(Series in high energy physics, cosmology and gravitation) ken j barnes group theory for the standard model of particle physics and beyond taylor francis (2010)

Since we are dealing with \'II'll1entary particles, we may as well think of conserved numbers carried on IIlight as well, for simplicity, start with the problem in classical physics and

'I.' In hll."" 11,1"" , (h/(/II/ 1!,1I1't'I.It)', III,

blw,,,,1 W I "Ih hI/III NII//ol/(t! AI'I',.I,.m/lir ' ,tlll/ml/ory USA

I 11" \l' II~"

II I hOIli< , LOve" II ,"PCU\ u( lheoreli ca l and ex perimental high energy physics,

«)"no lo!,y ,111(1 ,,1,IVil ce bctwecn llioll ;llld inre Ih e: rt:l them In recent years the fields of particle

ph )',,~, ,111.1 ,I\ lrophys ics have beco me increas ingly interdependent and the aim of this series i~ l() provide a library of books LO meet the needs of students and researchers in these fields

Olha ('('cent books in the series:

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Joint Evolution of Black Holes and Galaxies

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Gravitation: From the Hubble Length to the Planck Length

I Ciufolini, E Coccia, V Gorini, R Peron, and N Vittorio (Eds.)

Neutrino Physics

K Zuber

The Galactic Black Hole: Lectures on General Relativity and Astrophysics

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The Mathematical Theory of Cosmic Strings: Cosmic Strings in the Wire Approximation

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Geometry and Physics of Branes

U Bruzzo, V Gorini and, U Moschella (Eds.)

Modern Cosmology

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Gravitation and Gauge Symmetries

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Gravitational Waves

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Classical and Quantum Black Holes

P Fre, V Gorini, G Magli, and U Moschella (Eds.)

Pulsars as Astrophysical Laboratories for Nuclear and Particle Physics

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St:rit:s ill I ligh I':nt:rgy Physit.s a , Cos lllolo gy, nd (;ravilalion

the Standard Model '

c~ Taylor & Francis Group

Boca Raton London New York CRC Press is an imprint of the

A TAYLOR & FRANCIS BOOK

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6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742

No claim to original U.S Government works Printed in the United States of America on acid-free paper

1098 7654321 Internation al Standard Book Number: 978-1 -4200-7874-9 (Hardback) This book conta ins information obtained from authentic a nd hi ghly regarded sources Reasonable efforts have been made to publish reli able data a nd information , but the author a nd publisher ca nnot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have at te mpted to trace the copyrig ht holders of all material reproduced in thi s publication and apologi ze to copyright holders if permission to publish in this form ha s not been obtained If any copyright material has not been ack nowl edged please write and let us know so we may rectify in any future reprint

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Trademark Notice: Product or corporate names may be trade marks or registered trademarks, a nd a re used only for identification and explanation without intent to infrin ge

Library of Congress Cataloging-in- Publica tion Data Barnes, Ken ) , 1938-

Includes bibliographical refere nces a nd ind ex

ISBN 978-1-4200-7874-9

QCI74.17.G7B372010 539.7'25 d c22

http:// www.taylorandfrancis.com

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ss \'(/('() sitl' lit

ht I p :/I www.("·(' I)I '("s, ,(' uln

200902 1685

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( 'Oil tents

I '14'I,II 'l' " ix

\( I Illlwlcdgments xi

IIII r(lll uclion xiii

Symmetries and Conservation Laws 1

I ,'lgrangian and Hamiltonian Mechanics 2

Quantum Mechanics , 6

The Oscillator Spectrum: Creation and Annihilation Operators 8

' oupled Oscillators: Normal Modes 10

One-Dimensional Fields: Waves 13

The Final Step: Lagrange-Hamilton Quantum Field Theory 16

References 20

Problems 20

2 Quantum Angular Momentum 23

Index Notation 23

Quantum Angular Momentum 25

Result 27

Matrix Representations 28

Spin ~ 28

Addition of Angular Momenta 30

Clebsch-Gordan Coefficients 32

Notes 33

Matrix Representation of Direct (Outer, Kronecker) Products 34

~ 0 ~ = 1 EB 0 in Matrix Representation 35

Checks 36

Change of Basis 37

Exercise 38

References 38

Problems 38

3 Tensors and Tensor Operators 41

Scalars 41

Scalar Fields 42

Invariant functions 42

Contravariant Vectors (t ~ Index at Top) 43

Covariant Vectors (Co = Goes Below) 44

Notes 44

v

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v I PIIWIII',

·1i 'llsors ,I i)

Noles dnd I'ropl'rlil's 45

Rotations 47

Vector Fields 48

Tensor Operators 49

Scalar Operator 49

Vector Operator 49

Notes 50

Connection with Quantum Mechanics 51

Observables 51

Rotations 52

Scalar Fields 52

Vector Fields 53

Specification of Rotations 55

Transformation of Scalar Wave Functions 56

Finite Angle Rotations 57

Consistency with the Angular Momentum Commutation Rules 58

Rotation of Spinor Wave Function 58

Orbital Angular Momentum (~ x p) 60

The Spinors Revisited ~ 65

Dimensions of Projected Spaces 67

Connection between the "Mixed Spinor" and the Adjoint (Regular) Representation 67

Finite Angle Rotation of 50(3) Vector 68

References 69

Problems 69

4 Special Relativity and the Physical Particle States 71

The Dirac Equation 71

The Clifford Algebra: Properties of y Matrices 72

Structure of the Clifford Algebra and Representation 74

Lorentz Covariance of the Dirac Equation 76

The Adjoint 78

The Nonrelativistic Limit 79

Poincare Group: Inhomogeneous Lorentz Group 80

Homogeneous (Later Restricted) Lorentz Group 82

Notes 84

The Poincare Algebra 88

The Casimir Operators and the States 89

References 93

Problems 93

5 The Internal Symmetries 95

Rl'kl'en l'S • 105

l)rohlt'llls 105

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(, Lil' C"Otll' '11.'chllit llll·s for 1111' St.lIId •• rtl Motll'1 Lil' Croups ,., 1()7

I ~()(lb I1111 W\,lghl s , 108

~; lIllpll' I ~(l() l s "." " , , 11]

'I'll(' < ' 11'1.111 Moll ri x 113

l :indil1g !\lIlhc Rools 113

h'lld,lm ' nl(ll Weights 115

Tht' Wl'y l C roup 116

YO llng Tllbleau x 117

R"i s ing the lndices 117

The Classification Theorem (Dynkin) 119

1<('s ulL 119

Coincidences 119

Ref 're nces 120

Problems 120

7 Noether's Theorem and Gauge Theories of the First and Second Kinds 125

References , 129

Problems 129

H Basic Couplings of the Electromagnetic, Weak, and Strong Interactions 131

References 136

Problems 136

<.) Spontaneous Symmetry Breaking and the Unification of the Electromagnetic and Weak Forces 139

References 144

Problems 145

10 The Goldstone Theorem and the Consequent Emergence of Nonlinearly Transforming Massless Goldstone Bosons 147

References 151

Problems 151

I I The Higgs Mechanism and the Emergence of Mass from Spontaneously Broken Symmetries 153

References 155

Problems 155

12 Lie Group Techniques for beyond the Standard Model Lie Groups 157

References 159

Problems 160

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13 ThcSimplcSph·rc I()I

Problems 182

14 Beyond the Standard Model 185

Massive Case 188

Massless Case 188

Projection Operators 189

Weyl Spinors and Representation 190

Charge Conjugation and Majorana Spinor 192

A Notational Trick 194

5L(2, C) View 194

Unitary Representations 195

Supersymmetry: A First Look at the Simplest (N = 1) Case 196

Massive Representations 197

Massless Representations 199

Superspace 200

Threc-Dimensional Euclidean Space (Revisited) 200

ovariant Derivative Operators from Right Action 207

Superfic ld s 209

Sup 'rtransformations 211

Notcs 211

The Chiral Scalar Multiplet 212

5uperspace Methods 213

ovariant Definition of Component Fields 214

Supercharges Revisited 214

Invariants and Lagrangians 217

Notes 220

Superpotential 221

References 225

Problems 225

Index 229

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I'rcfacc

1111 ' 11 I )vp.lrlmcnt of Physics and Astronomy, University of Southampton,

1", l nn' I reLired It is hoped that this book will be appropriate for similar

,111111' may fi nd it difficult

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/\(,J :llowLedgments

1111 ', ll(lOk could never have been written without the consistently excellent

I,ll ',OIl1l' of the figures and general advice Dr Jason Hamilton-Charlton is

111.l11i<.t'd for his generosity in providing both LaTeX and English electronic

\ ,111 1 illlmi s upport and help when writing anything seemed quite impossible

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IlIlroduction

1111 : hoo" is dcfinitcly not a book on mathematics It is a book on the use

til " Illlllclrics, mainly described by the techniques of Lie groups and Lie

111 :, pl'ciill cases, the ideas are very firmly based on a lifetime of lecturing , l1l'ril'nce

xiii

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1

SYIJlmetries and Conservation Laws

,I lid there may be students for whom it is quite new Since we are dealing with

\'II'll1entary particles, we may as well think of conserved numbers carried on

IIlight as well, for simplicity, start with the problem in classical physics and illril to quantum mechanics later Well, the first thing is that it cannot simply

\' ,I n is h or appear Of course it can vanish by having equal but opposite charges ,II1I1ihiiate it (producing, for example, the photons of light), or it can appear III the reverse of this All other conserved quantities such as energy, and Iliwar and angular momentum must be conserved-in our picture carried on

111 (' photons Already we see that this must happen at the same time and at

JI.1rticles

You may well be familiar with the idea of conservation of charge being

oI ssociated with the four divergence ofthe current carrying that charge Calling

(1.1) 'I 'hen we have

at

'rhis means that the rate of increase of charge in the volume is equal to the rate

of flow of charge into the volume minus the rate of flow out of the volume A very natural feature of the model we use is where the charges are carried on the

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p"rtl\ ' I\'~ ' ()I (( lim,\" thi ~) \'IIII1' (·pllll'l·d :,:- ll ght dl.l11g111g ill thl ' wo rld of :-'Pl'Ci.l l rl'l.,livil

Li1l'r l' i:, ,1p pMl'nl CO il 11', 1 ' lioll of leng lhs lIlel dil.llion of limes

modi-fications are need ed, whi ch are yet further changed in qu antum ficld theo ry Butwe are getting too far ahead of ourselves Let us ask what syrnmetrieshave

to do with these conservation laws as our title of this chapter suggests There

pow-erful and extends to all types of description of the physics discussed earlier (Students note that Noether was a woman doing important work of this type

at a time when there were nowhere near as many women working in science.) The point that is necessary to understand at this stage is that all conserved quantities in physics are linked to symmetries in this way We shall meet examples of this later The mathematics underlying this structure is that of group theory, both discrete groups and continuous groups as described by Lie But for the moment we move on to simple examples in the next two chapters

Lagrangian and Hamiltonian Mechanics

Although it has been made clear that the reader is expected to be competent

in quantum field theory, an exception is made at this point to be sure that the readers really can cope

It is one of those curious quirks of history that long before quantum theory was developed this version of classical mechanics established a framework that was capable of treating both fields and particles in both classical and

approach to physics in terms of the "principle of least action," if you have not met it previously We shall approach the topic in a more pedestrian manner than Feynman, partly because I am not so brave a teacher and partly because

I want to get you calculating for yourself as soon as possible It is my firm belief that the best way to get on top of a subject like this is to lose your fear

of it by getting your hands dirty and actually doing the real calculations in detail yourself

Suppose we have a one-dimensional system-yes, it is going to be the

the spatial positions Then Newton's second law is replaced by the Lagrange [3] equation

", 111'1( ' II I ~ , III(' 111111' dc ' l'i \', III VI' ' l ,dg l" III) ; I,III, / , is T (11'1 ill Illl' dilll 'n' ll l'l'

1,, ' lw\'\'11 I Il l' ""1('1,(, l'l\('r); ('/') dlld Ihl' poll'nlidll'l1l'rgy (V), lh 'll is,

(1.5)

IIII' Ill i q OSl'S of pa rli a I differentiation, For the harmonic oscillator with mass

Now that we have a little experience with this formalism, we can take a look

.11 I he principle of least action You will have noticed perhaps that the concept (.1 force (which was primary in Newton's approach) has become secondary to

Ilw idea of potential The least action principle makes the equation of motion (l s(' lf something that is derived from the minimization of the action

where ti and tf are initial and final times The principle postulates that the Idual path (often alternatively called trajectory) followed by the particle is 111.)t which minimizes 5, Imagine that, given L as an explicit function of q

,Ind 1, you evaluate 5 for a few paths These are just fictitious paths and none (If them is likely to be the Newtonian one, I have drawn the three from the problem on the q-t diagram in Figure 1.1

These must start and finish at the same places and times According to the principle, only if one of these coincides with the Newtonian path will the v,) lue of 5 be the minimum possible, You need a calculus approach to get a general answer Notice, however, 5 is a function of the function q(t) We say

it is a "functional" of q(t) We need to find the particular function, qo(t), that minimizes 5

Suppose there is a small variation 8q(t) in a path q(t) from q(ti) to q(tf )'

When q(t) = qo(t), the variation 85 caused by this change 8q must vanish

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""11/' TIII 'II"/I"' 1111' :"n"t1nrn Wlflrtr'l (II n l { l I f I r , '"I~ '' ' '''''' '" 'f' '''''

which retrieves the Euler-Lagrange equation of motion The solution of this

As we shall see later, this formalism is well suited to treat systems of the many (indeed infinitely many) linked dynamical variables found in field theories But the transition from classical to quantum mechanics is made more transparent by considering the Hamiltonian formulation The idea, in the first

second order Euler-Lagrange equation by two linked first order equations Thi s piece of magic is performed by introducing

aL

IIH IIll\'nLlI nl , olS we Sh ,lli see.) Then th e Hamiltonian is introduced by the

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11 ' )',1 ' 111111 ' 11', 1I1 ~ , I()I'''l.lII()1l

('1.12) ,,,Ill 1111' hrler L,l grLl nge cquiJtion is replaced by the pair of equations

which iJre known as Hamilton's canonical equations To get a feel for this

for-""rI,ltion we return to our old friend the harmonic oscillator From Equation (I 7) we see that

Hamilto-"j,lIl is the total energy, T + V This is a very general feature, and provided

I h,l t time does not appear explicitly then

dH aH aH aHaH aH [ aH]

which reflects energy conservation In the present case the equations of lion, Equations (1.13) and (1.14), yield

I ion of the momentum, and on substitution into the second retrieves Equation

in-structive to solve the first order Equations (1.17) and (1.18) directly Consider the linear combination

(1.19)

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'tln"I' I "("PI II fT" '(fr"/ "'''(IIIn, fV'(lII'-' "I' ""n~r I " '!f' II f' '''''' "' ,'1"""

A = ae - i wl

(1 20)

(1.21)

Equation (1.19), we immediately find

The passage to quantum mechanics in this formalism is facilitated by

intro-d ucing the Poisson bracket nota tion The Poisson bracket of any two functions

and we see that

are alternative ways of writing Equations (1.13) and (1.14), the equations of

Icr, /il > il& , ~ I = - i(Ct~ - ~Ct) between the classical dynamical variables

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111.1 Illt ' ll' II , IIII ' 1i '111.111111111 d '( 'II.1II1 S IH 11\1 (', Op('I'.lIOI (,O ITl' lI)tI (' I1 l'l'S (We L1 Sl'

" 111 111I,d " 111111 :- w ith" I.) 111 p,lIli cul.ll', l:qLlclti27) y iclu s o n ( 1

dF (t)

d' I Il l' Il eisenberg equation of motion The time dependence has been

hili Ihe dynamical variables contain the time dependence,

I Ill' a lternative Schrodinger picture, in which the variables are time Ilt'l1tient, has the time dependence of state vectors given by the Schrodinger

(1.32)

!(',lture of this is that

pri nciple in the Schrodinger picture, In quantum field theory we shall find

th e Heisenberg picture very convenient

(1.34)

(1.35)

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n "ru"I' I flrPI II/fir IItr-I-' II,,,,,,,," IV III.'" " / ' , , , , , , , I '''I''' ' ,,,, ,, 'I H"'-

Wllh h ,ll llI01l (1 1 ) giving 11'1\/1.111 lhl' l'qll.l lil y of 111l'~l' 1Ilclll.lll ' form s

lhal

(1.36) (1.37)

so that we get

by combining these Now, please notice that this is not just the classical tion of motion (Equation (1.9» again What Equation (1.38) tells us is the

we take the expectation value of Equation (1.38) between (time independent) Heisenberg states, then we learn that the mean position of the particle does follow the classical path This is very reassuring, but there will be quantum fluctuations about the classical path, of course

The Oscillator Spectrum: Creation and Annihilation Operators

This subtopic is of such central importance later that it deserves a section all to itself You have no doubt all been exposed to this material before, but

theory (If you already know this method, it will at least serve as a review and

to establish notation.)

which to expand any general state, and thus must solve the time independent Schrodinger equation

HIE" > E"IE" > (1.39) for the eigenvalues and eigenvectors The Hamiltonian is given in Equation (1.34) as

(1.44) (1.45)

(1.46) (1.47) ( I n Equations (1.45)-(1.47) we now have the algebraic information in a suitable

form to find the spectrum I urge you to do Problem 1.14 before continuing.)

We are now in a position to see exactly why a and a i are so important They ll,lVe the magical property in that they take you from one energy eigenstate

Into another, rather than into some arbitrary linear combination of states To sec this, consider the effect of the Hamiltonian on an eigenstate that has been cha nged by the action of ai

= (wai +a t EII )IEn >

so we see that atlEIl > is indeed an eigenstate of H and (Ell + w) is the

e igenvalue In a similar way we can establish that a I Ell > is an eigenstate

w ith (Ell - w) as the eigenvalue this time Of course, you cannot lower the

e nergy until it becomes negative, so there must be a ground state of lowest

e nergy Eo with

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1\1 \ "Pllf' , "nll'! IP' If' -""frrH"H n 'flp r, "I' """ I r t "'/'''''' ,11" I" ,,""11

,l: il :- ddillill(lil 10 11Idllll.llll {'()Il:-.I:-I{'IH' (Ikw.lrl'! III r{'I,I[ivistil' physics s llch rl"l soning

will not be true ) Bul here you can prove it From Equ,llion (1.42)

Before we launch into an attack on the quantum field theory of infinitely many degrees of freedom, it is probably sensible to try a finite number of variables

I ,ct ' s s ln rt with the classical theory of two equal masses in a one-dimensional

: prin)\ l'Ollsldnl S' ,lilt! li t'd to fixed points by springs of spring constant k

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FIGURE 1.2

Three spring forces

from equilibrium, and the Lagrangian takes the form

L = "2m (ql + q2) - "2k (ql + q2) - "2g(q2 - ql) (1.56)

if none of the springs are stretched or compressed in the equilibrium position You can think of this as a model of a (very) small solid One advantage of the Lagrangian approach is that we never have to introduce the forces in the springs and then eliminate them again; constraints are handled very neatly

(1.57) (1.58) which are sufficiently simple that we do not need formal methods to solve them We spot the relevant combinations of variables by adding and subtract-

(1.59) (1.60)

which we recognize as uncoupled simple harmonic oscillators The solutions are then obvious We have one normal mode of oscillation with frequency

l1l1d Equation (1.60) is satisfied trivially by having the two displacements

eq ual The second normal mode has frequency

(1.62)

hu t opposite in sense The general solution is then obtained by superposition

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Notice that the frequency of the lowest mode is independent of g-naturally,

this mode is of zero frequency You can then think of a two atom molecule that

is free, and this mode corresponds to the free motion of the center of mass This is not really important for these lectures, but when you hear theorists worrying about zero modes and symmetries you will have some idea of their problems Zero modes can be a real pain for theorists as they usually need separate treatment, and the associated symmetry (here just translation) is not always easy to find

Now, how do we quantize a system like this? The key lies in the observation made earlier that we are just dealing with uncoupled harmonic oscillators in terms of

easy to work out the form

1 (2 2) k 2 (k ) 2

for the Hamiltonian So to quantize we simply have to put hats on the Q's and P's, and do the harmonic oscillator problems twice You should check, of course, that the commutation relations are

[Qi, Qj ] = a

[Qi , Pj ] = iOi j

Notice that the emphasis has now changed completely from "displacements

that there is no restriction (except the total energy available) on the number

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,Ii " i ILilIOll'o, I'V"I1 though Ill\' Illl111111'1' 01 is lIntil'l'l ing "di s pl.l c eJ11l'nts"

1111

Ihi s SI,lgl' Thi s dPIHod ch is ce ntral

to many body phys ics

\ 1!I'II'OIH' SIW,lKs 01 "phonons" ,1S the vibr,llion(J1 excitations (similarly

II,' ,11\' k<lliin g toward is a framework in which all elementary particles (' 1" '" ks, leptons, W ' , 20, photons, gluons) are excitations of underlying Iwilis Ilowever, we must first learn to handle a simple classical continuous .\ " ,'Ill

()nc-Dimensional Fields: Waves

\vh.l t is to be our generalization of the Lagrangian in Equation (1.56) when Illl'rl' is a continuum of "atoms" rather than just two? The sum over two ' ill'S becomes an integral over the position x, and presumably the last term I, '("() mes proportional to a spatial derivative We shall have to absorb dimen-

', lIlllS into the constants, of course, and we use ¢(x, t) rather than q(x, t) for 1"lure convenience, so that we write

L = 1/2 at - 2¢ - 2" ax pdx (1.68)

whe re p is a mass density, and f.1- and c are just convenient names for the Illodified constants Now what is the generalization of the Euler-Lagrange ,·quations? In this continuous case we define a Lagrangian density .c by

is the action to be minimized (I am being deliberately vague about the limits

Ilf the integrals You may think of a solid between x = 0 and x = L, or of a

I icld extending over all space.) The new feature in the continuous case is that [ depends not only on ¢ and ¢ = ~; but also on ~~ and all these must be varied Thus

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,/(1111/ '1"1/"(1/111')/ 1111 ' rli //n/l t r ' "'I" '''"

'lI1d w(' inlq~ r'll' by p.1rl s in I lor lhl' Ill,ddlc Icrlll, ,lIld by Pdfl s ill ,\ lor llll'

fi nd I l 'I'm, Lo geL

85 - f [dtdx -a£ - -a (a£) a - - - ( a£ )] 8

when the end-point (or boundary) terms are assumed to vanish Since 5 is arbitrary we get the Euler-Lagrange field equations

with

(1.74)

as the generalization to three spatial dimensions

Putting the expression for £ implicit in Equation (1.68) into Equation (1.73)

is that there are a few snags in its interpretation in relativistic physics For the moment we merely have to notice that we already know a lot about this equation If we ignore the J1,2 term, we have

in "natural" units Moreover, we are familiar with the idea of superimposing sinusoidal solutions to produce standing waves, when discrete frequencies arise from the boundary conditions, as in the case of sound waves in a tube

or transverse waves on a violin string Then a general solution can be written

Bullhi s is alrcady enoughhint to see what we need to do for the full equation 1':l.fllillion (1.75)- thc s inusoidal functions will still work, but the frequencies will Iw II '> dependenl !\ssume for simplicity thatthe end "atoms" in our linear

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11.1 111.11'\ ' l 'Il II S 'd lrdilll III Ill' ,11 1'1':;1 oi l ,

,

11 111

0

11 \' lI

s llrili

g Ihi

01 )" I' I1l'ra l solution, by superposition, may be written as

¢(x, t) = ~ Ar(t)SII1 L

r = l

(1.81)

)', Iven earlier So what we have discovered is that the Fourier series gives the

"lillie analysis The sinusoidal functions in X give exactly the correct

I ourier amplitudes, A r , act exactly as do normal coordinates To confirm this

\' I('W w e can construct the Hamiltonian for this system, and see if the lion (Equation 1.81) reveals a sum of uncoupled oscillators It is clear in our ' pression for the Lagrangian (Equation 1.68) that the first term is a kinetic 1.'rl11 and the remainder is potential, so that

solu-(1.82)

IS the form of the Hamiltonian Substituting Equation (1.81) and using the orthonormality of the sine functions in the region x = a to x = L reveals that

(1.83)

confirming the view we had formed The quantum version of this problem

is now obvious-this is exactly as previously shown except that there are

d n infinite number of oscillators, and therefore an infinite set of the I1r to be

s pecified as occupation numbers to define the state This does raise, however, the question of the zero-point energy problem We have now introduced an infinite number of oscillators each with minimum energy ~w, so that the total energy of our" continuous chain of atoms" is infinite The conventional view is

I ha t this is not a real problem, only the energy differences really matter (After

a It, there is no concept of destroying this" crystal" into an infinite number of

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II! (,rnllp I tiff' I I! {nr rllr ~ IfIlUIl/nt IIf ' ,,, II, I" "'(' " "",""',., I'

~ (I! co ntributions and l'lke a new refe re nce point for zero ' n ' rgy

The Final Step: Lagrange-Hamilton Quantum Field Theory

We now have just about enough experience to attempt the real problem The

models, later, this would be perhaps a multicomponent field, say the tromagnetic field or the electron field The excitations of the field will be identified with particles.)

elec-As a Lagrangian we take

L = f [~ (a¢) 2 _ ~ (a¢)2 _ J-L2 ¢2] dx

to quantize, we need to extend the idea of conjugate momentum so that we can work in Hamiltonian form Noting our treatment of the Lagrangian in

in direct analogy to Equation (1.11) Usually this is referred to as the

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"

1(1 r,() III Il l(' qlloll1ll / ,l'(/ V I ' I'~ I ( )fl III Il\'ld Ilwol'y wl'e l l'v.lil' lh 'objec ls </) d l1d

I ' " 1I 1ll' I lI 0 I'S II} .Inti l1 , 'l'he l1 w,' hd Vl' l() POS lUI ;) ll' a pprop ri a Le co mmuta to rs 1,,' IIVI'\'n ll w lll, (pil'asl' a L 0 Ll' 11 Lh Lhi s is a ne w a nd p os tul a tcd id ea We are

1\ 111 1 Il l); ill ,1nill ogy w iLh qU e nLum mcc ha nics, but this lack of commutation

I ", IIVI,,' a 1l Ii} nd it is no t a con seq uence of, for example, x having become a

' 1".t llllll11 o pera Lo r; o n th e co ntrary x is p erfectly classical here For this reason IIII' l'I' ill qu a nLi za ti o n is frequently referred to as "second quantization," and

III Ilh' v('rsio n we s hall propose "canonical second quantization.") Now

• II 1',' ,I as possible in the continuous case The generalization of the Kronecker

",/ w ilh lW O indices over which summation with an arbitrary vector, I, gives

1', 1i " the vec tor as

(1.90)

1," 111 reasonable functions I, as the defining property Again, we notice that

III I':qua tion (1.67), as previously in Equation 1.28, the commutation relations

,II,' between the time dependent operators at equal times, for example,

we simply make a Fourier expansion This time we shall not restrict ourselves

In a box of length L, but let x run from -00 to +00, and use running waves

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1M ( ,rO ll p I n rpri/ Tf'f IIf (" "'11 ( "''-'' IV lflll r l , '/ ' ''''If Ir , ".If' " '' ,,"1' f'f ,'/ 1'""

of them as factors, which allow a smooth comparison with wave function normalization.) When we quantize, this becomes promoted to

with corresponding expression for IT, and the crucial point is that the Fourier

commutation relations (Equation (1.91)) determine the commutators

for the mode operators

[a (k) , a (k' )] = 0

operators for this continuum case We need only substitute our expansions

then using Equation (1.99) we recognize the second term as the infinite sum

of zero point energies, which we have learned to discard Thus we can take

Iitlil w hi ch til (A) will crea te a quantum of frequency w in the now familiar

\\ IV

NI 111 n' the inte rpretation of the ground state as a vacuum with no particles I" tI This is c ntral to the interpretation of modern quantum field theory in

• 1"'111'1ll.1ry particle physics

10 );l'I ,1 littl e more feel for this new structure, consider the amplitude of ¢ 1,, '1 WI 'l'n th e vacuum and the one particle state of momentum p:

Ol</>(x, t)lp > = < Ol¢(x, t)a t (p) IO >

= < 01 f [a(k)e' dk 1 ' k ' X-IW/ +a t (k)e - ' 'k H'w/]a t (p)IO ' >

2Jr 2w

(1.103)

I ,,"11 th e conjugate of Equation (1.103) we see that the second term gives zero,

.llld with this trick in mind we rewrite the first term as

w hercw2 = j)-2+p2 now,and < 010 > = 1 has been assumed You will probably

I, 'l'ognize this as the wave function for this problem (We actually looked at

· 1.1nding waves earlier, which are superpositions of these But the j)-2 = 0 case ' hould be very familiar.) So this is where the old wave functions of quantum ll11'chanics appear; they are vacuum to one-particle matrix elements of the Iwld operators

Fi nally, think about the two-particle state

(1.107)

Ikca use of Equation (1.99), we see that this is symmetric under the kl ~ k2

1I1lcrchange There is no way to distinguish one quantum of energy (particle)

I rom another-we must be dealing with bosons Obviously, something will hdve to be modified later to handle fermions-and then the spin-and inter-.ll'lions between particles

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'II (,rnllp I I Ir'tl nl for rttr' , rrlllrlfl,-11 Nllmrl "I ' ,n,,,,, '"'F''' "'''',., 'I''''''

References

1 E Noether, Hachr Akad Wess Gotingernl Math.-Phys KF> II (19] 8): 235; M.A Tevel

2 RP Feynman, The Feynman Lectures on Physics, Vol 2 Addison-Wesley, Reading,

6 G Sterman Heisenberg Picture, Appendix A An Introduction to Quantum Field

7 G Sterman Schrodinger Picture, Appendix A An Introduction to Quantum Field

8 G Artken Fourier Series In Mathematical Methods for Physicists, 3rd ed., 1985, chapter 14

9 N Highan Kronecker delta Handbook of Writing for the Mathematical Sciences,

Society for Industrial and Applied Mathematics, 1998

10 P.AM Dirac Quantum Mechanics, 4th ed Oxford University Press, London

Problems

1.1 Solve x = -w 2 x to find x = xlI/axcos(wt + 8) where 8 and XII/ax are constants

1.2 Write down the Lagrangian for a body of mass m experiencing the

acceleration g due to gravity Hence, solve the problem of a body dropped from rest (Yes, it really is trivial.)

1.3 For the harmonic oscillator, evaluate S for the three paths (a) q =

for all three paths

1.4 Write out d H to check that the Legendre transformation really does

yield Hamilton's canonical equations

1.5 Show that if the kinetic term has the form

where i and j label the N particles of a system, and the m;j are generalized "masses" independent of the velocities if;, then H =

T + V whenever the potential is velocity independent (These cumstances do arise whenever the constraint equations [from the

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""""' " , , ' " ",,"" ", : , , ,

V llldhl,'~ yll ll 111 ~ ,1 II ',,' III Iii, ' 1/, 1 .111' IIHI.'I Wlllil'111 (\1 y lil1l l' Tr

ll1gll' pdrlicil' in tw o dimen s io n s (irsl

i ll C lrll' s.) s i,111 ,II11l Iilell ill a pOI,'1' l'oordin t'

I f, Chcck lila I I 1<1111 ilion's eq uillions for Probl em 1.2 do yield the same Sl'cond order equation on substitution Solve the first order equa- lio ns dire tty, a nd confirm the previous result

1.7 St<1l't fl' o l11 A = ax + f3p where a and f3 are constants and "design"

lh e forl11 in Equation 1.19 for yourself

I R Show that the two solutions are equivalent and find the

relation-s hiprelation-s between the conrelation-stantrelation-s of the two relation-solutionrelation-s

1.9 Check that the Poisson bracket equations of motion give the usual results for the particle falling under gravity

I JO Check tha t matrix elements of P (t) between Heisenberg states agree with those of P between Schrodinger states (Yes, these questions are trivial.)

1.11 Derive Equation (1.33) from Equation (1.28) 1.12 Derive Equations (1.36) and (1.37) from Equation (1.29)

1.13 If you are not sure about the meaning of Hermitian operators (like

q and p) please look up the idea As a check, show that p + -i ~

(in the Schrodinger representation) is Hermitian

1.14 The operator solution of the harmonic oscillator is of central tance To make sure that you have got the idea up to this point, close your notes and work it again starting with fI = x2 + p2 so that the constants are different

impor-1.15 Go on - do it!

1.16 Take the expectation value of fI = x2 + p2 for an arbitrary state, and use the hermiticity of x and p to show that you have a sum of moduli squared, hence, not negative

1.17 Assume C"IE,,+l > = ':PIE" >, where < E"IE" >= 1 Now use

< E,,+11 E,,+1 >= 1 to find e" is real Now show that Equation (1.51)

is correct Find the wave function for the first excited state, itly Hint: First use Equation (1.49) to find the ground state wave

explic-function

1.18 Derive Equations (1.57) and (1.58)

1.19 This is a question you can ignore if you like Try three equal masses

in a line joined only by springs (of constant g) between the middle one and each of the end ones and otherwise free You should get three equations of motion Try solutions in which all three masses have a single frequency (normal mode, of course), to get three al- gebraic questions Find the three values of the frequency that make these (homogeneous) equations compatible (You can put the de- terminant at zero Alternatively, pick out the zero frequency mode and the problem looks easy enough to guess the configurations of the other modes.)

1.20 Please work out Equation (1.66) for yourself

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1.'2 1 l'I \, v I :O> eiwei 1':qll ,llilltl ( 1.117) Sldll h y lil l' lI SlI,1I t'( lIlllIllll.ll o l'S lor

Ih e (\ ri g in ,,1 v ll's 1I'i "b a nd re l11('m ber lh,lll '7 1, rzl 0 'lIld so fOl'lh 1.22 Wh a t will th e w ave fun c ti o n look like? Write th e w ave fun cti o n

expli citly for 111 = 0 = n2

1.23 If this is not clear to you, try Problem 1.19 then think about Problem 1.22 if there are three masses

1.24 Go on - do it!

1.25 If you feel w eak, just verify that Equation (1.77) solves Equation (1.76) If you feel strong try to prove that this is the general solution (Change variables to x ± ct.)

1.26 Try superimposing two sine waves each of wavelength) and quency v but traveling in opposite directions If Va is the lowest frequency mode on a pipe of length L open at one end and closed

fre-at the other, whfre-at is the speed of sound?

1.27 Go on - do it! [cos2A - cos2B = 2sin(A + B)sin(B - A).]

1.28 Check this please

1.29 Check again please

1.30 Check that the dimensions work out for [¢(;r, t)ii:(y, t) = i8 d (! - y)]

-1.31 Actually it is not quite so straightforward-you need to be able to invert the Fourier transform But you can easily verify that Equa- tions (1.98) and (1.99) do give Equation (1.93), so please do this 1.32 Go on - it gets a bit messy - but you can do it!

1.33 Show this, please By now you really should be able to solve the harmonic oscillator by the operator method!

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2

() lIontum Angular Momentum

Illdt'x notation is the modern and easy way to work through problems such

I' I n lly easy to learn and takes only a little practice to become competent in Ii' li se Parts of this section will appear again later, sometimes as problems,

' 11 (he equations here will be numbered 11, 12, and so forth to emphasize the Iitli llt

Indi ces in this section are lowercase letters that can be attached as ',I I'ipts or superscripts to appropriate things, such as momentum or angular

"I'l'c ifying which direction of component is being treated Such an index,

Il1lp lies summation This is known as the Einstein convention For example,

(12)

IS, (A;)2 with the second component (A;)2 At this stage indices can appear I'it her as subscripts or superscripts The safe way to write things is to reserve

nu mbers for powers (or use brackets) There is a mathematical theorem to the dfect that there are two numerically invariant (i.e., not changing under, say,

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'., \ , /tllI/ ' / /l1 '/I/II/tI/ /1/1 ' ,~' "IIIIIII/ll 1/ 1/11'/ 1\ ,,/ / 111/1111' 111' /,'/1 1 111111 I W ' f,1/1 1i

" ro l.l l ion l 'v(' n if II, docs) tenso rs, 'J'lw sl fir is Ih t' Knll l t'ckcr dd!.1 8,/1 w hi '11

is sy mmetri

c ill

th e

O;j = 0 if i i= j

This has three important implications

which can be seen by writing out the sum as

which then yields, for example,

The second numerically invarianttensor is the Levi-Civita tensor £ijkt which

is totally antisymmetric in the indices with £123 = 1 by convention Obviously

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f ll'S, l/II'

IV,' dl'illll' .In dn g ular mome ntum by a se t of three operators, Ji, i = 1,2,3 or

l 1/ , w hi ch sa tis fy

[fi , JjJ = iCijkh J/ = Ji

(2.1) (2.2)

I • (l ,sl i nct from those appearing once, which is a free one for you to pick The

" III\(' index appearing three or more times is an error

I i('re Fijk is the Levi-Civita tensor (density), which is antisymmetric in any

Now the existence of this (Lie) algebra is very far from trivial and the

I (lllll'n t is very high indeed We will look at the latter aspect first Switch from

IIII' 'artesian basis to a spherical one by defining

(2.9)

(2.10) (2.11)

(2.12)

(2.13) www.pdfgrip.com

.' 0 \ ,/ fl lI/' I "" fl/ '/ I fl l /11 1' ," 1/1/ 1//11/1/ IVll li " ' / (II / '(1/ 1/1 'I' / '111/:,11 , II/II/ 1 / 11' /11 1111

(You sho uld writ e o ut l A, B ' I l'xp y li c itl to un dl.'rstdlld thi s po int )

= i Cijk (J jh + hJ j) = 0 by sy mme try, (2.14)

Such an operator is called a Casimir You can always use Casimirs as part

3-you will learn later that there can be no more-and set up the-eigenvalue problem as

tlf3 , m > = f31f3 , m >

hlf3, m > = mlf3, m > ,

(2.16) (2,17)

Then returning to our theme of the angular momentum we see that

hJ + If3, m > = (1+13 + J+)If3, m > (2.18)

(J +13 + 1+)If3, m > = (m + 1)J + 1f3, m > (2.19)

also

[ 2 1+1f3, m > = J +tlf3, m > (2.20) 1f3, m > = f3 1+1f3, m > (2,21)