UPPSALA UNIVERSITYBACHELOR 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 asthe Poincaré group and its non-relativistic version. In order to find the Noethercharges 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 particlein field theory. By promoting the Pauli-Lubanski spin vector to an operator in thequantized 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 onestring dimension together with one embedding dimension. As with the Pauli-Lubanski spin vector, we need to quantize this action to confirm that it is theaction for a smooth spinning particle.

Sammanfattning: Vi studerar punktpartiklar för att illustrera olika symemtriersom t.ex. Poincaré gruppen och dess icke-relativistiska version. För att hitta deNoether laddningar och Noether strömmar, vilka är bevarade under symmetrier,studerar vi Noether’s sats. Vi beskriver Pauli-Lubanksi spin vektorn, vilken har eninvarians under Poincaré gruppen och beskriver spin hos en partikel i fältteori.Genom att låta Pauli-Lubanski spin vektorn agera på ett tillstånd i kvantfältteoriser vi att den beskriver spin hos en partikel. Dessutom finner vi en verkan för enspinnande 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 spinnandepartikel.

BACHELOR PROGRAM IN PHYSICS

DEPARTMENT OF PHYSICS AND ASTRONOMY

DIVISION OF 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 Ilike to thank Susanne Mirbt and my friends for good advices, and my family forthe support they have given me. For given me something to look forward to, Iwould like to thank Shigeru Miyamoto.

2

CONTENTS

1. Introduction 4

2. Method 52.1. Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3. The Point Particle 73.1. The Non-Relativistic Action for a Free Massive Particle . . . . . . . . . . . . . . . 73.2. The Galilean Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3. The Relativistic Action for a Free Massive Particle . . . . . . . . . . . . . . . . . . 93.4. Equations of Motion for a Free Particle . . . . . . . . . . . . . . . . . . . . . . . . 10

4. The Lorentz Group 114.1. Boosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2. Rotations Around a Fixed Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5. The Poincaré Group 145.1. Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6. Noether’s Theorem 156.1. Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.2. Noether’s Theorem in Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 16

7. Pauli-Lubanski Spin 177.1. Pauli-Lubanski Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.2. Quantization of the Pauli-Lubanski Spin . . . . . . . . . . . . . . . . . . . . . . . 19

8. Smooth Spinning Particles 208.1. The Action of a Bosonic String . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.2. The Action of a Rigid Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.3. Classical Dynamics of a Rigid Particle . . . . . . . . . . . . . . . . . . . . . . . . . 238.4. 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 needto study not only the orbital angular momentum (as we do in the classical case) but also theinternal angular momentum of the particle. This internal angular momentum is what we callspin, and it determines in which state the particle is in. Spin is a conserved quantity andtherefore 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 mechanicalproperty 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 aquantity has a symmetry under a transformation, then this quantity is invariant under thistransformation, i.e. it is the same before and after this transformation. We have a practicalunderstanding of what a symmetry operation does, e.g. when a sphere is rotated an anglearound 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, suchas boosts, rotations and space-time translations, and internal symmetries such as the gaugetransformations in electromagnetism. As an example, consider that a physical system has atime translational symmetry. This would mean that performing an experiment before or aftera time translation for instance yields the same result. Mathematically this leads to invarianceof 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 systemand look at transformations under which the action is invariant. Through symmetries we willdescribe a quantity which describes spin in classical field theory, and an action for a smoothspinning 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 oneembedding 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 tostudy 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. Oneven smaller distances and higher energies one needs to apply string theory. However, bothquantum mechanics and string theory have their mathematical basis in symmetries, manyof which are relevant also classically. Therefore it is important to study these symmetries inthe 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 asimilar purpose, and also label the quantum numbers of a system. Classical symmetries havetheir 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 willalways study massive particles. This means we study particles having a mass, e.g. we studythe 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 wewould like to say that these quantities have spin in classical physics. It is important to studysuch 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 classicalphysics in one place. Often is information about symmetries and spin in classical physicswritten down in different kind of books. So in this thesis we will get a better understandinghow one can with symmetries describe spin in classical physics. Moreover will most of thecalculations 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 havebeen chosen based on their importance in the various areas which have been studied in thisthesis. 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 thetransformations in the Poincaré group (the transformations used in special relativity) and itsnon-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 asmooth spinning particle.

The first part consist of section 3,4 and 5. Here we will follow [1]-[4]. Reference [1] and [3] areabout special relativity and reference [2] and [4] are about quantum field theory. Since thePoincaré group are important in quantum field theory, [2] and [4] are two good referencesto follow when we want to prove the invariance of the relativistic action for a point particleunder the transformations in the Poincaré group.

The second part consist of section 6 and 7. Here we will follow [4]-[6]. Reference [5] is abook 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. In section 6 we studyNoether’s theorem. With Noether’s theorem we can find the conserved quantities which existunder a symmetry operation. In section 6 we also make the transition to field theory, andtherefore [4] and [6] are good references to follow since [4] and [6] are both about field theory.

The final part consist of section 7. In this section we follow [7]-[9]. Reference [7] is a bookabout string theory, and [8] and [9] are articles about rigid particles. Reference [8] is aboutfinding an action for a rigid particle, and reference [9] is about quantizing this action to no-tice that this action describes a smooth spinning particle. At the beginning of section 7 weexamine two classically equivalent actions for a bosonic string. To do this we follow [7]. Wewill then use [8] and [9] as guidelines to find an action for a smooth spinning particle and toensure that this action is indeed for a smooth spinning particle.

2.1. PROBLEM FORMULATION

The main question of this thesis is how symmetries and spin are introduced in the contextsof a one-particle system. This is later extended in order to describe symmetries and spin ofsystems consisting of several particles, and eventually to field theory. The purpose of this the-sis is to achieve a better understanding of the importance of symmetry operations, and howone can use symmetries to find conserved quantities, such as Noether currents and Noethercharges. We also want to know how one would describe a quantum mechanical property inclassical physics. In this thesis the quantum mechanical quantity spin is described in classi-cal physics. So the main purpose of this thesis is to understand symmetries, and then withsymmetries describe spin in classical physics.

6

3. THE POINT PARTICLE

3.1. THE NON-RELATIVISTIC ACTION FOR A FREE MASSIVE PARTICLE

We start by looking at the non-relativistic action, S, for a free massive particle and prove thatthe action is invariant under the non-relativistic version of the Poincaré group. That is, let usprove the invariance of the action for a free particle with inertial mass in systems where thevelocity, r ≡ ∣∣ ˙r

∣∣, is much smaller than the speed of light, c, under Galilean transformationsand translations. The action is defined as

S =∫

Ld t , (3.1)

and the Lagrangian, L, is defined as

L = T −U . (3.2)

Here T is the kinetic energy and U is the potential energy. For a free particle we have U = 0.Using (3.1) one gets the non-relativistic action for a free massive particle to be

S =∫

T d t = m

2

∫˙r 2d t . (3.3)

Here r is the position vector, and m is the inertial mass or simply mass.

It is important to keep in mind that there exist a relation between inertial mass and gravi-tational mass. Inertial mass is the mass parameter which gives a body its inertial resistanceto acceleration when it is exposed to a force, and gravitational mass is determined by thestrength of the force a body experiences while it is in a gravitational field. All experimentsconfirm the equivalence between the inertial mass and the gravitational mass, and AlbertEinstein developed the general relativity under the assumption that one cannot detect a dif-ference between inertial and gravitational mass through an experiment. [1]

3.2. THE GALILEAN TRANSFORMATIONS

The Galilean transformations are used to describe the correspondence between two frames,S = (t , x, y, z) and S′ = (t ′, x ′, y ′, z ′), in Newtonian physics, i.e. non-relativistic physics. Let oneof the frames be fixed and let the other move with a constant velocity, v , in the x-direction,where v is much smaller than the speed of light, i.e. v << c. At t0 = t ′0 = 0 are both of theframes at the origin, i.e. x0 = x ′

0 = 0.

7

Figure 3.1: Two coordinate systems.

The coordinates x, y, z, t in S′ as seen in S described through a Galilean transformation are [1]x ′ = x − v ty ′ = yz ′ = zt ′ = t

⇔

x ′ = x − vy ′ = yz ′ = z

. (3.4)

With this information one gets the action in the moving frame to be:

S′ = m

2

∫ tb

ta

˙r ′2d t = m

2

∫ tb

ta

(x ′, y ′, z ′)2 d t = m

2

∫ tb

ta

((x − v)2 + y2 + z2)d t =

= m

2

∫ tb

ta

(x2 −2xv + v2 + y2 + z2)d t =

= m

2

(∫ tb

ta

(x2 + y2 + z2)d t +

∫ tb

ta

(v2 −2xv

)d t

)=

= m

2

∫ tb

ta

˙r 2d t + mv

2(v (tb − ta)−2(x(tb)−x(ta))) = S +ζ .

(3.5)

Here ζ is a constant and is given by:

ζ= mv

2(v (tb − ta)−2(x(tb)−x(ta))) . (3.6)

Since ζ is fixed for constant endpoints, tb and ta , one can drop this term since it is not impor-tant in variational principles 1. Therefore the non-relativistic action (3.3) is invariant underthe Galilean transformations.

Note 1. The action (3.3) is also invariant under translations. This is proved later in section 5.1.

1Adding a constant to the action will give the same Lagrangian and therefore also the same equations of motion.

8

3.3. THE RELATIVISTIC ACTION FOR A FREE MASSIVE PARTICLE

For a system where the velocities are close to the speed of light the relevant transformationsare the Lorentz transformations [1]. Let us start by finding the action for such a system. Start-ing with the relativistic point particle, we define its action along a path being defined as pro-portional to the relativistic distance, d s, from two endpoints ta and tb

S =α∫ tb

ta

d s . (3.7)

Here α is a constant which is determined by the non-relativistic limit of this action. From themetric, ηµν, describing Minkowski space in special relativity, one can get an expression forthe relativistic distance. In this thesis, the Minkowski space will be given by 2

(gµν

)= (ηµν

)=−1 0 0 00 1 0 00 0 1 00 0 0 1

(3.8)

⇒ d s2 = d xµd xµ = d xµηµνd xν =−d t 2 +d x2 +d y2 +d z2 ⇒ (3.9)

⇒ S =α∫ tb

ta

√d xµd xµ = α

i

∫ tb

ta

i√

d xµd xµ =−iα∫ tb

ta

√−d xµd xµ =

=−iα∫ tb

ta

√d t 2 −d x2 −d y2 −d z2 =−iα

∫ tb

ta

√1−

(d x

d t

)2

−(

d y

d t

)2

−(

d z

d t

)2

d t =

=−iα∫ tb

ta

√1− ˙r 2d t =−iα

∫ tb

ta

√1− r 2d t .

(3.10)

In the non-relativistic limit the velocity, r ≡ ˙r , is much smaller than the speed of light, i.e.r << 1. Since r is small one can Taylor expand the action (3.10) 3

S ≈−iα∫ tb

ta

(1− 1

2r 2

)d t =−iα (tb − ta)+ iα

2

∫ tb

ta

r 2d t . (3.11)

The term −iα (tb − ta) is fixed for constant endpoints, tb and ta , and constant α. This meansthat one can drop this term since it is not important in variational principles. This gives thefollowing non-relativistic action

S = iα

2

∫ tb

ta

r 2d t = iα

2

∫ tb

ta

˙r 2d t . (3.12)

2We will always use Einstein summation and units such that the speed of light is equal to one, i.e. c = 1.3Taylor expansion is in appendix (A.1).

9

Comparing this action with the action (3.3) one obtains the relativistic action for a free mas-sive particle 4

S =−m∫ tb

ta

√−d xµd xµ =−m

∫ tb

ta

√−d xµ

dλ

d xµdλ

dλ . (3.13)

Here λ is any partameter of the particle’s path. If λ is time, t , one gets:

S =−m∫ tb

ta

√−x2d t . (3.14)

3.4. EQUATIONS OF MOTION FOR A FREE PARTICLE

Since one wants special relativity to hold for all velocities, one also wants the equations of mo-tion obtained from the relativistic action (3.14) to become the equations of motion obtainedfrom the non-relativistic action (3.3) at the low velocity limit. We know already that this isthe case since we have shown that the relativistic action (3.14) reduces to the non-relativisticaction (3.3) in the non-relativistic limit. However , it is instructive to see this explicitly at thelevel of the equations of motion.

The motion corresponds to a minimum of the action along a path, and by minimizing theaction one gets the Euler-Lagrange equations [1]

d

d t

∂L

∂qi− ∂L

∂qi= 0 . (3.15)

Here qi is the generalized coordinate and qi is the time derivative of qi .

Let us start with the equations of motion in the non-relativistic case. Using the non-relativisticaction (3.3), one gets the following equations of motion 5

mr j = 0 . (3.16)

Adding the three equations of motion for j = x, y, z one gets Newton’s second law when thesum of all forces, F , is zero, i.e. when there are no external forces acting on the particle

m ¨r = 0 ⇔ ¨r = 0 . (3.17)

For a relativistic particle the action is given by (3.13). If λ is time, one can see that the La-grangian for a free relativistic particle is 6

L =−m√

−x2 . (3.18)

4I.e. α=−i m.5Derivation is in appendix (A.2).6Comparing (3.14) with (3.1).

10

By using the Euler-Lagrange equations one will get the following equations of motion for afree relativistic particle 7

xα = (x x)

x2 xα (3.19)

xµ = (t , r ) ⇒

⇒ (0, ¨r ) =−˙r · ¨r

1− ˙r 2(1, ˙r ) .

(3.20)

In the non-relativistic limit is ˙r << 1 8

(0, ¨r ) =−˙r · ¨r

1− ˙r 2(1, ˙r ) →− 0 · ¨r

1− 02(1, ˙r ) = (0, 0)

¨r = 0 .(3.21)

Since this is the same as in the non-relativistic case, special relativity holds for low velocitiestoo.

4. THE LORENTZ GROUP

The Lorentz group is the group of transformations which transforms the four-vector, xµ ≡(t , x), under boosts and rotations. In the Lorentz group there exist six generators, three ofwhich come from boosts along an axis, and three from rotations around a fixed axis [2]. TheLorentz transformations 9 will be denoted as Λµν , and the four-vector under a Lorentz trans-formation is transformed in the following way

x ′µ =Λµνxν . (4.1)

Here x ′µ is the four-vector in the frame S’ as seen in the frame S, where S is a fixed frame andS’ is a frame which is either affected by a boost or a rotation (or both), i.e. S’ is affected by aLorentz transformation.

Let us take a closer look at the elements of the Lorentz group, and prove that the action (3.14)is invariant under Lorentz transformations.

4.1. BOOSTS

If the non fixed frame, S’, is affected by a boost, it is moving with a constant velocity, v , alongan axis 10. Unlike the Galilean transformation, the speed, v ≡ |v |, of S’ can now be close to the

7Derivation is in appendix (A.3).8 x << 1, y << 1, z << 19The transformations in the Lorentz group.

10See figure 3.1 for v being along the x-direction, i.e. the frame S’ being under a boost in the x-direction.

11

speed of light. The Lorentz transformation for a boost in the x-direction is given by

(Λµν

)=

γ −γβ 0 0−γβ γ 0 0

0 0 1 00 0 0 1

. (4.2)

Here γ is the Lorentz factor. β and γ are given by:β= v

c

γ= (1−β2)−1/2. (4.3)

Using (4.1) one finds that the four-vector in the boosted system is given by

x ′µ =

t ′

x ′

y ′

z ′

=

γ −γβ 0 0

−γβ γ 0 00 0 1 00 0 0 1

txyz

=

γt −γβx−γβt +γx

yz

=

γ

(t −βx

)γ

(−βt +x)

yz

⇒ (4.4)

⇒ x ′µ =

γ

(1−βx

)γ

(−β+ x)

yz

. (4.5)

With this information one can calculate the action using (3.14) for a free massive particle inthe boosted system

S′ =−m∫ √

−x ′µx ′µd t =−m

∫ √γ2

(1−βx

)2 −γ2(−β+ x

)2 − y2 − z2d t =

=−m∫ √

γ2(1−2βx +β2x2 −β2 +2βx − x2

)− y2 − z2d t =

=−m∫ √(

1−β2)−1 (

1−β2)(

1− x2)− y2 − z2d t =−m

∫ √1− x2 − y2 − z2d t =

=−m∫ √

−xµxµd t = S .

(4.6)

So the action for a free massive particle is invariant under boosts.

The Lorentz transformation for a boost in an arbitrary direction is given by

(Λµν

)=

γ −γβx −γβy −γβz

−γβx 1+ (γ−1)β2

x

β2 (γ−1)βxβy

β2 (γ−1)βxβz

β2

−γβy (γ−1)βyβx

β2 1+ (γ−1)β2

y

β2 (γ−1)βyβz

β2

−γβz (γ−1)βzβx

β2 (γ−1)βzβy

β2 1+ (γ−1)β2

z

β2

, (4.7)

12

where βi is given by

βi = vi

c, i = x, y or z . (4.8)

The action (3.14) is invariant under this Lorentz transformation too.

Note 2. The Lorentz transformation (4.7) reduces to the Lorentz transformation (4.2) if thevelocity, v , is in the x-direction.

4.2. ROTATIONS AROUND A FIXED AXIS

If the frame S’ has been rotated relative to the frame S with an angle, θ, around one axis, theframe S’ has been transformed under a rotation.

Figure 4.1: A rotation around the z axis.

For a rotation around the z axis with an angle θ (constant, i.e. θ is not time dependent) theLorentz transformation is given by

(Λµν

)=

1 0 0 00 cosθ −sinθ 00 sinθ cosθ 00 0 0 1

. (4.9)

Using (4.1) one finds that the four-vector in the rotated system is given by

x ′µ =

t ′

x ′

y ′

z ′

=

1 0 0 00 cosθ −sinθ 00 sinθ cosθ 00 0 0 1

txyz

=

t

x cosθ− y sinθx sinθ+ y cosθ

z

⇒ (4.10)

⇒ x ′µ =

1

x cosθ− y sinθx sinθ+ y cosθ

z

. (4.11)

13

With this information one can calculate the action using (3.14) for a free massive particle inthe rotated system

S′ =−m∫ √

−x ′µx ′µd t =−m

∫ √1− (

x cosθ− y sinθ)2 − (

x sinθ+ y cosθ)2 − z2d t =

=−m∫ √

1− x2 cos2θ+2x y cosθ sinθ− y2 sin2θ− x2 sin2θ−2x y cosθ sinθ− y2 cos2θ− z2d t =

=−m∫ √

1− x2(cos2θ+ sin2θ

)− y2(sin2θ+cos2θ

)− z2d t =

=−m∫ √

1− x2 − y2 − z2d t =−m∫ √

−xµxµd t = S .

(4.12)

So the action for a free massive particle is invariant under rotations.

A more general rotation in R3 is described by [3]

Λµν =

1 0 0 00 R11 R12 R13

0 R21 R22 R23

0 R31 R32 R33

. (4.13)

Under the rotation (4.13) is the action (3.14) again invariant.

5. THE POINCARÉ GROUP

We have proven the invariance of the action (3.14) under Lorentz transformations. Now, letus prove the invariance of the action (3.14) under the transformations in the Poincaré group.

The Poincaré group is the full group of isometries in the Minkowski space-time. It has 10generators: 3 from rotations around a fixed body, 3 from boosts along an axis and 4 fromtranslations [4]. Rotations and boosts together make up the Lorentz group. The Poincarégroup is thus the Lorentz group and the group of translations.

Let us take a closer look at the elements of the Poincaré group, and prove the invariance of theaction (3.14) under those transformations. Since the action (3.14) has already been proven tobe invariant under the Lorentz group, we only have to prove that the action (3.14) is invariantunder translations.

5.1. TRANSLATIONS

A translation is a displacement in both time and space:

14

Figure 5.1: A spatial translation (translation in space).

Mathematically a translation means adding a constant, aµ, to the space-time as follows

xµ→ xµ+aµ . (5.1)

Since aµ is a constant, it is easy to prove that the action for a free massive particle is invariantunder translations

S =−m∫ √

−xµxµd t →−m∫ √

− d

d t(xµ+aµ)

d

d t

(xµ+aµ

)d t =

=−m∫ √

−xµxµd t = S .

(5.2)

Therefore the action (3.14) is invariant under translations, and thus the action (3.14) is invari-ant under all Poincaré transformations 11.

Note 3. The non-relativistic action for a free massive particle is also invariant under transla-tions {

r → r + at → t +a0

. (5.3)

Here both a and a0 are constants.

S = m

2

∫˙r 2d t → m

2

∫ (d

d t(r + a)

)2

d t = m

2

∫˙r 2d t = S . (5.4)

6. NOETHER’S THEOREM

We have proven that the action (3.14) is invariant under the Poincaré transformations. Nowlet us find a quantity which will describe the spin of a particle if one quantizes it. To do thiswe need to have knowledge of Noether’s theorem in field theory. But first, let us study thenon-field version of Noether’s theorem.11The transformations within the Poincaré Group.

15

6.1. NOETHER’S THEOREM

Symmetries of classical theories correspond to conserved quantities. By using Noether’s the-orem one can find these quantities. Here we will follow [5], and exchange an arbitrary func-tion f ≡ f (x, y, y ′) with the Lagrangian L ≡ L(t , q, q) 12.

Assume that the Lagrangian is invariant under the transformation:{t → t = t +δtqk → qk = qk +δqk

. (6.1)

Here t is the time parameter and the qk ’s are the generalized coordinates, k ∈ {1, ...,n}. As-sume further that this transformation has infinitesimal generators ξ and η, i.e.:{

δx = εξδqk = εηk

. (6.2)

Here ε is a small constant. Then the following Noether current, j , is conserved 13

j = ∂L

∂qkηk +

(L− ∂L

∂qkqk

)ξ= pkηk −

(pk qk −L

)ξ= pη−Hξ . (6.3)

Here pk is the momentum associated with k th generalized coordinate, qk , and H is the Hamil-tonian.

6.2. NOETHER’S THEOREM IN FIELD THEORY

In principle the transition to field theory is achieved by letting the discrete generalized co-ordinates become continuous, and instead of using the time parameter, t , we use the four-vector, xµ. The continuous generalized coordinates are the fieldsφ. In field theory we use theLagrangian density 14, L ≡ L(xµ,φ,∂µφ), instead of the Lagrangian, L, where ∂µ denotes thederivative with respect to the four-vector, xµ.

Assume that the Lagrangian density is invariant under the transformation{xµ→ xµ = xµ+δxµ

φk (x) → φk (x) =φk (x)+δφk (x). (6.4)

Assume further that this transformation has infinitesimal generators ξ and η, i.e.{δxµ = εξµδφk (x) = εηk (x)

. (6.5)

12This can be done since the Lagrangian minimizes the action, i.e. since the Lagrangian is an extremal of theaction.

13Here the relation between momentum and Lagrangian, and the Legendre transform from a Lagrangian to aHamiltonian, have been used.

14Integrating the Lagrangian density, L , over xµ will give the Lagrangian, L.

16

Here ε is a small constant. Then the following Noether current, jµ, is conserved [4]

jµ = ∂L∂(∂µφk )

ηk −T µνξν

T µν = ∂L∂(∂µφk )

∂νφk − gµνL .(6.6)

Here T µν is the stress energy tensor, which is roughly described in the figure (6.1):

Figure 6.1: The stress energy tensor and explanations of its terms.

Moreover, in field theory we define the Noether charge, Q, which is also conserved

Q =∫

d 3x j 0 . (6.7)

7. PAULI-LUBANSKI SPIN

7.1. PAULI-LUBANSKI SPIN

We want to find a classical quantity which is invariant under the Poincaré transformations,and which describes the spin of a particle if one quantizes it. Let us start with defining thetotal angular momentum, Jνρ , as the Noether charge for the modified Noether current, J µ,νρ ,under a Lorentz transformation.

Let us consider the Lagrangian density being invariant under a Lorentz transformation

xµ→ x ′µ =Λµρxρ = (δµρ−ωµρ+O (ω2)

)xρ ≈ δµρxρ−ωµρxρ = xµ−ωµρxρ . (7.1)

Hereωµρ is infinitesimal. In appendix (A.4) it is proven thatωµρ is antisymmetric (in indices).

Note 4. If a tensor is symmetric one can exchange two indices without a change in sign, if atensor is antisymmetric one has to change the sign upon exchange of two indices{

Aµν = Aνµ , if Aµν is a symmetric tensor.Aµν =−Aνµ , if Aµν is an antisymmetric tensor.

(7.2)

Using Noether’s theorem in field theory, one finds that the following Noether current, jµ, isconserved under the Lorentz transformation (7.1)

jµ =−T µν(−ωνρxρ

)= 1

2

(T µνωνρxρ+T µνωνρxρ

)= 1

2

(T µνωνρxρ−T µνωρνxρ

)== 1

2ωνρ

(T µνxρ−T µρxν

).

(7.3)

17

Since 12ωνρ is a constant, one can neglect this and get the following modified Noether current

J µ,νρ = T µνxρ−T µρxν . (7.4)

This modified Noether current, J µ,νρ , is defined as the total angular momentum density, andthe Noether charge, Jνρ , for J µ,νρ is defined as the total angular momentum

Jνρ =∫

d 3xJ 0,νρ . (7.5)

So the total angular momentum is constructed to be invariant under Lorentz transforma-tions. However, it is not invariant under translations. Using a translation in space-time (5.1)one gets

J 0,νρ = xρT 0ν−xνT 0ρ → (xρ+aρ

)T 0ν− (

xν+aν)

T 0ρ == xρT 0ν−xνT 0ρ+aρT 0ν−aνT 0ρ = J 0,νρ +aρT 0ν−aνT 0ρ 6= J 0,νρ ⇒ (7.6)

⇒ Jνρ → Jνρ+aρPν−aνPρ 6= Jνρ . (7.7)

The momentum, Pµ, is defined as

Pµ =∫

d 3xT 0µ . (7.8)

So the total angular momentum, Jνρ , is not invariant under translations. Therefore one in-troduce the Pauli-Lubanski spin vector, Wµ, which is invariant under translations. The Pauli-Lubanski spin vector is defined as [6]

Wµ := 1

2εµνρσ JνρPσ . (7.9)

Here εµνρσ is the completely antisymmetric (in indices) Levi-Civita symbol defined as:

εi j kl :=

1 , if i , j ,k, l is an even permutation of (1,2,3,4).−1 , if i , j ,k, l is an odd permutation of (1,2,3,4).0 , otherwise.

(7.10)

A proof that the Pauli-Lubanksi spin vector is invariant under a translation is in the appendix(A.5) 15. In the proof the fact that the total angular momentum is antisymmetric (in indices)has been used

Jνρ =−Jρν . (7.11)

In appendix (A.5) W0 is found. In the same way can one find the rest of the components ofWµ to be:

W0 = J 12P 3 + J 31P 2 + J 23P 1

W1 = J 20P 3 + J 03P 2 + J 23P 0

W2 = J 01P 3 + J 30P 1 + J 13P 0

W3 = J 10P 2 + J 02P 1 + J 21P 0

. (7.12)

15It is only proven for W0, but the proof is the same for the other components of Wµ.

18

The momentum in the rest frame of a massive particle is given by

Pµ = (E0, 0) = (m, 0) . (7.13)

Here E0 is the rest energy of the particle and m is the rest mass of the particle. Using (7.12)and (7.13) one gets the Pauli-Lubanski spin in the rest frame of a massive particle to be

Wµ = (o,m J 23,m J 13,m J 21) = (0,m Jx ,m Jy ,m Jz ) = (0,m J ) . (7.14)

So the Pauli-Lubanski spin vector is proportional to the total angular momentum, J , in therest frame of a particle.

7.2. QUANTIZATION OF THE PAULI-LUBANSKI SPIN

We want to quantize the Casimir operator, W 2, for the Pauli-Lubanski spin and study if W 2

has spin eigenvalues. To do this we need to first define a Casimir operator for a quantity. Theformal definition of a Casimir operator is a bit involved. For our purpose we will define aCasimir operator as the following:

Definition 1. A Casimir operator, A2, for the set of operators Aα, α ∈ {1, ...,n} is defined as

A2 = AαAα . (7.15)

The Casimir operator, A2, commutes with every Aα, and moreover A2 is invariant under thesame transformations as Aα is invariant under.

Since the total Pauli-Lubanski spin,∑

n Wnµ, is conserved, so is the Casimir operator, W 2i , for

the Pauli-Lubanski spin, Wiµ, of particle i. So one can look in the center of mass (CoM) frameto find W 2

i for a massive particle. Using (7.14) one gets

⇒W 2i =W µ

i Wiµ =−0+m2 J 2i = m2 J 2

i . (7.16)

In the rest frame of a particle is

L = 0 ⇒ J = L+ S = S . (7.17)

Here L is the orbital angular momentum and S is the spin angular momentum. Using (7.17)one finds that W 2

i in quantum field theory has spin eigenvalues (quantum numbers)

⇒W 2i Ψ= m2si (si +1)Ψ . (7.18)

HereΨ represents the physical state and si is the quantum spin number of particle i.

Now we have seen that the Casimir operator W 2 describes the spin of a massive particlein quantum field theory, and since the Pauli-Lubanski spin is invariant under the PoincaréTransformations, and is proportional to the angular momentum in the rest frame of the par-ticle, the Pauli-Lubanski spin is a good label for spin in classical field theory.

19

8. SMOOTH SPINNING PARTICLES

Now we have seen a quantity which describes the spin of a particle in classical field theory.Let us move on to look at the action for a smooth spinning particle, i.e. let us look at the actionfor a particle whose quantized action introduces spin. One way to do this is by using the ac-tion for a bosonic string, and compactify one string dimension together with one embeddingdimension.

8.1. THE ACTION OF A BOSONIC STRING

The actions we will use for a bosonic string are the Nambu-Goto action and the classicallyequivalent Polyakov action. Let us study these actions before we perform a compactification.

Let us start by looking at the action for a bosonic string

S = T A = T∫

d A . (8.1)

Here T is the tension, and A is the total area of the world sheet spanned by the string 16. Thearea element, d A, of the world sheet spanned by a string can be written as

d A = d 2ξp−γ . (8.2)

Here γ is the determinant of the induced metric, γmn , on the string

γ= det(γmn)

γmn = ∂m X µ∂n Xµ

m,n ∈ {1,2} µ ∈ {0,1, ...,D −1,D} .

(8.3)

ξ = (τ,σ) is the parameters which span the world sheet of the string. σ is spacelike and τ istimelike. [7]

d 2ξ= dσdτ

X ′ ≡ d X

dσ, X ≡ d X

dτ.

(8.4)

The functions X µ(τ,σ) determines the shape of the world sheet. D is the number of spatialdimensions. 17. X (τ,σ) are the embedded coordinates of the world sheet

X (τ,σ) = (X 0(τ,σ), X 1(τ,σ), ..., X D−1(τ,σ), X D (τ,σ)

). (8.5)

By using (8.2) one gets the action for a bosonic string to be

S = T∫

d A = T∫

d 2ξp−γ . (8.6)

16The world sheet of a string describes how the string is embedded in spacetime. For further details see [7].17In a quantized theory requires bosonic strings 25 spatial dimensions and one time dimension, i.e. D = 25.

20

This action is called the Nambu-Goto action. Sometimes, one uses the Polyakov action in-stead, which is classically equivalent to the Nambu-Goto action. The Polyakov action for abosonic string is given by [8]

S = T

2

∫d 2ξ

p−g g mnγmn = T

2

∫d 2ξ

p−g g mn∂m X µ∂n Xµ . (8.7)

Where gmn is the world sheet metric

g = det(gmn) . (8.8)

Both the Nambu-Goto action and the Polyakov action are invariant under the Poincaré trans-formations and under reparametrizations [7]

X µ→ΛµνX ν+aµ

X µ→ X µ+Ξα∂αX µ

gmn = gmn +Ξα∂αgmn +∂mΞαgαn +∂nΞ

αgmα

, (8.9)

where Ξα are the parametrizations.

8.2. THE ACTION OF A RIGID PARTICLE

Let us start by using the Polyakov action and perform a compactification, and then do thesame with the Nambu-Goto action. We will study the two resulting actions to see what differsbetween them.

The embedded coordinates of a particle depend only on τ. So if one wants to go from theaction for a bosonic string to the action for a bosonic particle, one has to reduce X µ as follows

X µ(τ,σ) → X µ(τ) . (8.10)

To do this we will follow [8]. By using the reparametrization invariance (8.9), the followinggauge transformation can be fixed [8]

X D =σ σ ∈ [0,2πR] . (8.11)

Making the following two ansatzesX µ(τ,σ) = xµ(τ)+Rσyµ(τ) µ ∈ [0,1, ...,D −1]

g mn =[−1 0

0 e2

]⇒ gmn =

[−1 00 e−2

] , (8.12)

where R is called the compactification radius. The gauge (8.11) and the ansatz of X µ(τ,σ)(8.12) compactify one string dimension (X D ) together with one embedding dimension (σ).We now integrate over σ.

21

Using the Polyakov action for a bosonic string (8.7) and only keeping terms up to order R2,the resulting action for a bosonic particle will be 18

S =−πRT∫

dτ

e

[x2 +R2 (

2πx y −e2 y2)−e2] . (8.13)

Using the y- and e-field equations, one can eliminate y and e in the action (8.13) and get (upto boundary terms) [8]

S = 2πRT∫

dτ√−x2

[1− π2R2

2x2

(∂t

xp−x2

)2]=

=∫

dτ√−x2

[2πRT +π3R3T

1

−x2

(∂t

xp−x2

)2].

(8.14)

The extrinsic curvature, k, in the special case when one only has one curve is given by

k :=√

1

−x2

(∂t

xp−x2

)2

. (8.15)

Moreover, let us define the mass, m, and the rigidity parameter, α, as{m ≡ 2πRTα≡ (π3R3T )−1 . (8.16)

Using the definitions (8.16) and the extrinsic curvature (8.15) one gets the action

S =∫

dτ√

−x2(m +α−1k2) . (8.17)

This action is said to be the action for type 2 rigid particles, since its rigidity term 19 is propor-tional to k2.

Let us now instead integrate over σ using the Nambu-Goto action (8.6) 20. The resulting ac-tion will be [8]

S =−πRT∫

dτ√

−x2

(1+ R2

2x2

[x2 y2 +2πx y − (

x y)2

])=

=−1

2

∫dτ

√−x2

(2πRT + π3R3T

x2

[x2 y2

π2 +2x y

π−

(x y

)2

π2

]).

(8.18)

Using the definitions at (8.16) and let: xµpπ→ xµ

yµpπ→ yµ

, (8.19)

18Integration is in appendix (A.6).19The term including the rigidity parameter, α.20Using the same gauge (8.11) and the same ansatzes (8.12).

22

the action at (8.18) becomes

S =−1

2

∫dτ

√−x2

(m + 1

αx2

[x2 y2 +2x y − (

x y)2

]). (8.20)

This action is invariant under the following gauge transformation [8]

yµ(τ) → yµ(τ)+ v(τ)xµ(τ) . (8.21)

Where v(τ) are arbitrary functions. Since the Nambu-Goto action (8.6) is classically equiv-alent to the Polyakov action (8.7), so is the action (8.20) classically equivalent to the action(8.17).

The y-field equations are given by

yµ =xµx2 ⇒

⇒ x y = x x

x2 .(8.22)

Comparing this to the equation of motion for a free relativistic massive particle (3.19), it canbe seen that x y is the part which converges to zero in the non-relativistic limit 21. One will getback the action (8.17) from the action (8.20) by letting x y → 0 and eliminating the y by usingthe y field equations (8.22) [9]. So the action (8.17) is therefore the non-relativistic action fora rigid particle, while the action (8.20) is the relativistic action for a rigid particle.

8.3. CLASSICAL DYNAMICS OF A RIGID PARTICLE

We have seen that the action (8.20) is that of a relativistic rigid particle. Now let us find themomenta pµ, which are associated with xµ, and Pµ, which are associated with yµ, using thisaction. With pµ and Pµ we can find generators to some of the symmetries of the action (8.20).The generators and the Poisson bracket for these generators will be crucial when we laterquantize the action (8.20) to see that it describes a spinning particle. The momenta, pµ andPµ, are given by 22

pµ = dL

d xµ= mxµ

2p−x2

− 1

2α(−x2)3/2

[x2 y2xµ−2(x y)xµ+2x2 yµ+ (x y)2xµ−2(x y)x2 yµ

]Pµ = dL

d yµ= xµ

αp−x2

.

(8.23)

The momenta, pµ and Pµ, are conserved under translations{xµ→ xµ+aµyµ→ yµ+aµ

, (8.24)

21See (3.21).22Derivation is in appendix (A.7) and (A.8).

23

and the total angular momentum, Jµ, is conserved under rotations{xµ→ xν+Λµνxνyµ→ yµ+Λµνyν

, Λµν =−Λνµ . (8.25)

The total angular momentum, Jµν, is defined as

Jµν := Mµν+Sµν . (8.26)

Here Mµν is the angular momentum tensor, and Sµν is the spin tensor. They are defined asMµν = p[µxν] = 1

2

(pµxν−pνxµ

)Sµν = P[µyν] = 1

2

(Pµyν−Pνyµ

) . (8.27)

The canonical variables pµ and Pµ will have to satisfy the following two constraints [9]{φ1 = P 2 +α−2 = 0φ2 = P p + m

2α −SµνSµν = 0. (8.28)

φ1 is the generator of the gauge transformation (8.21), andφ2 is the generator of the world linereparametrizations (8.9), i.e. the constraints (8.28) are generators to some of the symmetriesof the action (8.20). One gets the Poisson bracket for φ1 and φ2 to be 23

{φ1,φ2

}= 0 . (8.29)

For a reparametrization invariant theory like this, the Hamiltonian, H , may be written as alinear combination of φ1 and φ2 [9]

H = v1φ1 + v2φ2 . (8.30)

The time-evolution of Sµν and Jµν are 24

Sµν = ∂Sµν

∂t +{Sµν, H

}= {Sµν, H

}= v2P[µpν] = v22

(Pµpν−Pνpµ

)Jµν = ∂Jµν

∂t +{Jµν, H

}= {Jµν, H

}= 0

. (8.31)

8.4. QUANTIZATION OF A RIGID PARTICLE’S MOMENTA AND SPIN

Now let us quantize the action (8.20). To do this, we need two operators, φ1 and φ2, to com-mutate with each other. There φ1 and φ2 is of similar form as the generators φ1 and φ2 inthe classical case. This is done by imposing commutation relations between xµ, pµ, yµ andPµ, such that φ1 commutates with φ2. After the quantization, we will look at the quantizedPauli-Lubanski tensor, Wµρσ, to see that the quantized version of the action (8.20) is that of a

23Derivation is in appendix (A.9).24Derivation is in appendix (A.10) and (A.11).

24

spinning particle.

Let us impose the following commutation relations between x, p, y and P{ [xµ, pν

]= [yµ,Pν

]= i gµν[xµ, yν

]= [xµ,Pν

]= [yµ, pν

]= [pµ,Pν

]= 0. (8.32)

Moreover, demand the following{φ1Ψ≡ (

P 2 +α−2)Ψ= 0

φ2Ψ≡ (P p + m

2α −SµνSµν)Ψ= 0

. (8.33)

Ψ represent the physical states. This will give the following relation 25

[φ1,φ2

]Ψ= 0 . (8.34)

So the demand (8.33) is consistent. When comparing (8.33) and (8.34) with (8.28) and (8.29)one can see that this is a suitable quantization.

The quantized Pauli-Lubanski tensor is given by [9]

Wµρσ = 1√−p2

p[µ Jρσ] = 1√−p2

p[µSρσ] = 1

2√

−p2

(pµSρσ−pσSρµ

). (8.35)

Using Heisenberg’s equation of motion we see that the momentum and the Pauli-Lubanskispin tensor are both conserved 26

{ [pµ, H

]= 0 ⇒ pµ = 0[Wµρσ, H

]= 0 ⇒ Wµρσ = 0, (8.36)

so the Poincaré group is still a symmetry, even in this quantized theory. Moreover, the Casimiroperator W 2 has integer eigenvalues if the wave functionΨ=Ψ(x, y) is single valued.

This shows that the action (8.20) will be an action for a spinning particle if one quantizes it.Therefore, the action (8.20) is an action for a smooth spinning particle.

25Derivation is in appendix (A.12).26Derivation is in appendix (A.13) and (A.14).

25

9. CONCLUSION

To illustrate the importance of symmetries in physical systems, we have shown that the ac-tion, S, for a point particle is invariant under the Poincaré group and its non-relativistic ver-sion. That is, we proved that the non-relativistic action for a point particle is invariant underGalilean transformations and translations.

The relativistic action was derived starting from the action for a point particle being definedas the relativistic distance, d s, between two times, tb and ta . By demanding that the non-relativistic limit of this relativistic action were to be the same as the non-relativistic action fora point particle, we found that the relativistic action for a point particle is given by

S =−m∫ tb

ta

√−x2d t . (9.1)

Here we already knew that by taking the low velocity limit of the equations of motion asso-ciated with the relativistic action (9.1) one would get the equations of motion for the non-relativistic action. However, this was done regardless since it is instructive to show that theequations of motion associated with the action (9.1) reduces to the equations of motion as-sociated with the non-relativistic action. As it turned out, this limit was important when welooked at the non-relativistic limit of the action for a rigid particle.

Noether’s theorem plays an important role in symmetries of physical systems. Under everysymmetry there exists a conserved quantity, and by using Noether’s theorem one could findthese quantities. Since these quantities are conserved they are interesting to study in ex-periments. Examples of quantities that are conserved are momentum, energy, and electriccharge. Let us look at an example of a quantity which is conserved under rotations.

Example 1. Let us consider a Lagrangian, L, that is invariant under a small rotation aroundthe nth axis, i.e.:

r → δθn × r . (9.2)

Here δθ is a small angle (constant). From Noether’s theorem one could then find that the con-served quantity, j , under this transformation is

j = (n × r ) ·∇q L = n · (p × r)= n · L . (9.3)

Where L is the angular momentum. So under a small rotation around the nth axis the nth

component of L is conserved.

We made the transition from classical theory to field theory to study Noether’s theorem infield theory. We found that the following Noether current, jµ, and Noether charge, Q, is con-served if there exist a symmetry of the Lagrangian density, L , under a transformation in both

26

the four-vector, xµ, and the fields, φ

jµ = ∂L∂(∂µφk )

ηk −T µνξν

Q =∫

d 3x j 0 .

(9.4)

T µν is the stress-energy tensor, ηk are the infinitesimal generators for the transformation inthe fields, φ, and ξµ is the infinitesimal generator for the four-vector, xµ. We studied thisNoether current, jµ, for a Lagrangian density being invariant under a Lorentz transforma-tion. After dropping some unimportant constants, we ended up with a modified Noethercurrent, J µ,νρ , which was defined as the total angular momentum density. The total angularmomentum tensor, Jνρ , was in field theory defined as the Noether charge for J µ,νρ . In otherwords, the total angular momentum was constructed to be invariant under Lorentz transfor-mations. However, it was shown that it was not invariant under translations and thereforenot under all Poincaré transformations. Thereafter, the Pauli-Lubanski spin vector, Wµ, wasintroduced

Wµ := 1

2εµνρσ JνρPσ . (9.5)

Here Pσ is the momentum, and εµνρσ is the Levi-Civita symbol. It was shown that the Pauli-Lubanski spin is invariant under the Poincaré group and that the Pauli-Lubanski spin wasproportional to the total angular momentum in the rest frame of a particle. We found thatthe Casimir operator, W 2, for the Pauli-Lubanski spin for a massive particle in quantum fieldtheory has the following eigenvalues (quantum numbers)

⇒W 2Ψ= m2s (s +1)Ψ . (9.6)

Ψ represents the physical states, and s is the spin quantum number. Since the Casimir op-erator, W 2, for the Pauli-Lubanski spin describes the spin of a massive particle in quantumfield theory, and since it is proportional to the total angular momentum in the rest frame of aparticle we would like the Pauli-Lubanski spin (9.5) to describe the spin of a massive particlein classical field theory.

In this thesis we have studied the Casimir operator for the Pauli-Lubanski spin vector. How-ever, Casimir operators has in general a big importance in physics. Let us take a look at twoimportant examples of Casimir operators.

Example 2. The Casimir operator, P 2i , for momentum, Pi , of particle i is very often used in

quantum field theory, e.g. when one calculates the scattering amplitude for a process.

Since the total momentum,∑

n Pµn , of the system is conserved, the Casimir operator, P 2

i , formomentum, Pi , of particle i will be conserved. Therefore one can look in the CoM frame fo findP 2

i for a massive particle

P 2i = Pµ

i Piµ =−E 2i + 02 =−

(√m2

i + 02)2

=−m2i . (9.7)

27

Example 3. In quantum physics one often uses the Casimir operator, L2, for orbital angularmomentum, L. L2 commutes with every component of L, and has the following eigenvalues

L2Ψ= l (l +1)Ψ⇒⇒ LΨ=

√l (l +1)Ψ .

(9.8)

Here Ψ represents the physical states, and l is the quantum number for orbital angular mo-mentum.

Next, it was considered the action for a classical bosonic string. Using the reparametrizationinvariance a gauge was fixed. This gauge together with an ansatz which was made, compacti-fied one string dimension together with one embedding dimension. The resulting action wasan action for a bosonic rigid particle

S =−1

2

∫dτ

√−x2

(m + 1

αx2

[x2 y2 +2x y − (

x y)2

]). (9.9)

Where m is the mass parameter, and α is the rigidity. It was found that this action in factdescribes a relativistic bosonic rigid particle since it reduces to the following action in the lowvelocity limit

S =∫

dτ√−x2(m +α−1k2) . (9.10)

Here k is the extrinsic curvature. It was seen that the quantized version of the action (9.9)is the action for a spinning particle, and therefore is the action (9.9) an action for a smoothspinning particle. It is important to remember that the action (9.9) only describes particleswith integer spin.

Since the action (9.9) only describes particles with integer spin, one would want to find anaction for smooth spinning particles with half-integer spin. This could be done by studyingsupersymmetry (SUSY), which is an extension to the Poincaré group. So in a future thesisone could study SUSY and supersymmetrize the action (9.9) to get an action for a smoothspinning particle with half-integer spin. In principle this is done by introducing a fermioniccoordinate, θ, on the world line such that 27

S =∫

dτL(τ) → S =∫

dτdθL(τ,θ) . (9.11)

The main purpose of this thesis was to understand symmetries, and then with symmetriesdescribe spin in classical physics. We gained knowledge of the importance of symmetry op-erations by proving that the action for a point particle is invariant under the Poincaré trans-formations and its non-relativistic version. Moreover did we look at the conserved quantitieswhich exist under a symmetry operation by studying Noether’s theorem. We used Noether’stheorem to describe the Pauli-Lubanski spin vector, which we quantized to see that it de-scribes the spin of a particle. Spin has no real classical meaning, but we did however see

27The world line is the path the particle travels in spacetime.

28

that if you want the total angular momentum to be invariant under the Poincaré transforma-tions one has to introduce spin even in the classical theory. Finally did we find an action fora smooth spinning particle with integer spin. To describe a smooth spinning particle withhalf integer spin one can study SUSY and supersymmetrize the action we found for a smoothspinning particle with half integer spin. So a main purpose for a future work could be tostudy SUSY and see how it is an extension to the Poincaré group, and then supersymmetrizethe action (9.9).

REFERENCES

[1] W. Rindler, "Special Relativity", 1960, Oliver and Boyd LTD., Edinburgh

[2] M. Srednicki, "Quantum Field Theory", 2007, Cambridge University Press, University ofCalifornia

[3] J. Minahan, "Joe’s Relatively Small Book of Special Relativity"

[4] S. Weinberg, "The Quantum Theory of Fields, Volume 1: Foundations", 1995, CambridgeUniversity Press, University of Texas

[5] B. van Brunt, "The Calculus of Variations", 2006, Springer, USA

[6] C. Itzykson, J-B. Zuber, "Quantum Field Theory", 2005, McGraw-Hill Inc., New York

[7] L. Brink, M. Henneaux, "Principles of String Theory", 1988, Plenum Press, New York

[8] J.Grundberg, J.Isberg, U.Lindström, H.Nordström, "On Smooth Particles and Strings", 31May 1989, University of Stockholm, http://inspirehep.net/record/287116

[9] J.Isberg, U.Lindström, H.Nordström, "Canonical Quantization of a Rigid Particle", 1989,University of Stockholm, http://inspirehep.net/record/288425

29

A.DERIVATIONS

f (x) =∞∑

n=0

f (n)

n!xn = f (0)

0!x0 + f ′(0)

1!x + f ′′(0)

2!x2 +O (x3) = [x << 1] ≈

≈√

1−x2∣∣∣

x=0+ 1

2(−2x)

1p1−x2

∣∣∣∣x=0

x −

( p1−x2− x2p

1−x2

1−x2

∣∣∣∣∣x=0

)2

x2 = 1− x2

2

(A.1)

L = m

2˙r 2 ⇒

⇒ d

d t

∂L

∂r j− ∂L

∂r j= 0

md

d tr j = 0

r j = 0

(A.2)

0 = d

d t

∂L

∂xα− ∂L

∂xα=−mc

d

d t

2xα

2√

xµxµ= 0

d

d t

xα√xµxµ

=xα

√xµxµ− xα 2xν xν

2p

xµ xµ

xµxµ= xαxµxµ− xαxνxν

(xµxµ)3/2= 0

xα = xνxνxµxµ

xα = (x x)

x2 xα

(A.3)

gµν =ΛαµΛαν =ΛαµgαβΛβν

(7.1)≈ (δαµ−ωαµ

)gαβ

(δβν−ωβν

)=

= (δαµgαβ−ωαµgαβ

)(δβν−ωβν

)=

= δαµgαβδβν−δαµgαβω

βν−ωαµgαβδ

βν+ωαµgαβω

βν =

= gµβδβν− gµβω

βν−ωβµδβν+O (ω2) ≈ gµν−ωµν−ωνµ

0 =−ωµν−ωνµωµν =−ωνµ

(A.4)

30

W0 = 1

2ε0νρσ JνρPσ =

= 1

2

(ε0123 J 12P 3 +ε0312 J 31P 2 +ε0231 J 23P 1 −ε0213 J 21P 3 −ε0132 J 13P 2 −ε0321 J 32P 1)=

= 1

2

(J 12P 3 + J 31P 2 + J 23P 1 − J 21P 3 − J 13P 2 − J 32P 1)=

= 1

2

((J 12 − J 21)P 3 + (

J 31 − J 13)P 2 + (J 23 − J 32)P 1) (7.11)= J 12P 3 + J 31P 2 + J 23P 1 (7.7)→

→ J 12P 3 +a2P 1P 3 −a1P 2P 3 + J 31P 2 +a1P 3P 2 −a3P 1P 2 + J 23P 1 +a3P 2P 1−−a2P 3P 1 =

[P i P j = P j P i

]=

= J 12P 3 + J 31P 2 + J 23P 1 +a2P 1P 3 −a1P 2P 3 +a1P 2P 3 −a3P 1P 2 +a3P 1P 2−−a2P 1P 3 = J 12P 3 + J 31P 2 + J 23P 1 =W0

(A.5)

S = T

2

∫d 2ξ

p−g g mn∂m X µ∂n Xµ(7.10)=

= T

2

∫d 2ξ

p−g g mn (∂m

(xµ(τ)+Rσyµ(τ)

)∂n

(xµ(τ)+Rσyµ(τ)

)+∂mσ∂nσ)=

= T

2

∫d 2ξ

p−g g mn (∂m xµ(τ)∂n xµ(τ)+R∂m xµ(τ)∂n

(σyµ(τ)

)++ R∂m

(σyµ(τ)

)∂n xµ(τ)+R2∂m

(σyµ(τ)

)∂n

(σyµ(τ)

)+∂mσ∂nσ)=

= T

2

∫d 2ξ

p−g(g mn∂m xµ(τ)∂n xµ(τ)+2Rg mn∂m xµ(τ)∂n

(yµ(τ)σ

)++ R2g mn∂m

(yµ(τ)σ

)∂n

(yµ(τ)σ

)+ g mn∂mσ∂nσ) (7.11)=

= T

2

∫dτ

√e−2

∫ 2πR

0dσ

(−x2(τ)−2Rx(τ)y(τ)σ+R2 (−y2(τ)σ2 +e2 y2(τ))+e2)=

=−T

2

∫dτ

e

(2πRx2 +4π2R3x y +R2

(8

3π3R3 y2 −2πRe2 y2

)−2πRe2

)=

=−πRTdτ

e

(x2 +2πR2x y + 4

3π3R4 y2 −R2e2 y2 −e2

)=

=−πRT∫

dτ

e

(x2 +R2 (

2πx y −e2 y2)−e2 +O (R4))

(A.6)

31

pµ = dL

d xµ=−1

2

d

d xµ

(√−x2

(m + 1

αx2

[x2 y2 +2x y − (

x y)2

]))=

=−1

2

(m

d

d xµ

√−x2 − d

d xµ

( p−x2

α(−x2

) [x2 y2 +2x y − (

x y)2

]))=

=−1

2

(m

1

2

1p−x2

(−δαµ xα− xαgαβδ

βµ

)− d

d xµ

(1

αp−x2

[x2 y2 +2x y − (

x y)2

]))=

=−1

2

(m

1

2

1p−x2

(−xµ− xµ)− 1

α

d

d xµ

[− −x2

p−x2

y2 + 2x y − (x y

)2

p−x2

])=

= mxµ

2p−x2

+ 1

2α

[− d

d xµ

√−x2 y2+

+(2yµ−2

(x y

)yµ

)(−x2)1/2 −

(2x y − (

x y)2

)(−xµ)(−x2

)−1/2

−x2

=

= mxµ

2p−x2

+ 1

2α

−y2

(−xµ)

p−x2

+ 2(yµ−

(x y

)yµ

)p−x2

+(2x y − (

x y)2

)xµ(−x2

)3/2

=

= mxµ

2p−x2

− 1

2α(−x2

)3/2

[(−1)

(y2 (

xµ)+2

(yµ−

(x y

)yµ

))(−x2)− (2x y − (

x y)2

)xµ

]=

= mxµ

2p−x2

− 1

2α(−x2

)3/2

[y2x2xµ+2x2 yµ−2

(x y

)x2 yµ−2

(x y

)xµ+

(x y

)2 xµ]

(A.7)

Pµ = dL

d yµ=−1

2

d

d yµ

(√−x2

(m + 1

αx2

[x2 y2 +2x y − (

x y)2

]))=−1

2

p−x2

αx2 2xµ =

=p−x2xµα(−x2)

= xµ

αp−x2

(A.8)

32

{φ1,φ2

}= {P 2 +α−2,P p + m

2α−Sbc Sbc

}=−

{P 2,Sbc Sbc

}=

=−({

Pa ,Sbc Sbc}

P a +Pa

{P a ,Sbc Sbc

})=

=−([

{Pa ,Sbc }Sbc +Sbc

{Pa ,Sbc

}]P a +Pa

[{P a ,Sbc

}Sbc +Sbc

{P a ,Sbc

}]){Pa ,Sbc } = ∂

∂Pi(Pa)

∂Sbc

∂yi− ∂

∂yi(Pa)

∂Sbc

∂Pi= δi

a∂

∂yi

(P[b yc]

)= δai P[bδc]

i = P[bδac] =

= 1

2(Pbδac −Pcδab) ⇒

⇒ {φ1,φ2

}=−1

2

([(Pbδac −Pcδab)Sbc +Sbc

(P bδa

c −P cδab)]

P a +

+ Pa

[(Pbδ

ac −Pcδ

ab)

Sbc +Sbc

(P bδac −P cδab

)])=

=−1

2

[(PbPc −Pc Pb)Sbc +Sbc

(P bP c −P c P b

)+

+ (PbPc −Pc Pb)Sbc +Sbc

(P bP c −P c P b

)]=−

[(PbPc −PbPc )Sbc +Sbc

(P bP c −P bP c

)]= 0

(A.9)

{Sµν, H

}= {Sµν,ν1φ1 +ν2φ2

}= ν1{Sµν,φ1

}+ν2{Sµν,φ2

}{Sµν,φ1

}= ∂Sµν∂yi

∂φ1

∂Pi− ∂Sµν∂Pi

∂φ1

∂yi= ∂

∂yi

(P[µyν]

) ∂

∂Pi

(PαPα+α−2)=

=(P[µδν]

i)(δα

i Pα+Pαgαβδβi)= 2P[µδν]

i P i = 2P[µPν] = PµPν−PνPµ == PµPν (1−1) = 0{

Sµν,φ2}= {

Sµν,P p + m

2α−Sbc Sbc

}= {

Sµν,P p}=

= ∂Sµν∂yi

∂

∂Pi

(Pαpα

)− ∂Sµν∂Pi

∂

∂yi

(Pαpα

)= P[µδν]iδα

i pα = P[µδν]i p i =

= P[µpν] ⇒⇒ {

Sµν, H}= ν2P[µpν]

(A.10)

{Jµν, H

}= {Mµν, H

}+{Sµν, H

}= ν1{

Mµν,φ1}+ν2

{Mµν,φ2

}+{Sµν, H

}{Mµν,φ1

}= {p[µxν],P 2 +α−2}= 0{

Mµν,φ2}= ∂Mµν

∂xi

∂φ2

∂pi− ∂Mµν

∂pi

∂φ2

∂xi= ∂

∂xi

(p[µxν]

) ∂

∂pi

(P a pa − m

2α−Sbc Sbc

)=

= p[µδν]i(P aδa

i − ∂

∂pi

(P[b yc]P

[b yc]))

= p[µδν]i P i = pµPν] ⇒

⇒ {J , H }(A.10)= ν2p[µPν] +ν2P[µpν] = ν2

(p[µPν] −p[µPν]

)= 0

(A.11)

33

[φ1,φ2

] (A.9)= −{([Pα,Sµν

]Sµν+Sµν

[Pα,Sµν

])Pα+

+Pα([

Pα,Sµν]

Sµν+Sµν[Pα,Sµν

])}[Pα,Sµν

]= [Pα,P[µyν]

]= [Pα,P[µ

]yν] +P[µ

[Pα, yν]

]=−P[µ[

yν],Pα]=−i P[µgν]α⇒

⇒ [φ1,φ2

]= i{(

P[µgν]αSµν+SµνP [µgν]α

)Pα+Pα

(P[µgν]

αSµν+SµνP [µgν]α)}P[µgν]αSµνPα = P[µPν]S

µν = [PαPβ = PβPα

]= 0 ⇒⇒ [

φ1,φ2]= 0

(A.12)

[pµ, H

]= ν1[pµ,φ1

]+ν2[pµ,φ2

]== ν1

[pµ,P 2 +α−2]+ν2

[pµ,P p + m

2α−SαβSαβ

]=

=−ν2

[pµ,SαβSαβ

]=−ν2

(Sαβ

[pµ,Sαβ

]+ [

pµ,Sαβ]

Sαβ)

[pµ,Sαβ

]= [pµ,P[αyβ]

]= 0 ⇒⇒ [

pµ, H]= 0

(A.13)

[Wµρσ, H

]= 1√−p2

[p[µSρσ], H

]= 1√−p2

(p[µ

[Sρσ], H

]+ [p[µ, H

]Sρσ]

) (A.13)=

= 1√−p2

p[µ[Sρσ], H

]= 1√−p2

p[µ(ν1

[Sρσ],φ1

]+ν2[Sρσ],φ2

])[Sab ,φ1

]= [Sab ,P 2 +α−2]= [

Sρσ],P 2]= [Sab ,Pc ]P c +Pc[Sab ,P c]=

=−([Pc ,Sab]P c +Pc

[P c ,Sab

]) (A.12)= −i(P[b gca]P

c +Pc P[b g ca]

)==−i

(P[bPa] +P[bPa]

)=−2i P[bPa] =[PαPβ = PβPα

]= 0[Sab ,φ2

]= [Sab ,P p + m

2α+Sdc Sdc

]= [

Sab ,P p]= [Sab ,Pα] pα+Pα

[Sab , pα

]==− [Pα,Sab] pα+Pα

[pα,Sab

] (A.13)= −i P[b gαa]pα =−i P[b pa]

⇒ [Wµρσ, H

]= 1√−p2

p[µν2[Sρσ],φ2

]=− iν2√−p2

p[µP{ρpσ}] =− iν2√−p2

P{ρp[µpσ]} =

=− iν2

2√−p2

P{ρ(pµpσ} −pσ}pµ

)= iν2

2√

−p2P{ρpµpσ} (−1+1) = 0

(A.14)

34