物理之美：对称性揭示的深奥原理

Symmetry

Physics

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XIX

A.1 Basis Vectors .......................... 250

A.2 Change of Coordinate Systems ............... 251

A.3 Matrix Multiplication ..................... 253

A.4 Scalars ............................. 254

A.5 Right-handed and Left-handed Coordinate Systems . . . 254

B Calculus 257

B.1 Product Rule .......................... 257

B.2 Integration by Parts ...................... 257

B.3 The Taylor Series ....................... 258

B.4 Series .............................. 260

B.4.1 Important Series ................... 260

B.4.2 Splitting Sums .................... 262

B.4.3 Einstein’s Sum Convention ............. 262

B.5 Index Notation ........................ 263

B.5.1 Dummy Indices .................... 263

B.5.2 Objects with more than One Index ......... 264

B.5.3 Symmetric and Antisymmetric Indices ...... 264

B.5.4 Antisymmetric

× Symmetric Sums ........ 265

B.5.5 Two Important Symbols ............... 265

C Linear Algebra 267

C.1 Basic Transformations .................... 267

C.2 Matrix Exponential Function ................ 268

C.3 Determinants ......................... 268

C.4 Eigenvalues and Eigenvectors ................ 269

C.5 Diagonalization ........................ 269

D Additional Mathematical Notions 271

D.1 Fourier Transform ....................... 271

D.2 Delta Distribution ....................... 272

Bibliography 273

Index 277

CONTENTS

Introduction

1.1 What we Cannot Derive

Before we talk about what we can derive from symmetry, let’s clarify

what we need to put into the theories by hand. First of all, there is

presently no theory that is able to derive the constants of nature.

These constants need to be extracted from experiments. Examples are

the coupling constants of the various interactions and the masses of

the elementary particles.

Besides that, there is something else we cannot explain: The num-

ber three. This should not be some kind of number mysticism, but

we cannot explain all sorts of restrictions that are directly connected

with the number three. For instance,

• there are three gauge theories

, corresponding to the three fun-

Don’t worry if you don’t understand

some terms, like gauge theory or

double cover, in this introduction. All

these terms will be explained in great

detail later in this book and they are

included here only for completeness.

damental forces described by the standard model: The electro-

magnetic, the weak and the strong force. These forces are de-

scribed by gauge theories that correspond to the symmetry groups

(1), SU (2) and SU( 3). Why is there no fundamental force follow-

ing from SU

(4)? Nobody knows!

• There are three lepton generations and three quark generations.

Why isn’t there a fourth? We only know from experiments

with

For example, the element abundance

in the present universe depends on the

number of generations. In addition,

there are strong evidence from collider

experiments. (See Phys. Rev. Lett. 109,

241802).

high accuracy that there is no fourth generation.

• We only include the three lowest orders in Φ in the Lagrangian

(Φ

, Φ

), where Φ denotes here something generic that de-

scribes our physical system and the Lagrangian is the object we

use to derive our theory from, in order to get a sensible theory

describing free (=non-interacting) ﬁelds/particles.

• We only use the three ﬁrst fundamental representations of the

double cover of the Poincare group, which correspond to spin 0,

 Springer International Publishing Switzerland 2015

J. Schwichtenberg, Physics from Symmetry, Undergraduate Lecture Notes in Physics,

DOI 10.1007/978-3-319-19201-7_1

4 physics from symmetry

and 1, respectively, to describe fundamental particles. There is no

fundamental particle with spin

In the present theory, these things are assumptions we have to

put in by hand. We know that they are correct from experiments,

but there is presently no deeper principle why we have to stop after

three.

In addition, there are two things that can’t be derived from sym-

metry, but which must be taken into account in order to get a sensi-

ble theory:

• We are only allowed to include the lowest-possible, non-trivial or-

der in the differential operator ∂

in the Lagrangian. For some the-

ories these are ﬁrst order derivatives ∂

, for other theories Lorentz

invariance forbids ﬁrst order derivatives and therefore second

order derivatives ∂

∂

are the lowest-possible, non-trivial order.

Otherwise, we don’t get a sensible theory. Theories with higher or-

der derivatives are unbounded from below, which means that the

energy in such theories can be arbitrarily negative. Therefore states

in such theories can always transition into lower energy states and

are never stable.

• For similar reasons we can show that if particles with half-integer

spin would behave exactly as particles with integer spin there

wouldn’t be any stable matter in this universe. Therefore, some-

thing must be different and we are left with only one possible,

sensible choice

which turns out to be correct. This leads to the

We use the anticommutator instead of

the commutator as the starting point

for quantum ﬁeld theory. This prevents

our theory from being unbounded from

below.

notion of Fermi-Dirac statistics for particles with half-integer spin

and Bose-Einstein statistics for particles with integer spin. Par-

ticles with half-integer spin are often called Fermions and there

can never be two of them in exactly the same state. In contrast, for

particles with integer spin, often called Bosons, this is possible.

Finally, there is another thing we cannot derive in the way we

derive the other theories in this book: Gravity. Of course there is

a beautiful and correct theory of gravity, called general relativity.

But this theory works quite differently than the other theories and

a complete derivation lies beyond the scope of this book. Quantum

gravity, as an attempt to ﬁt gravity into the same scheme as the other

theories, is still a theory under construction that no one has success-

fully derived. Nevertheless, some comments regarding gravity will be

made in the last chapter.

introduction 5

1.2 Book Overview

Double Cover of the Poincare Group

Irreducible Representations



 ++

(0, 0) : Spin 0 Rep

acts on



(

,0) ⊕ (0,

) : Spin

Rep

acts on



(

) : Spin 1 Rep

acts on



Scalars

Constraint that Lagrangian is invariant



Spinors

Constraint that Lagrangian is invariant



Vectors

Constraint that Lagrangian is invariant



Free Spin 0 Lagrangian

Euler-Lagrange equations



Free Spin

Lagrangian

Euler-Lagrange equations



Free Spin 1 Lagrangian

Euler-Lagrange equations



Klein-Gordon equation Dirac equation Proka equation

This book uses natural units, which means setting the Planck

constant h

= 1 and the speed of light c = 1. This is conventional in

fundamental theories, because it avoids a lot of unnecessary writing.

For applications the constants need to be added again to return to

standard SI units.

The starting point will be the basic assumptions of special relativ-

ity. These are: The velocity of light has the same value c in all inertial

frames of reference, which are frames moving with constant velocity

relative to each other and physics is the same in all inertial frames of

reference.

The set of all transformations permitted by these symmetry con-

straints is called the Poincare group. To be able to utilize them, the

mathematical theory that enables us to work with symmetries is in-

troduced. This branch of mathematics is called group theory. We will

derive the irreducible representations of the Poincare group

, which

To be technically correct: We will

derive the representations of the

double-cover of the Poincare group

instead of the Poincare group itself. The

term "double-cover" comes from the

observation that the map between the

double-cover of a group and the group

itself maps two elements of the double

cover to one element of the group. This

is explained in Sec. 3.3.1 in detail.

you can think of as basic building blocks of all other representations.

These representations are what we use later in this text to describe

particles and ﬁelds of different spin. Spin is on the one hand a label

for different kinds of particles/ﬁelds and on the other hand can be

seen as something like internal angular momentum.

Afterwards, the Lagrangian formalism is introduced, which

makes working with symmetries in a physical context very conve-

nient. The central object is the Lagrangian, which we will be able to

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物理之美：对称性揭示的深奥原理

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编写Point类，要求：具有属性：点的坐标（二维或三维）具有方法：输出X坐标最大的点的坐标。输出点关于原点的对称点。分别输出两点间距离最近和最远的距离。