.2 The Dirac Equation: a first-order relativistic wave equation . 26 4.2 The Clifford Algebra . 27 4.2.1 The Dirac Basis . 27 4.2.2 The Weyl Basis . 28 4.3 Properties of Spinors and the Dirac Equation . 29 4.3.1 Lorentz Covariance of the Dirac Equation . 29 4.3.2 Dirac Bilinears . 30 4.3.3 The Dirac Lagrangian . 30 4.4 Canonical Quantisation of the Dirac field . 32 4.4.1 Plane Wave solutions . 32 4.4.2 Chirality & Helicity . 33 4.4.3 Formulae required for Quantisation . 33 4.4.4 Quantising the field . 35 4.4.5 The Hamiltonian . 36 5 Introducing Supersymmetry 39 5.1 The Supersymmetry Algebra . 39 5.1.1 What is Supersymmetry? . 39 5.1.2 The Supersymmetry Algebra . 39 5.1.3 Extended Supersymmetry . 40 5.1.4 Forming the Algebra . 40 5.1.5 Interpretation of the Algebra . 42 5.2 Supermultiplets . 43 5.2.1 What are Supermultiplets? . 43 5.2.2 Constructing Massive Supermultiplets . 44 5.2.3 Constructing Massless Supermultiplets . 45 6 The Free Wess-Zumino Model 47 6.1 Introducing the Model . 47 6.1.1 A Supersymmetric Lagrangian . 47 6.2 Modifying the Model . 49 6.2.1 Commuting SUSY transformations . 49 6.2.2 On-shell and Off-shell SUSY . 49 6.2.3 Adding an Auxiliary Field . 50 6.2.4 Commuting modified SUSY transformations . 51 6.3 Obtaining the SUSY algebra . 52 6.3.1 The Supercurrent . 52 6.3.2 The Supercharges . 53 6.3.3 Recovering the SUSY Algebra . 54 7 Conclusion & Further Discussion 56 7.1 Further Discussion . 56 7.2 Conclusion . 56 Appendix A Spinor Indices and Identities 58 References 60 3 Chapter 1 Introduction 1.1 Preamble 1.1.1 Introduction Within the last century, the concept of symmetry has become fundamental to constructing an all-encompassing theory of our universe. The most successful theory we have to date, the stan- dard model, is essentially a theory of symmetries and how these lead to physical phenomena. The model hinges on an intrinsic property of fundamental particles, spin. It splits particles into two categories: fermions, with half integer spin, which make up matter, and bosons, with whole integer spin, which characterise particles that `carry' the fundamental forces. When working with the standard model, we utilise quantum field theory (QFT), which replaces the notion of particles with fields. The theory brings together special relativity and quantum mechanics. In this re- 1 port, we examine spin-0 and spin- 2 fields. We exclusively discuss free quantum field theory, and choose not to introduce interactions, which usually involves adding a potential term to our theory. Given that QFT is constructed with relativity in mind, Lorentz transformations become crucial in understanding why we observe the particles we do. These transformations form the Lorentz group, the representations of which correspond to different types of fields: the trivial representation refers to a spin-0 (bosonic) scalar field, the spinorial representation refers to spin- 1 2 (fermionic) spinor fields, whilst higher spin representations correspond to gauge bosons and other such particles. Supersymmetry (SUSY), is a relatively new theory that proposes a space-time symmetry (i.e. dependent on space-time co-ordinates) between fermionic and bosonic fields. The basic premise of the theory is the existence of supersymmetric generators, which turn a bosonic field into a fermionic field (and vice-versa). Thus, all physical particles must have `superpartners', which are coupled together in `supermultiplets'. As of writing, the theory is severely lacking in experimen- tal evidence, but its mathematical elegance has provided reason enough for it to become a large research focus. In 1975, Haag, Sohnius and Lopuszanski proved that a supersymmetric theory was the only possible space-time symmetry extension to an interacting quantum field theory. Further to this, SUSY can be shown to `solve' many of the problems with the standard model, such as the ‘fine-tuning problem', which we unfortunately do not have time to review. In this report, we focus on constructing a basic supersymmetric theory, the free massless 4 Wess-Zumino model, which consists of a single fermion and a single boson. We check that this transformation is a symmetry, and recover the associated supersymmetric algebra by applying results from QFT. 1.1.2 Conventions Throughout this report, we will be using index notation throughout with the space-time index µ = 0; 1; 2; 3 with Einstein's summation convention implied. For a 4-vector aµ we have: µ X µ a aµ = a aµ: (1.1) µ=0;1;2;3 Additionally, we choose to use natural units: those such that ¯h = c = 1. We will be exclusively using the Minkowski metric with the mostly-minus convention: 01 0 0 0 1 µν µν B0 −1 0 0 C g = η = B C : (1.2) @0 0 −1 0 A 0 0 0 −1 We define our 4-gradient by @ @ @ @ @ @ = @ = ; r = ; ; ; : (1.3) @xµ µ @t @t @x1 @x2 @x3 Using these conventions we write the 4-momentum, Pµ as Pµ = (E; −p); (1.4) where p is the 3-momentum. Once a system is quantised (as discussed later) we obtain the 4-momentum operator, @ P^ = (E;^ −p^) = i ; r = i @ : (1.5) µ @t µ 5 Chapter 2 Field Theory 2.1 Classical Field Theory 2.1.1 Background Field theory replaces the concept of a single particle generalised co-ordinate to that of a contin- uum of particles, where we may have several particles at any point in space-time. We denote a field, φ(x), as a function of space-time co-ordinates. The main quantity of interest in classical mechanics is a functional named the action, S, defined as Z Z S = L dt = (pq_ − H) dt; (2.1) where H is the Hamiltonian, H = T + V (the sum of kinetic and potential energies), L is the Lagrangian, given by L = T −V , and p and q are conjugate variables for generalised co-ordinates and momenta. Note the first half of this chapter follows closely to [1]. In field theory we opt to use the Lagrangian density, L = R L d3x, which we can write in terms of our fields. Hence we can write our action in terms of the Lagrangian density, L, as follows: Z 4 S = L (φ, @µφ) d x ; (2.2) @φ where @µφ = @xµ and µ is the space-time index µ = 0; 1; 2; 3. Throughout this report, we will use the term Lagrangian to refer to the Lagrangian density. Though this is not formally correct this is done for ease given the Lagrangian density is the main quantity we will be dealing with. 2.1.2 Equations of Motion This subsection follows closely to [2]. We find the classical equations of motion for a system by applying the principle of stationary action. For this we consider an arbitrary volume in space- time, Ω, and vary our field φ(x) ! φ0(x) = φ(x) + δφ(x) such that δφ = 0 on the volume surface, @Ω.