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 Poincarégroup 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 Poincaréalgebra, and the relations of the resulting Super-Poincaréalgebra 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 & Poincarégroups 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 PoincaréGroup ...... 24 3.3.1 The PoincaréGroup ...... 24 3.3.2 Casimirs of the PoincaréGroup ...... 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 standard 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 experimental 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:

1 0 0 0  µν µν 0 −1 0 0  g = η =   . (1.2) 0 0 −1 0  0 0 0 −1

We deﬁne our 4-gradient by

∂ ∂ ∂ ∂ ∂ ∂ = ∂ = , ∇ = , , , . (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, ∂ Pˆ = (E,ˆ −pˆ) = i , ∇ = 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. Hence, ∂L ∂L δL = δφ + δ(∂µφ) (2.3) ∂φ ∂ (∂µφ) ∂L ∂L = δφ + ∂µδφ. (2.4) ∂φ ∂ (∂µφ)

6 This gives variation in the action, Z δS = d4x δL (2.5) Z Z 4 ∂L ∂L 4 ∂L ∂L = d x − ∂µ δφ + d x ∂µ δφ + ∂µδφ , (2.6) ∂φ ∂ (∂µφ) ∂ (∂µφ) ∂ (∂µφ) where we have inserted an extra term which has been both added and subtracted, so that we can now perform integration by parts: Z Z 4 ∂L ∂L 4 ∂L δS = d x − ∂µ δφ + d x ∂µ ∂µδφ . (2.7) ∂φ ∂ (∂µφ) ∂ (∂µφ)

The second integral vanishes by Gauss’ Divergence Theorem; it is just a surface/boundary term. By inferring the action is stationary we have δS = 0. Applying the fundamental theorem of calculus of variations we find the classical equations of motion for our field, the Euler-Lagrange equations: ∂L ∂L − ∂µ = 0. (2.8) ∂φ ∂ (∂µφ) The above form of φ(x) is a real scalar field, i.e. it has a single (real) value for every space-time co-ordinate. In general, fields may also be complex, and not necessarily scalar; we will discuss Lagrangians involving spinor fields later in this report. Note that if φ(x) was a complex field, there would be two separate equations of motion for the system, one for φ(x) and the other for its conjugate, φ∗(x).

2.1.3 Noether’s Theorem We continue to follow [1,2], breaking down calculations in the following derivations and examples. A symmetry of a system is defined as a set of transformations that leaves the action invariant. Noether’s (first) theorem states that for every symmetry there is an associated conserved quantity, usually referred to as the Noether current. To find this quantity, we apply a similar process to that of before, considering a variation of S under a continuous transformation of our field,

φ(x) → φ0(x) = φ(x) + α∆φ(x), (2.9) where α is an inﬁnitesimal transformation parameter and ∆φ(x) is the shift in our ﬁeld. Total derivatives in the Lagrangian leave nothing but a surface term in our integral for the action, so we can write the shift in the Lagrangian as

µ L(x) → L(x) + α∂µJ (x), (2.10) where we are free to choose J µ arbitrarily. We then can look at what our shift in L should be, given it is a function of φ(x) and ∂µφ(x):

∂L ∂L α∆L = (α∆φ) + ∂µ(α∆φ) (2.11) ∂φ ∂(∂µφ) ∂L ∂L ∂L = α∂µ ∆φ + α − ∂µ ∆φ. (2.12) ∂(∂µφ) ∂φ ∂(∂µφ) | {z } =0

7 The second term in the above equation vanishes by the Euler-Lagrange equations. Comparing terms,

∂L µ ∂µ ∆φ = ∂µJ (2.13) ∂(∂µφ) ∂L µ ⇒ ∂µ ∆φ − J = 0. (2.14) ∂(∂µφ)

µ Hence have a conserved current satisfying ∂µj (x) = 0, where

∂L jµ = ∆φ − J µ. (2.15) ∂(∂µφ)

There is an associated Noether charge found by integrating the zeroth component as follows: Z Q = d3x j0. (2.16)

We will use a similar process to that discussed in order to ﬁnd our supercharges (supersymmetric generators) in later chapters.

2.1.4 An example of Noether’s Theorem The Lagrangian for a complex scalar ﬁeld is given by 1 1 L = (∂ φ)(∂µφ)∗ − m2φφ∗. (2.17) 2 µ 2 We can apply Noether’s theorem using the following transformation as our symmetry:

φ → eiαφ. (2.18)

This is an example of a U(1) symmetry. As stated before, we consider our transformation in the form φ(x) → φ0(x) = φ(x) + α∆φ(x), where α is an infinitesimal parameter. This means we can write eiα ≈ 1 + iα, and so find that α∆φ = iαφ. (2.19) The conjugate field, φ∗, transforms very similarly, but with a negative sign from the conjugation. As we now have two independent fields, we must modify our earlier definition for the Noether current as summing over the fields: Using the definition of our conserved current, we find

X ∂L jµ = ∆Φ − J µ ∂(∂µΦ) Φ=φ,φ∗ (2.20) = (∂µφ)∗(iφ) + (−iφ∗)(∂µφ) = i((∂µφ∗)φ − φ∗(∂µφ)), where we have chosen J µ = 0. This particular current is important in ﬁeld theory when dis- 1 µν cussing the Lagrangian for electromagnetic ﬁelds, L = − 4 Fµν F .

8 2.1.5 On-shell and Off-shell When using the term on-shell we mean to say that the classical equations of motion, the Euler- Lagrange equations, are obeyed. If this is not the case, we have an off-shell configuration. In a relativistic quantum system, where virtual particles are free to be created and destroyed, the term on-shell more specifically refers to the virtual particles satisfiying Einstein’s mass-energy relation, p2 + m2 = E2, (2.21) where p is the 3-momentum of the particle. Noether’s Theorem is an example of an on-shell result: as shown in finding an expression for the Noether current jµ, we use the Euler-Lagrange equations to infer the form of the current stated.

2.2 Moving to Quantum Field Theory 2.2.1 The Klein-Gordon Equation We can find the equations of motion for a real scalar field by quantising a system (to be discussed in further depth shortly). We use the standard quantisation, replacing physical observables with operators, written in natural units as: p → pˆ = −i∇, (2.22) ∂ E → Eˆ = i . (2.23) ∂t We use the standard mass-energy relation given above, and replace our observables with operators, noting that we must introduce a field given that we have quantised the system: pˆ2 + m2 φ = Eˆ2φ ∂ 2 (−i∇)2 + m2 φ = i φ, ∂t (2.24) ∂2 ⇒ − ∇2 + m2 φ = 0. ∂t2 Writing this in index notation we have

µ 2 (∂ ∂µ + m )φ = 0. (2.25) This is the Klein-Gordon equation: the simplest relativistic wave equation, which serves as the equation of motion for the field φ [2]. We can find the same expression by using the Euler- Lagrange equations, noting that the Lagrangian for a massive real scalar field is given by: 1 1 L = (∂ φ)(∂µφ) − m2φ2. (2.26) 2 µ 2 We then find ∂L = −m2φ, (2.27) ∂φ ∂L = ∂µφ, (2.28) ∂(∂µφ) which, when placed in the Euler-Lagrange equations, recover the Klein-Gordon equation, 2 µ µ 2 −m φ − ∂µ(∂ φ) = (∂ ∂µ + m )φ = 0. (2.29)

9 2.2.2 Conjugate Momentum and Hamilton’s Equations We continue to follow sources [1,2] closely for the rest of this section. In classical mechanics, we have an alternative description to the Lagrangian formalism for the motion of a system. This is the Hamiltonian formalism, where the Hamiltonian is obtained by performing a Legendre transform on our Lagrangian: H = pq˙ − L. Here we choose to use canonically conjugate momentum, given by ∂L p = . (2.30) ∂q˙ Instead of a single second order differential equation (the Euler-Lagrange equations of motion), we can describe a system using two first order differential equations, called Hamilton’s equations. These are: ∂H ∂H p˙ = − , q˙ = . (2.31) ∂q ∂p Analogous to classical mechanics, in field theory we define the conjugate momentum by

∂L(φ, ∂µφ) ∂L(φ, ∂µφ) π(x) = = = ∂0φ(x). (2.32) ∂φ˙ ∂(∂0φ) From this we can deﬁne our ﬁeld theory Hamiltonian by once again applying a Legendre transform: Z Z H = d3x H(φ, π) = d3x (πφ˙ − L), (2.33) where H is the Hamiltonian density.

2.2.3 The Stress-Energy Tensor An important symmetry in nature is that of space-time translations; translating our co-ordinates should not change the action. More speciﬁcally, the transformation can be written as xµ → xµ − aµ. (2.34) This means the ﬁeld transforms as µ φ(x) → φ(x + a) = φ(x) + a ∂µφ(x) (2.35) and similarly our Lagrangian as ν µ L → L + a ∂µ(δν L). (2.36) µ Comparing this the general form of our transformations, L(x) → L(x) + a∂µJ (x), we identify µ µ (J (x))ν = L δν . (2.37) µ So we have an associated conserved current, which we call the Stress-Energy Tensor, Tν , given by

µ ∂L µ Tν = ∂ν − L δν . (2.38) ∂(∂µφ) Integrating over the various parts of this tensor gives four diﬀerent conserved charges. These are the Hamiltonian (which is conserved for translations in time), and the components of physical 3-momentum (conserved for translations in space): Z Z H = T 00 d3x = H d3x, (2.39) Z Z i 0i 3 3 P = T d x = − π∂iφ d x. (2.40)

10 2.3 Canonical Quantisation of a Scalar Field 2.3.1 Poisson Brackets In classical mechanics, we can write the classical equations of motion in the Hamiltonian formulation by using Poisson Brackets. Using canonical co-ordinates we can deﬁne the Poisson Bracket of two functions, f(qi, pi) and g(qi, pi) as follows: X ∂f ∂g ∂f ∂g {f(q , p ), g(q , p )} = − . (2.41) i i i i ∂q ∂p ∂p ∂q i i i i i One reason for introducing Poisson Brackets is that they allow us to write a symmetric form of Hamilton’s equations: q˙i = {qi,H(qi, pi)}, p˙i = {pi,H(qi, pi)}. (2.42) We have the following important Poisson Bracket relations:

{qi, pj} = δij, (2.43)

{qi, qj} = {pi, pj} = 0. (2.44)

2.3.2 Quantisation In quantum mechanics we quantise a system by promoting our canonical variables (and other physical observables) to operators which act on a systems state. To signify this we place a hat on the operator, i.e.q ˆi. The general procedure involves introducing commutators, deﬁned as [A, B] = AB − BA, which take the same form as their Poisson bracket counterparts but with a factor of i¯h. In natural units we have:

[ˆqi, pˆj] = i δij, (2.45)

[ˆqi, qˆj] = [ˆpi, pˆj] = 0. (2.46) where as previously mentioned our 3-momentum becomes a diﬀerential operator:

p → pˆ = −i∇. (2.47)

In quantum ﬁeld theory we undertake a similar approach, utilising the conjugate momentum found earlier to write our commutators as h i φˆ(x), πˆ(y) = i δ(3)(x − y), (2.48) h i h i φˆ(x), φˆ(y) = πˆ(x), πˆ(y) = 0. (2.49)

Note how we now have a Dirac delta function, given we have a continuous system as opposed to a discrete one. The above equations are known as equal time commutation relations. For ease of notation, we will drop the hat from our operators throughout this report.

2.3.3 The Hamiltonian for the Klein-Gordon ﬁeld For this and the subsequent subsection, we follow [1] exclusively, and carry out all calculations explicitly. The Lagrangian for a Klein-Gordon ﬁeld can be written as 1 1 1 1 1 L = (∂ φ)2 − m2φ2 = φ˙2 − (∇φ)2 − m2φ2. (2.50) 2 µ 2 2 2 2

11 Using our earlier definition for the associated Hamiltonian is we find that Z H = d3x (πφ˙ − L) Z 3 ˙ 1 ˙2 1 2 1 2 = d x πφ − 2 φ − 2 (∇φ) − 2 m (2.51) Z 3 1 2 1 2 1 2 2 = d x 2 π + 2 (∇φ) + 2 m φ , where π = φ˙. Before going further, it is important to mention that the above field theory commutation relations, as well as our Hamiltonian, clearly treat spatial and time co-ordinates differently. Hence the Hamiltonian formulation is not ‘Lorentz invariant’, as well be discussed in greater depth shortly. However, this formulation makes quantisation far easier. Note in our commutation relations and thereafter we assume the Schrödingerpicture, in which the operators themselves are time-independent. We can use a Fourier expansion for our field φ(x, t): Z d3p φ(x, t) = eip.x φ(p, t), (2.52) (2π)3 where φ is real, and so φ∗(p) = φ(−p). Using this decomposition, the Klein-Gordon equation becomes: ∂2 + (p2 + m2) φ(p) = 0. (2.53) ∂t2 This equation is of the same form as the harmonic oscillator, where the natural frequency of the oscillator, ωp, is given by p 2 2 ωp = p + m . (2.54) From this we can say the general solution to the above equation is a linear superposition of oscillators with different frequencies, given by different momenta.

2.3.4 The Harmonic Oscillator in Quantum Mechanics The general Hamiltonian for SHO in quantum mechanics is given by

1 2 1 2 2 H = 2 p + 2 ω q , (2.55) where q and p obey commutation relations as previously outlined. We introduce ladder operators, a and a†, which as will be shown explicitly soon, act on a state to produce another state of lower or higher energy [3]. These are given by:

rω i a = q + √ p, (2.56) 2 2ω rω i a† = q − √ p, (2.57) 2 2ω which means we can write our canonical co-ordinates as 1 q = √ a + a† , (2.58) 2ω rω p = −i a − a† . (2.59) 2

12 Using the fact [q, p] = i, we can then ﬁnd the commutation relation for a and a†:

1 rω [q, p] = i = √ a + a† , −i a − a† 2ω 2 i = − a + a†, a − a† 2   i † † † † (2.60) = − [a, a] − a, a + a , a − a , a  2 |{z} | {z } =0 =0 i = − −2 a, a† 2 = i a, a† . Hence we have the commutator a, a† = 1. (2.61) Our Hamiltonian becomes: 2 1 rω 1 1 2 H = −i a − a† + ω2 √ a + a† 2 2 2 2ω ω 2 ω 2 = − (a2 + a† − aa† − a†a) + (a2 + a† + aa† + a†a) 4 4 (2.62) ω ω = (aa† + a†a) = (a, a† + 2a†a) 2 2 † 1 = ω(a a + 2 ). Using these results we can ﬁnd the commutation relations of the ladder operators with the Hamiltonian:

[H, a†] = ωa†, (2.63) [H, a] = −ωa. (2.64)

The rest of this subsection follows closely to [3]. We can use these commutation relations to act on a state with energy E, where H |Ei = E |Ei. We ﬁnd the energy eigenstates of states that have had a ladder operator applied on them as follows:

If the energy is bounded from below, such that we have a ground state |0i, then the ladder operator a must satisfy a |0i = 0. Hence when the Hamiltonian acts on the ground state we have

1 H |0i = 2 ω |0i . (2.67) If we deﬁne a state in the nth energy level as

|ni = (a†)n |0i , (2.68) then we ﬁnd applying the Hamiltonian to such a state gives

1 H |ni = (n + 2 )ω |ni . (2.69) 1 + Hence we see the spectrum of the SHO Hamiltonian can be given by (n + 2 ) where n ∈ Z .

13 2.3.5 Applying the Quantisation of the Harmonic Oscillator Moving back to the free scalar field, we make analogy to our definitions of q and p involving ladder operators by defining our field and conjugate momentum. The following calculations follow [1,3] but we show the steps in far greater detail. Once again expanding over all the different oscillator modes, Z d3p 1 φ(x) = a eip.x + a† e−ip.x , (2.70) 3 p p p (2π) 2ωp Z d3p rω π(x) = (−i) p a eip.x − a† e−ip.x , (2.71) (2π)3 2 p p † where ap and ap are our creation and annihilation operators, dependent on the 3-momentum index p we are summing over. The next part of the analogy involves finding commutators for the creation and annihilation operators. We use the ansatz † 3 (3) ap, aq = (2π) δ (p − q), (2.72) † † [ap, aq] = ap, aq = 0, (2.73) and then try to recover the known commutation relation for our fields: Z 3 3 r d p d q −i ωq ip.x † −ip.x iq.y † −iq.y [φ(x), π(y)] = 3 3 ape + ape , aqe − aqe (2π) (2π) 2 ωp Z 3 3 r d p d q −i ωq † i(p.x−q.y) † i(−p.x+q.y) = 6 ap, −aq e + ap, aq e (2π) 2 ωp Z 3 3 r d p d q −i ωq (3) i(p.x−q.y) (3) i(−p.x+q.y) (2.74) = 3 −δ (p − q)e − δ (q − p)e (2π) 2 ωp Z d3p i Z d3p = eip.(x−y) + eip.(y−x) = i eip.(x−y) (2π)3 2 (2π)3 = i δ(3)(x − y). Hence our ansatz is correct. Our Hamiltonian then becomes Z 3 1 2 1 2 1 2 2 H = d x 2 π + 2 (∇φ) + 2 m φ (2.75) 3 3 √ 1 Z Z d p d q ωpωq = d3x − a eip.x − a† e−ip.x a eiq.x − a† e−iq.x 2 (2π)3 (2π)3 2 p p q q 1 ip.x † −ip.x iq.x † −iq.x + √ i p ape − i p ape i q aqe − i q aqe (2.76) 2 ωpωq 2 m ip.x † −ip.x iq.x † −iq.x + √ ape + ape aqe + aqe . 2 ωpωq

R d3x i(k−k0).x (3) 0 We then integrate over x making use of the identity (2π)3 e = δ (k − k ). After the delta functions have been applied we have: Z 3 d p 1 n 2 † † † † H = 3 − ωp apa−p − apap − apap + apa−p (2π) 4 ωp † † † † (2.77) − p. − p apa−p − p.p apap − p.p apap + p. − p apa−p

2 † † † † o + m apa−p + apap + apap + apa−p .

14 2 2 2 Using ωp = p + m many terms then cancel:

Z d3p −ω Z d3p ω H = p −2 a† a − 2 a a† = p a† a + a a† (2π)3 4 p p p p (2π)3 2 p p p p Z d3p = ω a† a + 1 a , a† (2.78) (2π)3 p p p 2 p p Z d3p Z d3p ω = ω a† a + p δ(3)(0), (2π)3 p p p (2π)3 2 where we have use the commutation relations of creation and annihilation operators.

2.3.6 Infra-red and Ultra-violet Divergences This discussion follows [3]. Note that second integral above has noticeable convergence issues. Firstly, the integral diverges because we are integrating over an infinitely large space. The integral produces an infinity as we are looking to find the energy for an infinite volume; a more poignant question is to look for the energy density for a finite volume. As outlined in [3], we can take define our volume, V , as a box of length L, and say that

Z L/2 Z L/2 3 (3) 3 ip.x 3 (2π) δ (0) = lim d x e p=0 = lim d x = V. (2.79) L→∞ −L/2 L→∞ −L/2

So our energy density, ε0, can be written as:

E Z d3p 1 ε = 0 = ω . (2.80) 0 V (2π)3 2 p

The divergence due to the delta function is known as an infra-red divergence. However, there are still problems with the above integral given for energy density. We are integrating over infinitely large momentum modes, with the ωp term tending to infinity as momentum does. Hence our integral diverges. This divergence is an ultra-violet divergence. This is much more tricky to remove, with a maximum limit for our momentum, Λ, needed for the integral to converge. Instead, we choose to remove this term for our expression for the Hamiltonian, effectively setting our vacuum ground state to zero. As we are only interested in relative energies, this is an acceptable thing to do. Hence the Hamiltonian becomes

Z d3p H = ω a† a , (2.81) (2π)3 p p p which gives H |0i = 0 |0i as outlined.

15 Chapter 3

The Lorentz & Poincar´egroups

3.1 Lorentz Transformations 3.1.1 Deﬁning a Lorentz Transformation A Lorentz transformation is given by an O(3, 1) matrix which acts on space-time co-ordinates such that µ µ µ ν x → x˜ = Λ ν x , (3.1)

µ ∂x˜µ where we deﬁne our Lorentz transformation as Λ ν = ∂xν , which preserves the inner product of µ ν µ our co-ordinates, x · x = x gµν x = x xµ, such that: µ σ ν ρ σ ρ (Λ σx ) gµν (Λ ρx ) = x gσρ x . (3.2) We can then write this condition more compactly as µ ν σρ µν Λ σ Λ ρ g = g . (3.3) This section follows closely to [1, 3, 6]. This implies that we have some measure of space-time, µ usually denoted ∆ = x xµ, that is left invariant. The description O(3, 1) refers to the matrices belonging to the orthogonal group, implying they are real, have determinant ±1 with inverse matrix the same as their transpose matrix. Further to this we restrict our transformations to those which are proper and orthochronous. By ‘proper’ we mean these matrices which have determinant +1. By orthochronous, the transformations preserve the direction of time, i.e. 0 ↑ Λ 0 > 0 [4]. With this restriction, we write our new subgroup as SO+(3, 1) , referring to a special orthogonal group with these two conditions applied.

3.1.2 Lorentz Transformations on Scalar Fields Continuing on the same path set in the previous chapter, we would like to look at how scalar ﬁelds transform under a Lorentz transformation. The simplest way we can write is given by µ ˜ µ −1 µ ν φ(x ) → φ(x ) = φ((Λ ) ν x ). (3.4) Our intuition in writing this comes from the idea that the transformed ﬁeld takes the same value at a point that it would have taken before a Lorentz transformation was applied [5]. Note that from the condition given previously, our Lorentz transformations obey the identity −1 ρ −1 σ µν ρσ (Λ ) µ (Λ ) ν g = g , (3.5)

16 which we can use to check Lorentz invariance. A Lorentz-invariant quantity is one that remains unchanged under a Lorentz transformation (a combination of rotations and Lorentz boosts). For 1 2 1 2 2 example, in the Klein-Gordon ﬁeld Lagrangian, L = 2 (∂µφ) − 2 m φ , the second (mass) term will remain unchanged under a transformation. The ﬁrst (kinetic) term, transforms as

2 µν ˜ ˜ (∂µφ (x)) → g ∂µφ (x) ∂ν φ (x) µν −1 −1 = g ∂µ φ Λ x ∂ν φ Λ x ρ ρ = gµν Λ−1 x Λ−1 ∂ φ Λ−1 ∂ φ µ ρ ν σ ; (3.6) µν −1ρ −1ρ −1 = g Λ µ Λ ν Λ x (∂ρφ)(∂σφ) ρσ −1 = g Λ x (∂ρφ)(∂σφ) −1 2 = Λ x (∂µφ (x)) .

Hence we can say the Lagrangian transforms as a scalar,

L(φ(x)) → L(φ(Λ−1 x)). (3.7)

The above calculation is given in [1], but shown more explicitly here. Integrating over all space- time, we ﬁnd the action unchanged under this transformation and so we say the action is Lorentz invariant. Similarly, we can show the Lorentz invariance of the Klein-Gordon equation using the same technique:

2 2 ˜ −1ν −1σµ 2 −1 ∂ + m φ (x) → Λ µ ∂ν Λ ∂σ + m φ Λ x νσ 2 −1 (3.8) = g ∂ν ∂σ + m φ Λ x = 0 as desired.

3.1.3 Introduction to the Lorentz Group This subsection follows sources [6–8]. We can give the Lorentz transformation of any ﬁeld (be it a scalar, or a more complicated object as will be discussed shortly) using a matrix D(Λ), where

a a b −1 φ (x) → D(Λ) b φ (Λ x). (3.9) Here our ﬁeld φa is an object with n components, labelled by index a, and D(Λ) is an n × n matrix. By its deﬁnition, it can be seen that Lorentz transformations form a group, G [7]:

1. Closure: ∀ Λ1, Λ2 ∈ G, Λ1Λ2 ∈ G

2. Associativity: (Λ1Λ2)Λ3 = Λ1 (Λ2Λ3) = Λ1Λ2Λ3 ∀ Λ1, Λ2, Λ3 ∈ G 3. Identity element: Λ = 1 ∈ G 4. Inverse element: ∀ Λ ∈ G ∃ Λ−1 ∈ G such that Λ−1Λ = 1 This is the Lorentz group. Here, D(Λ) is a representation of the Lorentz group, realising the elements of the group by matrices, with the group operation given by matrix multiplication. The dimension of the matrices correspond to the dimension of the representation. The Lorentz group is a continuous group, and as such we can look at inﬁnitesimal transformations, which deﬁne a

17 vector space known as the group’s Lie algebra. This space has basis vectors which we call the generators of the algebra/group. An inﬁnitesimal Lorentz transformation can be written as

µ µ µ Λ ν = δ ν + ω ν , (3.10) where ω is the inﬁnitesimal parameter. We can then ﬁnd a condition on ω by applying the known condition for our Lorentz transformations, given in (3.3). In comparison to most texts on the area, we show this explicitly:

µ µ ν ν σρ µν (δ σ + ω σ)(δ ρ + ω ρ) g = g ν µρ ν µρ µ σν µν δ ρg + ω ρg + ω σg = g (3.11) gµν + ωνµ + ωµν = gµν ⇒ ωµν + ωνµ = 0, i.e. our parameter ωµν is antisymmetric in its indices, where we have used that because ω is small, we can neglect the O(ω2) term. As shown in [3], there are only 6 independent components in any antisymmetric 4 × 4 matrix, and we choose a representation of the Lorentz group such that these refer to 3 rotations and 3 Lorentz boosts. These matrices are the basis vectors of the aforementioned Lie algebra in this representation, which we shall label (Mρσ)µν . The ﬁrst two indices ρ and σ are antisymmetric, and have values ρ, σ = 1, 2, 3, giving for a total of 6 diﬀerent unique matrices. The choice of these indices infers the choice of boost/rotation involved. The other indices, µ and ν, are the standard space-time indices relating to the actual indices of the matrix itself. We now follow [3], and choose a representation of our matrices using our metric:

(Mρσ)µν = i (gρµgσν − gσµgρν ) . (3.12)

Lowering the space-time index ν so we can use the matrices to act on ﬁelds, we can write

ρσ µ ρµ σ σµ ρ (M ) ν = i (g δ ν − g δ ν ) . (3.13)

µ We can write our inﬁnitesimal parameter ω ν as a linear combination of our generators [3],

µ i ρσ µ ω ν = − 2 Ωρσ(M ) ν , (3.14) where we introduce a parameter of the Lorentz transformations Ωρσ, which is antisymmetric in its indices. For rotations this can be thought of the angle we are rotating around. This parameter details the type of transformation taking place, signifying the amount we are rotating ρσ µ or boosting by. As outlined in [3], we can look at the form of these (M ) ν by inserting indices. For example, boosts in the x1 direction take the form

0 1 0 0 01 µ 1 0 0 0 −i (M ) =   , (3.15) ν 0 0 0 0 0 0 0 0 whereas a rotation around the (x1, x2) plane takes the form

0 0 0 0 12 µ 0 0 −1 0 −i (M ) =   . (3.16) ν 0 1 0 0 0 0 0 0

18 The Lie algebra of the Lorentz group, which serves as commutation relations for elements, is then given as follows (shown for the representation we are using):

[Mρσ, Mτν ] = i (gστ Mρν − gρτ Mσν + gρν Mστ − gσν Mρτ ) . (3.17)

The full form of our inﬁnitesimal Lorentz transformations is

µ µ i ρσ µ Λ ν = δ ν − 2 Ωρσ(M ) ν . (3.18) For example, in the case where we have a rotation around the (x1, x2) plane, we have that Ω12 = −Ω21 = θ, and so can write our inﬁnitesimal transformation as 1 0 0 0 0 1 −θ 0 Λ12 =   . (3.19) 0 θ 1 0 0 0 0 1

If we want to look at ﬁnite Lorentz transformations (as opposed to inﬁnitesimal), we can do this µ by exponentiating our ω ν :

µ µ i ρσ µ Λ ν = exp(ω ν ) = exp(− 2 Ωρσ(M ) ν ). (3.20) The following step is made independently of the sources discussed in this subsection. By looking at the Taylor series of the above expression, namely

µ µ −i ρσ µ 1 −i 2 ρσ µ 2 1 −i 3 ρσ µ 3 Λ ν ≈ δ ν + 2 Ωρσ(M ) ν + 2! 2 (Ωρσ(M ) ν ) + 3! 2 (Ωρσ(M ) ν ) +..., (3.21) we see that in the case of our rotation around (x1, x2) plane, we can now write

1 0 0 0 0 cos θ − sin θ 0 Λ12 =   , (3.22) 0 sin θ cos θ 0 0 0 0 1 by using the deﬁnitions of the Taylor series for cos θ and sin θ. This is the form of a rotation we are used to; the new machinery we have introduced to explore Lorentz transformations has recovered the familiar result.

3.2 Representations and Spinors 3.2.1 Structure of the Lorentz Group We now choose to follow [6], but carry out all calculations in this subsection explicitly ourselves. So far we have very briefly mentioned different representations of the Lorentz group, discussing the form of transformations for scalar fields, and exploring another representation that is 4- dimensional, trying to recover the form of a rotation from the definition of this representations’ transformations. More generally, we write the Lie algebra for the Lorentz group as

[M ρσ,M τν ] = i(gστ M ρν − gρτ M σν + gρν M στ − gσν M ρτ ), (3.23) where in the representation discussed at the end of the last section we had M ρσ = Mρσ. To look at how transformations act on more complicated objects, it is useful to separate our generators

19 into 3 boosts and 3 rotations [6]. We deﬁne our rotation generators, Ji, and boost generators, Ki, as 1 J = M , (3.24) i 2 ijk jk Ki = M0i, (3.25) where ijk is the Levi-Civita tensor, antisymmetric in all its indices. Using these deﬁnitions, alongside the Lorentz group commutation relations in (3.23), our new commutation relations are:

[Ji,Jj] = iijkJk, (3.26)

[Ki,Kj] = −iijkJk, (3.27)

[Ji,Kj] = iijkKk. (3.28)

We are free to choose arbitrary combinations of these generators, and with intuition choose to deﬁne the particular combination, Ai and Bi, where:

1 Ai = 2 (Ji + iKi) , (3.29) 1 Bi = 2 (Ji − iKi) . (3.30) Once again we can then obtain another new set of commutation relations for these combinations, shown explicitly,

1 [Ai,Aj] = 4 ([Ji,Jj] + i [Ji,Kj] + i [Ki,Jj] − [Ki,Kj]) 1 = 4 (iijkJk + i(iijkKk) + i(−ijikKk) − (−iijkJk)) 1 = iijk · 2 (Jk + iKk) (3.31) = iijkAk, 1 [Ai,Bj] = 4 ([Ji,Jj] − i[Ji,Kj] + i[Ki,Jj] + [Ki,Kj]) 1 = 4 (iijkJk − i(iijkKk) + i(−ijikKk) + (−iijkJk)) (3.32) = 0.

Similarly, by analagous calculation we ﬁnd for Bi we have:

[Bi,Bj] = iijkBk. (3.33)

These relations expose two su(2) algebras (special unitary Lie algebras), where way can say the groups do not overlap given the commutator between the two diﬀerent types of generators vanishes. The relationship between the groups can be roughly given as

SO(3, 1) ' [SU(2) ⊗ SU(2)]/Z2, (3.34) where we have identiﬁed a ‘double covering’ and so must remove half of the elements as shown above. However, SU(2) ⊗ SU(2) is not an isomorphism to SO(3, 1) [6]. This report will not discuss the associated group theory in great detail, and aims only to identify representations of the Lorentz group. We label the quantum numbers associated with the SU(2) ⊗ SU(2) relation as (a, b). These numbers refer to the eigenvalues of Ai and Bi respectively. The standard form of 1 3 these eigenvalues, (given the su(2) Lie algebras), is given as a(a+1) = 0, 2 , 1, 2 ... and b(b+1) = 1 3 0, 2 , 1, 2 ... respectively. We shall discuss the various diﬀerent representations associated with

20 these quantum numbers, where the spin of the representation is given by J = a + b. The correct relation we have is that the complex linear combinations of the generators of the Lorentz algebra are isomorphic to the complex linear combinations of the Lie algebra of su(2) ⊗ su(2)’ [6]. Thus, the correct choice of covering, known as the ‘unique cover’, is the special linear group, SL(2, C), formed of complex 2 × 2 matrices with unit determinant, where

SO(3, 1) ' SL(2, C)/Z2. (3.35)

This relation between the groups is an isomorphism.

3.2.2 Representations of the Lorentz Group We follow [8] closely when discussing these representations, though the following diﬀers from this source in treating the more physical aspects of the representations of the group.

The Trivial Representation The trivial representation of the Lorentz group is given by (a, b) = (0, 0) and describes the transformation of scalar fields. The spin of this field is given by J = 0 + 0 = 0, which confirms this is a bosonic field. The generators of the transformation are described by M µν = 0, which, upon exponentiation, give Λ = 1, and so we have that

φ → Λ · φ = φ. (3.36)

However, we have thus far treated this as a finite-dimensional representation; as mentioned previously, our fields are a function of space-time co-ordinates, xµ, which themselves transform under the Lorentz transformation. This gives rise to an infinite-dimensional representation of the Lorentz group. Previously we had written

φ(x) → φ(Λ−1x), (3.37) which is the correct transformation for a scalar ﬁeld. Further to this, we can split the transformation of the co-ordinates and ﬁeld itself into two parts. The part of the transformation associated to the transformed co-ordinates is

−1 i µν φ(Λ x) = exp(− 2 Ωµν L ) φ(x), (3.38) where we can identify Lµν as the differential operator for angular momentum generalised to 4-dimensions, Lµν = i(xµ∂ν − xν ∂µ), (3.39) which satisfies the groups Lie algebra as given in (3.23). Note this is true for transformed co- ordinates of all types of fields, this not just a scalar as illustrated in (3.38). The transformation of the field itself, which we had previously labelled as D(Λ) when initially discussing the Lorentz group, is generated by Sµν , where

i µν D(Λ) = exp(− 2 Ωµν S ). (3.40) Similarly, Sµν satisﬁes the groups Lie algebra. The complete general Lorentz transformation of a ﬁeld (of any type/representation) is then

i µν φa(x) → exp(− 2 Ωµν M ) ab φb(x), (3.41)

21 where we have identiﬁed

M µν = Lµν + Sµν = i(xµ∂ν − xν ∂µ) + Sµν . (3.42)

For the scalar ﬁeld, Sµν = 0 (a 1 × 1 matrix with entry 0) and so the total transformation only involves that of the co-ordinates.

The Vector Representation 1 1 The vector representation is given by ( 2 , 2 ), which gives J = 1, and so the representation describes the transformation of a spin-1 (bosonic) ﬁeld. Note that the photon is such a ﬁeld. For this representation the transformation generator, Sµν , is given by

µν µν ρ µρ ν νρ µ S = (M ) σ = i (g δ σ − g δ σ) . (3.43) This is the representation discussed earlier when looking at a rotation in the (x1, x2) plane. The transformation takes the form of 4 × 4 matrices, which act on 4-vectors.

The Spinorial Representations When discussing this area we shall follow [6]. We can construct three further representations by looking at the isomorphism from SO(3,1) to SL(2, C). This group is an isomorphism to Spin(3, 1). 1 β The fundamental representation of this group, given as ( 2 , 0), are matrices Nα . The objects that transform by this representation are known as left-handed Weyl spinors, and are denoted by 1 β˙ ψ. The anti-fundamental (or conjugate) representation, given as (0, 2 ), are matrices Nα˙ . The objects that transform by this representation are known as right-handed Weyl spinors, and are denoted byχ ¯. Note that the introduction of dotted indices, and a bar above our spinor, is just helpful reminding notation that these indices or objects involve the conjugate representation. So we have: ψ1 χ¯1 ψα = , χ¯α˙ = , (3.44) ψ2 χ¯2 and can write the Lorentz transformation of our spinor ﬁelds as

0 β ψα → ψα = Nα ψβ, (3.45) 0 β˙ χ¯α˙ → χ¯α˙ = Nα˙ χ¯β˙ . Note that spinor (SL(2, C)) indices are not like regular (SO(3, 1)) indices - we differentiate between the two by choosing to use the α, β . . . as opposed to µ, ν . . . as for regular space-time (Lorentz) indices. For example, dotted and undotted indices cannot be contracted with each other as with regular indices, because they come from different representations. In addition, we do not lower or raise spinor indices using the space-time metric tensor; we must introduce a new tensor that does this job, called the unimodular antisymmetric tensor. We define this with the convention: 0 1 αβ = , (3.46) −1 0

αβ βγ γ where αβ = − , so that αβ = δα, i.e. to make sure we get back our original spinor after successively lowering and raising indices [6]. Note may refer to this as the ‘metric’ when dealing with spinor indices. Both the fundamental and anti-fundamental Weyl representations are 2-dimensional, though it is possible to construct a third reducible spinorial representation

22 1 1 by ‘summing’ the other two representations, i.e. ( 2 , 0) ⊕ (0, 2 ) [8]. By reducible we mean to say it is equivalent (related by a similarity transformation) to the direct sum of irreducible representations, which are in our case the fundamental and anti-fundamental representations of SL(2, C). The resulting representation involves 4-dimensional objects known as Dirac spinors, given by placing both the left- and right-handed Weyl spinors in a column vector:   ψ1 ψ ψ2 Ψ = =   . (3.47) χ¯ χ¯1 χ¯2

3.2.3 Transformations of Spinors We continue to follow source [6]. We want to look at how exactly these spinors transform, β β˙ exploring the form of matrices Nα and Nα˙ . SL(2, C) is spanned exactly by four matrices: the µ identity, and the usual three Pauli matrices. Hence we can deﬁne the Pauli operator, (σ )αα˙ :

1 0 0 1 0 −i 1 0 σ0 = , σ1 = , σ2 = , σ3 = . (3.48) 0 1 1 0 i 0 0 −1 αα˙ αα˙ αα˙ αα˙ Similarly, we can deﬁne the barred Pauli operator, by applying the metric twice:

µ αα˙ αβ α˙ β˙ µ (¯σ ) = (σ )ββ˙ = (σ0, −σ1, −σ2, −σ3) (3.49) i = (12, −σ ), where 12 is the 2 × 2 identity matrix. Intuition tells us that generators for the Lorentz transformations on Weyl spinors should be related to these matrices given they span SL(2, C). Our generators, M µν , have two space-time indices however. We can deﬁne two generators for our left- and right-handed Weyl spinors using pairs of the Pauli operator, where the following combinations have been chosen such that the Lorentz algebra is satisﬁed:

µν β i µ ν ν µ β (σ )α = (σ σ¯ − σ σ¯ )α , 4 (3.50) ˙ i ˙ (¯σµν ) β = (¯σµσν − σ¯ν σµ) β, α˙ 4 α˙ where we have [σµν , σρσ] = i (gνρσµσ − gµρσνσ + gµσσνρ − gνσσµρ) , (3.51) and similarly forσ ¯µ. These objects are the generators of Lorentz transformations for left- and right-handed Weyl spinors respectively. We can write this transformation as:

i µν β ψα → exp(− 2 Ωµν σ ) α ψβ, ˙ (3.52) i µν β χ¯α˙ → exp(− 2 Ωµν σ¯ ) α˙ χ¯β˙ . Using this we can ﬁnd the transformation for Dirac spinors very simply. We deﬁne the generator

σµν 0 Σµν = , (3.53) 0σ ¯µν

23 where our Dirac spinor transforms in the usual way,

i µν Ψ → exp(− 2 Ωµν Σ )Ψ. (3.54) Note that there are further representations of the Lorentz group than the ones outlined in this report (which refer to ﬁelds of higher spins) which we will not be discussing.

3.3 The PoincaréGroup 3.3.1 The PoincaréGroup The Poincarégroup, also known as the inhomogeneous Lorentz group, extends our Lorentz group to space-time translations. Hence it is the symmetry group of Minkowski space-time. When discussing this topic, we follow [6, 9, 10, 19]. In this group, our co-ordinates transform as

µ µ ν µ x → Λ ν x + a , (3.55)

µ µ where Λ ν is our standard Lorentz transformation and a is a 4-vector representing a translation. Group elements are labelled (Λ, a), where pure Lorentz transformations are given by (Λ, 0) and pure translations by (12, a) [6]. The unit element is given trivially by (12, 0). By applying two successive transformations [10], (Λ2, a2) and (Λ1, a1), we have

00 µ µ 0 ν µ µ ν ρ µ ν µ x = Λ2 ν x + a2 = Λ2 ν Λ1 ρx + (Λ2 ν a1 + a2 ). (3.56) From this we can write multiplication between group elements as

(Λ2, a2) · (Λ1, a1) = (Λ2Λ1, Λ2a1 + a2), (3.57) where elements under this multiplication do not commute. Generators of the Lorentz transformation are given as M µν , with the same definition as given previously. Generators of translations are given by the momentum operator, namely P µ = i ∂µ. From this we can define the Poincaré algebra:

[P µ,P ν ] = 0, [M µν ,P σ] = i (P µgνσ − P ν gµσ) , (3.58) [M µν ,M ρσ] = i (gνρM µσ − gµρM νσ + gµσM νρ − gνσM µρ) .

When considering Supersymmetry, we shall be extending this algebra to include the commutation (and anticommutation) relations for our supersymmetric generators. The resulting group is called the Super-Poincar´egroup.

3.3.2 Casimirs of the PoincaréGroup There are two different invariant quantities in the Poincarégroup that commute with all of the generators [10], and these are known as Casimirs. The first Casimir, C1 is related to mass invariance, and defined by µ C1 ≡ P Pµ. (3.59) The second is related to spin invariance, and defined by

µ C2 ≡ W Wµ, (3.60)

24 where we deﬁne the Pauli-Lubanski pseudovector, Wµ, given by 1 W = − M νρP σ. (3.61) µ 2 µνρσ

Here µνρσ is the 4-dimensional Levi-Civita tensor, totally antisymmetric in its indices. Note for massive ﬁelds we have µ 2 C2 = W Wµ = −m s(s + 1). (3.62)

25 Chapter 4

The Dirac Field

4.1 The Dirac Equation 4.1.1 An introduction to Fermions So far we have only discussed a field theory involving spin-0 fields, which are bosons (given they have whole integer spin) characterised by real or complex scalar fields. Fermions (particles with half integer spin) are described using the spinorial representation of the Lorentz group. We have discussed the Klein-Gordon equation, the equation of motion for spin-0 bosons, and now aim to 1 look for a suitable equation of motion for spin- 2 particles, which is known as the Dirac equation. For most of this section, we follow Paul Dirac’s original methodology from his landmark paper in 1928 [11]. We also follow [12].

4.1.2 The Dirac Equation: a first-order relativistic wave equation In 1928, Paul Dirac wrote a paper entitled “The Quantum Theory of the Electron”, which aimed at finding a wave equation, covariant under Lorentz transformations, and linear in P 0 (and hence space-time derivatives), that in the limit of ‘large quantum numbers’ resulted in the Klein- Gordon equation [11]. Dirac postulated such a ‘wave function’ (or field as we now know is the case) would be of the form: µ (iγ ∂µ − m)Ψ = 0, (4.1) where γµ is some 4-vector, m our mass constant, and Ψ our field, which is a spinor. This is the Dirac equation. Dirac multiplied this equation by its conjugate,

µ ν (iγ ∂µ + m)(iγ ∂ν − m) Ψ = 0 (4.2) µ ν 2 − γ γ ∂µ∂ν + m Ψ = 0 (4.3) 1 µ ν 2 2 {γ , γ }∂µ∂ν + m Ψ = 0, (4.4) where the curly brackets indicate an anticommutator

{γµ, γν } = γµγν + γν γµ, (4.5) that we are free to introduce as we can exchange the order of multiplication between the two sets of brackets in (4.2). Thus, to recover the Klein-Gordon equation, a condition must be placed on the anticommutator: {γµ, γν } = 2gµν . (4.6)

26 By the presence of this anticommutator, we have a Clifford algebra. For this to make sense, we need a representation of our γµ’s that are 4-dimensional matrices, which we call the gamma matrices. In the equations written above, the 4 × 4 identity matrix, 14, is implied where there are no components that indicate a matrix (i.e. the mass term in the Dirac equation and on the metric in the definition of the Clifford algebra). We introduce the Feynman slash notation, that for any four vector, aµ, contracted with a gamma matrix, we write:

µ a/ = γ aµ. (4.7)

Hence we can compactly write the Dirac equation as

i∂/ − m Ψ = 0. (4.8)

4.2 The Clifford Algebra 4.2.1 The Dirac Basis In Dirac’s original paper, he defines the Pauli matrices and tries to construct different combinations and convolutions of them to find a set of four 4 × 4 matrices that meet the defining relation for the Clifford Algebra. These matrices are then act on a 4-dimensional spinor, a Dirac spinor. In the ‘Dirac basis’ we have:

i 0 12 0 i 0 σ γ = , γ = i , where i = 1, 2, 3. (4.9) 0 −12 −σ 0 Given that these matrices are defined to anticommute with one another, and that for all matrices we have that (γµ)2 = ±1, there are only a finite number of unique objects we can define as combinations of gamma matrices [12]. These are given by γµγν , γµγν γρ and γµγν γργσ. Of the last combination in this list, there is only one such object that is unique and utilises all four gamma matrices. This matrix has its own name, γ5, and by convention is defined as:

γ5 = i γ0γ1γ2γ3. (4.10)

In the Dirac basis, this can be written as

0 1 γ5 = 2 . (4.11) 12 0 By permuting indices to find all unique combinations, we find there are a total of 16 matrices; these are related to the 16 different entries in the 4 × 4 matrix, D(Λ), that defines the Lorentz transformation for a Dirac spinor. Moreover, we can write the generators of Lorentz transformations for Dirac spinors in terms of our gamma matrices [12]: i Σµν = [γµ, γν ], (4.12) 4

i µν where, as given previously, our Dirac spinors transform as Ψ → exp(− 2 Ωµν Σ ) Ψ. From this 1 1 we can clearly see how the Dirac equation is the suitable equation of motion for the ( 2 , 0)⊕(0, 2 ) representation of the Lorentz group. Making use of the anticommutator {γ5, γµ} = 0, we can construct any product of three gamma matrices as a single gamma matrix contracted with γ5, for example, γ1γ2γ3 = −iγ0γ5 [12]. Using the relationship between Σµν and products of two gamma matrices, we realise there are 5 diﬀerent types of objects that make up the 16 unique

27 µ µν µ 5 5 matrices mentioned before. These are: Γi = {14, γ , Σ , γ γ , γ }, where i = 1, 2 ... 16. As written in [13], writing out all permutations in the γµ themselves we have

14 Γ1 0 1 2 3 γ , iγ , iγ , iγ Γ2 − Γ5 0 1 0 2 0 3 1 2 1 3 2 3 γ γ , γ γ , γ γ , γ γ , γ γ , γ γ Γ6 − Γ11 (4.13) 0 1 2 0 1 3 0 2 3 1 2 3 iγ γ γ , iγ γ γ , iγ γ γ , iγ γ γ Γ12 − Γ15 iγ0γ1γ2γ3 Γ16

2 where we define some components with a factor of i such that (Γi) = 1. Together, these form a complete basis for all 4 × 4 matrices, i.e. we can construct any 4 × 4 matrix using a linear combination of this set. From Dirac’s definition of the gamma matrices we form the Dirac basis. However, there are alternative ways in which define the gamma matrices, which lead to us defining a different basis using the products of gamma matrices mentioned above to span the space.

4.2.2 The Weyl Basis The Weyl basis is such a basis, which happens to be the natural choice when dealing with the left- and right-handed Weyl representations of the Lorentz group. We deﬁne gamma matrices as

0 σµ γµ = , (4.14) σ¯µ 0

µ 1 2 3 µ 1 2 3 where we utilise previously discussed Pauli 4-vectors, σ = (12, σ , σ , σ ) andσ ¯ = (12, −σ , −σ , −σ ). This gives us:

0 0 1 0  0 0 0 1  0 0 0 −i  0 0 1 0  0 0 0 0 1 1  0 0 1 0 2  0 0 i 0  3  0 0 0 −1 γ =   γ =   γ =   γ =   . 1 0 0 0  0 −1 0 0  0 i 0 0  −1 0 0 0  0 1 0 0 −1 0 0 0 −i 0 0 0 0 1 0 0

We use the same deﬁnition for γ5 as in (4.10), which gives us

−1 0 γ5 = 2 . (4.15) 0 12

We can then deﬁne projection operators,

1 − γ5 1 + γ5 P = 4 ,P = 4 , (4.16) L 2 R 2 which, in the Weyl basis, act on a Dirac spinor to expose its left-handed and right-handed components:

ψ 1 0 ψ ψ P Ψ = P = 2 = , L L χ¯ 0 0 χ¯ 0 (4.17) ψ 0 0 ψ 0 PR Ψ = PL = = . χ¯ 0 12 χ¯ χ¯

28 In the Weyl basis, considering the massless case, the Dirac equation reduces to µ µ 0 σ ∂µ ψ (γ ∂µ − m) Ψ = µ − 0 = 0 (4.18) σ¯ ∂µ 0 χ¯ 1 and so we arrive at the Weyl equation, the wave equation for massless spin- 2 particles, given for both left and right-handed spinors: µ σ¯ ∂µψ = 0, (4.19) µ σ ∂µχ¯ = 0. (4.20) Historically, this has been used as the equation of motion for neutrinos - electrically neutral particles, that have such small mass they had previously been assumed to be massless.

4.3 Properties of Spinors and the Dirac Equation 4.3.1 Lorentz Covariance of the Dirac Equation This section follows [13]. Using our knowledge of how Dirac spinors transform with an understanding of what our gamma matrices are, we can now check the Lorentz covariance of the Dirac equation. By this, we mean to check that given the transformation of the objects in the equation, the equation is of the same form for the transformed frame. In order to do this, we first must es- tablish the transformation properties of the derivative. For an arbitrary Lorentz transformation, µ 0µ µ ν x → x = Λ ν x , we have ∂ ∂xν ∂ ∂0 = = = Λ ν ∂ . (4.21) µ ∂x0µ ∂x0µ ∂xν µ ν By inserting the transformed derivative and transformed co-ordinates into our Dirac spinor, we have µ 0 = (iγ ∂µ − m) Ψ(x) (4.22) ν µ 0 −1 0 = (iΛ µγ ∂ν − m) Ψ(Λ x ). (4.23) Pauli’s fundamental theorem then states that if we have two sets of gamma matrices satisfying the Clifford algebra then they are related a nonsingular matrix D, where γ0µ = D−1γµD. This i µν matrix is nothing more than D(Λ) = exp(− 2 Ωµν Σ ) we found earlier for the transformation of the Dirac spinor. This implies 0µ µ ν −1 µ γ = Λ ν γ = D γ D. (4.24) Our transformed Dirac equation then becomes α µ 0 −1 0 0 = (iΛ µγ ∂ν − m) Ψ(Λ x ) −1 µ 0 −1 0 = (iD γ D∂µ − m) Ψ(Λ x ) −1 µ 0 −1 −1 0 (4.25) = (iD γ D∂µ − D D m) Ψ(Λ x ) −1 µ 0 −1 0 = D (iγ ∂µ − m)D Ψ(Λ x ). Defining the transformed field by Ψ0(x0) = D(Λ) Ψ(Λ−1x0) = D(Λ) Ψ(x) (4.26) we then can see that the Dirac equation is satisfied in the transformed frame, 0 0 0 (i∂/µ − m)Ψ (x ), (4.27) i.e. the Dirac equation is Lorentz covariant.

29 4.3.2 Dirac Bilinears This section follows closely to [12]. More generally we can use the gamma matrices to create hermitian (equal to its conjugate transpose) products that will allow us to form a Lagrangian for the Dirac field. These products, which we call bilinears, should also follow the correct transformation properties of the Lorentz group. When trying to find a product similar to the terms in the scalar field Lagrangian, one might think to define a bilinear by Ψ†γµΨ. However, this is not a correct choice - we have (γ0)2 = +1, with γ0 hermitian, but (γi)2 = −1, with γi anti-hermitian (equal to its negative conjugate transpose). This property can be written as (γµ)† = γ0γµγ0. (4.28) Instead it is useful to define Ψ¯ = Ψ†γ0, (4.29) which gives us Ψ¯ γµΨ hermitian as desired. Note that the bar here does not infer any relation to the right-handed field notation we have used previously; when placed above a Dirac field, Ψ, the bar indicates a quantity of the form shown above. Using (4.28) and our definition of the Σµν we can express the hermitian conjugate of Σµν as

i † (Σµν )† = (γµγν − γν γµ) 4 −i = γν†γµ† − γµ†γν† 4 i (4.30) = γ0γµγ0 · γ0γν γ0 − γ0γν γ0 · γ0γµγ0 4 i = γ0 (γµγν − γν γµ) γ0 4 = γ0 Σµν γ0. From this, we see the hermitian conjugate of a transformation on a Dirac spinor is similarly written as † 0 i µν 0 D(Λ) = γ exp(+ 2 Ωµν Σ )γ . (4.31) Hence we can write a transformed Ψ¯ as ¯ 0 0 † † 0 ¯ i µν Ψ (x ) = Ψ(x) D(Λ) γ = Ψ(x) exp(+ 2 Ωµν Σ ). (4.32) Contracting this with a transformed Dirac spinor we ﬁnd ¯ 0 0 0 0 ¯ i µν i µν ¯ Ψ (x )Ψ (x ) = Ψ(x) exp(+ 2 Ωµν Σ ) exp(− 2 Ωµν Σ )Ψ(x) = Ψ(x)Ψ(x), (4.33) and can say that Ψ(¯ x)Ψ(x) transforms as a Lorentz scalar. This is the simplest bilinear, and is of the correct form that it could appear in a Lagrangian for a Dirac ﬁeld. This is not the only bilinear that exists however - in reality we sandwich any of the 16 linearly independent products, µ Γi, such that Ψ(¯ x)ΓΨ(x) is a bilinear. For example, we say that Ψ(¯ x)γ Ψ(x) transforms as a Lorentz-vector, and Ψ(¯ x)Σµν Ψ(x) as a Lorentz-tensor.

4.3.3 The Dirac Lagrangian Given the form of the Dirac equation, and the form of Lorentz-covariant objects discussed above, we can identify the Dirac Lagrangian by inspection as

L = Ψ(¯ x) i∂/ − m Ψ(x). (4.34)

30 We can confirm this is correct Lagrangian by applying the Euler-Lagrange equations. We have a complex field Ψ, so treat Ψ and Ψ¯ independently when varying fields. For Ψ¯ we find ∂L ∂L / ¯ = i∂ − m Ψ(x); ∂µ = 0; (4.35) ∂Ψ ∂ ∂µΨ¯ ⇒ i∂/ − m Ψ(x) = 0. (4.36)

Similarly, the equations for the variation of Ψ give the same result after conjugation and multiplication with γ0 [12]. We now follow [3] and attempt to ﬁnd a conserved current for the Dirac Lagrangian by applying Noether’s theorem. The transformation Ψ → eiθΨ with Ψ¯ → e−iθΨ¯ leaves the Lagrangian L invariant. This symmetry is known as the ‘internal vector symmetry’, which has associated current jµ = Ψ¯ γµΨ. (4.37) We can conﬁrm this is conserved using the Dirac equation:

µ µ µ ∂µj = (∂µΨ)¯ γ Ψ + Ψ¯ γ (∂µΨ) = imΨΨ¯ − imΨΨ¯ = 0, (4.38)

µ µ where we have used that iγ ∂µΨ = mΨ and i∂µΨ¯ γ = −mΨ. Another such symmetry is ‘chiral’ 5 5 symmetry (also called axial symmetry), with transformation Ψ → eiγ Ψ and Ψ¯ → e−iγ Ψ.¯ As with the standard Noether procedure, we consider the inﬁnitesimal transformation Ψ → 5 ¯ ¯ 5 ¯ µ Ψ + iαγ Ψ and Ψ → Ψ − iαγ Ψ, which gives us ‘axial’ current ja :

µ ∂L 5 ∂L 5 ¯ ja = iγ Ψ − ¯ iγ Ψ ∂ (∂µΨ) ∂ ∂µΨ (4.39) = −Ψ¯ γµγ5Ψ.

Checking this is a conserved current we have

µ ¯ µ 5 ¯ 5 µ ∂µja = (∂µΨ)γ γ Ψ − Ψγ γ (∂µΨ) = imΨ¯ γ5Ψ + imΨ¯ γ5Ψ (4.40) = 2imΨ¯ γ5Ψ.

Hence this symmetry, which acts to ‘rotate left and right-handed fermions in opposite directions’ [3], is only realised for the massless case, m = 0. We can use the decomposition of the Dirac spinor, and the deﬁnition Ψ¯ = Ψ†γ0, to form a Lagrangian from Weyl spinors,

ψ L = Ψ(¯ iγµ∂ − m) Ψ = ψ† χ¯† γ0 (iγµ∂ − m) µ µ χ¯ µ † † 0 12 0 σ 0 12 ψ = ψ χ¯ i∂µ µ − m 12 0 σ¯ 0 12 0 χ¯ (4.41) µ † † iσ¯ ∂µ −m ψ = ψ χ¯ µ −m iσ ∂µ χ¯ † µ † µ † † = ψ iσ¯ ∂µψ +χ ¯ iσ¯ ∂µχ¯ − m ψ χ¯ +χ ¯ ψ .

It follows the equation of motion for this Lagrangian (in the massless case) is the Weyl equation discussed earlier.

31 4.4 Canonical Quantisation of the Dirac ﬁeld 4.4.1 Plane Wave solutions

This section follows [3,14] closely. Writing the Dirac equation with momentum operator pµ = i ∂µ, µ we have (γ pµ − m)Ψ = 0. Considering the ﬁelds as functions of momentum, we insert an ansatz of plane wave solutions into the Dirac equation, namely,

µ Ψ+ = us(p) e−ipµx , (4.42) µ Ψ− = vs(p) e+ipµx . (4.43) where us(p) and vs(p) are four component spinors dependent on the 3-momentum. Ψ+ gives 0 2 2 1 − positive frequency solutions, i.e. p = ωp = (p + m ) 2 > 0, and Ψ negative frequency solutions, p0 < 0. Thus, we have

µ s −ipx µ ± −m ±pµσ u (p)e (±γ pµ − m)Ψ = µ · s +ipx = 0, (4.44) ±pµσ¯ −m v (p)e where the curly brackets indicate the plus-minus sign referring to the top and bottom objects in the brackets. Note that u and v transform in the same way as Ψ under Lorentz transformations, so we have Lorentz scalarsuu ¯ andvv ¯ where we deﬁneu ¯ = u†γ0 andv ¯ = v†γ0 as usual [12]. We can decompose u and v as two component spinors and claim the following solves these equations:

1 ! 1 ! 2 s 2 s s (p · σ) ξ s (p · σ) η u (p) = 1 , v (p) = 1 , (4.45) (p · σ¯) 2 ξs − (p · σ¯) 2 ηs

µ r† s r† s rs where (p · σ) = p σµ, we have normalisation ξ ξ = η η = δ and s = 1, 2 is the spinor index relating to

1 0 1 0 ξ1 = , ξ2 = , and η1 = , η2 = . (4.46) 0 1 0 1

Here, ξs and ηs can be thought of the two spin states for each of the solutions. We must choose an axis for this spin to align to (usually taken as x3), then if we took the positive frequency solution then ξ1 and ξ2 are the spinors taken for the spin-up and spin-down ﬁelds along this axis respectively. We can check our ansatz is a solution by inserting it into (4.44). In contrast to [3, 14], we show this calculation slightly diﬀerently. For u(p) we have

1 !  1 1 s µ 2 s −m(p · σ) 2 + (p · σ)(p · σ¯) 2 ξ −m pµσ (p · σ) ξ 1 = µ s  1 1 s  pµσ¯ −m (p · σ¯) 2 ξ (p · σ¯)(p · σ) 2 − m(p · σ¯) 2 ξ

 1 1 1 1 s −m(p · σ¯) 2 + (p · σ) 2 (p · σ) 2 (p · σ¯) 2 ξ =  1 1 1 1 s  (p · σ¯) 2 (p · σ¯) 2 (p · σ) 2 − m(p · σ¯) 2 ξ (4.47)

 1 1 s −m (p · σ) 2 + (p · σ) 2 m ξ =  1 1 s  (p · σ¯) 2 m − m(p · σ¯) 2 ξ = 0

32 where we use the (square rooted) identity

µ ν 2 i i i j (p · σ)(p · σ¯) = σ pµσ¯ pν = p0 + p0pi(σ − σ ) − σ σ pipj 2 ij = (p0 − pipjδ ) (4.48) µ 2 = p pµ = m .

Using the same method for v(p) we can also conﬁrm this ansatz is correct.

4.4.2 Chirality & Helicity We continue to follow [3]. The property of a field (or particle upon quantisation) being left- or right-handed is known as chirality. This property follows naturally from the choice of spinorial representation previously discussed. Given that working in the Weyl basis we can split a Dirac spinor into two Weyl spinors of different chirality, this basis is also known as the chiral basis. For this reason, γ5 is often referred to as the chirality operator. Chirality is related to another physical property, helicity, which is defined as the projection of a particles spin (or, more correctly, its angular momentum) onto the direction of its linear momentum. The helicity operator is defined as 1 σi 0 λ = pˆ . (4.49) 2 i 0 σi The hat here indicates that we are dealing with a unit vector in the direction of p. A particle moving with spin acting clockwise to the direction of motion is said to have right-handed helicity, 1 λ = 2 , whereas a particle with spin aligned anti-clockwise to its motion is said to have left- 1 handed helicity, λ = − 2 . For massless particles, such as the photon, the properties of chirality and helicity are the same thing. This is not the case for massive particles, as we can perform a Lorentz boost such that in a chosen frame the particle is ‘overtaken’. Given a massless particle moves at the speed of light, this is not a problem as there is no such reference frame that can ‘overtake’ the particle.

4.4.3 Formulae required for Quantisation In leading up to quantisation, we follow [3] but carry out all calculations explicitly and add steps wherever possible. Given the relation ξr†ξs = ηr†ηs = δrs, we can construct further important inner and outer products for u and v that will be essential when quantising the Dirac ﬁeld. We can take the inner product of u with u† and also withu ¯. Although only the latter of the two is Lorentz invariant, the former is also required for quantisation [3]. We have:

1 ! 2 s r† s r† 1 r† 1 (p · σ) ξ u (p) · u (p) = ξ (p · σ) 2 , ξ (p · σ¯) 2 1 (p · σ¯) 2 ξs (4.50) r† s r† s r† s rs = ξ (p · σ)ξ + ξ (p · σ¯)ξ = 2p0 ξ ξ = 2p0 δ .

The latter inner product is given by

1 ! 2 s r s r† 1 r† 1 0 1 (p · σ) ξ u¯ (p) · u (p) = ξ (p · σ) 2 , ξ (p · σ¯) 2 1 1 0 (p · σ¯) 2 ξs (4.51) r† 1 1 s r† 1 1 s = ξ (p · σ) 2 (p · σ¯) 2 ξ + ξ (p · σ) 2 (p · σ¯) 2 ξ = 2m ξr†ξs = 2m δrs.

33 1 ! 2 s s (p · σ) η Similarly, for v (p) = 1 we ﬁnd − (p · σ¯) 2 ηs

r† s rs r s r† s rs v (p) · u (p) = 2p0δ andv ¯ (p) · v (p) = −2m η η = −2m δ . (4.52)

We can also take the inner product betweenu ¯ and v:

1 ! 2 s r s r† 1 r† 1 0 1 (p · σ) η u¯ (p) · v (p) = ξ (p · σ) 2 , ξ (p · σ¯) 2 1 1 0 − (p · σ¯) 2 ηs (4.53) r† 1 1 s r† 1 1 s = ξ (p · σ) 2 (p · σ¯) 2 η − ξ (p · σ) 2 (p · σ¯) 2 η = 0.

Similarly, we have thatu ¯r(p)·vs(p) = 0. Instead of evaluating the inner product of ur†(p)·vs(p) (and the combination with switched u and v), we choose to define this inner product with vs = vs(−p) as in [3]. For this, we choose to introduce a second momentum, p0µ = (p0, −p). We do this because this product will be useful in quantisation. The identity we have been using then needs to be modified, though we find a similar result:

0 0 i i 2 2 µ 2 (p · σ)(p · σ) = (p · σ¯)(p · σ¯) = (p0 + piσ )(p0 − piσ ) = p0 − p = p pµ = m . (4.54)

Hence we can evaluate the aforementioned inner product:

1 ! 0 2 s r† s r† 1 r† 1 (p · σ) η u (p) · v (−p) = ξ (p · σ) 2 , ξ (p · σ¯) 2 1 − (p0 · σ¯) 2 ηs (4.55) r† 1 0 1 s r† 1 0 1 s = ξ (p · σ) 2 (p · σ) 2 η − ξ (p · σ¯) 2 (p · σ¯) 2 η = 0.

Once again, we have the same for the inner product with u and v switched: vr†(p) · us(−p) = 0. The outer product of u and v is also useful in quantisation. For this we need to sum over spins. For u andu ¯ we ﬁnd

2 2 1 ! 2 s X s s X (p · σ) ξ 0 1 r† 1 r† 1 u (p)¯u (p) = 1 ξ (p · σ) 2 , ξ (p · σ¯) 2 2 s 1 0 s=1 s=1 (p · σ¯) ξ 2 1 ! 2 s X (p · σ) ξ r† 1 r† 1 = 1 ξ (p · σ¯) 2 , ξ (p · σ) 2 2 s (4.56) s=1 (p · σ¯) ξ m p · σ = p · σ¯ m µ = γ pµ + m = p/ + m,

P2 s s† where we have used s=1 ξ ξ = 12 alongside our previous identities, and written the ﬁnal two lines with implication the mass term is multiplied by the 4-dimensional identity, 14. Utilising a similar approach for v andv ¯ we ﬁnd

2 X vs(p)¯vs(p) = p/ − m. (4.57) s=1

34 4.4.4 Quantising the ﬁeld For the rest of this chapter, we continue to follow [3] but approach the subject with a more thorough treatment where possible, providing extra calculation steps. As before, we introduce canonically conjugate momentum, π, given as ∂L π(x) = = iΨ¯ γ0 = iΨ†. (4.58) ∂Ψ˙ We promote Ψ and Ψ† (which is our conjugate momentum when multiplied by i) to operators, and write

2 X Z d3p 1 ψ(x) = bs us(p)eip.x + cs †vs(p)e−ip.x , (4.59) (2π)3 p2ω p p s=1 p 2 X Z d3p 1 ψ†(x) = bs †us(p)†e−ip.x + cs vs(p)†eip.x , (4.60) (2π)3 p2ω p p s=1 p where we are expanding in terms of the plane waves we defined previously. Here we have intro- s † s † duced two further operators, bp and cp (and their conjugates), in analogy to quantisation for † the Klein-Gordon field, where we had creation and annihilation operators ap and ap. Applying s † s s † bp creates particles that take the form of u (p), and applying cp creates particles that take the form of vs(p). When promoting to operators it is necessary to apply commutation relations as before. However, given we are dealing with fermionic quantities, defining commutators for the theory will lead to some problems with the Hamiltonian being unbounded from below. The natural choice to solve this is to promote our Poisson brackets to anticommutators. The correct choice of anticommutators is

† (3) {Ψα(x), Ψβ(y)} = δαβ δ (x − y), (4.61) † † {Ψα(x), Ψβ(y)} = {Ψα(x), Ψβ(y)} = 0. (4.62) From this we can ﬁnd anticommutation relations for our new creation and annihilation operators:

{br , bs †} = (2π)3δrs δ(3)(p − q), p q (4.63) r s † 3 rs (3) {cp, cq } = (2π) δ δ (p − q). These are the only non-zero anticommutators, i.e.

r s r s r s † r s {bp, bq} = {cp, cq} = {bp, cq } = {bp, cq} = ··· = 0. (4.64)

As with the Klein-Gordon ﬁeld, we can prove these are the correct relations by evaluating the anticommutator of ﬁelds: Z 3 3 X X d p d q 1 † {Ψ(x), Ψ†(y)} = {br , bs †}ur(p)us(q) ei(p.x−q.y) + (2π)3 (2π)3 p p q r s 4ωpωq (4.65) r † s r s † −i(p.x−q.y) {cp , cq}v (p)v (q) e X Z d3p 1 = ur(p)¯us(p)γ0eip.(x−y) + vr(p)¯vs(p)γ0e−ip.(x−y) . (4.66) (2π)3 2ω s p

35 We can then apply the outer product formulae from the last section, and change the integration variable in the second term, so that p → −p and we can bring terms together:

Z 3 † d p 1 0 ip.(x−y) 0 −ip.(x−y) {Ψ(x), Ψ (y)} = 3 (p/ + m)γ e + (p/ − m)γ e (2π) 2ωp Z 3 d p 1 0 i 0 0 i 0 ip.(x−y) = 3 (p0γ + piγ + m)γ + (p0γ − piγ − m)γ e (2π) 2ωp (4.67) Z 3 d p 2p0 ip.(x−y) = 3 e (2π) 2ωp = δ(3)(x − y), where p0 = ωp, and so by applying the claimed anticommutation relation for bp and cp we have recovered the correct anticommutation relations for Ψ and Ψ†.

4.4.5 The Hamiltonian Continuing the analogy for quantising the Klein-Gordon ﬁeld, we can write the Hamiltonian density using a Legendre transform:

H = πΨ˙ − L (4.68) i = Ψ(¯ −iγ ∂i + m)Ψ. (4.69)

i As in [3], we shall breakdown this expression, and evaluate (−iγ ∂i + m)Ψ before ﬁnding the full expression, which we will integrate over to ﬁnd the Hamiltonian. We have that

2 Z 3 i X d p 1 s i s ip.x s † i s −ip.x (−iγ ∂i+m)Ψ = b (−iγ ∂i + m)u (p)e + c (−iγ ∂i + m)v (p)e . (2π)3 p2ω p p s=1 p (4.70) i When carrying out the diﬀerentiation, we must note eip.x = e−ipix , where the minus sign comes from the application of the Minkowski metric in the inner product.

2 Z 3 i X d p 1 s i s ip.x (−iγ ∂i + m)Ψ = b (−iγ · −ipi + m)u (p)e + (2π)3 p2ω p s=1 p (4.71) s † i s −ip.x cp (−iγ · ipi + m)v (p)e

2 Z 3 X d p 1 s i s ip.x s † i s −ip.x = b (−γ pi + m)u (p)e + c (γ pi + m)v (p)e . (2π)3 p2ω p p s=1 p (4.72)

We can simplify some of the terms in this expression using the Dirac equation for spinors. For u(p) we have

µ (γ pµ − m) u(p) = 0 0 i ⇒ γ p0 − γ pi − m u(p) = 0 (4.73) i 0 ⇒ γ pi + m u(p) = γ p0 u(p).

36 i 0 Similarly for v(p) we have γ pi + m v(p) = −γ p0 v(p). Once again using the fact p0 = ωp, we can write

2 Z 3 r X d p ωp (−iγi∂ + m)Ψ = γ0 bs us(p)eip.x − cs †vs(p)e−ip.x . (4.74) i (2π)3 2 p p s=1

Multiplying this by Ψ¯ = Ψ†γ0, using (γ0)2 = 1 and then integrating of space, we ﬁnd the Hamiltonian operator: Z 3 † 0 i H = d x Ψ γ (−iγ ∂i + m)Ψ Z 3 3 r X d p d q ωp h = d3x br †ur(q)†e−iq.x + cr vr(q)†eiq.x · (2π)3 (2π)3 4ω q q r,s q s s ip.x s † s −ip.x i bpu (p)e − cp v (p)e Z 3 r X d p ωp = d3q br †bs ur(q)†us(p) δ(p − q) − br †cs †ur(q)†vs(p) δ(−p − q) (2π)3 4ω q p q p r,s q r s r † s r s † r † s + cqbpv (q) u (p) δ(p + q) − cqcp v (q) v (p) δ(−p + q) X Z d3p 1 = br †bs ur(p)†us(p) − br †cs †ur(p)†vs(p) (4.75) (2π)3 2 p p p p r,s r s r † s r s † r † s + cpbpv (p) u (p) − cpcp v (p) v (p) X Z d3p 1 = br †bs ur(p)†us(p) − br †cs †ur(p)†vs(−p) (2π)3 2 p p p −p r,s r s r † s r s † r † s + cpb−pv (p) u (−p) − cpcp v (p) v (p) , where, as in [3], we have relabelled p → −p in the last step. We can then apply the inner product formulae previously evaluated, namely:

r † s r † s rs u (p) u (p) = v (p) v (p) = 2p0 δ , (4.76) ur(p)†vs(−p) = vr(p)†us(−p) = 0. (4.77)

With this, the Hamiltonian simpliﬁes to

2 X Z d3p H = ω bs †bs − cs cs † . (4.78) (2π)3 p p p p p s=1 The anticommutation relation (4.63) can then be written as

s s † s † s 3 (3) cpcp = −cp cp + (2π) δ (0), (4.79) which reveals a similar expression for the Hamiltonian as was found for the Klein-Gordon ﬁeld:

2 X Z d3p H = ω bs †bs + cs †cs − (2π)3 δ(3)(0) . (4.80) (2π)3 p p p p p s=1

37 The last term is another example of infra-red divergence, and can be removed as was the case with the Klein-Gordon ﬁeld. Looking at this expression, the need for anticommutators becomes more r r † 3 rs (3) obvious; if we had a commutator for this expression, namely cp, cp = −(2π) δ δ (p − q) (which is necessary to make the commutator between ﬁelds valid), then the Hamiltonian would become 2 X d3p H = ω bs †bs − cs †cs + (2π)3 δ(3)(0) , (4.81) (2π)3 p p p p p s=1 which is unbounded from below. This means states of increasingly lower energy could be achieved † by repeated application of cp [3]. This is unphysical, and so when dealing with fermionic quantities the introduction of anticommutators is essential.

38 Chapter 5

Introducing Supersymmetry

5.1 The Supersymmetry Algebra 5.1.1 What is Supersymmetry? The concept of Supersymmetry (SUSY) involves associating a ‘partner’ fermion to every boson in a physical system, and a ‘partner’ boson to every fermion. This is provided by introducing † supersymmetric generators (referred to also as supercharges), Qα, and its conjugate, Qα˙ . An informal deﬁnition for these generators is given as acting on a state and generating the partner state:

Qα |bosoni = |fermioni , (5.1) † Qα˙ |fermioni = |bosoni . (5.2) We shall be closely following source [15]. Note that our generators are left- and right-handed ¯ † Weyl spinors respectively, where we have Qα˙ = Qα˙ and once again use indices α andα ˙ as a reminding notation that we are dealing with the objects from the fundamental and anti- fundamental representations of the Lorentz group.

5.1.2 The Supersymmetry Algebra We can extend the Poincar´ealgebra discussed earlier to include the commutation relations of the supersymmetric generators. The following relations provide a more concrete deﬁnition of our † SUSY generators, Qα and Qα˙ . The anticommutators of generators are given as [15]:

† µ {Qα,Qα˙ } = 2σαα˙ Pµ, (5.3) {Q ,Q } = {Q† ,Q† } = 0, α β α˙ β˙

µ where as before σαα˙ is the Pauli operator and Pµ is the momentum operator. We also can deﬁne the commutation relations between our Supersymmetry generators and the generators of Lorentz transformations and translations. Combining these equations with the deﬁning relations for the

39 Poincarégroup (3.58), we find the Super-Poincaréalgebra (also referred to as the super-algebra):

[P µ,P ν ] = 0, [M µν ,P σ] = i (P µgνσ − P ν gµσ) , [M µν ,M ρσ] = i (gνρM µσ − gµρM νσ + gµσM νρ − gνσM µρ) , µν µν β (5.4) [Qα,M ] = (σ )α Qβ, µ [Qα,P ] = 0, β {Qα,Q } = 0.

This algebra details the nature of Supersymmetry, and these relations will be discussed in more detail shortly. In deﬁning our generators, there is a U(1) symmetry associated to our theory: shifting out generators by a phase eiθ leaves the SUSY algebra invariant,

˜ iθ † ˜† −iθ † Qα −→ Qα = e Qα; Qα˙ −→ Qα˙ = e Qα˙ . (5.5) We can check this manually as follows: 0 ˜ ˜ 2iθ : {Qα˙ , Qα˙ } = e {Qα˙ ,Qα˙ } = 0 (5.6) : 0 ˜† ˜† −2iθ † † {Qα˙ , Qα˙ } = e {Qα˙ ,Qα˙ } = 0 (5.7) ˜ ˜† iθ −iθ † −iθ † iθ µ {Qα˙ , Qα˙ } = e Qαe Qα˙ + e Qα˙ e Qα = 2σαα˙ Pµ. (5.8)

| {z † } | †{z } QαQα˙ Qα˙ Qα

This symmetry, denoted U(1)R, has an associated charge, called the R-charge, with the following commutation relations: [Qα,R] = Qα, [Qα˙ ,R] = Qα˙ . (5.9)

5.1.3 Extended Supersymmetry So far our discussion has only considered the case where we have one supersymmetric generator (as well as its conjugate), which we call N = 1 Supersymmetry. We can generalise our algebra for N generators as follows:

a † µ a {Qα,Qαb˙ } = 2σαα˙ Pµδb , a b {Qα,Qβ} = 0, (5.10) {Q† ,Q† } = 0, αa˙ βb˙ with a, b = 1,..., N . What was previously a U(1)R symmetry now becomes a U(N )R symmetry. Given this report is only an introduction to Supersymmetry, it will be focussing solely on the N = 1 theory.

5.1.4 Forming the Algebra In 1967, Coleman and Mandula published a paper detailing their ‘no-go theorem’, which stated the ‘impossibility of combining space-time and internal symmetries in any but a trivial way’ [16], referring to possible Lie group symmetries in an interacting quantum ﬁeld theory. This eﬀectively meant that theorising an additional space-time symmetry was not possible. In 1971, Russian

40 scientists Likhtman and Gol’fand released a paper detailing an extension of the Poincar´ealgebra to a graded Lie algebra [17], introducing SUSY generators and thus allowing the introduction of such a symmetry to evade the Coleman-Mandula theorem [15]. Further to this, it was later proven that Supersymmetry was the only possible extra symmetry exhibited by a graded Lie Algebra, when in 1975 Haag et. al. published a version of the Coleman-Mandula generalised to such an algebra [18]. The concept of this graded Lie algebra can be explained using operators. The remainder of this subsection follows closely to [19]. For a Lie algebra with operators Oa, we have that ηaηb e OaOb − (−1) ObOa = iCabOe, (5.11) where we introduce gradings ηa given as ( 0 for Oa a bosonic operator, ηa = (5.12) 1 for Oa a fermionic operator.

The introduction of ηa with this relation takes us to a graded Lie algebra. Note that in the above e relation, Cab are structure constants, which determine commutation relations between the group generators. For example, iijk are structure constants for SU(2). We can explain more about the origin of the four extra commutation/anticommutation relations we have introduced to form our super-algebra:

h µν i • Qα,M

Given Qα is a left-handed Weyl spinor, it should transform as β β 0 i µν i µν Qα → Qα = exp − Ωµν σ Qβ ≈ 1 − Ωµν σ Qβ. (5.13) 2 α 2 α We choose to consider inﬁnitesimal transformations, as denoted by the approximate sign. In this case, note we consider Ωµν small. However, we also known that Qα acts as an i µν operator, which more generally Lorentz transforms by D(Λ) = exp(− 2 Ωµν M ), with i i Q → Q0 = D†Q D ≈ 1 + Ω M µν Q 1 − Ω M µν . (5.14) α α α 2 µν α 2 µν

Evaluating the above equation neglecting second order terms in Ωµν (as it is small), we can compare this expression for the transformation of the operator to the transformation of a spinor: i i Q − Ω (σµν ) βQ = Q − Ω (Q M µν − M µν Q ) + O(Ω2). (5.15) α 2 µν α β α 2 µν α α This leads us the commutator we were looking for:

µν µν β ⇒ [Qα,M ] = (σ )α Qβ. (5.16)

h µ i • Qα,P We make an ansatz for this commutator by looking at its index structure: one lower spinor index, and one upper (Lorentz) space-time index. The only sensible commutator for these two objects is of the form µ µ α˙ † [Qα,P ] ∼ c · (σ )αα˙ Q (5.17)

41 where c is some constant we want to ﬁnd. Given this relation, we also have

α˙ † µ ∗ µ αβ˙ [Q ,P ] ∼ c · (¯σ ) Qβ. (5.18) µ ν Writing the Jacobi identity with P , P and Qα we have : 0 µ ν ν µ µ ν 0 = [P , [P ,Qα]] + [P , [Qα,P ]] +[Qα,[P ,P ]] ν µ α˙ † µ ν α˙ † = −c(σ )αα˙ [P ,Q ] + c(σ )αα˙ [P ,Q ] (5.19) 2 ν αα˙ µ αβ˙ 2 µ ν αβ˙ = |c| (σ ) (¯σ ) Qβ − |c| (σ )αα˙ (¯σ ) Qβ 2 ν µ µ ν β = |c| (σ σ¯ − σ σ¯ )α Qβ, where we have used the known Poincar´ealgebra commutator in the ﬁrst step, and ma- µ † † µ nipulated indices in the third step using (σ Q )α = (Qσ )α˙ . We however know that ν µ µ ν β (σ σ¯ − σ σ¯ )α 6= 0, and given this holds for arbitrary Qβ we must have that c = 0. Hence we have that µ α˙ † µ [Qα,P ] = [Q ,P ] = 0. (5.20)

n o • Qα,Qβ Using a similar index structure argument, we have that β µν β {Qα,Q } ∼ k · (σ )α Mµν . (5.21) µ µ Given that we have [Qα,P ] = 0, the left hand side of this equation commutes with P . However, as shown, M µν does not commute with P µ, so we must have that anticommutator in question vanishes, i.e. k = 0. We can easily lower the index of Qβ, and so can more generally write β {Qα,Q } = 0. (5.22)

n † o • Qα,Qα˙ † Similarly, for {Qα,Qα˙ } we make the following ansatz based on index structure: {Q ,Q† } ∼ t · (σµ) P . (5.23) α β˙ αα˙ µ We are free to choose an arbitrary value for t, as we have no constraints for this anticommutator. Convention gives t = 2, we means we have that {Q ,Q† } = 2(σµ) P . (5.24) α β˙ αα˙ µ 5.1.5 Interpretation of the Algebra This idea follows closely to [20]. An interesting interpretation of the defining equations above is related to the power of the derivative associated to Qα. If we think of the momentum operator (which generates translations) as related to the first order differential ∂µ, then we can say very roughly that our operator Qα is related to the square root of derivative. Looking back to our earlier discussion of the Klein-Gordon equation, which gave a description for spin-0 particles µ using a quadratic derivative, ∂µ∂ , we found that we effectively took the ‘square-root’ of this equation in order to derive the Dirac equation, an equation with first order ∂µ. By doing this we probed further algebraic structure, resulting in the gamma matrices and Clifford Algebra. In a similar way, the SUSY generators are related the square-root of the derivative.

42 5.2 Supermultiplets 5.2.1 What are Supermultiplets? When our supercharges act on a fermionic state, the partner state is said to be the ‘superpartner’. These states are grouped together in a ‘supermultiplet’, which is an ‘irreducible representation of the SUSY algebra’ [15]. We want to ﬁnd out how many fermions and bosons there are in a supermultiplet. In analogy to the gradings we introduced for the Lie algebra, we must introduce the fermion number operator, F, where

(−1)F |bosoni = +1 |bosoni , (5.25) (−1)F |fermioni = −1 |fermioni . (5.26)

From this we ﬁnd the anticommutation relations with our generators:

F F F {(−1) ,Qα} |bosoni = (−1) Qα |bosoni + Qα(−1) |bosoni = (−1)F |fermioni + Q |bosoni α (5.27) = − |fermioni + |fermioni = 0.

This means our anticommutator must be zero,

F {(−1) ,Qα} = 0. (5.28)

Applying the same method with Qα˙ ﬁnds an analogous result. Alternatively applying the anticommutator to |fermioni yields the same result. We then introduce a subspace of states |ii in a supermultiplet. These states have momentum pµ, an eigenstate of the momentum operator Pµ. They follow the completeness relation X |ii hi| = 1. (5.29) i

As in [15], we are interested in ﬁnding the number of bosonic and fermionic states in a supermultiplet, and so taking the trace of (−1)FP 0 ‘weights’ these states based on if they are bosonic or fermionic. We claim we can write 1 H = P 0 = Q Q† + Q†Q + Q Q† + Q†Q . (5.30) 4 1 1 1 1 2 2 2 2 We can check this using our anticommutator

† † † † µ µ Q1Q1 + Q1Q1 + Q2Q2 + Q2Q2 = (σ )11Pµ + (σ )22Pµ = 2(P0 − P3) + 2(P0 − (−P3)) (5.31) 0 = 4P0 = 4P .

43 Using this we can write the trace as ! X 1 X X Tr[(−1)FP 0] = hi| (−1)FP 0 |ii = hi| (−1)FQQ† |ii + hi| (−1)FQ†Q |ii 4 i i i   1 X X X = hi| (−1)FQQ† |ii + hi| (−1)FQ† |ji hj| Q |ii 4   i i j   1 X X = hi| (−1)FQQ† |ii + hj| Q(−1)FQ† |ji 4   i j   1 X X = hi| (−1)FQQ† |ii − hj| (−1)FQQ† |ji 4   i j = 0. (5.32)

Here we have inserted a complete set of states |ji and made use of the anticommutator given above. Given this trace is zero, we can say there as many bosonic states as fermionic states in any one supermultiplet, and so any supersymmetric model we construct must have the same number of bosonic degrees of freedom as fermionic degrees of freedom.

5.2.2 Constructing Massive Supermultiplets This subchapter continues to follow closely to [15] and [19]. As stated before, our supermultiplets are simply representations of the SUSY algebra. There are many diﬀerent categories of supermultiplets, which are characterised by their constituent particles being massive or massless. We label a massive single particle state by its mass, spin, and the third component of its spin, i.e. |m, s, s3i. We are free to choose a frame of reference, and so pick the rest frame with pµ = (m, 0). With this the SUSY algebra becomes:

† µ {Qα,Qα˙ } = 2(σ )αα˙ 0 = 2(σ )αα˙ p0

= 2m δαα˙ , (5.33)

{Qα,Qβ} = 0, {Q† ,Q† } = 0. α˙ β˙

We then deﬁne a vacuum state, |Ωsi called the ‘Cliﬀord vacuum’, which has a spin s. This state is constructed by acting on our particle state with both Q1 and Q2:

0 0 |Ωsi = Q1Q2 |m, s , s3i . (5.34)

Using the fact that Qα anti-commutes with itself, acting on the Cliﬀord vacuum with Qα annihilates the state: Q1 |Ωsi = Q2 |Ωsi = 0. (5.35) † Given this, we think of Qα˙ as a raising operator and Qα as a lowering operator [15]. A supermultiplet can then be constructed using a combination of these raising operators. The general

44 massive supermultiplet of spin s is given as:

|Ωsi † † Q1 |Ωsi ,Q2 |Ωsi † † Q1Q2 |Ωsi We then have diﬀerent types of massive multiplets based on the value of the spin. For s = 0, we form the massive chiral supermultiplet:

state s3 |Ω0i 0 † † 1 Q1 |Ω0i ,Q2 |Ω0i ± 2 † † Q1Q2 |Ω0i 0 The first and last of the four states here form a complex scalar field. The second and third form a Majorana fermion - a field that under charge conjugation does not change. This type of fermion holds the equivalent ‘amount of information’ as a Weyl spinor. Further to this, a Majorana spinor can be thought of as a Dirac spinor with the two Weyl spinor components conjugate to each other, i.e. ξ† =χ ¯ in the notation used in Chapter 3. We can always express a theory using Weyl or Majorana spinors, and in our case we choose to use Weyl spinors throughout. Historically, neutrinos have been postulated to be Majorana fermions. Moving on from this discussion, we can construct a different type of multiplet called a vector multiplet, with Clifford 1 1 1 vacuum with s = 2 . Given s3 = ± 2 for s = 2 we have:

state s3 1 |Ω0i ± 2 † † Q1 |Ω0i ,Q2 |Ω0i 0, 1, 0, −1 † † 1 Q1Q2 |Ω0i ± 2 The first and last set of states form two Majorana fermions, whilst the second and third sets form a real scalar field, and a massive vector field [15]. A vector field is a spin-1 field that corresponds to the vector representation of the Lorentz group mentioned in Chapter 3. An example is the electromagnetic vector potential Aµ.

5.2.3 Constructing Massless Supermultiplets For constructing massless multiplets, we follow [15] and choose to label individual states only by their energy and helicity: |pµ, λi. We choose to look at the frame with momentum eigenvalues given as pµ = (E, 0, 0,E). This means 4E 0 {Q ,Q† } = 2(σµ) P = 2E(σ0 + σ3) P = , (5.36) α α˙ αα˙ µ αα˙ µ 0 0 αα˙ which gives the SUSY algebra as :

{Q ,Q†} = 4E, 1 1˙ † {Q2,Q } = 0, 2˙ (5.37) {Qα,Qβ} = 0, {Q† ,Q† } = 0. α˙ β˙

45 Hence for the massless case there is only one raising operator, which acts to raise a states helicity. The Cliﬀord vacuum of helicity, λ, is given by 0 |Ωλi = Q1 |E, λ i . (5.38)

As before we have that Q1 and Q2 annihilate the state:

Q1 |Ωλi = Q2 |Ωλi = 0. (5.39) Following [15], we can sandwich the {Q ,Q†} anticommutator between a set of states, to give 2 2˙ † † hΩλ| Q2Q2 |Ωλi + hΩλ| Q2Q2 |Ωλi = 0. (5.40)

Using the fact that Q2 annihilates the Clifford vacuum, † hΩλ| Q2Q2 |Ωλi = 0, (5.41) † we say that Q2 creates ‘states of zero norm’ [15]. Hence the massless multiplet is made up of states: state helicity |Ωλi λ † 1 Q1 |Ωλi λ + 2 There are however further states that we need to include in the multiplet which come from invariance under the collective transformation of charge conjugation, parity (spatial inversion) and time reversal. This transformation is called CPT symmetry and is a fundamental symmetry in nature. This report does not have time to discuss this area in great detail, but the aforementioned invariance is required because of the CPT theorem, which states that ‘any Lorentz invariant local quantum field theory with a hermitian Hamiltonian must have CPT symmetry’ [21]. The effect 1 of this is that we must include states of −λ and −λ − 2 in addition to the ones above. These are given as: state helicity 1 |Ω 1 i −λ − −λ− 2 2 † Q |Ω 1 i −λ 1 −λ− 2 We can now set a value of helicity and construct a multiplet. For λ = 0, we have the massless chiral multiplet: state helicity |Ω0i 0 † 1 Q1 |Ω0i 2 1 |Ω 1 i − − 2 2 † Q |Ω 1 i 0 1 − 2 The first and last of the states corresponds to a complex scalar field, whilst second and third correspond to a Weyl fermion. This multiplet is the one of interest when we discuss the free massless Wess-Zumino model. We can similarly construct the massless vector multiplet: state helicity 1 |Ω 1 i 2 2 † Q |Ω 1 i 1 1 2 |Ω−1i −1 † 1 Q1 |Ω−1i − 2 The first and last states here correspond to a Weyl fermion, whilst the second and third to a massless spin-1 particle, a gauge boson [15].

46 Chapter 6

The Free Wess-Zumino Model

6.1 Introducing the Model 6.1.1 A Supersymmetric Lagrangian The simplest supersymmetric model is the massless free (non-interacting) Wess-Zumino model [22]. The model involves a chiral supermultiplet, which consists of a two-component (left-handed) Weyl fermion, ψ, and a complex scalar field, φ, both of which we have discussed at length in this report. Note that this section relies heavily on manipulating spinor indices and utilising spinor identities; these are given in detail in Appendix A. This whole chapter follows [15, 23], though almost all of the results stated in these sources are proved (or shown thoroughly) here. We can write our Lagrangian as L = Lscalar + Lfermion, (6.1) where the scalar and fermion Lagrangian’s take the form we have previously encountered: µ ∗ † µ Lscalar = (∂ φ )(∂µφ), Lfermion = iψ σ¯ ∂µψ. (6.2) We introduce an infinitesimal transformation for our fields, i.e. φ → φ + δφ, ψ → ψ + δψ. (6.3) Given a supersymmetric generator produces a fermionic state when acting on a bosonic state (and vice-versa), the SUSY transformation of our bosonic scalar field should involve the fermion field. The simplest transformation possible is given as α α β δφ = ψ = ψα = αβψ , (6.4) where is an infinitesimal small anti-commuting spinor. From this we can define the transformation of the conjugate complex field:

∗ † † †α˙ † †β˙ †α˙ δφ = ψ = ψα˙ = β˙α˙ ψ . (6.5) α Given we consider global Supersymmetry (i.e. we have that ∂µ = 0), the scalar Lagrangian then transforms as follows (shown in detail): µ ∗ µ ∗ δLscalar = ∂ (δφ )∂µφ + ∂ φ ∂µ(δφ) µ † † µ ∗ = ∂ ( ψ )∂µφ + ∂ φ ∂µ(ψ) µ † † µ ∗ (6.6) = ∂ ( ψ )∂µφ + ∂ (ψ)∂µφ † µ † µ ∗ = ∂ ψ ∂µφ + ∂ ψ∂µφ .

47 For Supersymmetry to actually be a symmetry, its transformation must leave the action invariant, i.e. δS = R δL d4x = 0. Thus δL must either vanish or be equal to a total derivative. Given this, we look at the scalar Lagrangian transformation alongside the fermion Lagrangian, and deduce that the fermion ﬁeld transformation, δψ, must be linear in †, φ and contain a single derivative [23]. Using this information, we claim the transformation is of the form

ν † δψα = −i(σ )α∂ν φ, (6.7) † ν ∗ δψα˙ = i(σ )α˙ ∂ν φ . From this, we can ﬁnd the form of the fermion Lagrangian transformation (shown explicitly):

† µ † µ δLfermion = i(δψ )¯σ ∂µψ + iψ σ¯ ∂µ(δψ) ν ∗ µ αα˙ † µ αα˙ ν † = i(i(σ )α˙ ∂ν φ )(¯σ ) ∂µψα + iψα˙ (¯σ ) ∂µ(−i(σ )α∂ν φ) ν ∗ µ αα˙ † µ αα˙ ν † = −(σ )α˙ ∂ν φ (¯σ ) ∂µψα + +ψα˙ (¯σ ) ∂µ((σ )α∂ν φ) (6.8) β ν µ αα˙ ∗ µ αα˙ ν †β˙ = − (σ )βα˙ (¯σ ) ∂ν φ ∂µψα + ψα˙ (¯σ ) (σ )αβ˙ ∂µ∂ν φ ν µ ∗ † µ ν † = −σ σ¯ ∂ν φ ∂µψ + ψ σ¯ σ ∂µ∂ν φ. This can then be further broken down by applying Pauli identities

µ ν ν µ β µν β µ ν ν µ β˙ µν β˙ (σ σ¯ + σ σ¯ )α = 2g δα and (σ σ¯ + σ σ¯ )α˙ = 2g δα˙ , (6.9) where using the fact we can swap the indices of the derivatives, we can rewrite the product of barred and unbarred Pauli operators as a mixture of the same product but with indices exchanged. Explicitly evaluating this calculation we have

˙ δL = −α 1 (σµσ¯ν + σν σ¯µ)β ∂ φ∗∂ ψ + ψ† 1 (¯σµσν +σ ¯ν σµ)β †α˙ ∂ ∂ φ fermion 2 α ν µ β β˙ 2 α˙ µ ν ˙ = −αgµν δβ∂ φ∗∂ ψ + ψ† gµν δβ†α˙ ∂ ∂ φ (6.10) α ν µ β β˙ α˙ µ ν µ ∗ † † µ = −∂ φ ∂µψ + ψ ∂µ∂ φ. Further to this, we can spot a clever trick: we can write the above as two terms plus a total derivative, the latter of which will vanish when integrated over. We claim that

µ ∗ † µ † δLfermion = − ∂ ψ∂µφ − ∂ ψ ∂µφ (6.11) µ ν ∗ µ ∗ † † µ + ∂µ σ σ¯ ψ∂ν φ − ψ∂ φ + ψ ∂ φ , which we can then check explicitly by expanding the derivative: µ ∗ † µ† µ∗ µ∗ = − ∂ ψ∂µφ − ∂ψ ∂µφ +∂µψ∂ φ +ψ∂ ∂µφ (6.12) µ∗ µ∗ † †µ † † µ − ∂µψ∂ φ −ψ∂ ∂µφ + ∂µψ ∂ φ + ψ ∂µ∂ φ µ ∗ † † µ = − ∂ φ ∂µψ + ψ ∂µ∂ φ. (6.13) Hence the claimed expression is correct. Not only this, but looking at the overall transformation of our Lagrangian, δL = δLscalar + δLfermion, we ﬁnd the terms nicely cancel to give µ ν ∗ µ ∗ † † µ ∂L = ∂µ σ σ¯ ψδν φ − ψ∂ φ + ψ ∂ φ , (6.14) which is a total derivative. This gives us stationary action as desired: Z δS = δL d4x = 0. (6.15)

48 6.2 Modifying the Model 6.2.1 Commuting SUSY transformations We continue to follow [15,23]. We have the requirement that the supersymmetric algebra should close, i.e. the commutator of two supersymmetric transformations should be a symmetry transformation. In order to show this, we introduce two SUSY transformations of the form previously discussed - each labelled with a different infinitesimal generator: 1 and 2. Evaluating the commutator acting on the scalar field we have

(δ2 δ1 − δ1 δ2 )φ = δ2 (δ1 φ) − δ2 (δ1 φ)

= δ2 (1ψ) − δ1 (2ψ) µ † µ † (6.16) = (−i1σ 2φ) − (−i2σ 1φ) µ † µ † = −i(1σ 2 − 2σ 1)φ. Thus, the commutator acting on the scalar ﬁeld does produce a symmetry transformation, which is in fact a translation, as can be seen by the presence of translation generator Pµ = i ∂µ in the above equation. This is a good sign, and implies the form of the SUSY algebra stated in the last chapter [23]. Similarly, we can evaluate the commutator acting on our fermionic ﬁeld:

(δ2 δ1 − δ1 δ2 )ψα = δ2 (δ1 ψα) − δ2 (δ1 ψα) µ † µ † = δ2 (−i(σ 1)α∂µφ) − δ1 (−i(σ 2)α∂µφ) (6.17) µ † µ † = −i(σ 1)α2∂µψ + i(σ 2)α1∂µψ. We can then use the Fierz identity given in Appendix A, for three spinors χ, ξ and η:

χα(ξη) = −ξα(χη) − (ξχ)ηα. (6.18)

Applying this identity (and manipulating indices as discussed in Appendix A) we ﬁnd

µ † µ † (δ2 δ1 − δ1 δ2 )ψα = − i(−2α(σ 1∂µψ) − (2σ 1)∂µψα) µ † µ † + i(−1α(σ 2∂µψ) − (1σ 2)∂µψα) µ µ = − i(1σ 2 − 2σ 1)∂µψα (6.19) µ † µ † + i(2α(σ 1∂µψ) − 1α(σ 2∂µψ)) µ µ = − i(1σ 2 − 2σ 1)∂µψα † µ † µ + i(1α2σ¯ ∂µψ − 2α1σ¯ ∂µψ). This calculation has been carried out in further detail to [15].

6.2.2 On-shell and Off-shell SUSY As mentioned in Chapter 5, the equation of motion for a (left-handed) Weyl spinor is given µ simply asσ ¯ ∂µψ = 0, and so the second term in the last line of the above equation vanishes if the equations of motion are obeyed. As previously discussed, this condition is known as being on-shell. If this is the case, we are left with the same expression as for the commutator acting on the scalar field. Hence only for this case we can say the SUSY algebra closes on-shell. However, we require that it closes both on- and off-shell and so must adjust our model to account for this. The reason why this problem occurs is linked to the number of degrees of freedom in each object

49 on- and off-shell. As shown in [15], we can see that number of degrees of freedom for the fermion field reduces by two when the equations of motion are upheld. For example, in the frame with pµ = (E, 0, 0,E), we have µ µ 0 3 0 0 ψ1 −iσ¯ Pµψ = −iσ¯ pµψ = (¯σ p0 − σ¯ p3)ψ = , (6.20) 0 2E ψ2 and so we lose a single degree of freedom. The second degree of freedom is lost considering the same example but with the conjugate field. Tabulating this we have that off-shell on-shell φ, φ∗ 2 d.o.f. 2 d.o.f. † ψα, ψα˙ 4 d.o.f. 2 d.o.f.

6.2.3 Adding an Auxiliary Field We continue to follow [15,23] and apply a ‘trick’ to solve the problem of degrees of freedom; we introduce a new complex scalar bosonic field, F , alongside its conjugate, F ∗, which together will add two degrees of freedom off-shell, but no degrees of freedom on-shell. Such a field is called an auxiliary field. The fields produces a contribution to the Lagrangian

∗ Laux = F F, (6.21) which, upon applying the Euler-Lagrange equations, gives trivial equations of motion, F = F ∗ = 0. (6.22) Hence the field and its conjugate have zero degrees of freedom on-shell. We are free to pick the Supersymmetry transformation of F and F ∗ in terms of the other fields making up the supermultiplet. With foresight, we choose to define this transformation as a multiple of the equation of motion for ψ [23]:

† µ δF = −i σ¯ ∂µψ, (6.23) ∗ † µ δF = i σ¯ ∂µψ, (6.24) Given this, the auxiliary Lagrangian transforms as

† µ † µ ∗ δLaux = i∂µψ σ¯ F − i σ¯ ∂µψ F . (6.25) With further foresight, we must add an extra term onto the transformation of the fermion ﬁeld:

µ † δψα = −i(σ )α∂µφ + αF, (6.26) † µ ∗ † ∗ δψα˙ = i(σ )α˙ ∂µφ + α˙ F . (6.27) Following [15], we denote the transformation of the fermion ﬁeld before the introduction of the old auxiliary ﬁeld as δ Lfermion, given as old µ ∗ † µ † µ ν ∗ µ ∗ † † µ δ Lfermion = −∂ ψ∂µφ − ∂ ψ ∂µφ + ∂µ σ σ¯ ψδν φ − ψ∂ φ + ψ ∂ φ . (6.28)

new We denote the transformation of the new fermion Lagrangian as δ Lfermion, which upon eval- uation can be written with the inclusion of a total derivative:

new old † µ ∗ † µ δ Lfermion = δ Lfermion + i σ¯ ∂µψF + iψ σ¯ ∂µF (6.29) old † µ ∗ † µ † µ = δ Lfermion + i σ¯ ∂µψF − i∂µψ σ¯ F + ∂µ(iψ σ¯ F ). (6.30)

50 The new Lagrangian transformation is given as δL = δLscalar + δLfermion + δLaux. Given the foresight in defining the new transformation of the fermion field, we have that the second and third terms in (6.30) cancel with the transformed auxiliary Lagrangian. Thus, Z 4 δS = (Lscalar + Lfermion + Laux) d x = 0, (6.31) and so the action is invariant under the modified SUSY transformation.

6.2.4 Commuting modified SUSY transformations We continue to follow [15] closely, though all calculations in this subsection are shown rather than being stated. To confirm the SUSY algebra closes we must follow the same procedure as before and evaluate the commutator of successive SUSY transformations acting on the modified set of fields. We have that

(δ2 δ1 − δ1 δ2 )ψα = δ2 (δ1 ψα) − δ2 (δ1 ψα) (6.32) µ µ = − i(1σ 2 − 2σ 1)∂µψα † µ † µ + i(1α2σ¯ ∂µψ − 2α1σ¯ ∂µψ) (6.33)

+ δ2 1αF − δ1 2αF. We can evaluate the last two terms in the above equation with our deﬁnition for the transformation of F : † µ † µ δ2 1αF − δ1 2αF = 1α(−i2σ¯ ∂µψ) − 2α(−i1σ¯ ∂µψ). (6.34) Hence we have that

µ † µ † (δ2 δ1 − δ1 δ2 )ψα = −i(1σ 2 − 2σ 1)∂µψα, (6.35) which is a symmetry transformation of the same form we found for the commutator acting on the scalar field. Hence for both on-shell and off-shell fermions the SUSY algebra closes. We then must consider the commutator acting on our newly introduced auxiliary field:

† µ † µ (δ2 δ1 − δ1 δ2 )F = δ2 (−i1σ¯ ∂µψ) − δ1 (−i2σ¯ ∂µψ) † µ ν † = − i1σ¯ ∂µ(−iσ 2∂ν φ + 2F ) † µ ν † + i2σ¯ ∂µ(−iσ 1∂ν φ + 1F ) (6.36) µ † µ † = − i(1σ 2 − 2σ 1)∂µF † µ ν † † µ ν † − 1σ¯ σ 2∂µ∂ν φ + 2σ¯ σ 1∂µ∂ν φ. We can then apply Pauli and spinor identities on the last two terms of (6.36):

† µ ν † † µ ν † † † µν † † µν −1σ¯ σ 2∂µ∂ν φ + 2σ¯ σ 1∂µ∂ν φ = 12g ∂µ∂ν φ + 21g ∂µ∂ν φ = −††∂µ∂ φ + ††∂µ∂ φ 1 2 µ 2 1 µ (6.37) † † µ † † µ = −12∂ ∂µφ + 12∂ ∂µφ = 0.

Thus, we have that µ † µ † (δ2 δ1 − δ1 δ2 )F = −i(1σ 2 − 2σ 1)∂µF, (6.38)

51 which is of the same form as the commutator acting on both the fermion and scalar fields (note we do not need to re-evaluate the commutator acting on the scalar field). Hence we can write the (off-shell) supermultiplet as the set X = {φ, φ∗, ψ, ψ†,F,F ∗}, (6.39) where for all members of this set we have that

µ † µ † (δ2 δ1 − δ1 δ2 )X = −i(1σ 2 − 2σ 1)∂µX. (6.40)

6.3 Obtaining the SUSY algebra 6.3.1 The Supercurrent Given we have shown that Supersymmetry in the free Wess Zumino-model is a symmetry, we can apply Noether’s Theorem to find the associated conserved current, which we call the supercurrent. By integrating over the first component of this current we will obtain the supercharges, which † are our generators Qα and Qα˙ . The supercurrent carries one space-time and one spinor index, and is obtained using the usual Noether proecedure. In comparison to the results in Chapter 2, as in [15] we can more generally express the conserved current as µ ∂L µ ∂µj = ∂µ δX − J , (6.41) ∂(∂µX) where our infinitesimal transformation parameter becomes what was previously given as α, and µ where δL = ∂µJ . In this equation we infer summation over all fields in the supermultiplet X. We have that both the supercurrent and its conjugate are conserved quantities, and so define these quantities by ∂L jµ + †j†µ = δX − J µ. (6.42) ∂(∂µX) Note that jµ carries a spinor index; although not shown, we have contracted this index with (and similarly for the conjugated term). In this expression, we have that

µ µ ν ∗ µ ∗ † † µ † µ J = σ σ¯ ψ∂ν φ − ψ∂ φ + ψ ∂ φ + iψ σ¯ F. (6.43) Further to that shown in [15,19], the remaining calculations in this chapter are shown explicitly here. Summarising all the diﬀerent expressions we need to evaluate in order to ﬁnd (6.42): ∂L • X = φ : = ∂µφ∗, δφ = ψ. (6.44) ∂(∂µφ)

∗ ∂L µ † † • X = φ : ∗ = ∂ φ, δφ = ψ . (6.45) ∂(∂µφ )

∂L † µ α ν † • X = ψα : = i(ψ σ¯ ) , δψα = −i(σ )α∂ν φ + αF. (6.46) ∂(∂µψα)

† ∂L † ν ∗ † ∗ • X = ψα˙ : † = 0, δψα˙ = i(σ )α˙ ∂ν φ + α˙ αF . (6.47) ∂(∂µψα˙ ) ∂L † µ • X = F : = 0, δF = −i σ¯ ∂µψ. (6.48) ∂(∂µF )

∗ ∂L ∗ † µ • X = F : ∗ = 0, δF = i∂µψ σ¯ . (6.49) ∂(∂µF )

52 Hence we have that

µ † †µ µ ∗ † † µ † µ ν † j + j = ψ∂ φ + ψ ∂ φ + iψ σ¯ (−iσ ∂ν φ + F ) (6.50) µ ν ∗ µ ∗ † † µ † µ − σ σ¯ ψ∂ν φ + ψ∂ φ − ψ ∂ φ − iψ σ¯ F µ ∗ † µ ν † µ ν ∗ = 2ψ∂ φ + ψ σ¯ σ ∂ν φ − σ σ¯ ψ∂ν φ . (6.51)

We can then apply a Pauli identity on the last term, and manipulate the order of spinors to form a more symmetric expression:

µ † †µ µ ∗ † µ ν † β β µν ν µ β ∗ j + j = 2ψ∂ φ + ψ σ¯ σ ∂ν φ − 2δαg − (σ σ¯ )α ψβ∂ν φ µ ∗ † † µ ν µ ∗ ν µ ∗ = 2ψ∂ φ + ψ σ¯ σ ∂ν φ − 2ψ∂ φ + σ σ¯ ψ∂ν φ (6.52) ν µ ∗ † † µ ν = σ σ¯ ψ∂ν φ + ψ σ¯ σ ∂ν φ.

The left and right hand side of this equation are now of the same form, and so we deﬁne the supercurrent and its conjugate as

µ ν µ ∗ jα = (σ σ¯ ψ)α∂ν φ , (6.53) †µ † µ ν jα˙ = (ψ σ¯ σ )α˙ ∂ν φ,

µ †µ each of which is conserved separately, i.e. ∂µjα = 0 and ∂µjα˙ = 0.

6.3.2 The Supercharges From the above equations we ﬁnd our conserved supercharges (which are our supersymmetric generators) by spatially integrating over the zeroth component of the supercurrent and its conjugate: √ Z √ Z 3 0 3 ν 0 ∗ Qα = 2 d x jα = 2 d x (σ σ¯ ψ)α∂ν φ , √ Z √ Z (6.54) † 3 †0 3 † 0 ν Qα˙ = 2 d x jα˙ = 2 d x (ψ σ¯ σ )α˙ ∂ν φ.

We then claim that these supercharges generate the transformations of the ﬁelds X: √ Q + Q†,X = −i 2 δX. (6.55)

We check this explicitly by applying equal-time commutation (and anticommutation) relations. This is not shown explicitly in the sources we have been following. In the case of the scalar ﬁeld φ, we have that √ Z † 3 h ν 0 ∗ † † 0 ν i Q + Q , φ = 2 d x σ σ¯ ψ(x)∂ν φ (x) + ψ (x)¯σ σ ∂ν φ(x) , φ(y) . (6.56)

We then insert the commutation relations

[φ(x), π(y)] = [φ∗(x), π∗(y)] = i δ(3)(x − y), [φ(x), φ(y)] = [π∗(x), π∗(y)] = 0, (6.57) [φ(x), π∗(y)] = [φ∗(x), π(y)] = 0,

53 ∗ ∗ where we deﬁne our conjugate momenta π(x) = ∂0φ (x) with π (x) = ∂0φ(x). Hence, √ Z † 3 0 0 ∗ † † 0 0 Q + Q , φ = 2 d x σ σ¯ ψ(x)[∂0φ (x), φ(y)] + ψ (x)¯σ σ [∂0φ(x), φ(y)] √ Z = − 2 d3x σ0σ¯0ψ(x)[φ(y), π(x)] + †ψ†(x)¯σ0σ0 [φ(y), π∗(x)] √ Z (6.58) = − 2 d3x σ0σ¯0ψ(x) · i δ(3)(y − x) √ = −i 2σ0σ¯0ψ(y) √ = −i 2 ψ, which successfully retrieves the result that δX = ψ for X = φ. We can show this is the case for the fermion ﬁeld by applying the anticommutation relations

† 0 (3) {ψα(x), ψα˙ (y)} = (σ )αα˙ δ (x − y), (6.59) with all other anticommutation relations vanishing. In doing this we require that any on-shell terms (notably those from the auxiliary field) vanish. Following a similar procedure as above we confirm that (6.55) holds for fermion fields.

6.3.3 Recovering the SUSY Algebra We continue to follow steps outlined in [15] but show them explicitly here. Introducing a pair of supersymmetric transformations, each with inﬁnitesimal generator 1 and 2, we can evaluate the commutator

h † † h † † ii h † † h † † ii 2Q + 2Q , 1Q + 1Q ,X − 1Q + 1Q , 2Q + 2Q ,X (6.60) √ √ h † † i h † † i = 2Q + 2Q , −i 2 δ1 X − 1Q + 1Q , −i 2 δ2 X

= −2 (δ2 δ1 − δ1 δ2 ) X µ † µ † = 2 1σ 2 − 2σ 1 i ∂µX, (6.61) where we have used results (6.40) and (6.55). We can then apply the Jacobi identity to (6.60), where we manipulate the identity as follows: [A, [B,C]] + [B, [C,A]] + [C, [A, B]] = 0 ⇒ [A, [B,C]] − [B, [A, C]] = − [C, [A, B]] (6.62) = [[A, B] ,C] .

† † † † Here we have that A = 2Q + 2Q , B = 1Q + 1Q and C = X, and so we write (6.60) as

hh † † † †i i µ † µ † 2Q + 2Q , 1Q + 1Q ,X = 2 1σ 2 − 2σ 1 i ∂µX. (6.63)

The commutator of the momentum operator with a ﬁeld X acts to translate the ﬁeld:

[Pµ,X] = i ∂µX. (6.64) Hence we can rewrite (6.63) as

hh † † † †i i µ † µ † 2Q + 2Q , 1Q + 1Q ,X = 2 1σ 2 − 2σ 1 [Pµ,X] . (6.65)

54 Given that this is true for any ﬁeld X (with on-shell terms vanishing) [23], we have that

h † † † †i µ † µ † 2Q + 2Q , 1Q + 1Q = 2 1σ 2 − 2σ 1 . (6.66)

Since inﬁnitesimal generators 1 and 2 are arbitrary [15], we choose to write

h † †i µ † 2Q, 1Q = 22σ 1Pµ, (6.67)

h † †i µ † 2Q, 1Q = −22σ 1Pµ, (6.68)

h i h † † † †i 2Q, 1Q = 2Q , 1Q = 0. (6.69)

By writing out (6.67) and manipulating the order of the spinors, we can ﬁnd a form of this equation that will allows us to extract 1 and 2:

µ † † † † † 22σ 1Pµ = 2Q1Q − 1Q 2Q α † †α˙ † †α˙ α = 2 QαQα˙ 1 − Qα˙ 1 2 Qα α † †α˙ α † †α˙ (6.70) = 2 QαQα˙ 1 + 2 Qα˙ Qα1 α † † †α˙ = 2 QαQα˙ + Qα˙ Qα 1 .

Removing 1 and 2 we are left with the anti-commutator

† µ {Qα,Qα˙ } = 2σαα˙ Pµ. (6.71) √ Note how the convention of defining the supercurrent and its conjugate with a factor of 2 leads to an an overall factor of 2 in the above commutator - if we had previously defined the anticommutator with a different overall factor we would be required to change the definitions of the supercurrent and its conjugate. Given that in (6.69) the commutators vanish, we can similarly manipulate the order of spinors to give the anticommutator

{Q ,Q } = {Q† ,Q† } = 0. (6.72) α β α˙ β˙ Hence, for the free massless Wess-Zumino model, with action given as Z µ ∗ † µ ∗ 4 S = ∂ φ ∂µφ + iψ σ¯ ∂µψ + F F d x (6.73) and transformations listed in (6.44) − (6.49), we have recovered the SUSY algebra.

55 Chapter 7

Conclusion & Further Discussion

7.1 Further Discussion

The natural progression from discussing the free massless Wess-Zumino model is the introduction of massive supermultiplets with an interaction term in the Lagrangian. This involves introducing a new quantity, W named the superpotential, which because of renormalizability takes a specific form. In this report we have exclusively discussed free field theory, and so further foundations of interacting field theory must be introduced before discussing this more complex model and reviewing a topic such as renormalizability. Note we have only discussed scalar and spinor fields, whilst studying other fields (such as vector fields etc.) will be necessary in order to describe more realistic SUSY models. Such a model is the Minimal Supersymmetric Standard Model (MSSM), which is the most well considered standalone SUSY model at present. This model is ‘minimal in the sense that it contains the smallest number of new particle states and new interactions con- sistent with phenomenology’ [24]. Another well considered area of research for SUSY is within String Theory.

Note that P 2 remains a casimir of the SUSY algebra. Thus, it commutes with SUSY generators, which implies the mass of both fermion and bosons in a supermultiplet must be the same. As mentioned in the introduction of this report, SUSY currently has little to no experimental evidence - we have not seen superpartners of the same mass as the particles we know make up the standard model. Hence Supersymmetry, if it exists, must be spontaneously broken, whereby the vacuum state itself is not supersymmetric. Given that (5.30) is positive semi-deﬁnite, in this case we have that the vacuum expectation value, h0|H|0i > 0, with Qα |0i= 6 0 [25].

7.2 Conclusion

This report began by introducing the concept of field theory and discussing the most basic example, the Klein-Gordon field. We quantised this field, making analogy to the simple harmonic oscillator. Problems with quantisation (the presence of infra-red and ultra-violet divergences) were then discussed. We then considered the Lorentz group, and saw that the scalar Klein- Gordon field was just one representation of many that the Lorentz group could take. This group was then extended to the Poincarégroup with the introduction of translations, and the full Poincaréalgebra reviewed. We then introduced the Dirac field, following Dirac’s methodology in his landmark 1928 paper. This field is a ‘sum’ of the two spinorial representations discussed in

56 Chapter 3. Subsequently, we looked at the Clifford algebra, and manipulated spinors in order to provide formulae for quantisation. Upon quantising, we found similar problems to that with the Klein-Gordon field. We then introduced the concept of Supersymmetry, examining its graded Lie algebra, and reviewing the extra relations gained in moving from the Poincaréto the Super- Poincaréalgebra. Next, we constructed massless and massive supermultiplets. We considered the massless free Wess-Zumino model, consisting of a free chiral supermultiplet. With some modification to the model, we confirmed that this model was a symmetry of the action, and that the SUSY algebra closed on- and off-shell. The report concludes with the recovery of this algebra for a quantised set of fields.

Acknowledgements

I would like to thank Prof. Wojtek Zakrzewski and Prof. Richard Ward for their help and useful suggestions throughout this project.

Declaration

This piece of work is a result of my own work except where it forms an assessment based on group project work. In the case of a group project, the work has been prepared in collaboration with other members of the group. Material from the work of others not involved in the project has been acknowledged and quotations and paraphrases suitably indicated.

57 Appendix A

Spinor Indices and Identities

The results in this appendix are discussed in [15], though where possible we try to show the origins of these results more explicitly. There are two different types of indices that we deal with throughout this report: Lorentz (space-time) indices, denoted µ, ν = 0, 1, 2, 3, and spinor indices, given as α, β = 1, 2 andα, ˙ β˙ = 1, 2 corresponding to fundamental and anti-fundamental representations of SL(2, C) from which left- and right-handed Weyl spinors originate. For spinor indices, the equivalent of a ‘metric’ that raises and lowers indices is given as αβ, an antisymmetric tensor defined as 0 1 0 −1 αβ = with = . (A.1) −1 0 αβ 1 0 αβ βγ γ where αβ = − , so that αβ = δα. The same definitions hold using dotted spinor indices. Raising and lowering indices we have

˙ ξ = ξβ, ξα = αβξ , χ† = χ†β, χ†α˙ = αβχ† . (A.2) α αβ β α˙ α˙ β˙ β˙ Only indices of the same type (i.e. two dotted or two undotted) can be contracted. Suppressed α undotted indices must be contracted ‘north-west’ to ‘south-east’: α. Suppressed dotted indices α˙ must be contracted ‘south-west’ to ‘north-east’: α˙ . This can be seen when taking the equivalent of an ‘inner-product’ using the ‘metric’:

α α β β α β α β ξχ = ξ χα = ξ αβχ = −χ αβξ = χ βαξ = χ ξβ = χξ. (A.3) Similarly, for dotted indices, looking at conjugated spinors

† † † †α˙ †β˙ †α˙ †α˙ †β˙ † †β˙ † † ξ χ = ξ χ = ξ ˙ χ = −χ ˙ ξ = χ ξ = χ ξ α˙ βα˙ βα˙ β˙ (A.4) = (ξχ)∗, where we note that the last line involves a complex conjugate as opposed to a hermitian conjugate. As can be seen, although a pair of spinors ξα and χα anticommute, their ‘inner-product’ using the ‘metric’ product is symmetric. Another useful identity we will need is the Fierz identity. For two-component spinors we have that

χα (ξη) = −ξα (χη) − (ξχ) ηα. (A.5) µ i i The Pauli operator is given as (σ )αα˙ = (12, σ ), where σ are the standard Pauli matrices. The barred Pauli operator is deﬁned as

µ αα˙ αβ α˙ β˙ µ i (¯σ ) = (σ )ββ˙ = (12, −σ ). (A.6)

58 From these deﬁnitions, then following Pauli identities automatically follow:

µ (σ )αα˙ (σµ)ββ˙ = 2 αβα˙ β˙ , (A.7) µ αα˙ ββ˙ α˙ β˙ αβ (¯σ ) (¯σµ) = 2 , (A.8) µ ββ˙ β β˙ (σ )αα˙ (¯σµ) = 2 δα δ α˙ , (A.9) µ ν ν µ β µν β (σ σ¯ + σ σ¯ )α = 2 g δα , (A.10) µ ν ν µ β˙ µν β˙ (σ σ¯ + σ σ¯ )α˙ = 2 g δ α˙ . (A.11) We can form an identity based on the contraction of the Pauli operator with spinors:

† µ † µ αα˙ †β˙ µ αα˙ δ ξ σ¯ χ = ξα˙ (¯σ ) χα = α˙ β˙ ξ (¯σ ) αδχ δ µ αα˙ †β˙ δ µ αα˙ †β˙ = −χ α˙ β˙ (¯σ ) αδξ = −χ β˙α˙ δα(¯σ ) ξ δ µ †β˙ (A.12) = −χ (σ )δβ˙ ξ = −χσµξ† = (χ†σ¯µξ)∗ = −(ξσµχ†)∗.

Similarly, for contracting two spinors with barred and unbarred Pauli operators we have that

† µ ν † † µ αα˙ ν †β˙ †δ˙ µ αα˙ ν β˙γ˙ † ξ σ¯ σ χ = ξα˙ (¯σ ) (σ )αβ˙ χ = α˙ δ˙ξ (¯σ ) (σ )αβ˙ χγ˙ † µ ατ˙ ρα ν β˙γ˙ †δ˙ = −χγ˙ α˙ δ˙(¯σ ) τρ (σ )αβ˙ ξ † µ ατ˙ ρα γ˙ β˙ ν †δ˙ † µ ν γρ˙ †δ˙ (A.13) = χγ˙ δ˙α˙ ρτ (¯σ ) (σ )αβ˙ ξ = χγ˙ (σ )ρδ˙(¯σ ) ξ † ν γρ˙ µ †δ˙ = χγ˙ (¯σ ) (σ )ρδ˙ξ = χ†σ¯ν σµξ†.

59 References

[1] Peskin, M. and Schroeder, 1995, An Introduction to Quantum Field Theory, HarperCollins Publishers [2] Dasgupta, M. 2009 An Introduction to Quantum Field Theory Lecture Notes, University of Manchester, accessed on 27th November 2015, http://www.ppd.stfc.ac.uk/PPD/resources/pdf/QuantumFieldTheory09.pdf [3] Tong, D. 2006, Lectures on Quantum Field Theory, Part III Cambridge University Mathe- matics Tripos, accessed on 17th February 2016, http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf [4] Boels R. and Br¨ummerF. 2012, Introduction to Supersymmetry and Supergravity Lecture Notes, University of Hamburg, accessed on 19th January 2016, http://www.desy.de/~fbruemme/SUSY/SUSY.pdf [5] Peeters K. and Zamaklar M. 2015, Advanced Quantum Theory Lecture Notes, University of Durham [6] Tanedo P. 2010, Fun and Supersymmetry, ... but mostly the latter, accessed on 26th March 2016, http://www.physics.uci.edu/~tanedo/files/notes/FlipSUSY.pdf [7] Rowland, T. and Weisstein, E. W. 2016, Group, From MathWorld - A Wolfram Web Re- source, accessed on 24th March 2016, http://mathworld.wolfram.com/Group.html [8] Maciejko J. 2013, Representations of Lorentz and Poincar´egroups, accessed on 25th March 2016, http://einrichtungen.ph.tum.de/T30f/lec/QFT/groups.pdf [9] Osborn H. 2016, Symmetries and Groups Michaelmas 2008 Lecture Notes (updated 12th February 2016), University of Cambridge, accessed 19th March 2016, http://www.damtp.cam.ac.uk/user/ho/GNotes.pdf [10] Schweber S. S. 2005, An Introduction to Relativistic Quantum Field Theory, Dover Publi- cations [11] Dirac P. A. M. 1928, The Quantum Theory of the Electron, Proc. Roy. Soc. Lond. A 117 (192) 610

[12] Zee A. 2003, Quantum Field Theory in a Nutshell, Princeton University Press

60 [13] Broida J. G. 2009, The Dirac Equation and the Lorentz Group, University of Colorado, accessed on 1st April 2016, http://physics.ucsd.edu/students/courses/fall2009/physics130b/Dirac_Lorentz.pdf [14] Louis J. 2011 Quantum Field Theory I Lecture Notes, University of Hamburg, accessed on 6th April 2016, http://www.desy.de/~jlouis/Vorlesungen/QFTI10/QFTI.pdf [15] Terning J. 2007, Modern Supersymmetry, Dynamics and Duality, Oxford: Oxford University Press [16] Coleman S. and Mandula J. 1967, All Possible Symmetries of the S Matrix, Physical Review 159 (5): 1251-1256 [17] Gol’fand Y. A. and Likhtman E. P. 1971, Extension of the Algebra of Poincare Group Generators and Violation of P Invariance, JETP Lett. 13, 323-326 [18] Haag R., Sohnius M. and Lopuszanski J. T. 1975, All possible generators of supersymmetries of the S-matrix, Nuclear Physics B 88: 257-274

[19] Quevedo F., Krippendorf S. and Schlotterer I, 2010, Cambridge Lectures on Supersymmetry and Extra Dimensions, accessed on 12th February 2016, arXiv:1011.1491 [hep-th] [20] Aitchison I. J. R. 2007, Supersymmetry in Particle Physics An Elementary Introduction, Cambridge: Cambridge University Press [21] Ohlsson T. 2011, Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory, Cambridge: Cambridge University Press [22] Wess J. and Zumino B. 1974, Supergauge Transformations in Four Dimensions, Nuclear Physics B 70: 39-50

[23] Martin S. P. 2011, A Supersymmetry Primer, accessed on 2nd December 2015, arXiv:hep-ph/9709356 [24] Baer H. and Tata X. 2006, Weak scale supersymmetry, from superﬁelds to scattering events, Cambridge: Cambridge University Press

[25] Cs´akiaC. and Tanedo P. 2016, Beyond the Standard Model, Lectures at the 2013 European School of High Energy Physics, accessed on 18th April 2016, arXiv:1602.04228v1 [hep-ph]