antonio zichichi (ed.), steven weinberg (auth.) - understanding the fundamental constituents of...
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Understanding
the
Fundamental Constituents
ofMatter
I. 1963
2.
1964
3. 1965
4. 1966
5. 1967
6.
1968
7.
1969
8. 1970
9.
1971
10. 1972
II .
1973
12.
1974
13. 1975
14.
1976
THE SUBNUCLEAR SERIES
Series Editor: ANTONINO ZICHICHI
European Physical Society
Geneva, Switzerland
STRONG, ELECTROMAGNETIC, AND WEAK INTERACTIONS
SYMMETRIES IN
ELEMENTARY PARTICLE PHYSICS
RECENT DEVELOPMENTS IN PARTICLE SYMMETRIES
STRONG AND WEAK INTERACTIONS
HADRONS AND THEIR INTERACTIONS
THEORY AND PHENOMENOLOGY IN PARTICLE PHYSICS
SUBNUCLEARPHENOMENA
ELEMENTARY PROCESSES AT HIGH
ENERGY
PROPERTIES OF THE FUNDAMENTAL INTERACTIONS
HIGHLIGHTS IN PARTICLE PHYSICS
LAWS OF HADRONIC MATTER
LEPTON AND HADRON STRUCTURE
N E W P H E N O M E N A I N S U B N U C L E A R P H Y S ~ S
UNDERSTANDING THE, FUNDAMENTAL
CONSTITUENTS
OF MATTER
Volume 1 was published by W. A. Benjamin, Inc., New York; 2·8 and 11-12 by Academic Press,
New York and London; 9-10 by Editrice Compositori, Bologna; 13-14 by Plenum Press, New York
and
London.
Understanding the
Fundamental Constituents
ofMotter
Editedby
Antonino Zichichi
European Physical Society
Geneva, Switzerland
PLENUM PRESS • NEW
YORK AND LONDON
Library of Congress Cataloging in Publication Data
International School of Subnuclear Physics, Erice, Italy, 1 976.
Understanding the fundamental constituents of matter.
(The Subnuclear series; 14)
Includes index.
1. Particles (Nuclear physics}-Congresses. I. Zichichi, Antonino. II.
Sicily (Region) III.
Rehovot,
Israel. Weizmann Institute of Science. IV.
Title. V. Series.
QC793.I5551976 539.7'21 78-2898
ISBN-13: 978-1-4684-0933-8 e-ISBN-13: 978-1-4684-0931-4
DOl: 10.1007/978-1-4684-0931-4
Proceedings
of the 1976 International School of Subnuclear Physics
(NATO-MPI-MRST Advanced Study Institute) held in Erice, Trapani,
Sicily, July 23-August 8, 1976 and sponsored by the Sicilian
Regional Government and the Weizmann Institute of Science
© 1978 Plenum Press, New York
Softcover reprint of the hardcover 1st edition 1978
A Division of Plenum Publishing Corporation
227 West 17th
Street, New York, N.Y. 10011
All rights reserved
No part of this
book may
be
reproduced, stored in a retrieval system,
or
transmitted,
in any form or by any means, electronic, mechanical, photocopying, microfilming,
recording, or otherwise, without written permission from the Publisher
PREFACE
During July and August of 1976 a group of 90 physicists from
56 laboratories
in
21 countries met
in Erice for the 14th Course
of the International School of
Subnuclear Physics. The countries
represented were Argentina, Australia, Austria, Belgium, Denmark,
the Federal Republic of Germany,
France,
the German Democratic
Republic, Greece, Israel , I taly, Japan, Mexico, Nigeria, Norway,
Sweden,
the United
Kingdom, the United States of America, Vietnam,
and Yugoslavia. The School was sponsored by the I ta l ian Ministry
of
Public Education (MPI), the Italian Ministry
of Scientif ic and
Technological
Research (MRST), the North Atlantic Treaty Organi
zation (NATO), the Regional Sici l ian Government (ERS), and the
Weizmann Inst i tute of
Science.
The program of the School was
mainly devoted to
the elucida
t ion and discussion of the progress achieved in the theoretical and
experimental understanding of the fundamental constituents of matter.
On
the theoretical front we had a series of remarkable lecturers
(C. N. Yang, S. Weinberg, G. C. Wick) attempting a description of
f ini te particles. group of
covered such
topics as the understanding
of the new particles (H. J . Lipkin),
whether
or
not je ts real ly exist (E.
Lillethun), and the unexpected
A-dependence of massive dileptons produced in high-energy proton-
nucleus coll isions (J. W. Cronin). Two other outstanding
questions
were covered
by
E. Leader and G.
Preparata respectively: whether
strong interactions
are s t i l l within the Regge framework, and i f i t
is real ly possible
to master strong interactions. A. J . S. Smith
convinced everybody that a large fraction of single inclusive lep
ton production in hadronic interactions can be accounted for by
pair
production. The
highlights of the School were the ( ~ - e ~ O ) events
presented
by W.
F. Fry. The program was completed
by an excellent
series of review lectures on the more class ical
f ie ld
of Subnuclear
Physics.
I hope the reader wil l enjoy this book as much as the students
enjoyed attending the lectures
and the discussion sessions, which
are the most a t t ract ive
features of
the School. Thanks to the work
of the Scientif ic Secretaries the discussions have been reproduced
as fai thful ly as
possible. At various stages
of my work I have
v
PREFACE
enjoyed the collaboration of many friends whose contributions have
been extremely
important for the School and are highly appreciated.
I thank them most warmly. A f inal acknowledgement to a l l those who,
in Erice, Bologna and Geneva helped me on so many occasions and to
whom I feel very much indebted.
A. Zichichi
January, 1977
Geneva
CONTENTS
THEORETICAL LECTURES
Crit ical Phenomena for Field Theorists • • •
S. Weinberg
Monopoles and
Fiber Bundles
C. N. Yang
1
53
Three Lectures on Solitons • • • • • • • • • • • • • • • •• 85
G. C. Wick
Can We Ever Understand Hadronic Matter?
A Proposal • • • • • • • • • •
G. Preparata
Can Pedestrians Understand the New Particles?
H. J . Lipkin
Are Strong Interactions
St i l l Within the
Regge Framework? • • • • •
E.
Leader
Hadronization of Quark Theories
H. Kleinert
115
179
255
289
Phenomenology of Neutral--Current Interactions • • • • • •• 391
J . J . Sakurai
REVIEW LECTURES
Weak
Currents and New Quarks • • • • • • •
M. Gourdin
Review of Massive Dilepton Production in
Proton-Nucleus Collisions
J . W. Cronin
vii
445
485
viii
CONTENTS
Are Jets Really There? • • • • • • • • • • • • • • • • • •• 507
E. Lillethun
Characteristics
f
~ - e ~ o Events Produced
by a
Neutrino Beam • • • • • • •
W. F. Fry
537
Hadron Physics a t FERMILAB • • • • • • • • • • • • • • • •• 555
T. Ferbel
A Review of
the ISR Results
G.
Valenti
611
The Highlights of the Tbil is i Conference • • • • • • • • •• 663
C. W. Fabjan
SEMINARS ON SPECIALIZED TOPICS
Hadron Nucleus Collisions in the Collective
Tube Model • • • • • • • • • •
G. Berlad
Production
of Dimuons by Pions and
Protons a t FERMILAB
A.
J . S. Smith
Physics with
the
Single Arm Spectrometer
a t FERMILAB • • • • • • • • •
D. Cutts
Azimuthal Correlations in Par t ic le Production
at Low p ~ • • • • • • • • • • • • •
G. Ranft
Monopoles ••••
P. Vinciarell i
Quarks, Color and Octonions
F. Buccella
Field Theory Approach to the Stat is t ical
Bootstrap • • • • • •
E. Etim
CLOSING
LECTURE
Fifty Years of Symmetry
Operators
E. P. Wigner
683
701
741
777
799
841
849
879
CONTENTS
Closing Ceremony •
List of
Participants •
Index
ix
893
895
905
CRITICAL PHENOMENA FOR FIELD THEORISTS
Steven Weinberg
Lyman Laboratory of Physics, Harvard University
Cambridge, Massachusetts 02138
1. INTRODUCTION
~ ~ n y of us
who are not habitual ly concerned with problems in
s ta t i s t i ca l physics have gradually been becoming aware of dramatic
progress
in that f ie ld . The mystery surrounding the phenomenon of
second-order phase transi t ions seems to have l i f ted , and theorists
now seem to be able to explain a l l sor ts of scal ing laws associated
with these t ransi t ions , and even (more or
less) to calculate the
"cr i t ical exponents"
of the scaling laws.
1 Furthermore, the methods
used to solve these problems appear to have a profound connection
wi th the methods of f ield theory - one overhears ta lk of
"re
normalization group equations", "infrared divergences", "ul t ra
violet cut-offs", and so on. I t is natural to conclude that f ie ld
theorists have a l o t to learn from thei r s ta t i s t ica l brethren.
For this reason,
I
s tar ted
a year
or so ago to try to learn
the
modern theory of cr i t ica l phenomena. I t has not been easy.
On one hand, there are a number of authors who use a language that
is unfamiliar to f ie ld theor is ts , involving concepts ( l ike block
spins, l a t t ice spacings, etc.) tha t refer specif ical ly to crystal
la t t ices . I t
is not so hard to rewri te the formulas in a con
tinuum language,
but the physical insight i s harder to translate.
On the other hand, there i s a school of theorists who follow very
closely the formalism of quantum field theory. This makes the
theory
even
harder to understand, because many of the formal de
vices they use (l ike coupling-constant and f ie ld renormalization)
were motivated in
f ie ld theory by the
need to deal with ul t ra-
violet
divergences, a problem that has l i t t l e to
do with cr i t ica l
phenomena. Above a l l , one wonders how renormalizable field theories ,
involving only a f in i te number of interact ion terms, can have any-
2
S. WEINBERG
thing
to do with the effect ive Hamiltonians of classical s ta t i s
t ical mechanics, which must surely involve terms o unlimited
complexity.
These lectures wi l l present what I have been able
to glean of
the
theory of cr i t ica l phenomena. After a brief review of the
f ie ld- theoret ic formalism of s ta t i s t ica l mechanics in Section 2,
the quali tat ive theory wil l be described in Sections 3 - 7, and
quanti tat ive methods wi l l be introduced in Sections 8 -11 . It hard
ly needs
to be said that almost none of the theory I describe in
these sections is original ly due to me. (For detai led
references,
consult
the
reviews l i s ted a t the back of these notes. I) The only
material which may possibly have originated with me is the proof of
the invariance of
the eigenvalues a t a fixed point in Section 7;
the "one-loop equations" presented in
Section 8;
and the use of
renormalization-group methods to deal with the Bloch-Nordsieck
problem in Section 3. Even here, I would not
be surprised to
be
informed by a kind reader that some or a l l of this material already
exis ts
in the published
In the l as t section I t ry to draw some lessons for f ield
theory from our study of cr i t ica l phenomena. The formalism used in
studying cr i t ica l phenomena guarantees that physical quanti t ies
are
cut-off independent for a l l theories , renormalizable or not. What
then determines which of the
inf ini te variety
of possible Lagrangians
in
f ie ld
theory is physically acceptable? Is renormalizabili ty
necessary? How does one handle phenomena
l ike
gravitat ion, where
symmetries seem to rule out any renormalizable theory? Some tenta
t ive answers are
offered, but the
questions
remain open.
2. STATISTICAL MECHANICS AND FIELD THEORY
This sect ion wil l present a very
condensed
review of the f ie ld
theoret ic
formalism of s ta t i s t i ca l mechanics. I
want especial ly to
explain why it i s that the s ta t i s t ica l physicists who study c r i t i ca l
phenomena
can l ive in a
three-dimensional world, unlike field
theorists , who need to work in four space-time dimensions.
Most of you probably know a l l about th is , but a t leas t th is dis
cussion wil l serve to f ix our notat ion.
The aim of quantum field theory i s to calculate S-matrix ele
ments. However, a t
a f in i te temperature there is
no
such thing
as
an
S-matrix; a f ini te temperature means that space i s f i l l ed with
debris l ike black-body radiat ion, so
any part icle that part icipates
in a coll is ion is scat tered again and again before it gets out to
in f in i ty . Instead of S-matrix elements, one wants to calculate a
par t i t ion function
Q = Tr
exp(-H/8)
(2.1)
CRITICAL PHENOMENA FOR FIELD THEORISTS
3
(H
is the Hamiltonian and 8 i s
the temperature in energy units , with
Boltzmann's constant equal to unity.) Of course, H may depend
on
a l l sorts of external currents, in which case Q is
a complicated
functional
of these currents, not a mere
number.
The "temperature
Green's functions" are the derivatives of R.n Q with respect to
these currents.
To calculate Q, one separates H into a f ree-par t ic le term Ho
and an interaction term V, and uses the thermodynamic version of
the Dyson formula:
00 f1
/
8
e
H
o
/8
e-H/8 = \ (_l)n
d d { ( ) ()}
L-- - 1:
1
••• T T VT
I
• . . VT
n. n T n
n=o
o
(2 .2)
where V(T) i s the "interact ion-representat ion" operator
(2 .3)
and
TT is the operator
creasing T-arguments.
with respect to 1/8.)
function i s
which orders the operators
in
order of de
(To prove this formula, simply dif ferent ia te
I t follows immediately that the par t i t ion
Q
00 f1
/
8
\ (_l)n . ~ -Ho/8 }]
L --I dT 1 • • • dT Tr
e T {V(T 1 )
•••
V(T
)
n=o n. n. T n
(2.4)
o
Both Ho and V can be expressed
in terms
of creation and anni
hilat ion operators as
usual. (This i s done even when there are
conservation laws which prevent actual par t ic le creat ion or annihi
lat ion, because
it
i s by far the most convenient way of
incorporat
ing the correct cluster-decomposition propert ies in
the
theory.)
Usually these operators appear n Ho and V in the form of various
fields, l inear combinations Ai(X) with simple translat ion propert ies .
In
the interact ion representation, V(T) i s then a functional of the
interact ion representation "fields"
(2.5)
-+
(The variable x may be continuous, as in f ie ld theory, or discrete,
as on a la t t ice . ) We can apply Wick's theorem to
Eq. (2.4), and
derive a set of Feynman rules which allow us to calculate Q
order
by-order in perturbat ion theory, jus t as in quantum f ie ld theory.
(Details can be found in the book of Fetter and Wa1ecka.
2
) The
con-
-+ -+
t ract ion of two f ields AI(XIT
I
) and A
2
(X
2
T
2
) yields a factor
~ 1 2 ( ~ I - ~ 2 , T I - T 2 ) = Tr[e-
Ho/8
T T { A I ( ~ I T I ) A 2 ( ; 2 T 2 ) } J
(2.6)
4
S. WEINBERG
with TT now defined with an extra
minus sign for T2 TI when Al
and
A2 are fermion f ie ld operators .
The only important
difference between th is formalism and tha t
of quantum theory i s
that we now
integrat ing the
"time"
iT over imaginary values from 0 to i /S ,
instead of over rea l values
from ~ to +00. In consequence, we can express the propagators as
Fourier integrals over momenta but Fourier sums over energies:
with w res t r ic ted
to
posi t ive or negative integer multiples of TIS.
(On a l a t t ice , the
p-integrals are cut off at momenta of the order
of an inverse l a t t i ce spacing, while in
a f ie ld theory they run over
a l l p.) Furthermore, the Green's functions have a remarkable per i
odici ty property: for T in
the range 0 < T <
l /S , we have
lI
12
(;:,T) - Tr[e-
Ho
/
S
TT{AI(;:,i) A
2
( 0 , i -
T
)}]
Tr[e-Ho/S eHo/S A
I
(;:) e-Ho/
S
e H O ( ~ - T ) A
2
(0) e-Ho(i -
T
)]
I 1 J-
TrLAI(;:) e-
HoT
A
2
(0) e-Ho(e-
T
)
Tr[A
2
(0) e-
Ho
/
S
e
HoT
A
I
(;:) e-HoTJ
Tr[e-
Ho
/
S
A 2 ( 0 , ~ ) AI(;:.,T)]
{
+ bosons
- fermions
(2 .8)
I t follows that the sum in Eq. (2.7) runs only over w-values with
w
{
TIS x even integer
TIS x
odd
integer
(bosons)
(fermions)
(2.9)
To calculate Q,
we
add up a l l diagrams with no
external l ines . The
momentum-space
rules
for these diagrams are the same as for the
vacuum amplitude in quantum f ie ld theory, except that every in
ternal energy i s replaced with a quantity iw sa t is fy ing the
quanti
zat ion conditions (2.9), and a l l
energy integrals are replaced
with
w sums:
(2.10)
CRITICAL PHENOMENA FOR FIELD THEORISTS
5
(2.11)
(2.12)
For instance, the Green's function for a scalar boson of mass m is
now
-i -+
6(p,w)
(21T) "
-i 1
(21T) " p
2
+w
2
+m
2
(2.13)
(I should
perhaps add here that Q is given by the sum of a l l
diagrams,
while £n Q i s given
by the sum of a l l connected diagrams.
The average value of any physical quantity A can be calculated by
introducing a term JA in V and noting that
(A)
TdA e -H/8}
T d e -H/8 }
d
8 dJ £n
Q
The
nth
derivative of £n Q with respect to external
currents i s
given
by the sum of a l l connected diagrams with n
external
l ines of
appropriate types. For systems with a
large
volume Q, the quantity
£n Q
wil l contain a factor
and derivatives of (£n
Q)/Q with respect to
various external cur
rents wi l l give the densi t ies of
the physical quanti t ies coupled to
these curren ts . )
A
second-order phase t rans i t ion occurs when one of the renor
ma1ized boson masses of the theory vanishes; the value to which the
temperature must be lowered to make the boson mass vanish i s known
as the cr i t i ca l temperature. (The renorma1ized
mass,
o r inverse
correlat ion length, i s defined in terms of the to ta l inverse
propa
gator a t p=w=O, including a l l "radiat ive" correct ions.
I t
is
therefore a function of the temperature.) Our experience in f ie ld
theory makes it
famil iar tha t a vanishing boson mass signals a
smooth
change in the propert ies of the
physical s ta tes , including thei r sym
metries. This i s to be
contrasted
with
a f i rs t -order phase t ransi
t ion,
in
which the propert ies of physical systems change
abruptly.
For instance, water a t sea
level pressure undergoes a f i rs t -order
phase t ransi t ion, with f in i te la tent heat , a t the boil ing point , but
water a t the
cr i t ica l pressure and magnets in
zero
external f ie ld
undergo second-order t rans i t ions a t the cr i t ica l temperatures.
I t is in a second-order phase t ransi t ion, when one of the boson
masses of the
theory vanishes, that the quantization of vir tual
energies has i t s most
important effect .
6
s. WEINBERG
A quantum f ie ld theory a t
zero temperature would a most have
logarithmic infrared divergences, unless the
Hamiltonian contained
super-renormalizable ~ 3 interact ions. On the other hand,
a t f in i t e
temperature we have only a three-dimensional momentum-space volume
element d
3
p instead of a four-dimensional element d
3
p dw available
to cut down the degree of
infrared divergences, and the Feynman
in tegra ls wil l
therefore have power instead of logarithmic diver
gences. (As far as the three-dimensional in tegra ls with w = 0 are
concerned, even ~ 4 interact ions are
superrenormalizable.) there
fore expect that a t
f in i te temperature there wi l l be dramatic
long
range correlat ions when any of the boson masses approaches zero.
The infrared divergences ar ise solely from the blowing up
of
the w = 0 terms in the sums over boson energies as p + 0; in a l l
other terms the
f in i te
value of w acts as an cut-off .
[See Eq. (2.13).] Therefore, in order to study the long-range ef
fects
which occur when
one of
the boson masses approaches zero, it
i s very convenient to emphasize only the w = 0 terms, by burying a l l
other terms in the sums over w in "black boxes". That i s , we
con
s t ruct an effec t ive three-dimensional f ie ld theory, in
which the
propagators consis t purely of the
w=O terms in (2 .7 ) , with a l l
terms having w'" 0 absorbed into the coupling constants of the e f
fect ive
Hamiltonian.
Of course, th is effec t ive Hamiltonian wil l
consis t not jus t
of the simple interact ions
that may have been
or ig inal ly present in V, but of
an in f in i te number of
interact ions,
a l l with temperature-dependent coeff icients . (For instance, in
~ 4 f ie ld theory, the effec t ive Hamiltonian wil l
contain ~ 2 n terms
produced by loops with n corners, in which al l the internal boson
l ines have w'"
0.)
As long as we keep track of a l l these in te r
act ions, the three-dimensional effect ive f ie ld theory i s supposed
to be ful ly equivalent to the original four-dimensional f in i te
temperature formalism.
In general, we would not expect to know
very
much about the
temperature-dependence of the parameters
in the effec t ive Hamil
tonian. The one thing we do know is that these parameters are
given by
sums over
diagrams with w'" O. Therefore, the effect ive
Hamiltonian i t s e l f does not
the inf luence of the
divergences, and the parameters in
th i s
Hamiltonian
are
expected to
be analyt ic functions of
temperature even near the cr i t ica l tempera
tures . Fortunately,
as we shal l see, th is i s a l l the
information
we need for most purposes.
In renormalizable theories with weak
couplings
and
high
tem
perature, it is possible
to say a good deal more about the
struc
ture of the effec t ive Hamiltonian.
3
(The material in the balance
of th is section i s
offered to readers who want to see some con
crete formulas for coeff icients in the effect ive Hamiltonian. I t
is not needed as a basis for the res t of the lec tures . ) I f the
couplings are weak, loop graphs are generally small , and most of
CRITICAL PHENOMENA FOR FIELD THEORISTS
7
the coeff icients in the effec t ive Hamiltonian simply
have approxi
mately the value they would have had in the original zero-temperature
theory. However, some of the W-sums in these loop graphs may con
t r ibute powers of the temperature to the coeff icients in the e f
fective Hamiltonian, and i f the temperature i s suff ic ient ly high,
these powers of temperature can compensate for powers of the cou
pling. For temperatures which are much la rger than any masses or
momenta, the temperature dependence of a one-loop diagram of dimen
s ional i ty D wil l be simply eD; here D i s jus t the degree
of diver
gence of the theory
a t
zero
temperature. Each loop also introduces
a coupling-constant factor , say f . Hence the leading contr ibut ion
to the effect ive Hamiltonian when e is large and f is small wil l
come from diagrams in
which each loop i s as divergent
as possible.
The worst divergences
in renormalizable theories are the quadratic
divergences in boson propagators, with D= 2.
Furthermore,
the only graphs in which loop
i s
quadrat ical ly divergent c o n ~
s i s t of jus t a s t r ing of
one-loop insert ions
in a
scalar
boson
propagator. Hence,
we can obtain the effect ive Hamiltonian to
lowest order in f but to a l l orders in fe
2
by simply calculat ing
the one-loop corrections to the scalar self-energies .
For instance, consider a theory involving a number of scalar
f ields ~ " with an interact ion
1.
The one-loop correct ion to the boson mass matrix in the effect ive
Hamiltonian i s
L \ M ~ , ( S ) =
1.J
We are only interested here in the leading terms when e i s much
greater than the zero-temperature mass M, so we can rewri te th is as
~ M ~
, ( e )
1.J
-TIe
(2TI)"
I t i s straightforward to calculate
that
for large e
2 -1 2
L\M
i j
(e) -+
24
e f
i jkk
There
are
also terms of f i r s t and zeroth order in e which are re
spect ively l inear ly and quadrat ical ly divergent, but we neglect
these because we now only are keeping terms in ~ M 2 of order fS2
Thus, to a l l orders in fe
2
but 'lowest order in f , the mass matrix
8
S. WEINBt:RG
in the effect ive Hamiltonian takes the form
We
expect the actual phase transi t ions to occur somewhere near the
temperatures where one of
the eigenvalues of this matrix vanishes.
As
expected, the mass matrix Mtj(8) i s perfect ly analyt ic near any
c r i t i ca l temperature.
Similar calculations have been carr ied out for arbi trary
re
normalizable gauge theories .
3
The general resul t is that
M ~ . ( 8 ) ~ M ~ . - 2l482{f"kk+6(e e)
. . + Tr[ r .y r . y ]}
1.J 1.J 1.J a. a. 1.J 1. " J "
where eO. are the
representations
of the gauge generators
on the
scalar f ie lds (including gauge coupling constant
factors) and the
r
i
are the matrices in the
Yukawa
couplinglJjril/JCPi of the
scalars
and spinors.
I t must be stressed that these
formulas only provide an ap
proximation to the "bare" mass matrix appearing in the effect ive
Hamiltonian. The "renormalized mass" (or inverse correlat ion
length) involves
a sum
over higher-order correct ions, which
become
more and more important near the cr i t ica l temperature.
In fact ,
as we shal l see, the renormalized mass has a dependence on temper
ature very different from that found here.
3. THE FLOATING CUT-OFF
The lesson of the l a s t
section
i s that cr i t ica l phenomena are
conveniently described
in terms of
an effect ive Euclidean three
dimensional f i e ld theory with t ~ m p e r a t u r e - d e p e n d e n t interact ions
of arbi t rary complexity. We can now forget the four-dimensional
theory from which we s tar ted , and
deal
with the three-dimensional
effect ive theory on i t s own terms.
We are interested in what happens when one of the boson masses
M in the effect ive Lagrangian becomes
small. Because the theory is
three-dimensional, the infrared divergences become so serious
in
th is
l imi t
that
perturbat ion theory necessari ly breaks down, no
matter how weak the coupling we s tar ted with may be. For instance,
i f the theory contains an in terac t ion ucp", then any graph with L
loops constructed solely from th is interact ion wil l contain
a fac
tor (8u)L. [See Eqs. (2.11) and (2.12).] But i t follows then by
simple dimensional reasoning that when a l l momenta are of the order
of the effect ive boson mass M, the loop integrals must also con
t r ibute
a factor MrL. (In
fact , th is is true for suff ic ient ly
large L
even
when the external momenta are much
la rger than M.)
CRITICAL PHENOMENA FOR FIELD THEORISTS
9
Hence the effect ive coupling constant contributed by each addit ional
loop i s 8u/M. No matter how small u may be, th is becomes large
when
M is suff ic ient ly small, and perturbation theory becomes useless .
What can we do about this? The problem ar ises from very small
vir tual momenta of order M, but the effect ive Hamiltonian we
are
working
with
involves a l l momenta, or a t l eas t a l l momenta less
than an inverse l a t t i ce spacing. Is it possible that
we are work
ing with an inappropriate effect ive Hamiltonian?
Suppose we choose
the
effect ive Hamiltonian to emphasize the
degrees of freedom that are rea l ly important near a phase t r ans i
t ion,
by introducing a momentum-space cut-off A, and integrat ing
only over momenta t with Ipl < A. (Eventually A wil l
be
allowed to
go to zero.) We do not want the introduction of the cut-off to
change the physics, so the effec t ive couplings must a l l be chosen to
depend on the cut-off , in such a way
as to leave the par t i t ion func
t ion A-independent. This condition
imposes a se t of different ia l
on couplings
in the effect ive Hamiltonian,
known
as
the renormalization-group equations.
The idea of
a f loat ing cut-off , introduced in such a way as
not to change the physics, i s actual ly very old. For an early ex
ample, l e t ' s return br ief ly to four dimensions, and consider the
classic
problem
of
Bloch and Nordsieck,4
the radiat ive corrections
to scat ter ing of an electron by an external Coulomb potent ia l .
In
each order of perturbat ion
theory,
we encounter infrared divergences
of the form £n nmy, where my
is
a
f ic t i t ious photon mass introduced
as an infrared cut-off . We a l l know that these infrared divergences
are cancelled by
real photon emission, but l e t ' s forget that for a
moment, and ask what i s the behaviour
of the purely e las t ic matrix
element as my -+O? To
answer th is ,
we introduce a ul t raviole t
cut
off A on the photon momenta, and use an effect ive Hamiltonian, in
which the interact ions are A-dependent
black boxes which include a l l
ef fec ts of "hard" photons with momenta I q I > A. (To define
the
cut-off , we can e i ther simply take Iql as the non-invariant square
q2 +
q
0
2
,
or use a regulator of mass A.) I t is assumed
that
the
photon mass my i s much smaller
than the smallest energy Echarac te r
i s t i c of the scat ter ing process. (Here E is me' or the in i t i a l
momentum
It . I, o r the final
momentum IPfl ,
whichever is smaller . )
We take A iff the range
my « A « E (3.1)
I t is easy to see that the only graphs which can produce leading
s ingular i t ies in my are those containing a single in terac t ion
of
the
(3.2)
plus any number
of
emissions and absorptions of
sof t photons with
Iql < A from the incoming or outgoing electron l ine . (See Fig. 1.)
10
S. WEINBERG
Furthermore,
since
A E, the external electron momenta never get
very far from the in i t i a l and final mass-shell values Pf and p.
(with P ~ = pi = - m ~ ) .
I t follows that the matrix element t a k e ~ the
form m
M(Pf,Pi,me,my) = F(Pf,Pi,me,lf)MA(Pf,Pi,m
e
)
(3.3)
where MA i s the mass-shell matrix element uf rA ui of rA' Note that
Figure 1. A typical
graph which can produce
leading singu
la r i t ies in my. Here wavy l ines are sof t photons; s t ra ight
l ines are electrons near
thei r mass shel l ;
and darkened
circles represent sums over graphs involving hard photons
and electrons
far from thei r mass shell .
HA does
not
depend on my and F depends on lily and A only in
the rat io
my/A, because we took A «E. For instance, the lowest-order radi
ative correction function may be
calculated
direct ly as
(3.4)
where
(3.5)
(3.6)
N o ~ it i s actual ly pret ty easy to calculate F to a l l orders in
a , and even to sum the series. Suppose however that we were
a l l
combinatoric cripples, and were unable to carry out the 'calculation
direct ly. The "renormalization group"
would do the job for us. We
know that the matrix element M must be independent of the float ing
cut-off , so the derivative of Eq. (3.3) with respect to A gives
-my
ClF
ClM
A
o =
M +
F -
A2
Cl(my/A) A
ClA
or in other words
ClMA
Cl,Q,n F
A-=
Cl
,Q,n(my/A)
MA
(3.7)
ClA
CRITICAL PHENOMENA FOR FIELD THEORISTS
11
But MA does not depend on ~ for my « E,
so the
derivat ive
on the
right-hand side must be independent of my/A.
and therefore F i s jus t a power of myl A
F ex: (my/A)A
in agreement with the lowest-order resul t (3.4) .
(3 .8)
(3 .9)
I t can be shown in precisely the same way that the infrared
divergences in any QED matrix element always sum up to a power of
my. (However, the resul ts are
different i f
the electron mass is
zero; for instance, the
lowest-order correct ion involves
~ n 2 m y
in
stead of ~ n my. The
reason
that our
renorma1ization-group argument
breaks down for massless QED i s
that the matrix element
MA
in this
case i s not independent of my, even though it
only involves "hard"
photons with
Iql
~ A » my; this is because a massless photon can
produce
an infrared divergence for any value of
Iql, when it is
emitted para l le l to the i n i t i a l or
final massless electron.
Similar
remarks apply to non-Abelian gauge theories .)
I f my \o ere so small that ~ n ( E / m y ) was of order 137, the ong1-
na1 perturbat ion theory in powers of a ~ n ( E / m y )
would have been use
less . However, by introducing a cut-off A with A » my but
a ~ n A/my « 1, we can use perturbation theory to calculate the
function F as accurately as
we l ike . True, we cannot then
also
calculate the matrix element MA, because a ~ n E / A i s
of
order unity,
but a l l the my dependence is in F, and th is
we can calculate . The
moral
i s clear:
when it i s infrared effec ts that inval idate per
turbat ion theory, the introduction o.f a f loat ing cut-off
may not
restore perturbat ion theory, but it does allow us to
say
useful
things about the infrared effec ts themselves.
4 RENORMALIZATION-GROUP EQUATIONS
The observations of the l as t two sections lead us to consider
the theory of one or more boson
fields
in three dimensions, with a
f loat ing cut-off A, and with cut-off dependent and temperature
dependent interact ions
of
arbi trary complexity, chosen subject to
the condition that the physics be cut-off independent. We wil l
take a single scalar f ie ld ¢(p) for simplici ty, but wil l work in a
Euclidean space with arbi trary dimensionality d, for reasons which
wil l become clear below. The effect ive Hamiltonian may be wri t ten
12
s. WEINBERG
\" (21T) d J d d + + + +
HA/8
= L -- ,- d Pl · · · d P u
(p . . . p ;A)c/>(p ) . . . c/>(p
)
H n n. n n 1 n 1 n (4.1)
We do not expl ic i t ly show the temperature-dependence of the Un;
temperature is now jus t one of the many parameters on which the
rea l symmetric coupling functions un may depend.
Each of the Un contains a momentum-conservation del ta function
+ + d + + _ + +
u (p
. . .
p ;A) = 0 (p + . . . +p )u
(p . . . p ;A)
n l n 1 nn l n
(4.2)
The usual cluster ing propert ies allow each of the Urt to be
expanded
as
a power series
in the momenta;
the coeff icients in these power
ser ies
are our coupling parameters.
In par t icular , the function u
2
may be wri t ten
(4.3)
We can think of the par t i t ion function and Green's functions as
being given by
a sum of Feynman diagrams with
propagators
-d + 1+1
21T) G(p;A)8(A-p) (4.4)
and wi
th vert ices
- (21T) d u
3
,- (21T) d u,,' . . . . [According to Eqs.
(2.11) and (2.12), the propagators and vert ices in the f in i te
temperature graphs in
d+l-dimensional
space-time are associated
with factors
-d- l
2i1T x -i(21T)
(21T)-d
d
-(21T)
respect ively.] The function 8 is taken here as
the usual s tep
function,
but our discussion could be eas i ly adapted to
deal with
a smoother cut-off function.
We are in teres ted in the behaviour of the Green's functions in
the infrared
l imi t ,
when a l l the momenta are
scaled
to zero together.
I f the coupling parameters were al l dimensionless and A-independent
th is would be a t r iv i a l problem, because A would be the only dimen
sional quantity in
the theory, and we demand
that the Green's
func
t ions are
A-independent. In th is case, as a l l momenta are scaled
together to zero, the Green's functions would simply
scale with
the i r
naive dimensionali ty. Of course, l i f e is not so simple, but
we t ry to use dimensional analysis for a l l i t ' s worth.
To th is end, l e t us define a new dimensionless momentum
t - piA (4.5)
CRITICAL PHENOMENA FOR FIELD THEORISTS
13
for which the cut-off i s
(4.6)
Also, we define a new f ie ld
X(£) =a(A)¢(p) (4.7)
with a(A) a constant to be chosen below. We can write the Hamil
tonian in terms
of new coupling
functions
with
-+ -+ -n nd -+ -+
g (9, • • • 9, ;A) =a(A) A u (PI . • . P ;A)
n nn n (4.9)
Again,
we
factor out a delta function
-+ -+
g
(9,
• • • 9, ;A)
n I n
od(£ + . . . +£ )g (£ . . .1 ;A)
Inn 1 n
(4.10)
and find that
- -+ -+ -n nd-d - -+ -+
g (9, • • • 9, ;A) = a(A) A u (p . . . p ;A)
n 1 n n 1 n
(4.11)
In choosing the f ie ld scale factor a(A),
we recal l that our
aim i s to
study the infrared
behaviour of the theory. For th is
purpose,
we must consider the low-momentum behaviour
of the propa
Before we would in general have
- -+ -+ 2 -+2-1
U
2
(p ,-p ;A) -+ m (A) + p Z (A) + . . .
(4.12)
and the re-sca1ed inverse propagator therefore has the behaviour
(4.13)
I t is a very great s implif icat ion to choose a(A) so that the coef
f ic ient of £2 here is a A-independent constant. (We wil l see in
the following sections the
price
that would have to be paid i f we
made any other choice.) While we are at i t , we may as well
pick
a(A) so that th is constant is unity,
i . e . ,
(4.14)
With this
defini t ion, g2
i s dimensionless, and since HAle i s
dimen
sionless , the f ie ld X and a l l the
coupling functions gn( l . . . n;A)
are also dimensionless.
Using (4.14) in (4.11), the coupling functions are now related
by
14
s. WEINBERG
g (t ... t ;A)
n 1 n
nd
- r - n - d n/2 - -+ -+
A Z (A)u (p .•• p ;A)
n
1
n
(4.15)
I f ui is the coefficient of a term of order N in the expansion of
un in powers of p, and gi is the corresponding term in the expansion
of ~ then
= Z(A)n/2
A
nd
/
2
-d-n+N u. (4.16)
gi 1
In par t icular , the mass has n
scaled coupling i s
8
2
(0)
2,
N =
0, so the corresponding re -
(4.17)
The gi [including 8
2
(0)] comprise the dimensionless
coupling param
eters of the theory.
We demand that the Green's
functions of the theory [for the
original f ields ¢(p)] should not depend on A. This imposes on the
couplings
ui a set of
different ia l
equations, giving dUi/dA in
terms of u and A. These equations can then be rewritten in terms
of the dimensionless couplings gi ' and must on dimensional grounds
take the form
(4.18)
(The calculation of the 8
i
wil l be taken up in Section 8.) Note
: ~ : ~ s ~ i b ~ ~ : ~ ~ : ~ ~ ~ e ~ ~ e ; : c ~ ~ c ~ ~ O ~ : e d : : : ~ : r ~ ~ u ~ : m ; : r : ~ : ; ~ : r e ~ ~ ~ a m -
only through the g(A); hence
the
condition that the Green's func
t ions be A-independent must be expressed in terms of the gi them
selves. That i s , the temperature and similar parameters enter the
theory only as " in i t i a l conditions", determining the values of the
g(A) at some arbitrary point A = Ao . Since the 8 do not depend on
any
dimensional
parameters except A, and the 8' s are dimensionless,
they also
cannot depend on A, except through the dimensionless
couplings gi(A). The 8
i
are in general non-zero for a l l interac
t ions, so even i f we s ta r ted with some simple (e.g. renormalizable)
theory, the introduction
of a cut-off would force us to include in
the effect ive Hamiltonian "all conceivable couplings consistent with
the symmetries of the theory.
In addition, the renormalization constants Z(A) or
a(A) sa t i s fy
renormalization group equati-ons, which must be l inear and homogene
ous. [If Z(A) is a solut ion, so must ~ Z(A) be, because we could
have s ta r ted with a f ie ld
¢/12 instead of ¢.] On dimensional
grounds, th is equation must then take the form
A d ~ Z(A) = Z(A)y[g(A)]
(4.19)
Now l e t ' s apply this formalism to the Green's functions of the
CRITICAL PHENOMENA FOR FIELD THEORISTS
15
theory. I f C ~ [ P I . . . PE;u(A) ,A] is a connected (not amputated or IPI)
Green's funct10n with E external l ines carrying momenta Pl· . . P
E
,
then the corresponding Green's function in the re-scaled theory is
D
- E / 2 - -+ -+
A Z(A) CE[p . . . PE;u(A),A]
I (4.20)
where
D = E + Ed - d
2
(4.21)
As always, a bar over r
E
or C
E
indicates
that
del ta function
Od(tl + . . . ) or od(P
I
+ . . . ) has been factored out of these Green's
functions; th is is the origin of the factor A-d in (4.20).
We do not include an expl ic i t A-dependence in r , because th is
Green's function i s d i m e n s i o n l ~ s s , and there are no ~ i m e n s i o n a l
parameters other than A on which it could depend. On the other hand,
-+
for any given theory, the Green's function C
E
must (at fixed p) have an ex-
p l i c i t A-dependence which
compensates for i t s dependence on A through
the coupling parameter u(A):
(4.22)
This i s what we
mean by the physics being A-independent.
The renormalization-group equation for the
Green's functions
can now be derived by
using
(4.20)
to
express fE in terms of C
E
,
then using (4.22) to change the cut-off , and then using (4.20)
again to express
C
E
back in terms of rEo For an arbi t rary momentum
scale parameter K, we have
-r [K
7
Kif .g ( ' ) ] - Z(,)-E/2 , D r ~ [ A K ! I . . . ,·u(A),A]
E )(, I • .. NE ' it - it t
E
E/2 -D - -k -k
= [Z(KA) /Z(A)] K rE [)(,l·· . X,E; g(KA)]
(4.23)
I t is convenient to
suppress the A-dependence, wri t ing
(4.24)
We can regard
giK
as
the solut ion of the
16
s. WEINBERG
(4.25)
with in i t i a l condition
for K 1
(4.26)
Equation
(4.23)
can be rewri t ten (using 4.19) as
(4.27)
We see in par t icular that the behaviour of the Green's for
low momenta i s determined by the solut ion
of Eq. (4.25) in the l imi t
K -+ O.
Each par t icular
physical theory with a par t icular value of the
temperature i s represented by a t rajectory in coupling-constant
space, along which (4.25) i s sa t i s f ied . Different points on a given
t rajectory do not represent different theories , but
only
different
Hamiltonians, corresponding to different cut-offs .
5. FIXED POINTS AND SCALING LAWS
In general , we would not expect the solutions of the renorma1-
ization-group
equations
to have any par t icular ly simple behaviour
for fI.-+O. For instance, i f m
2
(fI.) does not vanish as fI.-+O, then
(4.17) suggests that g2(0) would blow up l ike fI._2. In order to keep
the physics fl.-independent, the coupling constants g.(fI.) would then
also have to blow up for fI. -+0. However, it might b€ tha t for some
special t r a jec tor ies , m
2
(fI.) vanishes for fI. -+ 0, in which case the
various dimensionless
couplings
g.(fI.) might a l l remain well-behaved
for fI.-+O. We are going to see t h ~ t the t r a jec to r ies for which this
happens are jus t those corresponding to the
cr i t i ca l
temperatures
of the theory.
The simplest kind of non-singular behaviour i s for the g.(fI.)
to approach
fixed values gt for fI. -+ O. According to Eq. (4.18);
th is would require that , for a l l i ,
(5.1)
I f a t rajectory leads to such a fixed point for K + 0, then in th is
l imi t Eq. (4.27) gives
*
fE[K£l".K£E;g(fI .)] ex: K-Dp; (5.2)
where
(5.3)
In par t icular , the
two-point
function behaves l ike
CRITICAL PHENOMENA FOR FIELD THEORISTS
r 0::
K 2+y(g*)
2
This resul t is conventionally written in the form
and we see that the "cr i t ica l
exponent" n i s
*
= y(g )
17
(5.4)
(5.5)
(5.6)
The renorma1ized mass ~ (or inverse-correlat ion length 1 / ~ ) i s
defined by
~
/
2-j
d
f2
- l im f2 - - 2 -
K+O
d K
(5.7)
and (5.4) shows that th is vanishes, jus t as we expect at a phase
t ransi t ion.
Even i f
there
is a point g* a t which
8
vanishes, not a l l
t ra jector ies h i t
this point. In order to see what is involved,
consider t ra jec tor ies that pass close to g* The
renormalization
group equation (4.25)
can then be l inearized
M. .
1.J
== [d8
i
(g)]
dg.
J
g=g*
The solution can be wri t ten (barring degeneracies) as
* \' ( ~ ) A ~
giK - gi = l.. c e i K
~
(5.8)
(5.9)
(5.10)
where A ~ i s the
~ - t h
eigenvalue of M, and e ( ~ ) i s the corresponding
eigenvector (with a fixed but arbi trary normalization)
L
Mi ' e ~ ~ )
j J J
(5.11)
The eigenvectors are
class i f ied as infrared-at t rac t ive or infrared
repulsive, according as A ~
> 0 or A ~
< O. (The case A ~ = 0 is a
nuisance, and wil l
not be
considered here.) Clearly, the condition
for a t rajectory actual ly to hi t the fixed point is that c ~ = 0 for
a l l infrared-repulsive eigenvectors e ( ~ ) . The number of parameters
which have to be adjusted to achieve
th i s
is jus t the number of
infrared-repulsive eigenvectors.
For
a phase t ransi t ion of the
18 s. WEINBERG
usual type, in which
there i s jus t one parameter (the temperature)
tha t must be adjusted
to achieve a vanishing renormalized mass,
there must be jus t one repulsive eigenvector.
We wil l see in
Section 11 whether th is is
actual ly l ikely to be the case.
This approach yields useful information even when we are not
precisely
a t the cr i t ica l temperature.
In general, we would know
almost nothing about the temperature-dependence of the coupling
parameters. However, a coupling function
~ ( P l " ' P n ; A ) may be
thought of as a black box in which are buried al l effects of vir tual
par t ic les with momenta greater
than A (or energies w" 0; see Sec. 2)
so it is immume to infrared effects caused by vir tual par t ic les
very low momenta. That i s , unlike the functions, the
coupling parameters ought to be smooth functions of the temperature,
even near a cr i t ica l temperature.
I f there is jus t
one repulsive eigenvector e(R) , and i f 8
c
is a cr i t ica l temperature at which the trajectory hi ts the fixed
point , then the coefficient of th is eigenvector in (S.lO) must
vanish at 8
=
8
c
:
(S.12)
Therefore, by the above smoothness argument, we expect that for 8
near 8
c
'
C a:
(8 - 8 )
R c
(S .13)
As long as the gK are
suff ic ient ly close to g*, the couplings are
functions only of
g. _ g ~ a : ( 8 _ 8 ) K - 1 / V
1K 1 C
(S .14)
where V i s another c r i t i ca l exponent, defined in terms of the single
repulsive eigenvalue AR by
V = -lIAR> 0
(S .1S)
From
Eq. (4.27), we find that the Green's function with E external
l ines has infrared behaviour
*
K - ~ f [(8 - 8 )K-l/V]
E c
(S.16)
v ~ -V
(8 - 8
c
) - -I'.; FE
[K(8 - 8
c
) ]
or equivalently
(S .17)
with D ~ given by (S.3) and (S.6) as
D ~
= E ~ + 1 - ¥) -
d
(S .18)
and fE and FE unknown functions of a single
variable. Equation
CRITICAL
PHENOMENA FOR FIELD THEORISTS
19
(5.2) can be regarded as a
special case of the resul t (5.16), for
6 = 6
c
•
I t must be stressed that whether or not we are able actually
to calculate the c r i t i ca l exponents n and v, Eq. (5.16) or (5.17)
contains a remarkable quantity of information. Who would guess
that the Green's would take such a simple form, with scal
ing parameters n and
v that
do not depend on the number of
external
l ines?
One case of special in te res t i s provided by the l imi t K +0,
in which a l l momenta scale to
zero
together . For 6 + 6
c
there are
no
infrared divergences, so we expect FE in Eq. (5.17)
to be ana
ly t ic a t K = o. We can write the two-point function in th i s l imi t
as
with t;, a "correlat ion length". Comparison with Eq. (5.17) shows
that for 6 + 6
c
' t h ~ correlat ion length exhibi ts the
scal ing
be-
haviour
-v
t;, ex: (6 - 6 )
c
(5.19)
This
is the way V
is usually defined.
Note tha t the "renormalized
mass" l/t;, does vanish for
6 + 6
c
' showing again that th is rea l ly
is
a second-order phase t rans i t ion .
As a special subcase, consider the par t i t ion function i t se l f .
As we
have already remarked, R-n Q i s the
connected Green's function
with no external l ines . Since
we divided by a a-function in
defin
ing fE ' the function fo is jus t the free energy density
(5.20)
where Q is the volume of the system. Because there are no external
l ines , fo cannot depend on a momentum scale K, so (5.17) gives
*
ex: ( 6 - 6 )-VDO = ( 6 - 6 )+Vd
Q c c
(5.21)
The to ta l energy density per uni t volume
i s then
a [ ~ ] Vd-l
u = - a 1/6) Q ex:
(6
- 6c)
(5.22)
Finally, the speci f ic heat per uni t volume
i s
au -CJ.
c = as ex: (6 - 6c)
(5.23)
where CJ. is yet another cr i t ica l exponent
= 2-vd
(5.24)
20 S. WEINBERG
6. EXTERNAL-FIELD PROBLEMS
We can also apply the same formalism to study the effect of
external f ields. Suppose we add a "magnetic" perturbation to the
Hamil tonian
The
effect
is to
change the free energy density to
W(h)
00 hE
I
CE(D,D, . . . D) E
E=l
(6.1)
(6 .2)
But for zero external momentum, Eq. (5.17) gives
*
\ ~
\ d -\ d-H- . l) E
(8-8) (8-8) 2 2
E(D,D, . . . D) ~ fE(D,D, • . . D) ~ (8-8
c
)
c c
(6 .3)
Hence the free energy density takes the form
W(h) (6.4)
with S some unknown function of a single variable. The "magnetiza
t ion" (<P) :: M i s defined by the condition that
d
M = - 8 dh W(h)
\ ( ~ - 1 + J . ) f, - \ ( ~ + l - . l ) J
~ (8-8
c
)
2 2 s' L(8-8
c
)
2 2
(6.5)
In part icular , it may be possible to
have a spontaneous magnetiza
t ion: M I D for h = D.
In this case, Eq. (6.5) gives
M ~ (8-8 )8
c
(6.6)
where 8 is another cr i t ica l exponent
(6 .7)
More generally, the value of the
external f ield h required
to pro
duce any given magnetization is given by Eq. (6.5) as
\ ~ + l - . l )
h = (8-8) 2 2 x function of M(8-8 )-8
c c
or
equivalently
h
MO x function
of M(8-8 )-8
c
(6 .8)
(6.9)
CRITICAL PHENOMENA FOR FIELD THEORISTS
where 8
is one
more (the l a s t )
c r i t i ca l index
(d/2 + 1 - n/2)
8 = v(d/2 + 1 - n/2) /S = (d/2 _ 1 + n/2)
21
(6.10)
In f i e ld theory, it is very convenient to work with a potent ial ,
defined l ike the
Gibbs free energy as a Legendre transform
V(M) ~ W(h) -
hM
From
the above resul ts , it i s easy to see tha t
V(M) cr M
8
+
l
x
function of M(8-8 )-s
c
(6.11)
(6.12)
7. INVARIANCE OF THE EIGENVALUES:
CUT-OFFS VS. RENORMALIZATION
The discussion in Sections
5
and
6 has shown that
c r i t i ca l
phenomena are to a great extent governed by the number and the
values of the repulsive eigenvalues of the matrix (5.9) . This
natural ly ra ises the question whether the
eigenvalues are invariant
to
possible re-def in i t ions of the coupling parameters or the cut -off
procedure. In par t icular , would we obtain the same eigenvalues i f
we assumed the theory was renormalizable, and used a f loat ing re
normalization point (as in the original
work of Gell-Mann and Low
5
)
instead of a
f loat ing cut-off?
Suppose we introduce a new set of dimensionless coupling p a r a ~
eters ga(A).
They are defined in terms of a
dimensional quant i ty A,
which may be a new
kind of cut-off (perhaps smoother than a 8-func
tion) or a
f loat ing reQormalization point . I f the theory
i s non
renormalizable, or
i f
A i s some sor t of c u t - o f f ~ there
are as many
ga as gi . I f the theory i s
renormalizable and A is a
f loat ing re
normalization point there are only a f in i t e number of ga parameters,
and we are res t r i c ted to a finite-dimensional surface of t r a jec tor ies .
The old dimensionless
coupling parameters g i can only be
func
t ions of the
new parameters ga
and of the dimensionless ra t io A/A:
g. (A) = g. [g(A) ,A/ AJ
1 1
(7.1)
Further, the value of the
old cQupling parameters cannot depend on
how we choose the
new quant i ty A, so
- d '\ ago -
o = A -= g(A) = L ~ S
dA
a
ga
a
A
ago
+ _ 1
A a(fi.; A)
(7 .2)
-
where
S i s the new beta function
22
S. WEINBERG
(3 C)
g
(7.3)
The old beta function is then
or using (7 .2)
ag. _
(3. (g) = I a-
1
(3 (iD
1 a ga a
(7.4)
That is , (3 transforms l ike a covariant vector in the space of co
ordinate parameters._ One immediate consequence is the invariance
of fixed points: i f (3 vanishes a t ga = g:, then (3 vanishes a t the
corresponding point gi = g ~ . Now, how does the matrix (5.9) t rans
form? From
(7.4) , we have immediately
t
- ~
(3. ag
j
a g. ag. a(3 (7 5)
I ~ ---
=
I - :: S + ~ _a .
j ag ag
b
a ag}gb a
aga
ag
b
This is moderately complicated
(derivat ives of vectors are not
generally tensors) but it simplif ies a t a fixed point , where the
f i r s t term on the r ight vanishes. At such a point , (7.5) reads
I
I
-
M .
Sjb
S. M b
1J
1a a
j
a
(7 .6)
where
M .
-
[a(3i (g) la
g
j
1 * .
J
g=g
(7 .7)
-
[aSa(g)/a
g
b
]_ -*
ab
-
g=g
(7.8)
s.
=
[agi/agaJ
a
- -*
g=g
(7 .9)
-
I t follows then that i f e i s an eigenvector
of M with eigenvalue A,
I Mb
b a
e =
b
(7.10)
CRITICAL PHENOMENA FOR FIELD THEORISTS
then there is a corresponding eigenvector of M
with the same eigenvalue
e .
1
Is. e
1a a
a
I
M • e . =
Ae
. 1J J
J
23
(7.11)
(7.12)
The are therefore independent of the cut-off procedure.
Also, if_we constrained the theory to be renorma1izab1e, and cal
culated S using a f loat ing renorma1ization point instead of a f loa t
ing cut-off , the eigenvalues we would obtain would be some subset
of the eigenvalues obtained with
a
cut-off .
The use of renorma1iz
able f ie ld theories in s t a t i s t i ca l mechanics does not give a l l the
eigenvalues of M, but those it gives, it gives As i t
happens, one of the eigenvalues of M n a renorma1izab1e q," theory
is the repulsive eigenvalue AR' so renorma1izab1e theories can be
used to calculate
the
c r i t i ca l exponent v, as
done by
Brezin e t al.
1
8. THE ONE-LOOP
EQUATIONS
I t is remarkable tha t the functions Si(g) can be calculated
exactly in terms of one-100o diagrams. The derivation i s lengthy,
and wil l be out l ined
in an Appendix. Here,
we j u s t give the resul ts .
The one-loop equations ~ r e w.ritten in terms of a modified re
scaled coupling function ~ (Q, l ' . . Q,n;A) . [These are not precisely
the same as the coupling functions discussed up to now - the re la
t ion is explained in the Appendix. This modification wi l l not be
important in using t h ~ one-loop equations to study c r i t i ca l phenom
ena.] Let -(2TI)d
Ln
(Q,l . . . n;]J,A) be the sum of al l one-loop, one
par t ic le i r reducible g r ~ p h s , with n external l ines carrying out
going
rescaled momenta Q,l' . . In. These graphs are constructed with
vert ices
d d - + -t---t--t-
-(2TI) 0 (Q,1+"'+J(, )g (x, ••• x, ;A)
n n 1 n
(8.1)
and propagators with an infrared cut-off
(8.2)
where
(8.3)
Propagators are ngt i n c 1 ~ d e d on the
external
l ines
of Ln, and a
del ta-funct ion
O(Q,l+' " + ~ n ) i s factored out, as indicated by the
bar over L. The one-loop equations take the form
24
s. WEINBERG
d - -+ -+ -+ 7-
AdA g (J/, ... /, 1,-J/,1-"'-)(, l;A)
n 1 n- n-
For instance, l e t us suppose that our theory i s invariant
under a transformation ~ -+ - ~ , so that a l l
odd couplings are el imi
nated. For n = 2 and n = 4, the one-loop integrals are
_(2n)d (Q: -i·].1 - ~ J d d J / , 8 ( l i l - ] . 1 ) L ' 1 ( i ' A ) - g (i - i1 'A)
2 l ' 1" , .. ' , l ' l '
and (8.5)
-(2n)d L . (i
1
. .1 .. ;].1,A) :: - ~ J dd J/, 8( lil-].1)£\(t;A)g6 (i,-t,i1'" .1 . ;A)
+ ~ J dd J/, dd J/,' 8( Itl-].1)8( Itl-].1')£\(t;A)£\(t, ;A)
x [ ~ ( t - t ' . i 1 ,i2;A)g . ~ , - t t 3 , i . ;A) 8 ( t1+t2+t- t ' )
+ 2 permutations] (8.6)
where 1 .. :: - t1 - t2- t3 ' (See Figure 2.) In consequence, the one
loop equations for g2 :: L'1-
1
and g .. take
the form
A L'1- 1(1 . A)
dA l '
and
f
Y
-
2
+
t
l' a t J L'1-
1
(i
1
;A)
- ~ (2n) -df d J/, 8 ( I i i - I ) £\(t ;A)g . (1,-1,
t1
, - t 1 ;A)
(8 .7)
r
d
-
4
+
2Y
+ I J/,r' ~ l g .. ( t1 , • . . J . ;A)
~ r=l aJ/,t
- ~ ( 2 1 T ) -dI dJ/,8 ( Ii l - l )£ \ ( i A):g6 ( l , - t J l ' • •
t .. ;A)
+ ~ ( 2 1 T ) - d IJ/, ddJ/,' {8 ( I i i - I ) 8 1 ~ I - l ) + 8 (11'1-1) 8 (I i i - I )}
x
L'1(i
;A)£\(1i;A)[g . ( t - ~ .t
1
.t
2
;A)g . ( ~ ; - t ~ f % >
8 d ( 1 - 1 i + ~ + ~ )
+ g
. ( t , - ~ , i 1 J
3
;A)g . ( ~ , - t . t 2 .1 . )8
d
( t - t l+t
1
+ i
3
)
+ g . ( t , - t ' . t1 .1 . ;A)g ( t , , - t , i
2
.i
3
)8
d
( t - t l+t
1
+t")-11
(8.8)
CRITICAL PHENOMENA FOR FIELD THEORISTS
25
Since the fields are normalized
so that the coeff icient of
£2 in the
1
L2
=
4
2
4
L4 =
+
3
1
3
4 4
3
+ +
Figure 2. Graphs for the one-loop functions L2 and L
4
•
Here in ternal l ines represent the cut-off propagators (8 .2 ) ,
while the small circles
represent the interact ions gn.
power ser ies expansion of
/'>,-1(£1;11.) i s
unity,
Eq. (8.7) yields a
formula
for
y :
J d Q, 6( \£\
-1)1'1(£;II.)g4 (£'-£ '£1 '-£1 ;11.)
(8.9)
Note that the integrals in a l l these equations are taken over a
closed d-l-dimensional surface, and are therefore automatically
f in i te .
These equations are a l l exact. The "mistake" we 'make in drop
ping graphs with more than one loop i s
cancelled by the
"mistake"
we make in dif ferent ia t ing only the 8 ' s , not the
g ' s . However, the
fact that we have
exact
equations in closed form does not mean that
we can derive an exact solut ion. The equation for g2 involves g4;
the
equation for 84 involves g6; and so on. Only by using some
sor t of perturbation theory can we get useful resul ts .
26 S. WEINBERG
9. THE GAUSSIAN FIXED POINT
A fixed point i s characterized
by a set of coupling functions
~ ~ gt,
g ~ ,
. . . for which
the right-hand side
of Eq.
(8.4) vanishes.
There
is one obvious such fixed point , with
* -+ -+
g ( £ 1 " ' £ )
n n
°
(for n > 2)
(9.1)
[Equation (8.9) gives y = 0. ] This is known as the Gaussian fixed
point , because
i
corresponds to a free massless f ield theory, for
which f ie ld distr ibutions are Gaussians. The Gaussian fixed point
turns out to be re la t ive ly unimportant for s t a t i s t i ca l mechanics,
but the reasons why this is so are enlightening. Also, the
Gaussian fixed point wil l play an importan t par t for us in our dis
cussion of renormalizabi l i ty in Section 12.
To
find the eigenvalues of the M-matrix
at the Gaussian fixed
point , we note that there is a simple
basis in
which the matrix is
t r iangular . (That i s , the matrix has only zero elements above the
main diagonal.) Observe that there i s no term in Adgn/dA l inear
in gm for m< n, while the term l inear in gn i s simply
(9.2)
Hence H is t r iangular in a basis in which the vectors correspond to
interact ions i with defini te numbers n
i
of f ields and Ni of momentum
factors . The eigenvalues of a t r iangular matrix are jus t i t s d i
agonal elements, so
(9.2)
shows that the eigenvalues a t
the Gaussian
fixed pOint are the numbers
D.
1
n.d
1
2
(9 .3)
l.Je recognize
that
Di i s
jus t the "dimensionality" of renormaliza
t ion theory: an in terac t ion is jus t super-renormalizable i f D < 0,
jus t renormalizable i f D=O, and not renormalizable i f D>O.
For d > 2, the value of Di increases e i ther ni or Ni is in
creased. Here are the
eigenvalues for the interact ions with the
lowest values of n i
and Ni:
CRITICAL PHENOMENA FOR FIELD THEORISTS 27
Interact ion
n N D
</>2
2
0
-2
</>0
2
</>
2
2 0
¢ 0" </>, ( [J 2 </» 2
2
4
2
4
o d-4
4 2 d-2
6 o
2d-6
We see that for d = 3, there are two repulsive eigenvectors with
D < 0, corresponding to the super-t=en"ormalizable interact ions </>2 and
cp", and
one marginal eigenvector with D
= 0, corresponding to the
renormalizable interact ion </>6. (We do not include </>0
2
</> as an
"interact ion", because i t s coefficient
is
fixed by our f ie ld re
normalization convention.) Even for 3 < d < 4 ,
there are two repul
sive eigenvectors of M. Hence in order for the Hamiltonian to ap
proach the
Gaussian fixed point as A -+ 0 it would be necessary
to
adjust at leas t two free parameters
in the Hamiltonian to eliminate
the components
along both repulsive direct ions, not jus t the tempera
ture.
This i s what i s cal led a t r i c r i t i ca l
fixed point.
Tricr i t i ca l fixed points do occur in nature,
as
for instance
in antiferromagnets and He
3
-He" mixtures. However, the second-order
phase transi t ions considered here (as
in ferromagnets a t zero f ie ld ,
or water a t the c r i t i ca l pressure) occur a t a
cr i t ica l value of jus t
one parameter, the temperature, so they have nothing to do with a
t r i c r i t i ca l fixed point , l ike the Gaussian fixed point. We must
look for some other kind of fixed point.
10.
THE 'HLSON-FISHER FIXED POINT
We have already
noted
that the renormalization-group equations
cannot generally be solved without some sor t of perturbat ive ex
pansion. However, per turbat ion theory would
not generally
be ex
pected to work a t a l l near any fixed point except the Gaussian
fixed point. The S-function for an interact ion i involving ni
28
S.
WEINBERG
factors
of
<p and Ni
factors of momentum is
given
by Eq.
(8 .4)
as
f\
D.
g.
1 1
+ loop
terms
(10.1)
where
n.d
D.
1
d-n.+N. (10.2)
1
2-
1 1
and the "loop terms" arise from the y and L terms in (8 .4) . At a
fixed point g*, a l l 8i must vanish, so the f i rs t -order terms D i . g ~
must cancel the higher-order loop terms. In general, th i s would not
be possible for small non-zero values of the g ~ .
However, suppose that one
of the gi ' say gI ' has a very small
dimensionali ty D
I
, while DI1>DIII' . . . are a l l of order uni ty or
greater . I f we tentat ively suppose that gI i s small, then we can
find functions g ~ I ( g I ) ' g ~ I I ( g I ) ' . . . as power series solut ions
of
the equations
These power series can then be inserted
in the "loop terms" for 8
1
,
generating a series in powers of gI:
(10.4)
For small D
I
, the
equation 8
1
= 0 has a perturbat ive solution
DI b D ~
--+---
a a
3
(10.5)
Note incidental ly
that th is solution for g* may be useful even i f DT
is not so
small ,
provided that the coefficient a i s suff ic ient ly
large;
th i s is what actual ly seems to make the method work
in
pract ice.
Of course, for any integra l number d of spat ia l dimensions,
each dimensionality (10.2) is an integer or half- integer . Therefore,
in order to use the above expansions, we have
work
in
a mythical
world, in which d i s nearly but not quite
equal to a whole number.
To see how this works in pract ice, we return again to our
standard example, of a single scalar f ie ld <p, with a symmetry under
the transformation p + -<p o eliminate the interact ions
odd in <p.
Inspection
of the table in Section 9 shows that when d is near 4,
the dimensionality of the <p4 in terac t ion i s small, while a l l other
interact ions have dimensionalit ies of order unity or
greater . (We
exclude <p0
2
<p from the l i s t of interact ions, because i t s coupling i s
fixed by our field-renormalization convention to have the value
unity.) Since we are real ly in teres ted here in d=
3, we wil l there-
CRITICAL PHENOMENA FOR FIELD THEORISTS
29
fore set
d = 4 - £ ,
0<£«1 (10.6)
and hope that the expansion in powers of £ gives good results even
at £ =
1.
As we have seen, the f i r s t step in
finding the fixed
point
would be to express a l l other couplings in terms of the coupling
gr
of the ~ q interact ion. r t i s easy to
see that, apart from the ~ q
(and ~ 0 2 ~ ) interact ion, the coupling parameter gt(gr) of any in ter
action with n i factors of the field ~ wil l have a power series ex
pansion in
gr
which begins with a term of order
(10.7)
To check th is , we suppose that it true, and see i f it gives
consistent results a t a fixed point. Equation (10.7) shows that
g (p,p ') i s of order gr ' except
for
the zeroth order term p2 0
(p+p')
wfiich i s fixed by our field-normalization
conventions to have co
eff ic ient unity. Hence the propagator takes the form
~ ( t )
~ ;2
+
O(gr)
R.
(10.8)
A l s o ~ Eq. (10.7) shows that gq(tl t213tq) is of order g ~ , except for
the ~ - i n d e p e n d e n t term, which i s of course equal to gr by defini t ion.
A one-loop
graph
in Si
that i s bui l t entirely out ? 7 2 ~ q c o u p l i n g s
will have ni /2 vert ices, and hence be of order
gr
n1
,in agreement
with the order the term Digi in Si as
given by Eq. (10.7). These
terms therefore can cancel a t a fixed point satisfying (10.7).
(About the y term, see below.) On the
other hand, i f
we construct
the loop
from r ~ q vert ices and s other vert ices with m
1
,m
2
, . • . ,m
s
external l ines
apiece, then we must have
+ m - 2s + 2r
s
so the number of powers of gr in the graph wil l be
~ ( m + m + . . . + m ) + r = ~ n. + s
1 2 S 1
which i s greater t h a n ~ n .• This incidental ly shows that in lowest
order,
the one-loop grapfis for Si can be calculated using ~ q coup
l ings only, a fact that
wi l l prove useful l a te r .
Finally,
we must give special at tention to the ~ 0 2 ~ " interac
t ion".
This has D=O
(for
a l l d) and accordingly Eq. (8.9) gives
the fixed-point condition for this coupling as
30
S. WEINBERG
(The loop terms here are of order gi rather
than
gr '
because the
one-loop graph constructed from a ~ 4 interact ion is jus t a constant;
it therefore contributes to 8 for the ~ 2 interact ion, but
not to
the ~ 0 2 ~ or higher-derivative interact ions.)
The fact that y* is
of order gr2jus t i f ies
the
neglect of y in
the estimates made above.
Note
that i f we had not renormalized
our f ields, we would not have
the
term y avai lable to cancel the
one-loop terms in
the
8-function
for the ~ l J 2 ~ interact ion, and in consequence there could not be
a
fixed point of the type considered here. This is the whole motiva
t ion for field-renormalization
in s ta t i s t ica l mechanics - make a
fixed point possible.
Now we are in a posi t ion to do some actual calculat ions. From
Eq. (8 .8) , we see that the 8-function for the
~ 4 interact ion i s ex
actly given by
- ~ ( 2 7 f )
-dJd JI, 0 (
I i - I ) L'l(i ;1\)g6 (t,-t 0,0 ,0 ,0 ; 1\)
+-f(27f) -dJd JI, 0 ( Ill-1M
2
(i 1 \ ) g ~ (i -1 ,0,0; 1\) (10.9)
[The two terms in the curly brackets in (8.8) add up to jus t o( 111-1).
The factor 3 arises because there are three equal
terms
in the
l a s t integral in (8 .8 ) . ] We have seen that when the fixed-point
equations are used to express al l the other couplings
in
terms of
gr ' it turns out that
y
O ( g ~ )
g6
O ( g ~ )
g4
gr
+
O ( g ~ )
1:1-
1
=
12
+ O(gr)
Therefore in lowest order , we can simply drop y and g6' set 1:1 = 1/1
2
,
and replace g4 with the constant gr ' so that (10.9) gives
3 -d 2 3
8
r
(gr) = -e:g
r
+ z(27f) Sd gr + O(gr)
(10.10
where Sd i s the area of a unit
sphere
in d dimensions.· Solving
the
equation
8
r
= 0 then gives the Wilson-Fisher fixed point
CRITICAL PHENOMENA FOR FIELD THEORISTS
31
(10.11)
11.
CALCULATION OF THE EIGENVALUES
We have found a
of order € or less .
la te the eigenvalues
fixed point
in 4-€
dimensions with couplings g*
Now we shal l use perturbat ion
theory to calcu
of the matrix a Si/Cl gj a t th is fixed point.
From Eq. (10.1), we
see
that
Hij - (a s/a gj]g=g*
(11.1)
with ~ M i j
of
order € or less . In zeroth order, the j t h component of
the
i th eigenvector
of M is
jus t
0 i j ' so f i rs t -order perturbat ion
theory gives the eigenvectors of M as the "expectat ion value"
[
as 'J
.
'" D. + ~ M . . = ~
1 1 1 1 gi *
g=g
(11. 2)
to order €.
I t is convenient to consider three special cases:
(a) Any in terac t ion with Di « - €
< 0 corresponds to a negative
eigenvalue of M, and hence to
an infrared-repulsive eigenvector.
In our standard example of a single
rea l sca lar
f ie ld with a
¢ + -¢ symmetry, there i s jus t one of these eigenvectors, correspond
ing to the ¢2 in terac t ion, with D = -2.
(b)
Any in terac t ion with D i » € > O corresponds to a
posi t ive eigen
value of M, and hence to an inf rared-a t t rac t ive eigenvector. There
are an in f in i te number of these.
(c) Any interact ion with IDilof order € corresponds
to a borderline
eigenvalue of M, which might be posi t ive or negative, depending on
the value of the interact ion term ~ M . In our standard example,
there
is
jus t one of these, the interact ion ¢4, with DI = -€.
(Recall that ¢02¢ is not counted as an in terac t ion. ) This is re
pulsive at the Gaussian fixed point , but we cannot
t e l l whether it
i s repulsive or at t rac t ive a t the
Hilson-Fisher fixed
point without
taking interact ions
into account. From Eq. (10.10), we have
A
=
[ ~ ]
=
I ag
I
*
g
or,using (10.11),
(11. 3)
32
s.
WEINBERG
We see tha t th is eigenvector is infrared-a t t rac t ive ,
not repulsive,
so there is
jus t
one repulsive eigenvector, as required for an or
dinary second-order phase t ransi t ion. Now l e t us calculate the
c r i t i ca l index V to f i r s t order in £.
As shown in Section 5,
(11.4)
where All i s taken as the single repulsive eigenvalue
of M. This
eigenvalue i s jus t the one mentioned in case (a) above, correspond
ing to the interact ion ~ 2 . The B-function for th is in terac t ion is
given exactly by the
one-loop
equation
(8.7) :
Bn = (-2+Y)gn - ~ ( 2 1 T ) - d J d d Q , O ( l t l - l ) M t ; A ) g .. ( t , - t ,0,0;A) (11.5)
coupling f I I is defined as the constant term in
the inverse
propagator ~ - , so that
So far ,
th is
is
exact.
Now l e t us use the £-expansion.
At the fixed point , f i s
of
2 - * 2 A /2
order E ; g . equals gI plus terms of order E ; 0
equals 1 plus
terms of order £; so that to order E,
[
aBn] = _ 2
agn
g=g*
-d *
+ ~ ( 2 1 T ) S d gI
or , using (10.11)
(11.6)
The c r i t i ca l index (11.4) is then
1 £ 2
V = 2 + 12 + 0(£ )
(11.7)
Note the factor 1/12; th is makes the correct ion to the lowest-order
value of V rather small even for £ = 1. I f we
neglect higher-order
terms,
then (10.7) gives V = 0.5875 for e: = 1; the experimental value
i s
in the neighborhood of 0.6 to
0.7.
12. RENORMALIZABILITY AND ALTERNATIVES
As
I indicated
at the beginning, the
purpose
of these lectures
i s to present the theory of c r i t i ca l phenomena to f ie ld theor is ts
who want to apply th is machinery in quantum f ie ld theory.
We
now
turn to
one of these applicat ions. From now on, our
ef for t s are
CRITICAL PHENOMENA FOR FIELD THEORISTS
33
directed a t
phenomena in d = 4 dimensions, though we wil l again have
to consider formal variat ions in dimensionality. However, we wil l
continue to
work
in a Euclidean rather than a Minkowskian space,
leaving the analyt ic continuation to the physical region to be dealt
with separately.
A good deal of modern elementary par t ic le theory i s based on
the assumption that nature is described by a renormalizable quantum
f ie ld theory. However, the "f loat ing cut-off" formalism described
in these lectures raises ser ious questions
about
the physical s ig
nificance of the
renormalizabili ty requirement. In the formalism
described here, the effect ive Hamiltonian depends on the cut-off
in such a way that the
physics ( i . e . , the se t of Euclidean Green's
functions) is cut-off-independent for a l l theories , not only renor
malizable theories . In this formalism, a renormalizable theory
merely corresponds to a subset of t ra jec tor ies
(characterized
by
a
few renormalized coupling constants) for which
a l l but a few
of the
couplings vanish (at l eas t in perturbat ion theory) as A + 00 Why
should the real world correspond to such t ra jec tor ies?
I t might be
argued that renormalizable
f ie ld theories are dis
t inguished because they have only a
f in i te
number of f ree param
e te r s . This seems l ike a
rather unphysical requirement. We can
eliminate a l l free parameters
by demanding that a t A
=
1 MeV a l l
dimensionless coupling constants gi have the value 37. What i s
needed is not uniqueness
i t se l f , but a rat ionale for uniqueness.
Nor is experiment much help here. Non-renormalizable quantum
f ie ld theories always
inrolve a mass scale, such as Fn ~
190 MeV
for
chiral dynamics; G F - ~ ~ 300 GeV
for the Fermi theory of weak
interact ion; and GNgY.JTON ~ 2 · 1019
GeV
for general re la t iv i ty . At
energies which are much smaller than the character is t ic mass scale,
a non-renormalizable theory wi l l look as i f it were
renormalizable.
( I f symmetries do not allow any renormalizable in terac t ions , it
wil l look l ike a f ree f ie ld theory.) The experimental success of
quantum electrodynamics only shows
that any non-renormalizable in
teract ions have a character is t ic scale larger than a few GeV. The
success (so far) of renormalizable gauge theories of weak and elec
tromagnetic interact ions only indicates that any non-renormalizable
interact ions
have a
character is t ic
scale greater than 300 GeV. We
need theoret ical guidance to t e l l us whether
physics wi l l continue
to look renormalizable a t rea l ly high energies, l ike 1 0 ~ GeV. And
i f not , then what does pick out the trajectory corresponding to
the rea l world from the inf ini te number of possible theories?
I know of only one promising approach to th i s problem. Random
ly chosen quantum f ie ld theories tend to develop unphysical singu
l a r i t i e s i f extended to suff ic ient ly high energies. (In the Euclid
ean region, any singular i ty is unphysical.) For instance,
suppose
some
coupling
constant
obeys the very simple renormalization-group
34
equation
II. dg(lI.)
dll.
The solut ion i s , for arbi t rary K,
g(KII.)
g(lI.)
l-ag(II.) /,nK
S. WEINBERG
(a> 0)
For g(lI.) >0, th i s develops a
singulari ty at /,nK=I/ag(II.), and Eq.
(4.27) then
suggests that some s ingular i ty occurs in Green's func
t ions a t f in i te Euclidean momenta. Nature must pick out t ra jec tor ies
which avoid s ingular i t ies of th is type.
to this requirement is to that the
Hamiltonian l i e on a t rajectory
which hi t s a fixed point
for II. +00 .
Equation (4.27) shows
that
in
th i s case, the Green's functions
simply behave l ike powers o
K when the momentum scale K goes to
inf ini ty , and do not develop unphysical s ingular i t ies . [In par t ic
ular , th i s is the case for our example above i f we s t a r t with a
negative coupling, g(lI.) < 0.] Theories with this property, tha t the
t ra jec tory hi t a fixed point for II. + 00, wil l be called asymptotically
safe.
Of course, in order to be asymptotically safe, the trajectory
near the f ixed point must have no components along eigenvectors o f
the M-matrix,that are ul t raviole t - repuls ive . (Since II. now goes to
in f in i ty
instead of zero, ul t raviolet-repulsive eigenvectors are
those with posi t ive eigenvalues.) But in a l l cases that have been
studied, it turns out that there are only a f in i te number of ul t ra
vio le t -a t t rac t ive eigenvectors ( i . e . , negative eigenvalues), and
a l l the res t are ul t raviole t - repuls ive . Therefore, the demand that
a theory
be asymptotically
safe imposes an in f in i te number of con
s t ra in t s on the coupling parameters, leaving only a f in i te number
of
free parameters, i . e . , the of the t ra jec tory a t
the fixed
point along the f in i te number of ul t raviole t -a t t rac t ive eigenvectors .
Thus, asymptotic safety can provide a rat ionale for picking physical ly
acceptable
quantum f i e ld theories , which may e i ther explain renormal
izabi l i ty , or
else replace i t .
To see how
this works in prac t ice , l e t us consider the theory
of a single
rea l sca lar f ie ld in f ive dimensions.
Let us ask whether
or not i t is possible for such a theory to achieve asymptotic safety
by hi t t ing the Gaussian fixed point . As we saw in Section 9, the
eigenvalues a t
the Gaussian
fixed point are simply equal to the
dimensionali t ies
of the various
interact ions:
for each interac t ion
with n ~ - f a c t o r s and N momentum factors, there i s an eigenvalue
given by (9.3),
which
for
d = S reads
D = -S+tn+N
CRITICAL PHENOMENA FOR FIELD THEORISTS 35
I f we do not impose the symmetry under </> + -</>, then there
are 2 ul t ra
vio le t -a t t rac t ive eigenvectors, corresponding to the super-renormal
izable interact ions
</>2, with D
=
-2, and
</>3, with D
= - ~ . (As usual,
we do not include </>02</> as an interact ion,
and we do not include a </>
in terac t ion because such an interact ion can always be eliminated by
shi f t ing </> by a
constant.)
Thus there i s a two-parameter set
of
t ra jec tor ies which h i t the Gaussian fixed point , and are therefore
asymptotically safe.
I t is easy to see that th is two-dimensional surface of t ra jec
tor ies simply corresponds to the
super-renormalizable theories of a
scalar
f ie ld
in
five dimensions. (Working in these super-renormal
izable theories, can calculate
the
matrix of derivat ives of the
Gell-Mann-Lowbetafunction
5
at the Gaussian fixed point , and check
that the eigenvalues of th is matrix are -2 and - ~ . ) The two param
eters
needed
to describe the t ra jec tor ies which h i t the Gaussian
fixed point are jus t the renormalized mass and </>3 coupling. Thus
in th is
case, asymptotic safety is achieved by requir ing renormaliz
abi l i ty . More generally, the Hamiltonian wil l approach
the Gaussian fixed point for A+00 i f it corresponds to a super
renormalizable theory or an asymptotically free renormalizable theory.
However, it is not c lear that nature rea l ly does
choose t ra jec
tor ies which
hi t the Gaussian fixed point for A+ 0 0 . The notorious
problem i s gravitat ion: no
one
has been able to think of a sa t i s
factory theory of gravitat ion which is renormalizable. Is i t pos
s ib le
that nature achieves asymptotic safety by aiming the
t r a jec
tor ies of the
effect ive Hamiltonian a t some fixed point other than
the Gaussian fixed point?
As an example of
what i s possible, l e t ' s return to our example
of a
scalar
f ie ld in f ive dimensions, but l e t ' s now impose
the sym
metry under the transformation </> + -</>. This plays a role here similar
to that of
general covariance the theory of gravitat ion - it
eliminates the only interact ion </>3 that is renormalizable. There
s t i l l is one ul t rav io le t -a t t rac t ive eigenvector, corresponding to
the "interact ion" </>2, but a t rajectory that reaches the
Gaussian
fixed point along th is direct ion simply corresponds to a free f ie ld
theory.
An in terac t ing asymptotically safe
theory with th is sym
metry must h i t some
other fixed point.
What about the Wilson-Fisher fixed point? We are rea l ly in
terested
(in this example) in five-dimensional space, but in order
to do calculations,
l e t ' s
work in 4+£ dimensions. The eigenvalues
can then be calculated from the resul ts of Sec. 11, by simply
changing E to
-E.
We see that there are two
ul t rav io le t -a t t rac t ive
eigenvectors a t the Wilson-Fisher point , with eigenvalues
36
s.
WEINBERG
\
A = - 2 - ~ + 0(1':2)
I I 3
and an inf ini te number of ul traviolet-repulsive eigenvectors , with
posi t ive eigenvalues. I f
we assume tha t the eigenvalues do not
change sign
for 0 < 1 ' : ~ 1 , we can conclude tha t there are jus t two
ul t rav io le t -a t t rac t ive eigenvectors in five dimensions. With
this
assumption,
there is a two-parameter se t of
asymptotically safe
f ie ld theories
in
five dimensions associated with the Wilson-Fisher
fixed
point . These theories are not renormalizable in the usual
sense - the symmetry under ~ +
- ~ rules out the possibi l i ty of any
renormalizable theory in five dimensions. However, they are in ter
act ing theories with no
unphysical s ingular i t ies a t high energy.
(In fact , th i s theory does
have unphysical features, but of a
dif ferent kind. Changing I': to -I': in Eq.
(10.11)
shows that the ~ ~
coupling
constant
a t the Wilson-Fisher fixed point i s negative in
4+E dimensions. This means
that the potent ia l (6.11) goes to ~
for I ~ ) I + 00, so the energy is not bounded below.
However, th is
is an accident of th i s par t icular model. Any theory wil l have non
t r i v i a l
u l t rav io le t -a t t rac t ive
fixed
points in d+E dimensions i f it
is renormalizable and asymptotically free in d dimensions. I t is
well known that asymptotic freedom can only be achieved for a ~ ~
theory in four
dimensions by giving the coupling constant an un
physical negative value. However, there are plenty of other
asymptotically
free
renormalizable theories , such as
chira l
dynamics
in two dimensions
6
and non-Abelian gauge theories
in four dimensions,
which are not plagued
by negative-energy problems.)
The asymptotically safe
theories of a
scalar
f ie ld
in f ive
dimensions are characterized
by
two free parameters: one dimension
less parameter picks out a par t icular t rajectory
in the two
dimensional surface of t ra jec tor ies which hi t the
Wilson-Fisher
fixed point , and one parameter
with the dimensions of mass gives
the value of
the
cut-off
A a t which any given point along th is
t rajectory i s reached. In order to perfect the analogy with general
re la t iv i ty , we can eliminate the f i r s t parameter
by demanding that
the theory have zero boson mass. This i s done simply by assuming
that the theory l i e s on a t rajectory which not only h i t s the Wilson
Fisher fixed point
for A + 00, but also hi t s
the
Gaussian fixed point
for A + O. (To see that there is l ike ly t be
such a t ra jec tory ,
note tha t in d > 4 dimensions the symmetry ~ + - ~ el iminates a l l
infrared-repulsive eigenvectors a t the Gaussian fixed point except
for the one corresponding to the "interact ion" ~ 2 . Hence it i s only
necessary to
adjust one
dimensionless parameter to hi t the Gaussian
fixed point for A + 0, and we do have one free parameter a t our dis
posal . ) This theory is then described by a
single free parameter,
with the dimensions of a mass, jus t l ike general re la t iv i ty .
CRITICAL PHENOMENA FOR FIELD THEORISTS
37
Perhaps gravitat ion works this way. I t may be that general
covariance rules
out any renorma1izab1e theory of gravitat ion in
four dimensions, jus t as the symmetry under ~
~ - ~
rules out any
renorma1izable theory of a scalar f ie ld
in 5 dimensions. Neverthe
less ,
nature may achieve asymptotic safety anyway, by picking out
t ra jector ies which hi t
some fixed point other than the Gaussian
fixed point for A ~ 0 0 . For this to be possible, there must be some
non-tr ivial fixed point with a t l eas t one ul t raviole t -a t t ract ive
eigenvector.
Unfortunately, it i s not so easy to check that th is i so.
General re la t iv i ty becomes formally renormalizable
in d = 2 dimen
sions, so one might try to study the fixed points in four dimensions
by working in d = 2 + £ dimensions and expanding in E:. However,
general re la t iv i ty actual ly becomes a t r iv ia l
theory
in two dimen
sions (because R ~ v - ~ g ~ v R vanishes identical ly) and it i s not
c lear
how to
expand
around d = 2. Work
on
this is in progress.
In
any case, I think it is wrong to
hope that we will learn how
to make sense out of arbitrary non-renorma1izable field theories.
I t would be a disas ter for theoret ical physics i f it were found that
the inf ini te variety of physical theories with a l l possible couplings
were a l l equally acceptable. We would then have no guide in under
standing how nature picks out the part icular theory that describes
our world. What we need instead i s some principle, l ike asymptotic
safety,
which
picks out a very limited class of physically acceptable
theories - perhaps renorma1izable, perhaps not .
Acknowledgments
I am very grateful to Paul Martin and David Nelson for frequent
enlightening discussions on
the
theory of cr i t ica l phenomena through
out the l a s t year. I would not have been able to prepare these
lec
tures
without thei r help in learning
th i s
subject . I also wish to
thank Edouard Brezin, Sidney Coleman, Leo Kadanoff, Philippe Nozieres,
Kenneth Wilson, and Edward Witten for useful conversations, and
Antonio Zichichi
for his kind
hospi ta l i ty
a t
Erice.
38
s. WEINBERG
Appendix
DERIVATION OF THE
ONE-LOOP
EQUATIONS
\ole
will now derive the renormalization-group equations sat is
fied by the coupling functions. For the present ,
it will be con
venient to
work with the original functions ~ ( P I •••Pn;A) rather
than the re-scaled functions gn(tl ••. tn;A). I t
will also be con
venient to t reat the Q4adratic coupling U
z
as much as possible l ike
any other kind of interaction. In order to accomplish this , l e t us
write the original function uz,OLD appearing in the Hamiltonian as
u (-+ -+ •A) = U (-+ -+ •A) + - I -+ d -+ -+
z,OLD
PI'PZ' - z,NEW PI'PZ' GNEW(p)o (PI + p ) (A.l)
where G
NEW
is arbitrary but A-independent. (We
can take G
NEW
of
the form (apZ+b)-I, but this is not necessary.) The propagator can
then be taken as
(2n)-d
GNEW
(p)6(A-lpl) (A.2)
and U
z
NEW is now regarded as jus t another interact ion. From now
on, we' drop the label "NEW": G and U
z
are to be understood as GNEW
and uz,NEW unti l further notice.
We
consider
the set f al l connected Green's
functions
C(PI . • . P
n
) , with
outgoing
momenta
PI ••• Pn. These functions are
calculated with
vert ices - (2n)d u
Z
' - (2n)d u
q
' etc . , and with
propa
gators (2n,d G
(p)6(A-lpl),
except that the 6-function i s omitted on
external l ines. (We do not assume that the external momenta are
below the cut-off .) Our demand on the couplings i s that these
Green's functions be independent of
the cut-off:
(A.3)
for a l l n and a l l momenta. When we different ia te C with respect to
A, we encounter terms of three different kinds, shown in Figure 3:
(a) The derivative d/dA
may act on one of the A-dependent coupling
functions in the graphs for
C. Such terms
may be writ ten
d \' J did -+ -+ -I (mn) -+ -+ 1-+ -+
- (2n) L d k . . • d k dA u (k . . . k ; A) r
A
(k . . . k P . . • p )
I m ml mImI n
m . -
(mn) -+ -+ 1-+ -+
Here r A (k •.• k P . • • p ) is a sum of graphs having m external
HI
ml
n -+-+
l ines carrying incoming momenta kl . . . k
m
n external l ines carry-
ing outgoing momenta PI . . .Pn. I t i s defined with cut-off propaga
tors for the external k-l ines but propagators without cut-offs on the
p-l ines, except that i f a k-l ine turns into a p-l ine with no in te r
action,
there i s jus t one propagator without cut-off . Since we
CRITICAL PHENOMENA FOR FIELD THEORISTS
d
dt\
+
( b)
( a )
+
( c)
Figure 3. An example of the variat ion of a connected Green's
function
with cut-off . Here the shaded ci rc le
represents the
to ta l connected Green's
function with four external l ines ;
the
circ le with a ver t ica l bar
represents the sum of a l l graphs
such that every l ine leaving on the r ight is connected to a t
leas t one l ine enter ing
on the
l e f t ; the darkened ci rc les re
present various interact ions Un; the x in the
f i r s t
term on
the r ight
represents the derivative of un with
respect to A;
and the x in the following two terms represents She replace
ment of a cut-off by a a-funct ion a(lql-A). For
the sake of clar i ty , the only
graphs
shown are those with s ix
l ines
enter ing the barred ci rc le from the l e f t .
39
want to include a l l connected diagrams in C, fA consis ts only of
the graphs
in which each p-l ine is connected by
some path to
a t
leas t one
k-l ine , so that the in terac t ion urn can connect the whole
graph together. Apart from th is proviso, f l
rnn
) may contain dis
connected par ts .
(b) The derivative d/dA may act
d i rec t ly on
one of
the
cut-off
func
t ions 6(A-lql) associated with anyone of
the in ternal l ines of C.
The two ends
of
th is internal
l ine may terminate e i ther in the same
vertex, or in two different
vert ices. I f they terminate in the
same
vertex, there must be some
other
set of m ~ 2 l ines with
out
going momenta ~ l ••• m attached to th is vertex. Also, s ince C i s
connected, each p-l ine wil l be connected by some path to a t leas t
one of the
k- l ines . The contribution of such graphs to dC/dA is
then
40
s. WEINBERG
(c) Final ly, when d/dA acts on
the cut-Dff function associated with
an in ternal l ine which ends in two different ver t ices , there wi l l
be two sets
of l ines with outgoing momenta I t ~ . . . t ~ and I t ~ ' . . . t ~
respect ively, attached to these two ver t ices . Again, since C i s
connected, each p- l ine is connected by some path to a t leas t one
It'- or I t"- l ine. The contr ibut ion of such graphs to dcl dA is
then
~ ( 2 7 f ) d l: ( r ; s ~ J i q i k ~ . . . ik' ik'; . . i k"o (A- lq l )
r . s . r s
r , s
(
-+) (-+ -+, -+" ( -+ -+" -+"
X q,k . . .
1 1 r S l 1 S
x r
A
r+s ,n)
(k-+' . . . +k' -+" -+" 1-+
-+ )
k
. . .
k
p • • • p
1 r 1 sIn
We can now put this a l l together, and wri te the
resul t
as
d -+ -+ d ~ J d-+ d-+ -+ -+ (mn) -+ -+ 1-+ -+
~ ( p . . . p ) = -(27f) L d k . . . d k BA (k ... k )r
A
(k . . . k p . . . p )
n m=2 1 m 1 m 1 mI n
where (A.4)
-+ -+ d -+ -+
B, (k . . . k ) - d ' u (k . . . k ; A)
1 '1m it m 1 m
1 ' \ f f 1-+1 -+ -++ -+, -+-+-+
- '2 L d qo(A- q )G(q)u (q,lC . . . k ;A)u + ( - q , k ' ~ . . k";A)
k-+k '&k" r,+l 1 r s lIS
(A.5)
The sum in the l as t term runs
over
a l l values of r a n d s with r + s =
m, and over a l l m /r s ways of par t i t ioning the momenta It
l
. . .
t
m
into subsets I t ~ . . . t ~ and It'; . . . ~ . We see that for C to be indepen
dent of A, it i s
suff ic ient that
-+ -+
B, (k
...
k ) = 0
it 1 m
(A.6)
-+ -+
for a l l m and al l k
l
. . . k
m
. This i s
one form of renormalization-group
equation. (See Figure 4.)
We note further that the kernel
r ~ m n ) ( k l . . . I p l
. . .
P ) general ly wil l have an inverse, because in the
absence of interact ions i t i s simply proportional to
-+ -+ -+ -+ -+ -+
o o(k -p ) . . . o(k -p )G(k ) . . . G(k )
mn 11 m m 1 m
Therefore we expect (A.6) to be necessary as well as suff ic ient .
[Equation (A.6) i s similar to a
set of equations derived in
quite
a dif ferent way
by
Wegner and Houghton.
7
They dif fer in
that
G appears instead of u
2
1
, and that the sum in the l as t
term includes
CRITICAL
PHENOMENA FOR
FIELD THEORISTS
o
Figure 4. The Wegner-Houghton equations for u
6
• These are
derived here by
demanding that the A-derivative in Figure 3
vanish.
41
terms with r or s equal to one. The difference ar ises
because
our
cut-off procedure i s
different
for
external
l ines . ]
These equations
have a well-known drawback: the l a s t term in
Eq.
(A.5)
i highly discontinuous in the external momenta, con
t r ibuting only when the
to ta l
momentum of some subset with an odd
number of l ines has an absolute value equal to A. Also, these
equations mix terms with different numbers of loops, so they are
not very convenient for use in
i tera t ion schemes. To
avoid these
problems, we introduce a new se t of coupling functions.
The reason for the discontinui t ies in the l a s t term in (A.5)
is obvious: in calculat ing Green's functions we must include tree
graphs, whose in ternal l ines contain cut-off functions 8(A-lql) .
Even though
a l l external l ines of such graphs l i e below the
cut
off A, the to ta l
momentum
of
subsets
of the external l ines may be
above th i s
cut-off , and the contribution of
such graphs wi l l con
ta in discontinui t ies in the external momenta. The to ta l Green's
functions certainly do not contain such discontinuit ies (they
are
A-independent ) so in order to cancel them out the Un
must contain
A-dependent discontinui t ies of some sor t . These are automatically
generated by the l a s t term in Eq. (A.5).
The diagnosis immediately suggests a cure. Let us introduce
a new se t of coupling functions v n ( ~ l ... n;A), by the prescript ion
that -(2n)d
un
i s to be
constructed as a sum of a l l possible
tree
graphs, with
vertex functions -(2n)d
vn
, and with in ternal- l ine
factors
(2n)-d
G
(q)8(lql-A) (A.?)
When we take the derivat ive of Un with respect to
A, we then en
counter terms of two
different types (see Figure 5):
(a) The derivative d/dA may act on one of the cut-off functions
8(lql-A) associated
with an internal l ine of
the t ree , changing it
to -o(lql-A).
The two ends of
th is
internal l ine must be connected
to
separate
trees bui l t out of v-vert ices.
(b) The
derivative
d/dA may
act
on
one of the vertex factors ,
a v
m
. Since
the whole graph is a
t r ee , each of the m l ines at tached
to th is vertex must i t s e l f be connected to a separate t ree bui l t out
of v-vert ices and propagators (A.7).
42 S. WEINBERG
1K ~ 0_£
+ x ~ + X < + X ~
+
...
Figure S. A-derivative of
a
Un-function expressed in terms
of A-derivatives of vn-functions. Here dark c i rc les repre
sent
Un-functions; l ight
c i rc les represent vn-functions;
the x in the f i r s t term on the r ight represents the
re
placement of a cut-off 8-function with a a-funct ion; and
the x ' s in the other terms represent A-derivatives act ing
on Un-functions.
When
inser ted into the equations of
Figure 4, th is
yields
the one-loop equations.
When we
inser t these resul ts for dUrn/ciA in Eq. (A.S), we find
that term (a) cancels the l a s t term in (A.S), so the sum of (b) and
the second
term
of (A.S) must vanish.
NOW,Urn+2 in
Eq.
(A.S) i s i t s e l f a sum of tree graphs bui l t
out
of v-ver t ices . There must be a unique path from the l ine with
momentum q to the l ine with momentum -q, and each v-vertex along
th is path i s at tached to a t ree bui l t out of v-ver t ices . The
v-vert ices in term (b) are also connected to such t rees, so
the
trees may be divided out. leaving us with
o =
(A.8)
The
las t
sum i s over a l l values of m
l
, m
2
, ••• ,m£, with
and
over a l l m /m
l
m
2
. . .
m£ ways of par t i t ioning the ~ . ~ o m e n t q
into £ se ts , with the j - th se t consis t ing of momenta
k ~ J ) . . . ~ J ) .
I
I t should be noted that the
in which any of the mj are zero.
the i r
ef fec t is simply to change'
l as t sum in (A.8) includes
terms
These can be t r iv ia l ly summed:
G(cl) into
CRITICAL PHENOMENA FOR FIELD THEORISTS
-+ -+ -+ -+ -+ -+
G'(q;A) = G(q) - G(q)V
2
(q,-q;A)G(q) +
-+ -+ -+ -+-1
G(q) [1 + V
2
(q,-q;A)G(q)]
where v
2
is v
2
with the o-function factored out:
- + - +
_-+-+ d-+-+
V
2
(ql ,q2;A) = V
2
(q1,q2;A)o (q1 +q2)
This us to
introduce a new v
2
function
-+ -+ -+ d-+ -+ -+ -+
V ~ ( q 1 , q 2 ; A )
=G-
1
(q1)O (q1+
q
2) + v
2
(q1,q2;A)
which has the same A-derivative as v
2
d v ~ dV
2
dA
= dA
but which yields the corrected propagator
-+ -+ 1-+ d-+-+
V;(Q1,Q2;A) = G '- (Q1;A)o (Q1+
q
2)
That i s , Eq. (A.8) s t i l l holds i f we replace G with G', v
2
with v; ,
and res t r ic t the sums so that mj > 0 for a l l j. From now on we wil l
drop the primes on v
2
and G, ana interpret (A.8) to include only
terms with mj > O.
We now make the transi t ion
to
the re-scaled variables. Define
new coupling functions:
-+ -+
g ( ~ ••• Pn.A)
n
A
A'
-
(A.9)
and a new propagator
(A.
10)
so that
(A. 11)
Insert ing (A.9) and (A.lO) in (A.8) gives
44
S. WEINBERG
x
-+ -+ -+( 1 ) -+( 1)
17 I 7
g +·2 (9.,-9,1,9:
1
.•. , ;1I.)8()(,1 -1)L'I()(,1;1I.)
m
1
m
1
x •••
7 7 -t;(k) -+(k)
g + ()(,k 1 ,-)(',9, • • • 9: ;A)
m9, 2 - 1 ~
(A.12)
After
factoring out a momentum-conservation del ta function, th is
becomes jus t the general one-loop equation (8.4) .
REFERENCES
1. For surveys of the modern theory of
c r i t i ca l phenomena, including
references to the original
l i t e ra ture , see the following reviews:
K. G. Wilson and J. Kogut, Physics Reports l2C, No.2 (1974);
M. E. Fisher, Rev. Mod.
Phys.
46, 597
( 1 9 7 4 ~ E .
Brezin,
J . C. Le Guillou, and J. Zinn-Just in , in Phase Transitions and
Critical
P h e n o m e n a ~
ed. by C. Domb and M. S. Green (Academic
Press, New
York, 1975), Vol. VI;
F. J . Wegner, in Trends in
Elementary Particle Theory (Springer-Verlag, Berlin, 1975),
p.17l; K. Wilson, Rev. Mod. Phys. 47, 773 (1975); Shang-Keng
Ma,
Modern Theory o f Critical Phenomena-CWo A. Benjamin, Inc . ,
Reading, Mass., 1976).
2.
A.
L. Fetter and J . D. Walecka, Quantum Theory o f Many-Particle
Systems (McGraw-Hill, Inc . , New York, 1971), Chapter
7.
3. S. Weinberg, Phys. Rev. D ~ , 3357 (1974); L. Dolan and
R. Jackiw,
Phys. Rev. D ~ , 3320 (1974);
D. A. Kirzhnits and A. D. Linde,
Zh. Eksp. Teor. Fiz. ~ , 1263 (1974); C. W. Bernard, Phys. Rev.
D ~ ,
3312 (1974).
4. F. Bloch and A. Nordsieck, Phys. Rev. ~ , 54 (1937). For a
version in modern notation, see S. Weinberg, Phys. Rev. 140,
B5l6 (1965).
5. M. Gell-Mann and F. E. Low, Phys. Rev. 95, 1300
(1954).
6.
The fixed point in the nonlinear a-model in 2 + E: dimensions
has
been under intensive study la te ly ; see W. A. Bardeen, B. W. Lee,
and R. E. Shrock, Ferrnilab-Pub-76/33-THY, March 1976; E. Brezin,
J . Zinn-Justin, and J . C. Le Guillou, Saclay preprints , May 1976.
The motivation of
these studies appears to be quite different
from that described here.
7. F. J . Wegner and A. Houghton, Phys. Rev. A.§., 401 (1973).
CRITICAL PHENOMENA FOR FIELD THEORISTS
DISCUSSION 1
FERBEL:
D I S
C U S S I O N S
CHAIRMAN: Prof. S. Weinberg
Scientific Secretary: F. Posner
45
Could you elaborate on the question of Higg's boson production
relat ive
to
W production in hadronic collisions?
WEINBERG:
Higg's production has been discussed in detai l in a recent paper
by El l i s , Gaillard, and Nanopoulos. I believe that they do the cal
culations you ask about, but I do not know the answer. I wil l make
one point though: Higg's bosons' couplings are proportional to mass,
so Higg's bosons tend to be emitted from internal l ines of heavy par
t ic les . A good place to
look for Higg's bosons therefore is in neu
trino reactions, where they are eIT.itted from the exchanged W l ine.
Ell is , Gaillard, and Nanopoulos, and LoSecco have calculated the
probability
for Higg's production
near threshold in neutrino-nucleon
reactions to be about 10-
5
•
FERBEL:
I f charm violating processes occur, might one expect f inal s ta te
correlations such as K+K+
or
K+e+?
WEINBERG:
I have been assuming that the neutral currents do conserve charm.
I f there is a milliweak ~ C = 2 neutral Higg's exchange, then DO-Dooos
ci l la t ions will be
much fas ter than DO decay.
The resul t would be
that an incoherent mixture of
D ~
and/or
D ~
would be produced, each de
caying equally into Ks or
leptons
of ei ther charge.
46
s. WEINBERG
PARSONS:
How easy would i t be to introduce more flavours into your
model?
WEINBERG:
This model needs four quarks because i f there were
more, this
would be l ikely to introduce CP violat ion into the W-exchange process.
To include a bottom quark b, one requires
as the appropriate multiplet
for weak interactions where
a ~
+
a ~
+ a ~ = 1 .
The experimental data on S decay provides information on u + d.
The experimental data on A decay provides information on u + s. An
over-al l normalization can be fixed by
~ decay. This sets
t ight
limits on al and a2, so that we must have la31 $ 10-
3
• Such small
numbers in the theory are rather unsatisfactory.
I t i s , however, possible to
introduce extra doublets into the
theory which do not mix with u, d, s , and c. These can be used to
cancel anomalies due to heavy leptons. The real problem
would be i f
neutrinos found to cause u and b quarks,
for instance, thus indicating a non-zero mixing angle a3.
CRONIN:
In
order
to be sure of your ideas about the nature of CP viola
t ion, i t is real ly insufficient to have the electr ic dipole moment
of the neutron come out
~
2 X
10-2q
e-cm. Thus one real ly needs to
see
the CP-violating
effects
of the Higg's scalar . What is the nature
of the CP-violating
effects of the Higg's scalar , i f i t could be iso
lated?
WEINBERG:
Let
me
f i r s t answer the question I thought you were going to
ask: What experimental evidence
could be used to check that Higg's
exchange is responsible
for CP non-conservation? Since Higg's bosons
couple essential ly to mass, we would expect CP-violating amplitudes
of order 10-
3
in K ~ 3 decay, but no corresponding effect in S decay.
As to the way that CP violations would show up in Higg's boson
decay, I have not worked i t
out. I t would be a good exercise
for a
student. The Higg's decays should violate CP strongly.
CRITICAL
PHENOMENA FOR FIELD
THEORISTS
47
LEADER:
You have offered us a beautiful and natural mechanism for pro
ducing a small number, namely 10-
3
, in amplitude. However, you rely
for this on a knowledge of the
quark and the Higg's masses. Could
you explain what i t is that gives you such confidence in your knowl
edge of these
masses?
WEINBERG:
As the quark masses, I jus t going along with al l the
standard ideas on quarks and constituent models. As
for the
Higg's
mass, i t is expected to be of
order I f x 300
GeV, where f is the ¢4
coupling constant. I f f is of order a, as generally supposed, then
the Higg's mass is of the order of the intermediate vector boson mass.
Linde and I have recently shown that
there is an effect ive lower
bound of order a
Z
• In the simplest SU(2) x U(l) model,
the Higg's
mass
is greater than 3.72 GeV.
POSNER:
A rather elementary question: Why does a CP and P non
conserving milliweak interact ion imply a detectable
e lec t r ic
dipole
moment for the neutron?
WEINBERG:
I f CP is broken, but not P, then
there is no moment. I f the
neutron is real ly two d and one u
quarks, then the moment
equals
4h (the moment of
the
d) - Y
(the moment of the u). The moments
arise from the vir tual processes
u + H+ + S
(or d) + u
d + H + c (or u) + d
MARCIANO:
Is the statement that the CP violat ion is
due
to the H+ propa
gator and not the W propagator, a gauge-dependent statement? Could
you clar i fy your
statement that you
feel that
there wi l l be CP vio
lat ion even for the case ¢l = ¢z?
WEINBERG:
The sum of
the W-propagator and the part
of the H
propagator
corresponding to a
Goldstone boson is gauge invariant and conserves
CPo The
remaining
part
of the H propagator is then
also gauge in
variant , and violates CPo Thus the amount of CP violat ion is gauge
invariant .
48 S. WEINBERG
Even i f ¢1 = ¢2, there can be a CP violat ion in the interact ion
of these
Higg's
bosons with other Higg's bosons that cannot couple to
quarks. I have not t r ied to calculate such effects .
FREEDMAN:
What classes
of
theories permit Higg's bosons which do not vio
la te CP? Huw
do these couplings differ from those discussed in this
morning's talk? What, i f any, characteris t ics are different between
the two, mass, e tc . , and can one incorporate both in a single model?
WEINBERG:
Such theories are certainly possible. One can always impose
CP on L. Then Crs would have to be real , or have phases that could
be made real . However,
for a f in i te range of parameters there wil l
s t i l l be a spontaneous
breaking of
CPo
DISCUSSION 2
YOON:
Does the fact that cr i t ica l phenomena in boson systems has i t s
origin in the infrared behaviour of the system ref lect the physical
picture of phase transi t ions aris ing from long-range correlations?
How does one understand cr i t ica l phenomena in fermion systems where
there are no infrared divergences?
WEINBERG:
Second-order phase transi t ions can occur in systems composed
purely of fermions,
because the fermions can have bosonic collect ive
excitat ions. That is a subject, which I wil l not go into in
these lectures.
ALVAREZ:
In ¢4 quantum field theory,
the two-point function is
ul t raviole t
quadratically divergent. In the f in i te temperature theory, the three
momentum integral is logarithmically divergent. When the energy sum
is
performed, do the logarithms sum up to power behaviour?
WEINBERG:
Of
course, they had bet ter , because the divergences must be
the
same at
f in i te temperature
as at zero temperature. The Lagrangian
does not know what the temperature i s , so the counter-terms available
to
absorb in f in i t ies are necessarily temperature-independent. However,
i t is d i f f icu l t
to do the energy sums as you suggest af ter doing the
CRITICAL PHENOMENA FOR FIELD THEORISTS
49
momentum integrals .
f i rs t by converting
tum integral .
I have found i t much easier to do the energy sum
i t into a contour integral , and then do the momen-
PAULI:
In the s ta t i s t ica l mechanics formalism, i f you take the fourth
derivative of the par t i t ion function, you get a Feynman-like graph
with four external legs. We know what this corresponds to in quantum
f ield theory vis-a-vis the S matrix and differential cross-sections.
We also know is f ield theory how to include symmetries. What physi
cal and formal mathematical changes occur when we do this
in a s ta
t i s t ica l mechanical form at f ini te temperatures?
WEINBERG:
Symmetries l ike rotation or isospin invariance govern the
temp
erature Green's functions just as they govern the S matrix or the
Green's functions in quantum f ield theory. The only symmetry that
is really fouled up by a f ini te temperature is Lorentz, or Galilean,
invariance.
KLEINERT:
I f you pass the cr i t ica l
point of a ferromagnet, however, do
the 4-point Green's functions not change symmetry?
WEINBERG:
No, the Green's functions as I have defined them have the ful l
symmetry of the underlying theory, i .e . of the Hamiltonian.
The
symmetry-breaking affects the
various
expectation values calculated
using these Green's functions.
POSNER:
A useful and powerful formulation of quantum f ield theory is by
means of integrals . Are there any part icular pi t - fa l l s in
dealing with
s ta t i s t ica l
mechanics by functional techniques? In field
theory functional integrals , one rotates from t1inkowskian
to Euclidean
space to avoid ambiguities. In s ta t i s t ica l mechanics one has iw in
stead of po. How does this affect the functional integrals of s ta
t i s t ica l mechanics and their possible ambiguities?
WEINBERG:
Your classmate a t Harvard, Claude Bernard, has worked
out a very
nice formulation of s ta t i s t ica l mechanics by using functional integrals .
He has derived Feynman
rules
at a f ini te temperature in a gauge theory.
50
s. WEINBERG
Because we are now in Euclidean space, the troubles with cut-offs due
to the metric are gone. Things work very well.
MARCIANO:
What plays the role of an infrared regulator in
this
formalism?
WEINBERG:
The part i t ion function is not well-defined a t the cr i t ica l temp
erature. The temperature e regulates
these divergences since for
e +e
c
' the par t i t ion function Q is f ini te
and well-defined.
DISCUSSION 3
(Soientifio
S e o r e t a ~ : O. AZvarez)
MARCIANO:
What does the renormalization
group have to
do
with infrared
divergences?
WEINBERG:
I t is unfortunate that Wilson, probably out of modesty, called
these
equations renormalization group equations, because of the formal
appearance to the equation
of Gell-l1ann and Low. The Wilson work
con
cerns infrared behaviour while Gell-Hann and Low considered ultra
violet behaviour. The "floating renormalization point" introduced by
Gell-llann and Low has a formal similari ty to the floating cut-off
used by Wilson, but the purpose is
entirely different .
GARCIA:
Following your discussion of the Bloch-Nordsieck problem, can you
find a similar approach for coupled massless
fields?
WEINBERG:
I believe there is a renormalization group argument, but I have
not been
able to complete i t .
GARCIA:
I do not clearly see the connection between your use
of the re
normalization group and the idea
of
"thinning" out of the degrees of
freedom as used by Wilson and Kadanoff.
WEINBERG:
This is an example of the
trouble
with translat ing solid s ta te
language into the language of field theorists . Kadanoff's idea of
block spins, or changing the la t t ice
spacing by integer multiples
CRITICAL PHENOMENA FOR FIELD THEORISTS
51
would be equivalent to changing the cut-off by integer multiples.
In quantum f ie ld theory we usually change the cut-off continuously,
but we could change i t by discrete steps.
PAULI:
When you compared the work of Brezin et a l . with
the work of
the
Wilson school, you stated that every eigenvalue of M is an eigenvalue
of M, but not al l
eigenvalues of U are eigenvalues of M. How do you
know that Brezin' s method will give the repulsive and the "important"
eigenvalues of MZ
WEINBERG:
The only way to check this is to calculate
the eigenvalues,
using
an approximation scheme such as the £ expansion. Brezin et al . work
in 4 - £ dimensions, and find one at t ract ive and one repulsive eigen
value. They cannot show, within a s t r ic t ly renormalizable theory,
that a l l ignored eigenvalues are at t ract ive, but this seems reason
able, and can be
shown by introducing non-renormalizable
perturbations.
PAULI:
You stated that i f you have more
than
two repulsive eigenvalues
in three
dimensions, the £ expansion
is necessary. Why can you not
vary temperature and say
magnetic field? Are there alternatives to
the
£ expansion?
WEINBERG:
I must have been unclear. We believe that
ordinary second-order
phase transi t ions are
associated
with fixed
points that have only one
repulsive eigenvector, because these transi t ions can be brought about
by adjusting only one free parameter, the temperature. Even with a
magnetic
f ie ld , the Gaussian fixed
point cannot describe an
ordinary
second-order phase transit ion, because
introduction
of the field des
troys
the ~ ~ - ~ symmetry, and there are
therefore three repulsive
eigenvectors at the Gaussian fixed point.
One alternative to the £ expansion is an expansion in liN,
where N
is the number of fields. I will not discuss this in these
lectures.
FREEDMAN:
We learned that in three dimensions i t was not
possible
to do
perturbation theory about the Gaussian fixed
point because of two
repulsive eigenvectors. In order to circumvent this we go to 4 - £
dimensions where only one repulsive eigenvector exists . Although
we can perform our calculations now,
what
can we learn about
the
52
s. WEINBERG
physics
in our original three
dimensional problem?
WEINBERG:
The hope is that the eigenvalues do not change sign between
£ « 1 and £ = 1. The quali tat ive features of the physics wi l l then
remain.
PHAM QUANG HUNG:
To which physical si tuat ions corresponds the l imit A O?
WEINBERG:
The Gaussian fixed point has a zero eigenvalue corresponding to
the interact ion ~ 6 . However, i t is only possible to reach this fixed
point i f two parameters are adjusted to eliminate components of the
trajectory along the two repulsive eigenvectors which correspond to
the ~ 2 and ~ ~ interact ions.
MONOPOLES AND FIBER BUNDLES
Chen Ning Yang
State University of New York
Physics Department, Stony Brook, N.Y. 11794
Magnetic
Monopole and Need to Introduce Sections
The magnetic monopole is the magnetic charge. While the idea
of magnetic monopoles must have been discussed in class ical
physics
early in the history of e lec t r ic i ty and magnetism, modern discussions
date back to 1931 in the
important paper of Dirac
l
in which he point
ed out tha t magnetic monopoles in quantum mechanics exhibit some
extra and subtle features. In par t icular , with the existence of a
magnetic monopole of strength g,
electr ic
charges and magnetic
charges must necessari ly be quantized, in quantum mechanics. We shal l
give a new derivation of this resul t in a few minutes.
I f one wants to describe the wave function of an electron in
the f ie ld of a magnetic monopole, i t is necessary to find the
vector potential X around the monopole. Dirac chose a vector
potential which has a s t r ing
of s ingular i t ies . The necessity of
such a s t r ing
of s ingular i t ies is obvious i f we prove the
following theorem
2
.
Theorem. Consider a magnetic monopole of strength g ~ 0 a t
the origin and consider a sphere of radius R around the origin.
There does not exis t a vector potential
X
or
the
monopole
magnetic f ie ld which i s
singulari ty free on the sphere. This
theorem can be eas i ly proved in the following way. I f there were
a
singulari ty free Awe consider the loop integral
around a paral le l on the sphere as indicated in Figure 1. By
53
54
C.N.YANG
FIGURE 1. A sphere of rad ius R with a magnetic monopole
a t i t s cen te r . The
p a r a l l e l divides the sphere
i n to two caps a
and S.
Stoke ' s theorem t h i s loop i n t eg ra l i s equal to the to ta l magnetic
f lux
through the cap alpha:
y A ~ d x ~ = Q
a
.
(1)
Simi l a r i l y we can apply Stoke 's theorem to cap S obtaining
j A ~ d x ~ = ~ S . .
(2)
~ a and ~ S the upward magnetic f lux through the caps
a and S, both of which are bordered by the para l l e l . Sub t rac t ing
these two equat ions we obta in
o = ~ A - ~ B '
(3)
which i s equal to
the to ta l f lux out of
the
sphere , which in tu rn
i s equal to 4ng I
O.
We have thus reached a cont rad ic t ion .
Having proved t h i s theorem, we observe t ha t R i s a rb i t r a ry .
Thus one concludes t ha t there must be a s t r ing of s ingu la r i t i e s
or s t r ings of s i n g u l a r i t i e s in the vector po ten t i a l to descr ibe the
monopole f i e ld . Yet
we know
tha t the magnetic f i e ld around the mono
pole i s s ingu la r i ty free . This suggests
t ha t
the s t r ing of
s i n g u l a r i t i e s i s not a r ea l physical d i f f i c u l t y . Indeed the
s i tua t ion i s reminiscent of the problem t ha t one faces when one wants
to f ind a paramet r iza t ion of the sur face of the globe . The
coordinate system t ha t we usua l ly use, the l a t i t u d e and the long i tude ,
i s not s ingu la r i ty
free . I t has s ingu la r i t i e s a t the north
pole
and
a t the south pole . Yet the surface of the globe i s evident ly
without
s i n g u l a r i t i e s . We deal wi th t h i s s i tua t ion usual ly in something
l i ke the way i l l u s t r a t e d
in f igure 2. We consider a rubber sheet
with n ice ly def ined coordinates and s t r e t ch and wrap it down onto the
globe so t ha t it covers more than the nor the rn hemisphere. Simi la r ly ,
MONOPOLES AND FIBER BUNDLES
r l rr7,
LIIJ7 t
lO
y/ rr l J
r lTTJ
FIGURE 2. Hethod of parametrizing the globe.
FIGURE 3. Division of space outside
of
monopole g into
overlapping regions
Ra and R
b
·
55
we consider another rubber sheet with nicely defined coordinates and
s t re tch and
wrap i t upwards so that i t covers more than the southern
hemisphere. We
now
have
a
double system of coordinates to describe
the points on the globe.
The
description i s analyt ic in the domain
covered by each sheet , i f we had done no violence in the stretching
and wrapping. In the overlapping region covered by both sheets ,
one has two coordinate systems which are transformable into each
other by an analyt ic non-vanishing Jacobian. This double
coordinate system i s an ent i re ly sa t i s fac tory way to parametrize
the globe.
Following th i s idea
we shal l now t ry to exorcise the s t r ing
of s ingular i t ies in the monopole problem by dividing space into
two regions. We shal l
ca l l
the points
outs ide
of the origin,
above the lower cone in f igure
3, region Ra. Similar ly, we shal l
ca l l
the points outs ide of the origin, under the upper cone, Rb.
56
c. N. YANG
The union of these two regions gives a l l points
outs ide
of
the
origin. In Ra we shal l choose a vector potent ial for
which
there
is
only one non-vanishing component of A, the azimuthal component:
r s ~ n 8 (1 - cos 8),
(4)
I t
is
important to not ice that th is vector potent ial has no
s ingu la r i t i e s anywhere in Ra. Similarly
in
~ we
choose the
vector
potent ia l
(5)
which no in
Rb. I t i s to
that
the
cur l
of ei ther of these two potent ia ls give correct ly the
magnetic
f ie ld of the monopole.
In the region of overlap, since both of the two se ts
of vector
potent ia ls
share the same cur l , the difference between them must be
cur l less and therefore must be a gradient .
Indeed a simple
calculat ion shows
(6)
where <P i s
the azimuthal angle. The
Schrodinger equation for an
electron in the monopole f ie ld
is thus
1
2
2m
(p-eA ) lj
+
Vlj = Elj in
R
a '
a a a
1 2
2m
(p-eA
b
) 1jJb + V1jJb =
E1jJb'
in
R
b
,
where lj and 1J h are respect ively the wave functions in the two regions.
The f a c ~ that the two vector potent ials in these two equations are
dif ferent by a gradient t e l l s us, by the well known gauge pr incip le ,
that lj a and 1J b are related
by
a phase factor transformation
1jJa =
S1J b' S
= exp ( iea) ,
(7)
or 1jJa = [exp (2iq<P)]1jJb' q = ego
(8)
Around the equator which is ent i re ly in
R ,1jJ i s single valued.
Similar ly, s ince the equator i s also ent ifelya
in R
b
,
1jJb is single
valued
around the equator. Therefore, S must return to i t s original
value when
one
goes around the equator. That implies Dirac 's
quant izat ion condition:
2q = integer .
(9)
MONOPOLES AND FIBER BUNDLES
57
Hilbert Space of Sections
Two
~ ' S ,
~ a and ~ b ' in R
a
, and Rb respectively, that sa t i s fy
the condition of t rans l t ion (8) in the overlap region, is cal led
a
sect ion
by the mathematicians. We see that around a monopole the
electron wave function
i s
a sect ion and not an ordinary function.
We shal l ca l l these wave sections.
Different wave
sect ions (belonging to different
energies, for
example), c learly sa t i s fy the same condition of t ransi t ion (8) with
the same q. Thus we need to develop3 the concept of a
Hilbert
space
of sect ions. To do th is we define the scalar product of two
sect ions ~ , n (for the same q) by
( n , ~ ) = n * ~ d 3 r . (10)
(The question of convergence a t r = 0 and r
Notice that in the overlap
00 is ignored here) .
( n a ) * ~ a = ( n b ) * ~ b
so that (10)
i s well defined.
( l l )
I t
is clear tha t i f ~ i s
a section, then x ~ i s
also a sect ion,
s ince
Thus x i s an operator in the Hilbert space of sect ions. Similarly
we prove that the components of
( ~ - e A )
are operators , but those
of p are not. Furthermore ~
and p-eA are both Hermitian.
Following Fierz
4
we shal l now t ry to construct angular momentum
operators . Define
-+
-+
L
-+ -+ -+
r x (p
- eA)
~
r
(12)
I t is
clear that L , L
y
' L
z
are Hermitian operators on the Hilbert
space of sect ions.
x
The
following
commutation
rules can be eas i ly
verif ied:
[L ,x] 0,
[Lx'Y]
iz ,
[L ,z]
=
- iy ,
x x
[Lx'px -
eA ]
x
0,
[L ,p
x y
- eA ]
y
= i (p - eA ) ,
z z
[Lx'pz -
eA ]
-
i (p
-
eA ) .
z
y
y
(13)
I t
follows
from these that
[L ,L ] = iL , etc .
x y z
(14)
58
c. N. YANG
EQ. (13). together with
i t s consequence
(14). show that Lx. Lv'
L
z
are the angular momentum operators. We emphasize that neither
the Hilbert space, nor these
operators, possess any "s ingular i t ies" .
(The singulari t ies of Aa and
Ab are not
real s ingular i t ies
because
they occur outside of Ra and Rb'
respect ively.)
Monopole
harmonics
Y n
q ..... ,m
2 -+ 2 -+
Since ~ r , L] = 0, we can diagonalize r and
study operator L
for fixed r . I .e . we shal l study sections of the form
2 2
15 ( r - rOH:,
where ~ i s a section dependent only on angular coordinates Sand
<p. t operates then
on
"angular sections".
Eq.
(14)
shows that [L2, L
z
] = O. Simultaneous diagonalization
produces the familiar
multiplets
with eigenvalues R.,(R., + 1) and m
L2y n' = R.,(R.,+l)Y n ; LY a =mY n
q, .... ,m q, .... ,m z q, .... ,m q, ... ,m,
(15)
where
R., = O , ~ , 1 . . • and, for each value of R.,. m ranges from -R., to
3
+R., in in tegra l steps of increment. The Yq.R., are the eigensections
which we shal l ca l l monopole harmonics. We AWall show l a te r that
the allowed values of R., and
mare
Iql, Iql + 1, Iql + 2, . . . ,
m
-R." -R., + 1, . • . ,R."
(16)
and that each of these R. m combinations occur exactly
once.
We
shal l choose each Y normalized so that
7T 27T 2
r inSdS flY R., I d<P = 1. (17)
o 0 q, ,m
(Notice. that in R
ab
, kYq,t,m)aI2 = kYq,R."m)bl 2.) Different Yq,R."m
for a flxed q) are orthogonal, a fact one easi ly proves in the
usual way f r o ~ (15).
We shal l choose
the phases of
Yq,R."m
such
that the matrlx elements of L
z
, L • L
z
between the Y's conform
to
the convention adopted in ch. 2 of Edmonds' bookS. In
part icular
(L +
iL)Y n
= ( R . , - m ) ~ ( R . , + m + l ) ~ Y n
m+l
x y q, ...
,m q ..... ,
(18)
These monopole harmonics wil l
be expl ic i t ly exhibited. Each i s
analytic. That i s , R., ) is analyt ic R a n d R.,
)b
i s
analytic
i n ~ . The s ~ t
o ~ ~ l l monopole h a r m o ~ i c s for
q
• ,m
a fixed q forms a complete set of
sections. as we shal l see.
MONOPOLES AND FIBER BUNDLES
Eq.
Expl ic i t express ions for Y
q,&,m
Sta t ing from (12) one eas i ly ve r i f i e s
mY
q,&,m
L Y
( - ia
q)Yq &
m'
in
z q,&,m <p
,
,
mY
q,&,m
L Y ( - ia
+
q)Yq & m'
in
z q,&,m
<p
, ,
(20) shows
t ha t
Y
q,&,m
8 (e)ei(m+q)<p
q,
&,m
in
R
a '
Y
q,£,m
8 (e)ei(m-q)<p
q,&,m
in
R
b
·
59
(19)
R
a '
R
b
·
(20)
(21)
The condi t ion for a sec t ion shows tha t
[8 (e)]
=[8
n (e ) ]b
q,&,m a q,-<-,m
in the over lap.
They are , in fac t , the same funct ion. Apply'
(19)
to Y An exp l i c i t evaluat ion of the operator [ rx(p- eA)]2
.q,&,m
act1ng on Y gives
q,&,m
[&(& + 1) - q2]8
=[_ 1 ~ s ine ~ + ~
,£,m sine
ae ae
s in e
(m + q cos e) 2J 8 n •
q, C,m
(22)
Writ ing cose = x,
t h i s gives
2 1 .2
- ( l - x )0 +2x8 + l -x
2
(m
+ qx) 8 ,
-1 ~
x
~ 1 ,
(23)
where prime means d i f fe ren t i a t ion with respect t x. This
equat ion can
be t rea ted in the usual way, through analyzing the
ind ica l equat ions a t x = ±l . We sha l l , however, pursue a
d i f fe ren t method which y ie lds the normal iza t ion constant and phase
fac tor automat ical ly .
Before proceeding we note t ha t s ince Y i s s ing le valued in each
region, (21) shows tha t
m - q = in teger .
Thus
& - q in teger .
(24)
60
c. N. YANG
Now
(19)
shows that
2
£(£ + 1) ~ q •
(25)
Eqs. (24) and (25) show that the allowed values of £ are among those
given in (16).
We shal l now show that each value of £ in (16) i s allowed, by
construct ing, for each
of them, the expl ic i t function e 0
q, )(',m
e N
J l · ~ £ - q
11 +x
Hq
, £ - I I = integer
~
0,
(26)
q,£,-£ q, C
where
N
q,£
> o.
(27)
To show th is
one subst i tutes (26) into (23) and ver i f ies tha t the
l a t te r i s sat isf ied. The factor N £ is
inserted so that
Y 0 _0 i s normalized
in the senseqof (17).
q,)(', C
Repeated application of (18) onto the monopole harmonics
Yq £._£ (given by (21)
and (26)) leads to, (for £,m sat isfying
(+6» the expl ic i t expression for Y £ given
3
below. (As
stated above, th is method leads to a a t o r n ~ t i c a l l y normalized
Y
n
s ta r t ing from normalized
Y 0
_0).
q, C,m q,)(', )(,
(Y ) = M
(1_X)a/2(1+X)S/2
p
a 'S(x)e
i
(m+
q
)¢,
q,£,m a q,£,m n
(Yq,£,m)b=
(Yq,£,m)ae-2iq¢,
where
a =
-q -m, S = q - m, n = £ + m, x = cosS,
,;
M _ 2
m
2£+1 (£-m) (£+m) 2
q,£,m- 4n (£-q) (£+q) '
and pa,S(x) are the Jacobi polynomials,
n
(28)
(29)
(30)
n n
pa,S(x)= ~ ( l _ X ) - a ( l + x ) - S __d _ [(l_x)a+n(l+x)S+n], (31)
n 2nn dx
n
which are defined i f
n,n + a, n + Sand n + a + S are a l l integers
~ o.
(32)
MONOPOLES AND FIBER BUNDLES
61
Completeness
of Monopole Harmonics
For a given q(q may be negative) the set of
Y .
with
~ , m sa t
isfying (16) form a complete set of orthonormal
s e ~ t i ~ s . I .e . every
continuous section ( i .e . a section satisfying (9), with ~ a and ~ b
being continuous
in Ra and ~ ) can be expanded as a series
E a Y
~ , m
~ , m q , ~
Proof: Y. can be expressed
3
in terms of pial. lsi (x)
q,"',m \l
ow for fixed q = integer or half- integer, and q + m = integer,
there are four possible cases:
a 2 .
0,
~
0,
so
that
-m 2 .
Iql
and
\ l = 2 + m,
a ~
0,
S
~
0,
so that
Iml
s -q, q ~ O
and \ l =
Q. +
q,
a "5:
0, S
~
9,
s6 that
Iml
~ q,
q:..0
and
\ l
2 - q,
a ~
0, S
~
0,
so that m
~ Iql
and \ l = 2 -
m.
In case (33), the allowed values of ~ according to (16), are
g,
= 1m I, 1m
I
+ 1,. . . which are precisely
\ l = 0, 1 ,
2,
(33)
(34)
(35)
(36)
(37)
In case (34), the allowed values of
g, according to (16) are g, = q,
-q + 1, •. . which are
also
precisely (37). Continuing this
way we
conclude that given q integer or half- integer, q +
m = integer,
the allowed values of ~ according to (16) are always precisely those
given by (37).
Now for
fixed Ia I , Is ,
the Jacobi polynomials P I l l sl , (v=0,
1,
\ • A.(m+ ) \ l
2, . . • ) form a complete set . The exponential functions e ~ ~ q
m+q = a l l integers) also form a complete
set . I t can be proved from
these
results that Y Om forms a complete
set
of section for fixed q.
q, ..,..
Examples and Analyticity of Y
q,g"m
For
the
case q = 0, a = S, and (31) shows
that
(
_l)m ~ : 2 m/2
p-m,-m < (1 )
pm
RTm =
~
(g,+m): -x 2
(38)
62 C.N.YANG
where pR. is the associated Legendre
function. Substitution of (38)
m
into (28) shows that
yO.
= usual spherical harmonics Yo .
,"',m N,m
We tabulate in table 1 a few of the monopole harmonics for
q = ~ , 1 , these examples i l lus t ra te
the fact that Y 0 is analytic
q,N,m
everywhere. I .e . , (Y 0 ) is analytic
in R a n d (Y 0 )b
i s
q,,,,,m a a q,N.m
analytic i n ~ . For e x a m p 1 e , ( Y ~ ) a is clearly analytic in Ra'
which
includes
the
point S=O, and
= I 1 - cosS/ ~ (39)
is clearly analytic
in ~ which includes
the
point S=n.
Schrodinger Equation
I t is simple
to
show by exp1ici.t evaluation, and with
the
aid
of (19)
that
(p -
eA)2
1
a
(r
2
...2.)
1
2
- 2ar
+ -[r
(p- eA)]
ar 2
r
r
1 a
(r
2
L)
+ --. .[L
2
2
(40)
- 2" ar
ar
2
- q
] .
r
r
The Hamiltonian thus
eigenfunctions of
H
and
L . I .e . we take
commutes with L2
and
L .
Hence in solving for
we can choose s p e c ~ f i c eigenvalues for L2
z
tfJ = R(r)Y
0
q,,,,,m.
(41)
obtaining
2
1 a (r2...2.) + R.(H1)-q + v _ E]R =
o.
(42)
-
-2 ar ar 2
mr mr
For the
case that V = 0 this equation was solved by Tamm who found
that R is a Bessel function,
i f E ~ Q ,
1
R = - J (kr)
,
IiU
~
(43)
where
MONOPOLES AND FIBER BUNDLES 63
k = v'zmE •
(44)
I f E 0, (42) has no meaningful solution.
Table 1
Examples of
fuY in a region
9.
R.
m
(Y'4'iTY9. R, 2m) a
~
~ ~
_e
i
4>/1_x
£
~
e
0
/1+x
3
3
2·4>
2"
2"
1312e 1. l1+x(l - x)
3
~
_ ~ i 4 > 11-x(l+3x)
2
3
-
~
_ ~ e o 11+x(l-3x)
2
3
3
13/2e-
i
¢/1-x(l+x)
2"
-2"
1
1 1
.f3/4e
2i
4> (l-x)
1 0
-h /2e
i
4> h_x
2
1
-1 h/4e
o
(l+X)
x
= cose.
To obtain Y
q,R"m
in 11, apply (8).
Dirac E9.uation
Using the monopole harmonics discussed above, one can also
dis
cuss the motion of a
Dirac electron in the field of a magnetic mono
pole. This was done in references 6 and 7 where bound states were
found.
64
C. N. YANG
Remarks
(A) I t i s important to
real ize that the above-described way
of using (A) and (A)b together to
describe
the
magnetic f ie ld of
a monopole h ~ s an addltional advantage: I t gives the
magnetic f ield
H correctly everywhere. In older papers one oftentimes took a single
A with a s t r ing of
s ingular i t ies . Since by defini t ion
11· (lIxA) 0,
the magnetic f ie ld described by IlxA must have continuous flux lines.
Thus i t s
flux lines consist of
the dotted l ines of Figure 4, plus
the bundle l ines described by the l ine ,
so
as to make
net flux a t
the
origin zero. Thus, IlxA
does
not correct ly describe
the magnetic
field of the monopole, a point already
emphasized by
Wentzel. 8
(B) For ordinary spherical harmonics there
are
a number
of
important theorems such as the
spherical harmonics addition theorem,
the decomposition of
products
of spherical harmonics using Clebsch
Gordon coefficients , etc. These theorems can be
9
generalized to
monopole harmonics.
(c) I t is instructive
to
go back
to
the reasoning concerning
Figure
1 and t ry
to repeat the steps
for
the combined A , A descrip
t ion
of the
magnetic f ie ld . Choose the paral le l
to
be ~ h e gquator.
Then
, b
jl(Afl\dX nS·
Thus 41Tg na - nS = Jr ( ~ ) a - ( ~ ) 8 ] dl
which i s , by (6), equal to the increment of a around the equator
i . e . 2g(21T) = 41Tg.
We
have
arrived a t
an
identi ty. My reason for going
through this
simple argument i s that it embodies exactly the
gis t of the
proof
of the famous Gauss-Bonnet-Allendoerfer-Weil-Chern theorem and the
Chern-Weil theorem which seminal in contemporary
mathematics. As a matter of fact , gauge f ie ld , of which electro
magnetism i s the simplest example, is conceptually identical to some
mathematical concepts of f iber bundle theory. Table 2 gives
2
a
t ranslat ion
table for the terminologies used by physicists on the
one
hand and mathematicians on the other. We notice that in part icu
l a r Dirac 's monopole quantization (9) is identical to the mathemati
cal concept of class i f icat ion of U(l) bundles according to
the f i r s t
Chern class.
MONOPOLES AND FIBER BUNDLES
Table 20 Translation of Terminology
Gauge Field terminology
gauge (or global gauge)
gauge type
gauge potential b
k
11
S Eq.(8)
phase factor ~ Q P
f ie ld strength fk
a k l1V
source J
l1
electromagnetism
isotopic spin gauge field
Dirac's monopole quantization
electromagnetism without monopole
electromagnetism
with monopole
a
I
•
e
. , e lectr ic source
65
Bundle terminology
principal coordinate bundle
principal f iber bundle
connection on a principal
f iber bundle
transit ion function
paral le l displacement
curvature
?
connection on a U
I
bundle
connection on a SU
2
bundle
class i f icat ion of U
I
bundle
according o
firsE Chern
class
connection on a t r iv ia l
U
I
bundle
connection on a nontrivial
U
I
bundle
66
"
\
I
/
/
--- . .
\
\
C. N. YANG
FIGURE 4. Magnetic Flux l ines due to A. Since V' (VxA) = 0,
flux l ines
are everywhere continuous. Hence there
is "return flux" along sol id l ine .
TRIVIAL BUNDLE
NONTRIVIAL BUNDLE
(MOEBIUS STRIP)
FIGURE 5. Examples of a t r i v i a l and a nontr ivial f iber bundle.
MONOPOLES AND FIBER BUNDLES
67
The las t two entries of the table identif ies electromagnetism
with and without magnetic monopoles with connections on t r iv ia l
and
nontrivial U(l) bundles. We
can gain some understanding of these
facts by looking at (i) a paper loop and ( i i ) a Moebius s t r ip
(Figure 5). I f they are cut along the dotted lines, each would break
into two pieces. Looking a t the resultant pieces we cannot di f
ferentiate between cases (i) and ( i i ) . The two cases are different
only in the way the "resultant pieces" are put together. In case
( i i ) ,
a twist of one of the "resultant pieces" is necessary.
Thus case (i) corresponds to
Wa = SW
b
, where 5 =
1, (no twist) ;
and case ( i i ) corresponds to (q f 0),
Wa = SW
b
, where S = exp ( 2 i q ~ ) , ( twist) .
A bundle where the transi t ion function S is necessarily
from 1 is cal led nontrivial , because a twist is needed.
electromagnetism with a megnetic monopole i s nontrivial
and (8)] .
FOOTNOTES
1
P.A.Mo Dirac, Proco Roy. Soco A133, 60 (1931).
different
Hence
[cf . (7)
2Tai Tsun Wu and Chen Ning Yang,
Phys.
Rev. D12, 3845 (1975).
3
Tai
Tsun Wu and Chen Ning Yang, Nuclear Phys. Bl07, 365 (1976).
4
M
• Fierz, Helv. Phys. Acta 17, 27 (1944).
5
A.R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton,1960).
6Yoichi Kazama, Chen Ning Yang and Alfred So Goldhaber, to appear in
Phys. Rev. D.
7Yoichi Kazama and Chen Ning Yang, to
appear in Phys. Rev. D.
8
G. Wentzel,
Prog. Theor. Phys. 5uppl. 37-38, 163 (1966).
9
Tai
Tsun
Wu and Chen Ning Yang - to be published.
68
c. N. YANG
D I S C U S S
I O N
S
CHAIRMAN: Prof. C.N. Yang
Scientific Secretaries: N. Parsons and B. Jancewicz
DISCUSSION 1
PHAM QUANG:
Could
you comment on the differences between Dirac's quantiza
t ion relat ion and Schwinger's quantization relat ion which has twice
the value of Dirac?
YANG:
We agree with Dirac's quantization relat ion for g and found no
reason for Schwinger's. Schwinger required two str ings for
reasons
of additional symmetry. In our approach, no str ings are required at
a l l , provided we keep the vector potentials A ~
and A ~ in the allowed
regions Ra
and Rb' respectively. One can deal with the
singulari t ies
in the forbidden regions arbi t rar i ly . In our opinion, the quantiza
t ion rule of Schwinger is groundless.
BERLAD:
Can one construct operators which wil l cause transi t ions between
s ta tes of different q?
YANG:
Maybe, but one does not know how to do i t and be physically
meaningful.
KLEINERT:
From a purely group theoret ical approach, such operators are
easi ly constructed. Your Yqlm for fixed q form a representation of
MONOPOLES AND FIBER BUNDLES 69
0(3.1). The quantum number q seems to
be analogous to the hel ic i ty
of a rotating top so that the angular momentum, of course, has to
be
greater than the in t r ins ic hel ic i ty of such a top. The quantum num
ber q seems to
f i l l precisely the same role here. For different
values of q, the Yqlm seem to form representations of a larger
group,
for example 0(4.1), in
which
you can construct these raising and
lowering operators.
YANG:
You may be
right .
The electron-monopole problem is certainly
more complex than that described by two coordinates e,¢ for fixed
r ,
since
the electromagnetic f ield has a momentum density distr ibution
and therefore has iner t ia . A complete clarif ication of this type of
problem is yet to be made.
Could you clar i fy
the
motivation for introducing the extra term
in the
expression for the angular
momentum commutation relat ions in
the presence of a singular
potent ial?
YANG:
I t was f i r s t shown by Saha that i f you have a monopole of
strength g and an electron charge e, then everywhere in space you have
crossed E and H f ie lds . You wil l have a non-zero Poynting vector
•
•
9
e
The system has cylindrical symmetry about the l ine joining the elec
tron and the monopole. Hence the Poynting vector, and the f ie ld
momentum, points an azimuthal with respect to the axis
of symmetry, giving r ise to an angular momentum - q ~ / r
from the elec
tron to the monopole.
On
the other hand, Fierz,
in
1944,
observed
that the term
-qr / r
is needed to obtain the correct commutation relat ions
for the to ta l
angular momentum. Both points of view should be equivalent in a
f ield theory of electrons and monopoles in interact ion with the elec
tromagnetic f ield. However, such a f ie ld theory is s t i l l to be worked
out.
70
C.N.YANG
There i s , however, a problem in the paper of Fierz owing to
boundary conditions which were not definable due to str ing singu
la r i t ies . In our section approach, we have "opened up" the singu
lar i t ies instead
of bundling them up into a str ing. In this way,
we are able to handle them rigorously.
JONES:
In the Dirac approach, the total flux leaving a sphere surround
ing a monopole is zero because a l l the flux comes back in along the
str ing, even though you cannot see where the str ing is . In your
approach, is the total flux leaving the charge equal to 4ng?
YANG:
Yes. There i s , in our scheme, no "return flux"
at
a l l . In
Dirac's scheme, the vector potential has zero curl. This inevitably
leads to a divergenceless magnetic field. In
other words, there
must be a return flux, as
you said. This return flux in the past
was supposed to be cancelled by the "Dirac veto", and led to much
confusion. In our approach, this confusion is removed from the
beginning by not considering a str ing of "return" flux at a l l .
LIPKIN:
To describe one monopole, you
have divided space into two re
gions. Would you need more
regions to
describe systems with several
monopoles?
YANG:
One
needs more regions when one has more monopoles of whatever
sign.
LIPKIN:
Is there a simple relat ion between the number of regions and
the
wrapping number discussed
in Wick's lectures?
WICK:
need only regions for one monopole, no matter
what the
value of q = ge is .
Therefore, I do not think there is a connection
with my wrapping number. However, there may
be a somewhat
different
way
of formulating the question, to that i t makes sense.
MONOPOLES AND FIBER BUNDLES
71
WIGNER:
I f one writes down the ordinary Dirac equation for two oppositely
charged monopoles, the Hamiltonian is not self-adjoint - - i t shows a
mathematical pathology
as discussed by von Neumann. Does this dif f i
culty
manifest i t se l f also in quantum f ie ld theory, and i f so, how?
YANG:
This same difficulty
exists. However, the diff icul ty does not
imply that quantum f ie ld theory equations for magnetic monopoles are
necessarily meaningless. I t merely means that one cannot interpret
the "stat ic approximation". I believe the
si tuation
is similar to
the case for ordinary QED in the th i r t ies , or even now: the equa
tions
for QED are
probably correct, but
have be understood cor
rect ly. With monopoles included,
the
renormalization theory, par
t icularly of the large
of becmes completely open
question.
WIGNER:
One thing that I am not clear about is whether a
theory with
a
coupling constant of the order of g ~ 137 can be useful because of
the diff icul t ies of convergence.
YANG:
I agree, but there is one thing I would l ike to add. I t is also
not known whether QED forms a convergent series in a; in fact , there
are many guesses that i t does not converge in a.
WIGNER:
That is true, but QED is s t i l l useful; whether i t would s t i l l
be useful with a equal to 137 rather than 1/137 is less clear.
YANG:
Yes, I agree completely; i t is 137
2
less clear
WIGNER:
What are the experimental indications for the
existence of the
monopole?
YANG:
Last summer, 1975, Price and collaborators
published
porting on experimental evidence for a magnetic monopole.
generated many discussions. I t is generally regarded now
conclusive.
a paper re
The report
as not