(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)

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11" \l' II~"
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Ih e:
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ph )',,~, ,111.1 ,I\ lrophys ics have beco me increasingly interdependent and the aim of this series
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Olha ('('cent books in the series:

Particle and Astroparticle Physics
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Joint Evolution of Black Holes and Galaxies
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Gravitation: From the Hubble Length to the Planck Length
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The Galactic Black Hole: Lectures on General Relativity and Astrophysics
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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 a
I':nt:rgy
, Cos
Physit.s
lllolo

gy, nd (;ravilalion

Group Theory for 0:'?~:!
the Standard Model '

of Particle Physics
and Beyond

Ken J. Barnes
University ofSouthampton
School ofPhysics 6- Astronomy
United Kingdom

0

c

c~

CRC Press
Taylor & Francis Group
Boca Raton London New York

CRC Press is an imprint of the
Taylor &: Francis Group, an informa business

A TAYLOR & FRANCIS BOOK

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Library of Congress Cataloging-in- Publica tion Data
Barnes, Ken ). , 1938Group theory for the standard model of par ticl e ph ys ics and beyond I Ken ). Barnes.
p. cm. -- (Series in high energy physi s. cos mology .•1 lid gl.lV ilalion)
Includes bibliographical references a nd ind ex.
ISBN 978-1-4200-7874-9
1. Group theory. 2. Quantum th eo.ry.""rlil'l
phy,ics)
u:1r,IlIgl' l'

(N dl'ar
I. Title,
QCI74.17.G7B372010
539.7'25--dc22

200902 1685

Visit the Taylor & Francis Web s ite lit
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ss \'(/('()
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ht I p :/I
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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

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v


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'
llll·s'11.'chllit
for 1111' St.lIId •• rtl Motll'1 Lil' Croups ,., .... 1()7
.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 sing the lndices ... ... ...... ............................. . ...... 117
The Classification Theorem (Dynkin) . ...... ... ........ ... .. . .. .... 119
1<('s ulL ... . ..... . ..... ... .. . ........................................ 119
Coincidences ................. ... ...... ...... ........ ... . .... . .... . 119
I ~()(lb

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

II

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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VIII

13 ThcSimplcSph·rc ............................................. I()I
Rcfcrcn es ..... ....... . ....... ... . ... .................. . .......... I g 1
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
Superficlds . .. . ................. . .................................. 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
IIII', hook emerged au t of lectures to first year postgraduate students at the
1111 ' 11 I )vp.lrlmcnt of
1", l nn' I reLired. It is

Physics and Astronomy, University of Southampton,
hoped that this book will be appropriate for similar
),, " 111 ps of readers in many other institutions across the world. Experimenters
111 Illis s ubject would probably gain much from reading this book, although
,111111' may fi nd it difficult.

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/\(,J..:llowLedgments

1111 ', ll(lOk could never have been written without the consistently excellent
114'11' 01 Mrs. Ilannah Williams, who handled LaTeX, figures, and packaging
"I' I', "ll
II\' y
with case. My son, Dr. Geoffrey W. Morton, is also thanked
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
, " Iill'S pf Illy supersymmetry notes. Finally, I thank my wife, Jacky, for her
\ ,111 1illlmi s upport and help when writing anything seemed quite impossible.

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Xl


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IlIlroduction

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

Illlllclrics, mainly described by the techniques of Lie groups and Lie
Although no proofs of theorems and the like are given, except
111 :, pl'ciill cases, the ideas are very firmly based on a lifetime of lecturing
,. l1l'ril'nce.
til "

, iI )', \'111"<1 5.

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xiii


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1
SYIJlmetries and Conservation Laws

~Illi may a lready be familiar with the ideas of conserved quantities, such as
\ I"cIq!, in electromagnetism, but it will not hurt to go through this once more,
,Ilid 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
1'.Irti Ics, and indeed we will start with the charge e on the proton. If we
\ lliisider the charge of the electron (-e), which carries electric currents, what
,II I we mean by "it is conserved" and what consequences might this have? We
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
I he same spatial point, but this is natural when the charges are carried on the
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
/" the current carried by an electron (of charge ( - )e) we can write

(1.1)

'I'hen we have

apat + Y.i = 0

(1.2)

where p is the time component of j" and j is the spatial part of this current.
Ir we integrate over a fixed volume we find

ap
- +
at

.

flow of current normally mto the volume

- flow of current normally out of the volume

= O.

(1.3)

'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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1


'1'l f "/ ~

1 " I""f/l "' 1 "f!

.1 11

.... . " .... . . ~ .... , ./ . . . . . . . . . . . ..., ~ ..

I~

........

o- - • • , __ . ~ ...

p"rtl\ ' I\'~ ' ()I (( lim,\" thi ~)

y w he

rl'l.,livil
in differenl

I'

\'IIII1' (·pllll'l·d
ll ght
:,:-.
dl.l11g111g ill thl ' wo rld Li1l'r
of :-'Pl'Ci.l l
re
l' i:, ,1p pMl'nl CO il 11',1 ' lioll of leng lhs .lIlel dil.llion of limes
·ference frelmes. y imilarl in quanlulTl m' hani s furlher modi-

fications are needed, 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
is a theorem by E. Noether [1] to the effect that this is precisely what happens.
It is not appropriate to prove this theorem at this stage, but it is very powerful 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
quantum aspects. You are strongly urged to read Chapter 19 of Volume II of The
Feynman Lectures on Physics [2] as an introduction to the deep and fascinating
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
harmonic oscillator. We shall call the displacement from equilibrium q (t)
rather than x(t) because later on we shall want "displacements of the fields"
at various points x and we do not wish to confuse the" displacements" with
the spatial positions. Then Newton's second law is replaced by the EulerLagrange [3] equation
d aL
dt

aq

aL
aq

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(1.4)


'r ' • • , , ., , •. ,

", 111'1( '

II

. ' • • •• . , ... .• • , ' • ••• ' .

~

•. ,'

I ~ , III(' 111111' dc 'l'i \','IIIl VI'
,dg l"T
(11'1
III); I,III,ill / ., is

1,, ' lw\'\'11 I Il l' ""1('1,(, l'l\('r);

Illl' dilll 'n' ll l'l'

('/') dlld Ihl' poll'nlidll'l1l'rgy (V), lh 'll is,

(1.5)

be reg.Hdcd aas funclion of th e independent variables q and q for
of pa rli a I differentiation, For the harmonic oscillator with mass

.1111 1 s pring consta nt k we have

011111 I ~ 10

IIII' Ill i q OSl'S
III

V=
\ , 11\'1'('

{() 2

k

mu}

2

2q = - 2- q

2

= .t
, So that
III
m ·
mu} 2
L = - q2 - - q

2

01111 1

2

(1.7)

th e Euler-Lagrange equation yields
d.

dt (mq)
dlld

(1.6)

=

2

-mw q

(1.8)

we retrieve

if =

_w

2

q

(1.9)

vxpccted,
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
.I ',

5=

i

Ii

l!

L(q,q)dt

(1.10)

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{lIf Ir,

'"I ~ '' '

'''''' '" 'f' '''''

'i

rr /2w

FIGURE 1,1

q- t diagram.

Now we can work out the change of action for any path as
05

i
=i
=

tf

t;

I;

tf

(aL
aL) dt
-oq + --:aq
aq
(aL
aq oq + dtd [aL]
aq -

= i tf oq
I,

d
dt

[aL]
aq oq ) dt

( aL _ !£ [aL]) dt + [a~ Oq]tf
aq d t aq
aq I,

where we used oq = ftoq in the second step, But we are considering paths
with fixed end points, so that oq(ti) = a = oq(tf) for any variation, and the
final term vanishes. Hence, since 05 must vanish for arbitrary oq, we need

d
dt

aL aL
aq = aq '

which retrieves the Euler-Lagrange equation of motion. The solution of this
is the qo(t), which gives the path actually followed by the particle.
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
place, is to find a change in variables (from q and q) which will replace the
second order Euler-Lagrange equation by two linked first order equations.
Thi s piece of magic is performed by introducing
p=

aL
aq

(1.11)

"gl'lwril li 7.cd momentum conj ugate to the generalized coordinate q."
(W lwil 1/ is IIH IIll\'nLlI nl , olS we Sh,lli see.) Then th e Hamiltonian is introduced by the

.1S ,1

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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

aH

q= ap
.

(1.13)

aH

p=--,

aq

(1.14)

which re
iJ 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

p=

aL

.

aq = mq,

(1.15)

whi ch is reassuring, and we can then see that from Equation (1.12)

p2

mui

= 2m +-2-q

2

I ~ , I he

form of the Hamiltonian in the new variables. Notice that the Hamiltois the total energy, T + V. This is a very general feature, and provided
I h,l t time does not appear explicitly then
"j,lIl

dH

aH

aH

aHaH
aH

[

aH]

at = aqq + apP = aqap+ ap -aq

= 0,

(1.16)

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

.
p
q =-

(1.17)

P= -muiq

(1.18)

m

when Equation (1.16) is used directly. The first of these reconfirms the definiand on substitution into the second retrieves Equation
(1 .9) as the second order equation of motion. It turns out, however, to be instructive to solve the first order Equations (1.17) and (1.18) directly. Consider
the linear combination
I ion of the momentum,

(1.19)

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(,

'tln"I'

I "("PI

II fT"

'(fr"/

"'''(IIIn, fV'(lII'-'

A= -

"I'

""n~r

I

".'!f' II f' '''''' "' ,' 1"""

iw/l

(1 .20)

A = ae - i wl

(1.21)

as the obvious solution, where a is constant. Taking the complex conjugate of
Equation (1.19), we immediately find
1 1
.
.
x = --(A+ A*) = _ _(ae - 1wt +a *e 1wt ) ,
.J2mw
.J2mw

(1.22)

which is equivalent to the previous solution.

Quantum Mechanics
The passage to quantum mechanics in this formalism is facilitated by introd ucing the Poisson bracket nota tion. The Poisson bracket of any two functions
f and g, of q and p, is simply

{j,g}

of og ofog
== - - - - oq op opoq

(1.23)

= it

(1.24)

{p , H}=p

(1.25)

and we see that

{q,

H}

are alternative ways of writing Equations (1.13) and (1.14), the equations of
motion. Moreover, if F is any function of q and p, then

of
of

dt=aqit+opP
dF

of
= -{q,
oq

H}

of
op

+ -{p, H}

= {F, H}

(1.26)

while

{q,q}=O
{p , p} = 0
{q , p}

=1

(1.27)

follow directly from the definition (Equation (1.24)) of the Poisson bracket.
The tran sition to quantum mechanics is now effected by the correspondence

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

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· '1'" .. ,. I '" ' '''

fI'

,

.. "

. ..

,

.

...

'II,.."

,

'f I '

•

111.1 Illt ' ll' II'1i, IIII

d'111.111111111
'( 'II.1II1
S IH 11\1

" 111 111I,d "

111111 :-.

w ith"

I.)

111

(',

Op('I'.lIOI (,O ITl'

p,lIlicul.ll',
27) yoiclu
n ( l:qLlclti
s

lI)tI (' I1 l'l'S.

(We

L1 Sl'

1.

1t7(t), P(t) 1= i

(1.28)

, I" ('SS ll1 g tlH.' I leisenberg uncertainty principle [5], and Equation (1.26) gives

dF (t)
-;It =

-i[F(t), H]

(1.29)

Il eisenb
erg equation of motion. The time dependence has been exhibto uraw the reader's attention to the fact that this is quantum mechanics
" I""l'ssed in the Heisenberg picture [6], where states are time independent
hili Ihe dynamical variables contain the time dependence,
I Ill' a lternative Schrodinger picture, in which the variables are time indeIlt'l1tient, has the time dependence of state vectors given by the Schrodinger
"l(lldtion
d'.

I Ill'

III·.!

HI1/I(t)

>

= i ~ 11/I(t)

at

>

(1.30)

wi th the formal solution
(1.31)

where we have identified the Schrodinger state at time zero with /1/1(0) >,
wi th /1/1 > the time independent Heisenberg state, Of course, Equation (1.31)
1', just a unitary transformation between the two pictures, with
(1.32)

the corresponding transformation between operators. The important
!(',lture of this is that

.IS

[q , p] = i

(1.33)

lollows immediately from Equation (1.28) as an expression of the uncertainty
pri nciple in the Schrodinger picture, In quantum field theory we shall find
th e Heisenberg picture very convenient.
In the quantum case we have the operator version of Equation (1.15)
(1.34)

(1.35)

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n

"ru"I'

Wllh h ,ll .llI01l 1(1 ..

I flrPI II/fir

IItr-I-' II,,,,,,,,"

) giving 11'1\/1.111

IV III.'"

"/

'

,,,,,,,.

I

'''I''' '

,,,, ,, . . . 'I H"'-.

lhl' l'qll.l lil y of 111l' ~l' .1Ilclll.lll
'
form s.

From th ' 1il'i senbl'rg l'l]ucllion of motion
lhal

( I ~ qllation

( 1.29» nWl'
l'clsily
ca

~.'

(1.36)
(1.37)
so that we get
4(t)

= -muiq(t)

(1.38)

by combining these. Now, please notice that this is not just the classical equation of motion (Equation (1.9» again. What Equation (1.38) tells us is the
behavior of the operator with time, not where the particle can be found . If
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
I want to stress the operator treatment that we shall see again in our field
theory. (If you already know this method, it will at least serve as a review and
to establish notation.)
We seek a set of states lEn >, n = 0, 1, ... , to serve as a complete basis in
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
~2

mw2

2m

2

P
H =-+-q

~2

but our classical treatment suggests Equation (1.19)

a=

~ (q011W+ i Plmw)

(1.40)

fit =

~ (qJmw-ip lmw)

(1.41)

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,,"1""

.". ' ",., . '- •,-"

-'"

, ••• -,. , •• "

17 It7 :

1I11t!)

't

2

1
1

2 111 (0

1'> 2 +

i

- [r) , PJ
2

= ~ ( ~ + JIIu} q2)
\\' 11('1'('

l ~qLl a tion (1.33)

2

2m

w

_ ~2

is used in the last term. Hence we have
i

w

H=wa a+A

(1.42)

2

(1.43)

H = ~(a i a +aa i )

(1.44)

2

[a,a i ] = 1

(1.45)

l\lliow by adding and subtracting. Notice that (from Equation (1.42)),
[H, ail

= wa i [a, ai l

[H , ail

= wa i

(1.46)

= -wa .

(1.47)

[H, a]

( 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 ai 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 nge
d
by the action of ai

= (wa i +a t EII )IE n
=

(Ell

+ w)a i lEIl

>

>

(1.48)

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 aI 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

alEo

> = 0

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(1.49)


\ "Pllf' , "nll'! IP'

1\1

,l:--

il :-

If'

---""frrH"H n --'flp r,

"I' """ Ir

t "'/''''''

.,11"

I" ,,""11

10 11Idllll.llll {'()Il:-.I:-I{'IH' . (Ikw.lrl'!
ch
III r{'I,I[ivistil' physics s ll
not be true .) Bul here you can prove it. From Equ,llion (1.42)

ddillill(lil

rl"l soning
we SCI.' lhal

1

HIEo> = 0 + 2wlEo >

!w

establishing Eo =
as the ground state (or zero point) energy. Then, by
raising, we see that the energy spectrum is

n

= 0, 1, ...

(1.50)

and the corresponding eigenstates are given by
(IP)"

IE" > =

r;:;

lEo >

(1.51)

'" n!
where the exact factor follows from the requirement
<

E"IE" > =1

(1.52)

of normalization. It is now natural to speak of a vacuum rather than a ground
state, and then to envisage the" creation of particles" (or" excitation of quanta")
into that vacuum. Indeed if we define a number operator
(1.53)

to conform to our notation in Equations (1.42) and (1.50), then the change of

notation to

Nln > = nln >
Hln > = E"ln

>=

(1.54)

(n + 1/ 2)wln >

(1.55)

becomes irresistible.

Coupled Oscillators: Normal Modes
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
Sp,)l' l' (e .g., in ,I s lraislot
g ht
on a horizontal table) tied together by a spring of
:-- prin)\ l'Ollsldnl S' ,lilt! li t'd to fixed points by springs of spring constant k.

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• ',If""'"

"'

.1

.. ••••••

..-•• _ ... . . . . .

r..- • ..-

FIGURE 1.2
Three spring forces.

I have in mind the picture in Figure 1.2, where ql and q2 are displacements
from equilibrium, and the Lagrangian takes the form
L

1 .2 .2
1
2
1 .2 .2
= "2 m (ql + q2) - "2 k (ql + q2) - "2 g (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
in this formalism. The Euler-Lagrange equations yield

(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 subtracti ng to obtain
(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
Wl

=

If

(1.61)

l1l1d Equation (1.60) is satisfied trivially by having the two displacements
eq ual. The second normal mode has frequency
W2

=

jk:2g

(1.62)

nd Equation (1.59) is satisfied trivially by the two displacements being equal
hu t opposite in sense. The general solution is then obtained by superposition

.1

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