UPPSALA UNIVERSITY BACHELOR THESIS 15 HP Symmetries of the Point Particle Author: Alexander Söderberg Supervisors: Ulf Lindström Subject Reader: Maxim Zabzine June 23, 2014 Abstract: We study point particles to illustrate the various symmetries such as the Poincaré group and its non-relativistic version. In order to find the Noether charges and the Noether currents, which are conserved under physical symme- tries, we study Noether’s theorem. We describe the Pauli-Lubanski spin vector, which is invariant under the Poincaré group and describes the spin of a particle in field theory. By promoting the Pauli-Lubanski spin vector to an operator in the quantized theory we will see that it describes the spin of a particle. Moreover, we find an action for a smooth spinning bosonic particle by compactifying one string dimension together with one embedding dimension. As with the Pauli- Lubanski spin vector, we need to quantize this action to confirm that it is the action for a smooth spinning particle. Sammanfattning: Vi studerar punktpartiklar för att illustrera olika symemtrier som t.ex. Poincaré gruppen och dess icke-relativistiska version. För att hitta de Noether laddningar och Noether strömmar, vilka är bevarade under symmetrier, studerar vi Noether’s sats. Vi beskriver Pauli-Lubanksi spin vektorn, vilken har en invarians under Poincaré gruppen och beskriver spin hos en partikel i fältteori. Genom att låta Pauli-Lubanski spin vektorn agera på ett tillstånd i kvantfältteori ser vi att den beskriver spin hos en partikel. Dessutom finner vi en verkan för en spinnande partikel genom att kompaktifiera en bosonisk sträng dimension till- sammans med en inbäddad dimension. Som med Pauli-Lubanski spin vektorn, kvantiserar vi denna verkan för att bekräfta att det är en verkan för en spinnande partikel. BACHELOR PROGRAM IN PHYSICS DEPARTMENT OF PHYSICSAND ASTRONOMY DIVISIONOF THEORETICAL PHYSICS 1 ACKNOWLEDGEMENT I would like to give many thanks to my supervisor, Ulf Lindström, for the intro- duction to this subject, and for many interesting discussions. Moreover, would I like to thank Susanne Mirbt and my friends for good advices, and my family for the support they have given me. For given me something to look forward to, I would like to thank Shigeru Miyamoto. 2 CONTENTS 1. Introduction 4 2. Method 5 2.1. Problem Formulation . 6 3. The Point Particle 7 3.1. The Non-Relativistic Action for a Free Massive Particle . 7 3.2. The Galilean Transformations . 7 3.3. The Relativistic Action for a Free Massive Particle . 9 3.4. Equations of Motion for a Free Particle . 10 4. The Lorentz Group 11 4.1. Boosts . 11 4.2. Rotations Around a Fixed Axis . 13 5. The Poincaré Group 14 5.1. Translations . 14 6. Noether’s Theorem 15 6.1. Noether’s Theorem . 16 6.2. Noether’s Theorem in Field Theory . 16 7. Pauli-Lubanski Spin 17 7.1. Pauli-Lubanski Spin . 17 7.2. Quantization of the Pauli-Lubanski Spin . 19 8. Smooth Spinning Particles 20 8.1. The Action of a Bosonic String . 20 8.2. The Action of a Rigid Particle . 21 8.3. Classical Dynamics of a Rigid Particle . 23 8.4. Quantization of a Rigid Particle’s Momenta and Spin . 24 9. Conclusion 26 A. Derivations 30 3 1. INTRODUCTION When we study the total angular momentum of a particle in quantum mechanics we need to study not only the orbital angular momentum (as we do in the classical case) but also the internal angular momentum of the particle. This internal angular momentum is what we call spin, and it determines in which state the particle is in. Spin is a conserved quantity and therefore it is important to study this quantity, e.g. in particle colliders is spin of big impor- tance. In this thesis we will try to get a better understanding of the quantum mechanical property spin by studying its classical counterpart. This will we do by using symmetries. We shall take the classical particle as our starting point for a discussion of symmetries. If a quantity has a symmetry under a transformation, then this quantity is invariant under this transformation, i.e. it is the same before and after this transformation. We have a practical understanding of what a symmetry operation does, e.g. when a sphere is rotated an angle around an axis through its center. In physics we are interested in symmetries of the equa- tions of motion. There are two main categories of symmetries. Space-time symmetries, such as boosts, rotations and space-time translations, and internal symmetries such as the gauge transformations in electromagnetism. As an example, consider that a physical system has a time translational symmetry. This would mean that performing an experiment before or after a time translation for instance yields the same result. Mathematically this leads to invariance of the equations of motion, or equivalently of the action for the system, under the mathemat- ical transformation. From the action of a system one can find the equations of motion associated with this sys- tem. Therefore, we study the action of systems. We will proceed from the action of a system and look at transformations under which the action is invariant. Through symmetries we will describe a quantity which describes spin in classical field theory, and an action for a smooth spinning particle. A smooth spinning particle has spin once it is quantized. We will find an ac- tion for a smooth spinning particle by compactifying one string dimension together with one embedding dimension. We will only consider strings with integer spin, i.e. bosonic strings, and therefore we will only find an action for smooth spinning particles with integer spin, i.e. bosonic smoth spinning particles. In general, to see if a quantity describes spin, one need to study whether its quantized version introduce spin or not. Not all systems in physics can be described by classical models. When one is considering sys- tems with smaller distances and higher energies, one needs to use quantum mechanics. On even smaller distances and higher energies one needs to apply string theory. However, both quantum mechanics and string theory have their mathematical basis in symmetries, many of which are relevant also classically. Therefore it is important to study these symmetries in the classical case. Symmetries of classical theories correspond to conserved quantities that are essential in de- scribing e.g. the motion of a system. It is often these conserved quantities we study in ex- 4 periments, e.g. the energy and the momentum. Symmetries in quantum mechanics have a similar purpose, and also label the quantum numbers of a system. Classical symmetries have their quantum mechanical counterpart, but the opposite is not always true. Spin is one ex- ample of a quantum mechanical property which is not fully covered in the classical theory. The classical counterpart is best described by the Pauli-Lubanski vector. In this thesis we will always study massive particles. This means we study particles having a mass, e.g. we study the Pauli-Lubanski spin for a particle with mass and not for a photon. The total angular momentum in classical physics is identical with orbital angular momen- tum. However, in quantum mechanics one also has to consider the spin angular momentum. The spin angular momentum is what describes spin in quantum mechanics. It has no mean- ing in classical physics, but since some classical quantities introduce spin when quantized we would like to say that these quantities have spin in classical physics. It is important to study such quantities since they give us a better understanding of quantum mechanical properties. 2. METHOD In this thesis we will gather important information about symmetries and spin in classical physics in one place. Often is information about symmetries and spin in classical physics written down in different kind of books. So in this thesis we will get a better understanding how one can with symmetries describe spin in classical physics. Moreover will most of the calculations be done so one can follow the mathematical steps. This project is a literature study. The references in this thesis are mostly to books. They have been chosen based on their importance in the various areas which have been studied in this thesis. We will use all of the references as guidelines and do most of the calculations. We can in principal divide this thesis into three parts. In the first part we will illustrate the im- portance of symmetries by proving that the action for a point particle is invariant under the transformations in the Poincaré group (the transformations used in special relativity) and its non-relativistic version. In the second part we will describe a quantity in classical field the- ory which describes the spin of a particle. Finally in the third part, we will find an action for a smooth spinning particle. The first part consist of section 3,4 and 5. Here we will follow [1]-[4]. Reference [1] and [3] are about special relativity and reference [2] and [4] are about quantum field theory. Since the Poincaré group are important in quantum field theory, [2] and [4] are two good references to follow when we want to prove the invariance of the relativistic action for a point particle under the transformations in the Poincaré group. The second part consist of section 6 and 7. Here we will follow [4]-[6]. Reference [5] is a book about some mathematics which exists in physics, and reference [6] is another book of 5 quantum field theory. We follow reference [5] since we want to know the mathematical ver- sion of Noether’s theorem and why we can apply it to physical systems.