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UPPSALAUNIVERSITY BACHELORTHESIS15HP

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.

BACHELORPROGRAM INPHYSICS

DEPARTMENT OFPHYSICS ANDASTRONOMY

DIVISION OFTHEORETICALPHYSICS

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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.

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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

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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-

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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

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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 study Noether’s theorem. With Noether’s theorem we can find the conserved quantities which exist under a symmetry operation. In section 6 we also make the transition to field theory, and therefore [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 book about string theory, and [8] and [9] are articles about rigid particles. Reference [8] is about finding 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 we examine two classically equivalent actions for a bosonic string. To do this we follow [7]. We will then use [8] and [9] as guidelines to find an action for a smooth spinning particle and to ensure that this action is indeed for a smooth spinning particle.

2.1. PROBLEMFORMULATION

The main question of this thesis is how symmetries and spin are introduced in the contexts of a one-particle system. This is later extended in order to describe symmetries and spin of systems 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 how one can use symmetries to find conserved quantities, such as Noether currents and Noether charges. We also want to know how one would describe a quantum mechanical property in classical 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 with symmetries describe spin in classical physics.

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3. THE POINT PARTICLE

3.1. THENON-RELATIVISTICACTION FOR AFREEMASSIVEPARTICLE

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

¯ ˙r¯¯

¯, is much smaller than the speed of light, c, under Galilean transformations and translations. The action is defined as

S = Z

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 =

Z

T d t =m 2

Z

˙¯r2d 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 resistance to acceleration when it is exposed to a force, and gravitational mass is determined by the strength of the force a body experiences while it is in a gravitational field. All experiments confirm the equivalence between the inertial mass and the gravitational mass, and Albert Einstein 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. THEGALILEANTRANSFORMATIONS

The Galilean transformations are used to describe the correspondence between two frames, S = (t, x, y, z) and S0= (t0, x0, y0, z0), in Newtonian physics, i.e. non-relativistic physics. Let one of 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= t00 = 0 are both of the frames at the origin, i.e. x0= x00 = 0.

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Figure 3.1: Two coordinate systems.

The coordinates x, y, z, t in S0as seen in S described through a Galilean transformation are [1]

x0= x − v t y0= y z0= z t0= t

˙

x0= ˙x − v

˙ y0= ˙y

˙ z0= ˙z

. (3.4)

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

S0=m 2

Z tb

ta ˙¯r02d t =m 2

Z tb

ta ¡ ˙x0, ˙y0, ˙z0¢2

d t =m 2

Z tb

ta ¡( ˙x − v)2+ ˙y2+ ˙z2¢ d t =

=m 2

Z tb ta

¡ ˙x2− 2 ˙xv + v2+ ˙y2+ ˙z2¢ d t =

=m 2

µZ tb

ta

¡ ˙x2+ ˙y2+ ˙z2¢ d t + Z tb

ta

¡v2− 2 ˙xv¢ d t

=

=m 2

Z tb

ta

˙¯r2d t +mv

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

(3.5)

Hereζ is a constant and is given by:

ζ =mv

2 (v (tb− ta) − 2(x(tb) − x(ta))) . (3.6) Sinceζ is fixed for constant endpoints, tband ta, one can drop this term since it is not impor- tant in variational principles1. Therefore the non-relativistic action (3.3) is invariant under the 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.

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3.3. THERELATIVISTICACTION FOR AFREEMASSIVEPARTICLE

For a system where the velocities are close to the speed of light the relevant transformations are 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 taand tb

S = α Z tb

ta

d s . (3.7)

Hereα is a constant which is determined by the non-relativistic limit of this action. From the metric,ηµν, describing Minkowski space in special relativity, one can get an expression for the relativistic distance. In this thesis, the Minkowski space will be given by2

¡gµν¢ = ¡ηµν¢ =

−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

(3.8)

⇒ d s2= d xµd xµ= d xµηµνd xν= −d t2+ d x2+ d y2+ d z2 (3.9)

⇒ S = α Z tb

ta

qd xµd xµ=α i

Z tb

ta

iq

d xµd xµ= −i α Z tb

ta

q

−d xµd xµ=

= −i α Z tb

ta

q

d t2− d x2− d y2− d z2= −i α Z tb

ta

s 1 −

µd x d t

2

µd y

d t

2

µd z

d t

2

d t =

= −i α Z tb

ta

p1 − ˙¯r2d t = −i α Z tb

ta

p1 − ˙r2d 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 ≈ −i α

Z tb

ta

µ 1 −1

2r˙2

d t = −i α(tb− ta) +iα 2

Z tb

ta

˙

r2d t . (3.11) The term −i α(tb− ta) is fixed for constant endpoints, tband ta, and constantα. This means that one can drop this term since it is not important in variational principles. This gives the following non-relativistic action

S =iα 2

Z tb ta

˙

r2d t =iα 2

Z tb ta

˙¯r2d 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).

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Comparing this action with the action (3.3) one obtains the relativistic action for a free mas- sive particle4

S = −m Z tb

ta

q

−d xµd xµ= −m Z tb

ta

s

d xµ dλ

d xµ

dλdλ . (3.13)

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

S = −m Z tb

ta

p− ˙x2d t . (3.14)

3.4. EQUATIONS OFMOTION FOR AFREEPARTICLE

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 obtained from the non-relativistic action (3.3) at the low velocity limit. We know already that this is the case since we have shown that the relativistic action (3.14) reduces to the non-relativistic action (3.3) in the non-relativistic limit. However , it is instructive to see this explicitly at the level of the equations of motion.

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

d d t

∂L

∂ ˙qi ∂L

∂qi = 0 . (3.15)

Here qiis the generalized coordinate and ˙qiis the time derivative of qi.

Let us start with the equations of motion in the non-relativistic case. Using the non-relativistic action (3.3), one gets the following equations of motion5

m ¨rj= 0 . (3.16)

Adding the three equations of motion for j = x, y, z one gets Newton’s second law when the sum 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). Ifλ is time, one can see that the La- grangian for a free relativistic particle is6

L = −mp

− ˙x2. (3.18)

4I.e.α = −im.

5Derivation is in appendix (A.2).

6Comparing (3.14) with (3.1).

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By using the Euler-Lagrange equations one will get the following equations of motion for a free relativistic particle7

¨

xα=( ˙x ¨x)

˙

x2 x˙α (3.19)

xµ= (t , ¯r) ⇒

⇒ (0, ¨¯r) = − ˙¯r · ¨¯r

1 − ˙¯r2(1, ˙¯r ) . (3.20)

In the non-relativistic limit is ˙¯r << ¯18 (0, ¨¯r ) = − ˙¯r · ¨¯r

1 − ˙¯r2(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 velocities too.

4. THELORENTZGROUP

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 of which come from boosts along an axis, and three from rotations around a fixed axis [2]. The Lorentz transformations9will be denoted asΛµν, and the four-vector under a Lorentz trans- formation is transformed in the following way

x= Λµνxν. (4.1)

Here xis the four-vector in the frame S’ as seen in the frame S, where S is a fixed frame and S’ is a frame which is either affected by a boost or a rotation (or both), i.e. S’ is affected by a Lorentz 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, along an axis10. Unlike the Galilean transformation, the speed, v ≡ | ¯v|, of S’ can now be close to the

7Derivation is in appendix (A.3).

8x << 1, ˙y << 1, ˙z << 1˙

9The 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.

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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 0

0 0 0 1

. (4.2)

Hereγ is the Lorentz factor. β and γ are given by:

β =vc

γ = (1 − β2)−1/2

. (4.3)

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

x=

t0 x0 y0 z0

=

γ −γβ 0 0

−γβ γ 0 0

0 0 1 0

0 0 0 1

t x y z

=

γt − γβx

−γβt + γx y z

=

γ¡t − βx¢

γ¡−βt + x¢

y z

(4.4)

⇒ ˙x=

γ¡1 − β ˙x¢

γ¡−β + ˙x¢

˙ y z˙

. (4.5)

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

S0= −m Z q

− ˙xx˙0µd t = −m Z q

γ2¡1 − β ˙x¢2− γ2¡−β + ˙x¢2− ˙y2− ˙z2d t =

= −m Z q

γ2¡1 − 2β ˙x + β2x˙2− β2+ 2β ˙x − ˙x2¢ − ˙y2− ˙z2d t =

= −m Z q

¡1 − β2¢−1

¡1 − β2¢ ¡1 − ˙x2¢ − ˙y2− ˙z2d t = −m Z q

1 − ˙x2− ˙y2− ˙z2d t =

= −m Z q

− ˙xµx˙µd 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)ββ2x2 (γ − 1)βxββ2y (γ − 1)βxββ2z

−γβy (γ − 1)ββyβ2x 1 + (γ − 1)β

2y

β2 (γ − 1)ββyβ2z

−γβz (γ − 1)ββzβ2x (γ − 1)ββzβ2y 1 + (γ − 1)ββ22z

, (4.7)

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whereβiis 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 the velocity, ¯v, is in the x-direction.

4.2. ROTATIONSAROUND AFIXEDAXIS

If the frame S’ has been rotated relative to the frame S with an angle,θ, around one axis, the frame 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 angleθ (constant, i.e. θ is not time dependent) the Lorentz transformation is given by

¡Λµν¢ =

1 0 0 0

0 cosθ −sinθ 0 0 sinθ cosθ 0

0 0 0 1

. (4.9)

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

x=

t0 x0 y0 z0

=

1 0 0 0

0 cosθ −sinθ 0 0 sinθ cosθ 0

0 0 0 1

t x y z

=

t x cosθ − y sinθ x sinθ + y cosθ

z

(4.10)

⇒ ˙x=

1

˙

x cosθ − ˙y sinθ

˙

x sinθ + ˙y cosθ

˙ z

. (4.11)

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

S0= −m Z q

− ˙xx˙µ0d t = −m Z q

1 −¡ ˙x cosθ − ˙y sinθ¢2¡ ˙x sinθ + ˙y cosθ¢2− ˙z2d t =

= −m Z q

1 − ˙x2cos2θ + 2 ˙x ˙y cosθ sinθ − ˙y2sin2θ − ˙x2sin2θ − 2 ˙x ˙y cosθ sinθ − ˙y2cos2θ − ˙z2d t =

= −m Z q

1 − ˙x2¡cos2θ + sin2θ¢ − ˙y2¡sin2θ + cos2θ¢ − ˙z2d t =

= −m Z q

1 − ˙x2− ˙y2− ˙z2d t = −m Z q

− ˙xµx˙µd t = S .

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

A more general rotation inR3is described by [3]

Λµν=

1 0 0 0

0 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, let us 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 10 generators: 3 from rotations around a fixed body, 3 from boosts along an axis and 4 from translations [4]. Rotations and boosts together make up the Lorentz group. The Poincaré group 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 the action (3.14) under those transformations. Since the action (3.14) has already been proven to be invariant under the Lorentz group, we only have to prove that the action (3.14) is invariant under translations.

5.1. TRANSLATIONS A translation is a displacement in both time and space:

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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 aµis a constant, it is easy to prove that the action for a free massive particle is invariant under translations

S = −m Z q

− ˙xµx˙µd t → −m Z

s

d

d t(xµ+ aµ) d

d t¡xµ+ aµ¢d t =

= −m Z q

− ˙xµx˙µd 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 transformations11.

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

½ r → ¯r + ¯¯ a

t → t + a0 . (5.3)

Here both ¯a and a0are constants.

S =m 2

Z

˙¯r2d t →m 2

Z µd

d t( ¯r + ¯a)

2

d t =m 2 Z

˙¯r2d t = S . (5.4)

6. NOETHERSTHEOREM

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

11The transformations within the Poincaré Group.

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6.1. NOETHERSTHEOREM

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, y0) with the Lagrangian L ≡ L(t, q, ˙q)12.

Assume that the Lagrangian is invariant under the transformation:

½ t → ˜t = t + δt

qk→ ˜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 generatorsξ and η, i.e.:

½ δx = ²ξ

δqk= ²ηk . (6.2)

Here² is a small constant. Then the following Noether current, j, is conserved13 j = ∂L

∂ ˙qkηk+ µ

L − ∂L

∂ ˙qk

˙ qk

ξ = pkηk³

pkq˙k− L´

ξ = pη − Hξ . (6.3)

Here pkis the momentum associated with kt hgeneralized coordinate, qk, and H is the Hamil- tonian.

6.2. NOETHERSTHEOREM IN FIELDTHEORY

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 the Lagrangian density14,LL(xµ,φ,∂µφ), instead of the Lagrangian, L, where ∂µdenotes the derivative with respect to the four-vector, xµ.

Assume that the Lagrangian density is invariant under the transformation

½ xµ→ ˜xµ= xµ+ δxµ

φk( ¯x) → ˜φk( ¯x) = φk( ¯x) + δφk( ¯x) . (6.4) Assume further that this transformation has infinitesimal generatorsξ and η, i.e.

½ δxµ= ²ξµ

δφk( ¯x) = ²ηk( ¯x) . (6.5)

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

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

14Integrating the Lagrangian density,L, over xµwill give the Lagrangian, L.

(17)

Here² is a small constant. Then the following Noether current, jµ, is conserved [4]

jµ= L

∂(∂µφk)ηk− Tµνξν

Tµν= L

∂(∂µφk)νφk− gµνL.

(6.6)

Here Tµνis 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 =

Z

d3x j¯ 0. (6.7)

7. PAULI-LUBANSKI SPIN 7.1. PAULI-LUBANSKISPIN

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 the total angular momentum, Jνρ, as the Noether charge for the modified Noether current,Jµ,νρ, under a Lorentz transformation.

Let us consider the Lagrangian density being invariant under a Lorentz transformation xµ→ x= Λµρxρ=¡

δµρ− ωµρ+ O (ω2)¢ xρ≈ δµρxρ− ωµρxρ= xµ− ωµρxρ. (7.1) Hereωµρis infinitesimal. In appendix (A.4) it is proven thatωµρis antisymmetric (in indices).

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

½ Aµν= Aνµ, if Aµνis a symmetric tensor.

Aµν= −Aνµ, if Aµνis an antisymmetric tensor. (7.2) Using Noether’s theorem in field theory, one finds that the following Noether current, jµ, is conserved under the Lorentz transformation (7.1)

jµ= −Tµν¡−ωνρxρ¢ =1

2¡Tµνωνρxρ+ Tµνωνρxρ¢ =1

2¡Tµνωνρxρ− Tµνωρνxρ¢ =

=1

2ωνρ¡Tµνxρ− Tµρxν¢ .

(7.3)

References

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