Download The Search for Unity: Notes for a History of Quantum Field Theory

The Search for Unity: Notes for a History of Quantum Field Theory
Author(s): Steven Weinberg
Source: Daedalus, Vol. 106, No. 4, Discoveries and Interpretations: Studies in Contemporary
Scholarship, Volume II (Fall, 1977), pp. 17-35
Published by: The MIT Press on behalf of American Academy of Arts & Sciences
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STEVEN WEINBERG
The Search for Unity: Notes
Field Theory
for aHistory
of Quantum
Introduction
field theory
is the theory of matter and its interactions, which
Quantum
out
and special relativity in the late
of
the
fusion
of
grew
quantum mechanics
in the fol
suffered frequent fluctuations
1920s. Its reputation among physicists
came
so
close to
at
that
field
low
times
quantum
theory
dropping
lowing years,
be abandoned altogether. But now, partly as a result of a series of striking suc
cesses over the last decade, quantum field theory has become the most widely
framework for attacks on the fundamen
accepted conceptual and mathematical
a set of ultimate laws of nature were to
If
like
tal problems of physics.
something
next
in the
be discovered
few years (an eventuality by no means expected), these
in the language of quantum field
laws would probably have to be expressed
theory.
like a full his
there does not exist anything
To the best of my knowledge,
tory of the past fifty years of quantum field theory. Existing histories of modern
cover special and general relativity pretty thoroughly,
and they take us
physics
through
the
early
years
of
quantum
mechanics,
but
their
treatment
of quantum
the triumph of the statistical
mechanics
ends with
interpretation
generally
a
nature of
not
1927.
This
around
is
only because of the fundamental
pity,
an
but
because
its
offers
also
quantum field theory,
history
interesting insight to
the
nature
of
scientific
advance.
It iswidely
supposed that progress in science occurs in large or small revolu
tions. In this view, the successes of previous revolutions tend to fasten upon the
a body of doctrine, from which he must
scientist's mind a language, amind-set,
to
break free in order
advance further. There is great debate about the degree to
are brought about by the individual scientific genius,
which
these revolutions
to
of dis
able
transcend the fixed ideas of his times, or by the accumulation
there seems to
crepancies between existing theory and experiment. However,
be general agreement that the essential element of scientific progress is a deci
sion to break with the past.
Iwould not quarrel with this view, as applied to many of the major advances
in the history of science. It certainly seems to apply to the great revolutions
in
in
and
of
of
this
the
century:
quantum
physics
development
special relativity
of quantum field theory since 1930 pro
mechanics. However,
the development
in which the essential element of progress has
vides a curious counterexample,
17
WEINBERG
STEVEN
18
been
the
realization,
tum mechanics
again
and
that a revolution
again,
were
in the
revolutions
and
relativity
is unnecessary.
sense
the
of
If quan
French
Revo
lution of 1789 or the Russian Revolution
of 1917, then quantum field theory is
more of the order of the Glorious Revolution
of 1688: things changed only just
so that they could stay the same.
enough
I will try to tell this story here without
To some extent,
using mathematics.
Iwill let the history do the job of explication?I
will go into some of the histori
more
cal developments
fully than others, because they help to introduce ideas
which are needed later. Even so, this is not an easy task?there
is no branch of
natural
is so
which
science
nature
how
as
behaves,
so
abstract,
field
quantum
far
removed
from
notions
everyday
of
theory.
cannot be a story of physics in one country. As we shall see, quantum
and Britain, and was
field theory had its birth in Europe, especially inGermany
revived after World War II by a new generation
in
of theoretical physicists
and
center
the
United
States.
The
has
United
the
States
been somewhat
Japan
of the intense activity of the last decade, but physicists
from many countries in
are
and
Asia
have
made
essential
And
contributions.
Europe
although there
a
national
in
in
discernible
this
styles
they have played only minor role
physics,
or the social or the cultural
history. It is not the national
setting that has deter
mined
the direction of research in physics,
but rather the logic of the subject
itself, the need to understand nature as it really is.
This article is not a history of quantum field theory, but only "notes for a
This
historians of
history." A great deal of work needs to be done by professional
I
science in uncovering
the story of the last half-century
of theoretical physics.
if this article were to spur someone to take on this overdue
would be delighted
task.
Prehistory:
Field Theory and Quantum
Theory
Before quantum field theory came field theory and quantum
theory. The
has been told again and again by able histo
history of these earlier disciplines
rians of science, so Iwill do no more here than remind the reader of the essential
points.
field theory was based on Newton's
theory of
was a
not
Newton
himself
did
of
fields?for
him,
gravitation
gravitation.
speak
"ac
in the universe,
force which acts between every pair of material particles
matter which they contain and propagates on all
to
solid
of
the
cording
quantity
as the inverse square of the dis
sides to immense distances, decreasing
always
of
the eighteenth
tances."1 It was the mathematical
century who
physicists
The
first successful
to
it convenient
found
classical
replace
this
mutual
action
at
a distance
with
a
grav
itational field, a numerical quantity (strictly speaking, a vector) which is defined
at every point in space, which determines
force acting on any
the gravitational
at
from
all the material
which
contributions
that
and
receives
point,
particle
particles
parler?it
one
said
at
every
other
really made
that
the
earth
point.
attracts
this
However,
no difference
was
in Newton's
the moon,
or
a mere
mathematical
fa?on
theory of gravitation
that
the
earth
contributes
field, and that it is this field at the location
general gravitational
which acts on the moon and holds it in orbit.
de
whether
to
the
of the moon
QUANTUM
FIELD
THEORY
19
an existence of their own with the devel
really began to take on
of
the theory of electromagnetism.
in
the
nineteenth
Indeed,
century
opment
It
in
1849.
into physics by Michael
the word "field" was introduced
Faraday
to consider electromagnetic
was still possible for Coulomb
forces
and Ampere
or electric currents, but it
as acting
directly between pairs of electric charges
became very much more natural to introduce an electric field and a magnetic
of space, produced by all the charges and currents in the
field as conditions
Fields
and
universe,
on
turn
in
acting
every
charge
current.
and
This
interpretation
that elec
demonstrated
after James Clerk Maxwell
almost unavoidable
waves travel at a finite
which
the
of
The
force
speed,
speed
light.
tromagentic
is not produced by the
acts on electrons in the retina of my eye at this moment
became
electric
currents
on
in atoms
the
sun
at
the
same
but
moment,
an
by
electro
was produced by these currents about eight
light wave, which
magnetic
now
reached my eye. (As we shall see, an
minutes
and
which
has
ago,
only
was
to account
in
1945
Richard
and John Wheeler
made
attempt
by
Feynman
a
wave,
this
for
work,
of
retardation
forces
electromagnetic
in an
action-at-a-distance
but the idea did not catch on, and they both went
frame
on to more
promising
work).
idea of a field as an indepen
Maxwell
himself did not yet adopt the modern
dent inhabitant of our universe, with as much reality as the particles on which it
acts. Instead (at least at first) he pictured electric and magnetic
fields as distur
bances
in an
medium?the
underlying
in
only
This
one
have
would
special
frame,
had
aether?like
tension
in a rubber
mem
one
obvious
experimental
implication?the
waves would
on the
of
observed
electromagnetic
depend
speed of the
speed
observer with respect to the aether, just as the observed speed of elastic waves in
a rubber membrane
on the
depends
speed of the observer relative to the mem
brane. Maxwell
himself thought that his own field equations were in fact valid
brane.2
at rest
with
respect
to
the
aether.3
The
notion
of
an
aether that underlies the phenomena of electromagnetism
persisted well into the
to discover
twentieth century, despite the repeated failure of experimentalists
as
it
of
the
motion
the
earth
aether
of
the
effects
revolves
about the
any
through
sun.
the idea of a field as an
But even with the problem of the aether unresolved,
own
in
in
its
minds.
Indeed, it became
entity
physicists'
right grew stronger
a
to
matter
that
itself
is
manifestation
of electric and
suppose
popular
ultimately
a
in
the
in
theme
of
the
theories
electron
fields,
magnetic
developed
explored
1900-1905 by Joseph John Thomson,
Wilhelm Wien,4 M. Abraham,5
Joseph
in 1905
Larmor, Hendrik Antoon Lorentz,6 and Jules Henri Poincar?.7 Finally,
the essential element needed to free electromagnetic
from
the
need
for an
theory
aether was supplied by the special theory of relativity of Albert Einstein.8 New
rules were given for the way that the observed space and time coordinates of an
event change with
changes in the velocity of the observer. These new rules were
so that the observed
specifically designed
speed of a light wave would be just
in
that speed calculated
Maxwell's
the velocity of the observ
theory, whatever
er. Einstein's
the effects of motion
theory removed any hope of detecting
through the aether, and although the aether lingered on in theorists' minds for a
it eventually died away, leaving electromagnetic
fields as things in them
while,
selves?the
As
tension
it happens,
in
the membrane,
it was
the study
but
without
the membrane.
of electromagnetic
phenomena
that gave
20
STEVEN
WEINBERG
rise to the quantum
nineteenth
century,
theory as well as to special relativity. By the end of the
it was clear that the classical theories of
electromagnetism
were incapable of
and statistical mechanics
the energy of electro
describing
radiation at various wavelengths
emitted by a heated opaque body.
magnetic
too much energy at very high
The trouble was that classical ideas predicted
so much energy in fact that the total energy per second emitted at
frequencies,
turn out to be infinite! In a paper read to the German
all wavelengths
would
on
December
15, 1900, a resolution of the problem was pro
Physical Society
for us to con
posed by Max Karl Ernst Ludwig Planck.9 It will be worthwhile
centrate on Planck's proposal for a moment,
not only because it led to modern
but also because an understanding
of this idea is needed in
quantum mechanics,
order to understand what quantum field theory is about.
Planck supposed that the electrons in a heated body are capable of oscillating
back and forth at all possible frequencies,
like a violin with a huge number of
or
of
all
Emission
possible
absorption of radiation at a given
strings
lengths.
occurs
frequency
when
the
electron
at
oscillations
that
frequency
give
up
ener
field. The amount of energy
gy to or receive energy from the electromagnetic
an
at
radiated
second
therefore de
any frequency
per
opaque body
by
being
on
amount
at
in
of
oscillations
that
the
electron
average
energy
pends
particular
frequency.
It was
in calculating this average energy that Planck made his revolutionary
that the energy of any mode of oscillation
is quan
suggestion. He proposed
an
not
to
set
with
it
oscillation
tized?that
that
is
is,
any desired
possible
going
as
with
allowed
in
but
certain
distinct
values
classical
mechanics,
energy,
only
that
Planck
assumed
the
difference
between
of the energy. More
specifically,
any two successive allowed values of the energy is always the same for a given
mode
of
oscillation,
and
is
equal
to
the
frequency
of
the
mode
times
a new
of nature which has come to be called Planck's constant. It follows that
the allowed states of the modes of oscillation of very high frequency are widely
a great deal of energy to excite such amode
separated in energy, so that it takes
at all. But the rules of statistical mechanics
tell us that the probability of finding
a great deal of energy in any one mode of oscillation
falls off rapidly with in
constant
creasing energy; hence the average energy in oscillations of very high frequency
must fall off rapidly with the frequency,
and the energy radiated by a heated
must
off
with
the
also
fall
rapidly
frequency of the radiation, thus avoiding
body
the catastrophe of an infinite total rate of radiation.
to radiation
Planck was not ready to apply the idea of energy quantization
is
has described Planck's view as follows: "Radiation
itself. (George Gamow10
or
store
to
can
in
the
be
returned
which
like butter,
quar
grocery
only
bought
the butter as such can exist in any desired
although
ter-pound packages,
amount.") It was Einstein11 who in 1905 proposed that radiation comes in bun
to the
later called photons, each with an energy proportional
dles of energy,
frequency.
were
together by Niels
Bohr in his theory of atomic spectra.12 Like the hypothetical modes of oscilla
tion in Planck's work, atoms in Bohr's theory are supposed to exist in distinct
an
states with certain definite energies, but not generally equally spaced. When
a
a
definite
excited atom drops to a state of lower energy, it emits
photon with
In 1913 the ideas of Planck
and Einstein
brought
FIELD
QUANTUM
THEORY
21
energy, equal to the difference of the energies of the initial and final atomic
to a definite frequency, and it is
states. Each definite photon energy corresponds
that we see vividly displayed when we look at the bright lines
these frequencies
crossing
the
spectrum
a fluorescent
of
lamp
or
a star.
after, was
early quantum theory,
its
brilliant
ad
hoc
mathematical
manipulation,
justified by
inspired guesswork,
success in explaining
the behavior of atoms and radiation. Quantum
theory
became the coherent scientific discipline known as quantum mechanics, through
Max Born, Pascual Jordan,
the work of Louis de Broglie, Werner Heisenberg,
in 1925
and
Erwin
Adrian
Maurice
Paul
Dirac,
Pauli,
Schroedinger
Wolfgang
1926.13 Armed with this formalism, theorists were able to go back to the prob
the allowed energy levels of material systems, and to repro
lem of determining
its origins in the
duce the successful results first found by Bohr. But despite
a coherent way
still
in
mechanics
dealt
of
thermal
radiation, quantum
theory
not
with
in
atoms?and
radiation itself.
only with material particles?electrons
from Planck
This
to Bohr and for a decade
The Birth ofQuantum Field Theory
of 1925-1926 to fields
The first application of the new quantum mechanics
came
one
rather than particles
in
of the founding papers of quantum mechanics
and Jordan14 turned their attention to the
itself. In 1926, Born, Heisenberg,
field in empty space, in the absence of any electric charges or
electromagnetic
currents. Their work can best be understood by an analogy with Planck's 1900
theory of thermal radiation.
Planck, itwill be recalled, had treated the motion of the electrons in a heated
an idealized
picture, in which the electrons were replaced with
body in terms of
an unlimited number of modes of
like a violin with a huge
simple oscillation,
further
number of strings of all possible
He
had
that the
lengths.
proposed
allowed energies of any one mode of oscillation were separated by a definite
quantity,
equal to the frequency of the mode times Planck's constant. One of
the products of the new quantum mechanics
of 1925-1926 was a confirmation of
it was proved that the energies of a simple oscillator,
Planck's proposal:
like a
violin string, were indeed quantized,
in just the way that Planck had guessed.
The essential feature of the dynamics of simple oscillations,
used in obtaining
this result, is that the energy required to produce any given displacement
in the
we pull a violin
to the square of the displacement?as
oscillator is proportional
it becomes harder and
string farther and farther from its equilibrium position,
harder to produce any further displacement.
But essentially
the same is true of an electromagnetic
field?the
energy in
one
to
mode
of
oscillation
of
the
field
is
the
of
the field
any
square
proportional
strength?in
a sense,
to
the
its
of
square
from
"displacement"
the
normal
state
of field-free empty space. Thus,
field the
by applying to the electromagnetic
same mathematical
methods
that they had used for material oscillators, Born et
al. were able to show that the energy of each mode of oscillation of an electro
allowed values are separated by a basic unit of
magnetic field is quantized?the
the
of
the mode times Planck's constant. The physi
energy, given by
frequency
cal interpretation
of this result was immediate. The state of lowest energy is
radiation-free
empty
space,
and
can
be
assigned
an
energy
equal
to zero.
The
STEVEN
22
WEINBERG
times
then have an energy equal to the frequency
can
and
Planck's constant,
be interpreted as the state of a single photon with
that energy. The next state would have an energy twice as great, and therefore
would be interpreted as containing two photons of the same energy. And so on.
to the electromagnetic
the application of quantum mechanics
field had at
Thus,
idea of the photon on a firm mathematical
foundation.
last put Einstein's
next
state must
lowest
and Jordan had dealt only with the electromagnetic
field
Born, Heisenberg,
it did not lead to any
in empty space, so although their work was illuminating,
The first "practical" use of quantum field
predictions.
important quantitative
a 1927 paper of Paul Adrian Maurice Dirac.15 Dirac was
was made
in
theory
an old
problem: how to calculate the rate at which atoms in
grappling with
radiation and drop into states of lower
excited states would emit electromagnetic
was
so
not
correct
in deriving an answer?the
The
much
energy.
difficulty
formula had already been derived in an ad hoc sort of way by Born and Jordan16
this guessed-at formula
and by Dirac17 himself. The problem was to understand
as
a mathematical
crucial
importance,
one
in which
sists
of
an
consequence
of
because
the process
"particles"
excited
atom,
are
quantum
actually
whereas
after
This
mechanics.
of spontaneous
Before
created.
the
event,
the
of
of
of radiation
an
is
con
the
event,
it consists
was
problem
emission
system
atom
in a state
If quantum mechanics
could not deal with
of lower energy, plus one photon.
not
an
it
could
be
and
of
creation
destruction,
processes
physical
all-embracing
theory.
can best be understood
theory of such processes
quantum-mechanical
to
In the absence of any
and
fields
oscillators.
the
between
analogy
by returning
field is like an ensemble of com
interaction with atoms, the electromagnetic
energy is given to any mode of oscilla
pletely isolated violin strings; whatever
The
tion,
or
equivalently,
whatever
the
number
of
photons
of
a
particular
if an atom did not interact
it will stay the same forever. Similarly,
frequency,
state itwas placed. But
in whatever
with radiation, itwould remain indefinitely
atoms
do
interact
with
radiation,
because
electrons
carry
an
electric
charge.
So
the true analogy is with a set of violin strings that are weakly coupled together,
as by the violin soundboard.
knows what happens when one
Every musician
feed energy into the other modes of
will gradually
oscillator
is set going?it
this cannot happen
oscillation until they are all excited. In quantum mechanics
so instead the probability
are
the
because
gradu
quantized,
energies
gradually
was
atom will be found
originally stored in the
ally increases that energy which
that a photon will have been
other words,
field?in
in the electromagnetic
created.
Dirac's
firmed
successful
the
universal
treatment
character
of
the
of quantum
spontaneous
mechanics.
emission
However,
of
radiation
con
the world
was
and
to be composed of two very different ingredients?particles
still conceived
in
terms
but
were
to
of
in
both
be described
fields?which
quantum mechanics,
con
were
and
like electrons
protons
very different ways. Material
particles
state of a system, one had to de
ceived to be eternal; to describe the physical
for finding each particle in any given region of space or
scribe the probabilities
a
range of velocities. On the other hand, photons were supposed to be merely
and
of an underlying
manifestation
field,
entity, the quantized electromagnetic
could be freely created and destroyed.
QUANTUM
FIELD
THEORY
23
It was not long before a way was found out of this distasteful
dualism,
toward a truly unified view of nature. The essential steps were taken in a 1928
and then in a pair of long papers in 1929
paper of Jordan and Eugene Wigner,18
1930 by Heisenberg
and Pauli.19 (A somewhat different approach was also de
veloped in 1929 by Enrico Fermi.20) They showed that material particles could
as the quanta of various fields, in just the same way that the
be understood
is
the
field. There was supposed to be
quantum of the electromagnetic
photon
one field for each type of elementary particle. Thus,
the inhabitants of the uni
verse were conceived
to be a set of fields?an
electron field, a proton field, an
were reduced
in status to mere epiphe
field?and
particles
electromagnetic
nomena. In its essentials, this point of view has survived to the present day, and
forms the central dogma of quantum field theory: the essential reality is a set of
all else is
fields, subject to the rules of special relativity and quantum mechanics;
a
as
derived
of the quantum dynamics of these fields.
consequence
to matter had an immediate
This field-theoretic
implication: given
approach
just as pho
enough energy, it ought to be possible to create material particles,
tons are created when an atom loses energy. In 1932 Fermi21 used this aspect of
quantum field theory to formulate a theory of the process of nuclear beta decay.
in 1896 that a crystal containing uranium salts
Ever since Becquerel discovered
was known that nuclei were
would fog a photographic
subject to vari
plate, it
ous kinds of radioactive decay. In one of these modes of decay, known as beta
the
decay,
nucleus
emits
an
electron,
and
changes
its own
chemical
properties.
the 1920s it was believed that the nuclei are composed of protons
Throughout
so there was no great
and electrons,
paradox in supposing that every once in a
while one of the electrons gets out. However,
in 1931 Paul Ehrenfest and Julius
a
Robert Oppenheimer22
presented
compelling
though indirect argument that
nuclei do not in fact contain electrons, and in 1932 Heisenberg23
in
proposed
stead that nuclei consist of protons and the newly discovered neutral particles,
the neutrons. The mystery was, where did the electron come from when a
nucleus
much
suffered
the
same
a beta
place
as
decay?
the
Fermi's
photon
answer
in the
was
radiative
that
decay
the
of
electron
an
excited
comes
from
atom?it
is created in the act of decay, through an interaction of the field of the electron
with the fields of the proton,
and a hypothesized
the neutron,
the
particle,
neutrino.
problem remained to be solved after 1930, in order for quantum field
to
take its modern form. In formulating
the pre-field-theoretic
theory
theory of
individual electrons, Dirac24 in 1928 had discovered that his equations had solu
to electron states of
tions corresponding
negative energy, that is, with energy
less than the zero energy of empty space. In order to
explain why ordinary
electrons do not fall down into these
states, he was led in 1930
negative-energy
to propose25 that almost all these states are
already filled. The unfilled states, or
"holes" in the sea of negative energy electrons would behave like particles of
positive energy, just like ordinary electrons but with opposite electrical charge:
plus instead of minus. Dirac thought at first that these "antiparticles" were the
protons, but their true nature as a new kind of particle was revealed with the
discovery26 of the positron in cosmic rays in 1932.
Dirac's theory of antimatter allowed for a kind of creation and annihilation
of particles even without
introducing the ideas of quantum field theory. Given
One
STEVEN
24
WEINBERG
a
electron can be lifted up into a positive
enough energy,
negative-energy
to the creation of a positron (the hole in the negative
energy state, corresponding
energy sea) and an ordinary electron. And of course the reverse annihilation
process can also occur. Dirac himself had always resisted the idea that quantum
in
field theory is needed to describe any sort of particle but photons. However,
1934 a pair of papers by Wendell
and by Pauli and
Furry and Oppenheimer27
Victor Weisskopf28
showed how quantum field theory naturally
incorporates
the idea of antimatter, without
particles of negative
introducing unobserved
energy, and satisfactorily describes the creation and annihilation of particles and
and particles and anti
antiparticles. For most theorists, this settled the matter,
now seen as
are
various
of
the
quantum fields.
particles
coequal quanta
It is important to understand
that quantum field theory gave rise to a new
view not only of particles but also of the forces among them. We can think of
two charged particles interacting at a distance not by creating classical electro
fields which act on one another, but by exchanging
photons, which
magnetic
to
one
other
kinds of force
from
the
other.
pass
particle
Similarly,
continually
can be produced by
These
other
kinds
of
particle.
exchanged parti
exchanging
cles are called virtual particles, and are not directly observable while they are
because
exchanged,
being
their
as
creation
real
(e.g.,
particles
a free
electron
a
an electron) would violate the law of conservation
of
turning into photon and
that
dictates
the quantum-mechanical
energy. However,
uncertainty
principle
the energy of a system that survives for only a short time must be correspond
so these virtual particles can be created in intermediate
ingly highly uncertain,
states of physical processes,
but must be reabsorbed again very quickly.
From this line of reasoning, one can infer that the force produced by the
a range (the distance beyond which
a
it
exchange of
given type of particle has
to the mass of the exchanged par
falls off very rapidly) inversely proportional
ticle. Thus,
the photon, which has zero mass, gives rise to a force of infinite
the
range,
familiar
in an
neutrons
and
a million
predict
a few
inverse-square
atomic
nucleus
force
of Coulomb.
was
known
between
force
The
a
to have
range
of
a little
protons
less than
so Hidekei Yukawa29 was able in 1936 to
of a centimeter,
millionth
an
new
kind of particle, the meson, with amass
the existence of
entirely
hundred
times
sumes that the energy
for uncoupled
simple
values of the different
that
of
the
In
electron.
calculating
these
forces,
one
as
a
not just a sum of squares of fields, as
density at point is
but also contains products of the
mechanical
oscillators,
fields (and their rates of change) at that point. These
are the unknowns
that have to be sought by our theo
interactions
multifield
of quantum field theory,
efforts. From the viewpoint
retical and experimental
all questions about the particles of which matter is composed and the forces that
act
among
them
are
only
are the fundamental
what
means
quantum
to
an
end?the
fields,
real
and what
problem
is to determine
are the interactions
among
them.
The Problem of Infinities
the early days of quantum field theory as if itwere a grand
to triumph. This has been a somewhat distorted picture,
from
progress
triumph
the theory was thought to be subject to a grave
for almost from the beginning
I have described
internal
inconsistency.
QUANTUM
FIELD
THEORY
25
a 1930 paper of Oppenheimer,30
who was
problem first appeared in
to calculate the effect on the energy of an atomic electron produced by its
trying
field. Just as the exchange of vir
interaction with the quantum electromagnetic
an
two
electrons produces
tual photons between
energy of interaction between
so
also in quantum field theory the emission of virtual photons and their
them,
same electron produces a self-energy, which might depend
reabsorption by the
as an
on the atomic orbit
occupied by the electron, and which might show up
discov
shift in atomic energy levels. Unfortunately,
observable
Oppenheimer
of
the
electro
the
ered that the energy shift predicted
quantum
by
theory
field was infinite!
magnetic
The infinity here arises because when an electron in an atom turns briefly
The
into
a
photon
the original
tron
involves
and
an
a sum
electron,
these
two
in an infinite variety
electron
over
all
the ways
that
particles
of ways.
can
The
the momentum
share
the momentum
self-energy
can
be
of
of the elec
shared
out,
and
can be, this sum turns out to
because there is no limit to how large the momenta
be infinite. It was not obvious that this would have to happen; after all, there are
series in which one adds up an infinite number
many examples of mathematical
1 + Vi + Va + Vs + . . .).How
of terms and gets a finite result. (For example,
of
found
that
the
the electron behaved more like
ever, Oppenheimer
self-energy
and could hardly be interpreted as a finite
the series 1 + 1 + 1 + 1 + ...,
quantity.
a bit a few years later, when
WeisskopP1
a virtual electron,
processes
positron, and pho
empty space, with the positron and photon then being
over as a
the original electron,
leaving the new electron
state.
to
This contribution
the self-energy cancelled the
real particle in the final
worst part of the
but the self-energy
original infinity found by Oppenheimer,
was left in the form of a sum over virtual momenta which behaved like the series
. . ., and which
1 + Vi + Vi + lA +
still could not be interpreted as a finite
The problem was
included the effects of
ton are created out of
annihilated along with
ameliorated
in which
quantity.
such as the "polarization of
Similar infinities were found in other problems,
and
the
the vacuum" by applied electric fields,32
scattering of electrons by the
electric fields of atoms.33 (One of the few bright spots in this otherwise dis
couraging picture came in the treatment of infinities associated with photons of
it was shown in 1937 by Felix Bloch and Arnold Nord
very low momenta;
sieck34 that these infrared infinities all cancel in the total rates for collisions.) Of
course,
if one
uses
a
theory
to calculate
an
observable
quantity,
and
finds
that
the answer is infinite, one concludes either that amathematical
mistake has been
the 1930s, the ac
made, or that the original theory was no good. Throughout
was that quantum field
no
theory was in fact
cepted wisdom
good, that itmight
be useful here and there as a stopgap, but that something radically new would
have to be added in order for it to make sense.
The problem of infinities in fact has provided
the single greatest impetus
toward a radical revision of quantum field theory. Some of the ideas which were
tried out in the 1930s and 1940s are listed below:
a
1. In 1938 Heisenberg35
proposed that there is fundamental unit of energy,
and that quantum field theory works only at scales of energy that are small
compared with this energy unit. The analogy was with the other fundamental
comes
constants, Planck's constant and the speed of light. Quantum mechanics
STEVEN
26
WEINBERG
as
into play when ratios of energies and frequencies
approaches values as small
Planck's constant, whereas special relativity is needed when velocities approach
values as large as the speed of light. In the same way, Heisenberg
supposed,
some
entirely new physical theory might be needed when energies exceed the
fundamental unit, and some mechanism
in this theory might wipe out the con
tributions
of virtual particles with such high energies,
thus avoiding the prob
lem of infinities.
in
idea
drew
the 1930s from the
support
(Heisenberg's
observation
that the showers of charged particles produced by high energy cos
in quantum
mic rays did not behave as expected
This dis
electrodynamics.
of new particles,
the
crepancy was later realized to be due to the production
and not to a failure of quantum field theory.)
mesons,
2. John Archibald Wheeler36
in 1937 and Heisenberg
in 194337 indepen
a
to
which
would
have replaced
dently proposed
physics,
positivistic
approach
a
sort
with
different
field
of
sometimes
called an "S
quantum
theory
theory,
matrix theory," which would
involve only directly measurable
quantities. They
do not actually allow us to follow what happens to
reasoned that experiments
electrons in atoms or in collision processes.
Instead, it is only really possible to
measure
the energies and a few other properties of bound systems like atoms,
for various collision processes. These quantities obey cer
and the probabilities
such as reality, conservation
of probabilities,
tain very general principles,
smooth
principles
energy
conservation
dependence,
that were
supposed
laws,
to replace
etc.,
it was
and
the assumptions
these
general
of quantum
field
theory.
3. Dirac38 in 1942 suggested that quantum mechanics
ought to be expanded
as the initial or
not
to include states of
which
could
appear
negative probability,
final state of any physical process, but which would have to be included among
the
states
intermediate
introduced
into
the
in these
sums
over
processes.
the ways
In
that
this manner,
the
minus
intermediate
signs
states
might
share
be
out
of the system, so that a finite answer would be obtained,
just as
lA + . . . is a finite quantity (the natural logarithm of 2)whereas
lA + . . . is infinite.
4. As already mentioned,
and John Wheeler39
Richard Feynman
in 1945
of abandoning field theory altogether,
considered
the possibility
replacing the
interaction among particles with a direct action at a distance.
field-mediated
Some of these ideas have survived and are now part of the regular equipment
In particular,
the idea of a pure S-matrix
of theoretical physics.
theory has
the momentum
?
?
1
Vi + Vi
1 + Vi + Vi +
involve
in the development
of so-called "dispersion relations" which
flourished
are now a
states
of
observable
Also,
only
negative probability
quantities.40
device, useful especially for dealing with the polarization
handy mathematical
none of these ideas
of virtual photons.41 However,
proved to be the key to the
of
infinities.
problem
than most theorists had
The solution turned out to be far less revolutionary
an
had
been
found by Oppenhei
the
of
electron
that
Recall
energy
expected.
to receive an infinite contribution
from the emis
and Weisskopf
mer, Waller,
sion and reabsorption of "virtual" photons. An infinite self-energy of this sort
in orbit in an atom, but also when
appears not only when the electron ismoving
it is at rest in empty space. But special relativity tells us that the energy of a
mass by the famous formula E = mc2. Thus the
particle at rest is related to its
QUANTUM
FIELD
THEORY
27
electron mass found in tables of physical data could not be just the "bare" mass,
the quantity appearing in our equations for the electron field, but would have to
be identified with the bare mass plus the infinite "self mass," produced by the
interaction of the electron with its own virtual photon cloud. This suggests that
the bare mass might
itself be infinite, with an infinity which
just cancels the
a
mass
to
in
finite
total
the self mass, leaving
be identified with the mass
infinity
that is actually observed. Of course, it goes against the grain to*suppose that a
field equations
like the bare mass, which appears in our fundamental
quantity
we
can
never
turn
off the electron's virtual photon
could be infinite; but after all,
cloud
to measure
the
bare
mass,
so no
paradox
arises.
Similar
remarks
apply
to
other physical parameters,
like the charge of the electron. Is it then possible that
masses
and bare charges cancel the infinities found in
the infinities in the bare
not
quantum field theory,
only when we calculate the total masses and charges,
as well? This method,
of eliminating
but in all other calculations
infinities by
a redefinition of
come to be called
has
into
them
parameters,
physical
absorbing
renormalization.
in 1936,42 and again
idea was suggested by Weisskopf
in the mid 1940s. However,
it was far from obvious that the idea
In order to eliminate
infinities by absorbing them into redefini
tions of physical parameters,
the infinities must appear in only a special way, as
The
renormalization
by Kramers43
would work.
corrections
to
the
observed
values
of
these
parameters.
For
instance,
in order
to
absorb the infinite shift in atomic energy levels found by Oppenheimer
into a
redefinition of the electron mass, the infinite part of the self-energy would have
to be the same for all atomic energy levels. The mathematical
available
methods
in the 1930s and early 1940s were simply inadequate for the task of sorting out
to see if they could
all the infinities that might appear in all possible calculations
was
even
more
all be eliminated by renormalization.
important, there
Perhaps
no
reason
to
were
no
do
so?there
data
which
forced
experimental
compelling
theorists to come to grips with these problems. And of course, physicists had
other things on their minds from 1939 to 1945.
Revival
at Shelter Island
on the foundations
On June 1, 1947, a four-day conference
of quantum
a
near
at
mechanics
Shelter
small
island
of
the
end
Island,
opened
Long Island
in New York State. The conference brought together young American
physi
cists from the new generation who had started their scientific work during the
war at Los Alamos and the MIT Radiation
as well as older
Laboratory,
physi
cists who had been active in the 1930s. Among
the younger participants was
Willis Lamb, an experimentalist
then working
in the remarkable group of physi
cists founded at Columbia University
I.
I.
the re
Rabi. Lamb announced
by
sults of a beautiful experiment,
in which he and a student, R. C. Retherford,
an effect of the
had for the first time measured
self-energy of the electron in the
hydrogen
atom.44
The existing theory of the hydrogen atom had been first advanced by Niels
Bohr in 1913,12 then put on a sound mathematical
foundation by the quantum
mechanics
of 1925-1926,13
and finally corrected to include effects of relativity
and the spin of the electron by Heisenberg
and Jordan,45 C. G. Darwin,46
and
STEVEN
28
WEINBERG
In the final version of this theory, in particular as formulated by Dirac,
certain pairs of excited states of the atom were expected to have exactly equal
to the two different ways that the spin
energy. (These pairs of states correspond
of the electron and the angular momentum
associated with its revolution around
the nucleus could combine to give a definite total angular momentum.)
But this
own
with
all
effects
of
the
of
the
electron
its
electro
interaction
theory ignored
to
field?the
effects
that
had
tried
calculate
when
he
magnetic
Oppenheimer30
discovered
the infinities. If such effects were real they would presumably
shift
the energies of these pairs of states, so that they would no longer be exactly
Dirac.24
equal.
and Retherford
found. By using the new techniques of
radiation that had come out of wartime work in radar, they
that the energies of the first two excited states of hydrogen
were
and
(the 2sy2
supposed to be equal according to the 1928
2pV2states), which
is now
Dirac
differed
by about 0.4 parts per million. This
theory, actually
known as the Lamb shift.
at Shelter Island en
in part by Lamb's results, the participants
Stimulated
of the underlying
tered into an intense discussion
theory. I was not at Shelter
Island (having just entered high school) and I cannot trace the historical devel
of quantum field theory that developed
opment of the different reformulations
most
It
be
valuable
for a historian of science to gather
around that time.
would
at
Shelter Island and succeeding
confer
the recollections
of the participants
This
is what
Lamb
handling microwave
were able to show
ences,
the
read
papers
that were
written
at
that
time,
and
put
together
a coher
ent account. I will sketch here only a few of the products of this period.
The Lamb shift itself was first calculated by Hans Bethe,47 I believe on the
to eliminate
train ride back from Shelter Island. Using mass renormalization
infinities,
he
obtained
a
in
result
reasonable
agreement
with
the
value
an
this was a
nounced by Lamb. However,
by Bethe himself,
acknowledged
were
not
that
fully consistent with
rough calculation,
involving approximations
the Special Theory
of Relativity.
of quantum field theory
There were at least three general reformulations
were
out
in the late 1940s that
relativistic and that were
worked
thoroughly
a
treatment
to
of the infinities.
allow
systematic
sufficiently
simple and elegant
well
the Shelter
been
before
One of these approaches had actually
developed
in
and his colleagues
Island Conference,
Japan, but I
by Sin-Itiro Tomonaga48
as
States in the
believe that their work had not yet become known in the United
at
summer of 1947. The two other approaches were contributed by participants
and Richard Feynman.50
Shelter Island, Julian Schwinger49
one to associate
a
Feynman's work led to set of pictorial rules which allowed
a definite numerical
to each picture of how the momentum
and energy
quantity
could flow through the intermediate states of any collision process: the probabil
sum of these individual quan
ity for the process is given by the square of the
rules were very much more than a handy calculational
tities. The Feynman
an essential feature of quantum field theo
algorithm, because they incorporated
a
and
between
symmetry
antiparticles. Each line in Feynman
particles
ry?the
a
one
at
can
of
the
and de
end
line
created
either
represent
particle
diagram
or
an
It
is
at
this
the
other
the
other,
way.
equal
antiparticle
going
stroyed
treatment
of
particles
and
antiparticles
that
ensures
that
the
quantities
calcu
QUANTUM
FIELD
THEORY
29
lated by Feynman diagrams are independent of the velocity of the observer, as
at every stage of the calculation.
of Relativity,
required by the Special Theory
states involving
intermediate
As had been shown long before by Weisskopf,31
a crucial role in
down
the
of
infinity, from
antiparticles
play
cutting
degree
. . ., to
more
like
disasters
like 1 + 2 + 3 +
something
manageable
1 +
+
Vi
Vi
. . . .The
Feynman
rules
ensured
automatically
the
cancellations
of the worst infinities, leaving over the more manageable
infinities, which could
be eliminated by renormalization.
Before the end of 1947 Schwinger51 had used his approach to carry out what
I believe was the first calculation of another effect of the electron's cloud of
virtual
umphs
the
the
photons,
of Dirac's
electron,
anomalous
a number
which
moment
magnetic
1928 theory24 was
of
its prediction
characterizes
the
the
One
electron.
of the magnetic
strength
of
the
of
the
tri
moment
electron's
of
inter
action with magnetic
field. How
fields, and the strength of its own magnetic
at Columbia
in 1947 had revealed
that the magnetic
ever, experiments52
moment of the electron is actually a little larger than the Dirac value, by 1.15 to
1.21 parts per thousand. By absorbing the infinite effects of virtual photons into
a renormalization
of the charge of the electron, Schwinger was able to calculate
a finite
moment
that was larger than the Dirac value by just 1.16 parts
magnetic
per thousand!
Of course, both
fects like the Lamb
the experimental
and the theoretical determinations
of ef
shift and the anomalous magnetic moment
of the electron
ex
have been enormously
improved since 1947.53 For instance, right now the
moment
value
of
the
of
the
electron
is
than
the
Dirac
perimental
larger
magnetic
value by 1.15965241 parts per thousand, whereas
the theory gives this anoma
as 1.15965234 parts per thousand, with uncertainties
lous magnetic moment
of
about 0.00000020
and 0.00000031
parts per thousand,
cision of the agreement between theory and experiment
The pre
respectively.
here can only be called
spectacular.
in 1949 showed that the formalisms of Schwinger
Finally, Freeman Dyson54
and Tomonaga
would yield the same graphical rules that had been found by
also carried out an analysis of the infinities in general Feyn
Feynman. Dyson
man
and
sketched out a general proof that these infinities are always
diagrams,
of precisely the sort which could be removed by renormalization.55
As a gradu
ate student in the mid-1950s,
I learned the new approach to quantum field
lucid papers.
theory by reading Dyson's marvelously
I should emphasize
that the theory of Schwinger,
Feynman,
Tomonaga,
and Dyson was not really a new physical theory. It was
simply the old quantum
field theory of Heisenberg,
Pauli, Fermi, Oppenheimer,
Furry, and Weisskopf,
but cast in a form far more convenient
for calculation,
and equipped with a
more realistic definition of
masses
like
and charges. The
physical parameters
continued vitality of the old quantum field theory after fifteen years of attempts
to find a substitute
is truly impressive.
This raises an interesting historical question. All the effects that were calcu
lated in the great days of 1947-1949 could have been estimated
if not actually
calculated at any time after 1934. True, without
the renormalization
idea, the
answers would have been
at
but
it
least
would
have
been
infinite,
formally
to
the
order
of
of
like
the
Lamb
shift and
guess
possible
magnitude
quantities
STEVEN
30
WEINBERG
the correction to the magnetic moment
of the electron.
(An infinite series like
1 + Vi + Vi + !4 + . . . grows very slowly; after amillion
terms, it is still less
than 14.4.) Not only was this not done?most
theorists seemed to have believed
that these quantities were zero! Indeed, some evidence for what was later called
in 1938, but to the best of my
the Lamb shift had actually been discovered56
see whether
to
no
of this
checked
the order of magnitude
theorist
knowledge,
was
or
a
more
in
less
would
be
what
quan
expected
reported energy splitting
tum field theory.
was quantum field theory not taken more seriously? One reason is the
Why
tremendous prestige of the Dirac theory24 of 1928, which had worked so well in
for the fine structure of the hydrogen
spectrum without
including
accounting
more
the
of infinities dis
Even
effects.
appearance
important,
self-energy
credited quantum field theory altogether inmany physicists' minds. But I think
that may not have been
that the deepest reason is a psychological
difficulty,
is a huge apparent
of
There
science.57
historians
by
sufficiently
appreciated
distance between
the equations that theorists play with at their desks, and the
practical
of
reality
atomic
and
spectra
collision
a certain
It takes
processes.
cour
age to bridge this gap, and to realize that the products of thought and mathemat
ics may actually have something to do with the real world. Of course, when a
branch of science is well under way, there is continual give and take between
and one gets used to the idea that the theory is about
theory and experiment,
the pressure of experimental
real.
Without
data, the realization
something
comes harder. The great thing accomplished
the
discovery of the Lamb shift
by
our
us
was not so much
to
that it forced
theories, as that it
physical
change
us
to
take them seriously.
forced
Weak
and Strong
Interactions
for quantum field theory was at a
1949, enthusiasm
soon lead to an understanding
it
that
would
expected
not
the
only
dynamics of photons, electrons, and
phenomena,
For a few years after
theorists
high level. Many
of all microscopic
fidence?shares
there
it was
However,
positrons.
began
not
in quantum
a second
depression,
long
field
before
theory
which
was
there
tumbled
was
to
last
another
collapse
on the physics
for
almost
in con
and
bourse,
twenty
years.
of the renormaliza
Part of the problem arose from the limited applicability
a
of
to
renormalization
be eliminated
tion idea. In order for all infinities
by
to
infinities
masses
for
these
and charges, it is necessary
physical parameters like
arise
in
only
a limited
number
of ways,
as corrections
to masses,
charges,
etc.,
but not otherwise. Dyson's work54 showed that this would be the case for only a
small class of quantum field theories, which are called renormalizable theories.
and positrons
The simplest theory of photons,
(known as quantum
electrons,
in this sense, but most theories are not.
electrodynamics) is renormalizable
was one important class of physical phenomena which
there
Unfortunately,
field theory. These were
not be described by a renormalizable
apparently could
the weak interactions, which cause the radioactive beta decay of nuclei mentioned
Fermi21 had in
Field Theory.
above in the section on the Birth of Quantum
in 1932 which, with a few modifications,
vented a theory of weak interactions
FIELD
QUANTUM
approximation,
culations
all weak
described
adequately
of
that is, including
rates.
transition
THEORY
31
interaction phenomena
in the
a
only
single simple Feynman
as
However,
as
soon
this
diagram
was
theory
order of
lowest
in cal
to the
pushed
next order of approximation,
it exhibited infinities which could not be removed
a
of
redefinition
by
physical quantities.
The other major problem had to do with the limited validity of the approxi
mation techniques used in 1947-1949. Any physical process is represented by
an
sum
infinite
quence
ous
of
Feynman
states consisting
of intermediate
To
types.
is the
process
each
square
diagram
of the
we
sum
these
numbers
a numerical
quantities.
a
representing
of definite
associate
of
one
each
diagrams,
quantity;
in
Now,
se
particular
of particles
the
rate
of vari
of
the
electro
quantum
are very small;
the quantities
associated with complicated
dynamics
diagrams
for each additional photon line there is one additional factor of a small number
known as thefine-structure constant, roughly 1/137. In the Fermi
theory of weak
factor is even smaller?at
the typical energies of
interactions, the corresponding
it is 10"5 to 10~7. It is the rapid decrease of the
elementary particle physics,
contributions
associated with complicated
Feynman
diagrams
(together with
the renormalizability
of the theory) that makes it possible to carry calculations
in quantum
However,
electrodynamics
in addition
to
to
such
a fantastic
and
electromagnetic
degree
weak
of
accuracy.
there
interactions,
is one
in elementary
other class of interaction
known as the strong
particle physics,
interactions. It is the strong interactions that hold atomic nuclei
together against
the electrostatic
repulsion of the protons they contain. For strong interactions,
to the fine structure constant is
the factor corresponding
roughly of the order of
one instead of 1/137, so
are
complicated Feynman diagrams
just as important as
ones.
course
are
of
is
these
interactions
al
(This
simple
why
strong.) Thus,
though attempts have been made time and time again to use quantum field
theory in calculations of the nuclear force, it has never really worked in a con
vincingly quantitative
mesons
and hyperons
and
then
in accelerator
kinds of strongly interacting particles called
way. New
were being discovered
from 1947 on, first in cosmic rays
laboratories,
and
quantum
field
theory
was
at first
en
thusiastically used to study their strong interactions, but again with little quan
titative success.
It was not that there was any
in thinking of
difficulty
renormalizable
quantum field theories that might account for the strong inter
was just that
actions?it
having thought of such a theory, there was no way to
use it to derive reliable
and to test if it were true.
quantitative
predictions,
The nonrenormalizability
of the field theory of weak interactions and the
uselessness
of the field theory of strong interactions
led in the early 1950s to a
with quantum field theory. Some theorists turned
disenchantment
widespread
to the study of symmetry
and conservation
laws, which can be ap
principles
detailed dynamical
calculations. Others
plied to physical phenomena without
and Heisenberg,
and worked to
picked up the old S-matrix theory of Wheeler
of
interaction
that
would
involve
develop principles
strong
physics
only observ
able quantities.
In both lines of work, quantum field
was used heuristi
theory
but not as a basis for quantitative
cally, as a guide to general principles,
calculation.
Now,
as a result of work
by many
physicists
over the last decade,
quantum
STEVEN
32
WEINBERG
field theory has again become what it was in the late 1940s, the chief tool for a
are quantum
detailed understanding
of elementary
particle processes. There
field theories of the weak and strong interactions, called gauge theories, which are
not subject to the old problems of nonrenormalizability
and incalculability,
and
which are to some extent even true! We are still in the midst of this revival, and
I will not try to outline its history, but will only summarize how the old prob
in the new theories.58
lems of quantum field theory are surmounted
The essence of the new theories is that the weak and the strong interactions
are described
in a way that is almost identical to the successful older quantum
field theory of electromagnetic
interactions. Just as electromagnetic
interactions
so
are
of
the weak
the
among charged particles
produced by
photons,
exchange
interactions are produced by the exchange of particles called intermediate vector
bosons and the strong interactions by the exchange of other particles called gluons.
vector bosons,
All these particles?photons,
and gluons?have
intermediate
certain
and
have
interactions
symmetry prin
spin,
by
powerful
governed
equal
as gauge
states that the
known
(A
gauge
symmetry
principle
ciples
symmetries.
fundamental equations do not change their form when the fields are subjected to
certain
whose
transformations,
theories
these
are
so
similar
varies
effect
to
quantum
with
and
position
electrodynamics,
they
time.)
share
Because
its funda
of being renormalizable.
Indeed, the relation between weak
and electromagnetic
interactions is not merely one of analogy?the
theory uni
vector
fies the two, and treats the fields of the photon and the intermediate
of a single family of fields.
bosons as members
like the photon, but in
The
intermediate vector bosons are not massless,
stead are believed to have perhaps 70 to 80 times the mass of a proton or neu
the
tron. This huge mass
between
is not due to any essential dissimilarity
vector
from
the
and
boson
but
instead
arises
the
intermediate
fields,
way
photon
field theory breaks down when the field
that the symmetry of the underlying
are solved. The family of intermediate vector bosons, of which
the
equations
one heavy charged particle and its
a
to
is
is
contain
believed
member,
photon
mental
property,
called
antiparticle,
the W+
and W~,
and
one
even
heavier
neutral
particle,
called
like nuclear
the Z?. Exchange of the W produces the familiar weak interactions,
a new kind of weak
Z?
would
of
the
beta decay, whereas
produce
exchange
in which
the participating
interaction,
particles do not change their charge.
in 1973, and are found to have just
Such neutral current processes were discovered
in
All of the intermediate vector
theories.
these
about the properties
expected
too heavy to have been produced with existing accelerator
bosons are much
them with colliding beams of
facilities, but there are great hopes of producing
too
before
and
protons
long.
antiprotons
the strong interactions may well have
In contrast, the gluons which mediate
zero
mass.
Such
theories
with
massless
gluons
have
a
remarkable
property
as asymptotic freedom?at
the
very high energy or very short distances,
now
it
is
In
decreases.
of
the
interactions
consequence,
strength
gradually
gluon
out detailed calculations of strong
possible to use quantum field theory to carry
it has been pos
interaction processes at sufficiently high energy. In particular,
a
some
in
to
account
the
observed
features
for
of
sible
process such as high
known
energy
electron-nucleon
scattering.
FIELD
QUANTUM
THEORY
33
Just as the gluon interactions become weak at high energy and short dis
tances, they also become strong at low energy and long distances. For this rea
son, it iswidely believed (though not yet proved) that particles which carry the
quantity called color with which gluons interact (in the same sense that photons
interact with electric charge) cannot be produced as separate free particles. The
colored particles
include the gluons themselves,
which
is presumably
why
as real
never
have
been
observed
The
colored
gluons
particles.
particles are also
believed to include the quarks discussed
in this double issue o? Daedalus in the
article by Sidney Drell. The observed strongly interacting particles such as neu
trons,
protons,
and
mesons
are
to be
believed
compound
states,
of
consisting
no net color. This
a
quarks, antiquarks, and gluons, but with
picture represents
over
matter:
the
field
the
view
of
the
funda
of
nearly complete triumph
particle
mental entities are the quark and gluon fields, which do not correspond to any
particles that can be observed even in principle, whereas the observed strongly
interacting
particles
are
not
elementary
at all,
but
are mere
consequences
of
an
quantum field theory.
are
a unified gauge
and
hopes of
theory of weak, electromagnetic,
intermediate vector bosons, and gluons would
strong interactions. The photon,
then form part of a single family of fields. However,
in order for this to be
to
to
in
there
would
have
be
other
fields
this
possible,
family, corresponding
mass.
one
to
the
estimate,
particles of extraordinarily
expected
high
According
mass of these new
particles would be 1017 (a hundred thousand million million)
proton masses. These masses are so high that it is no longer possible to ignore
as done almost
the gravitational
fields of these particles,
else in
everywhere
particle physics.
strenuous
to the present,
efforts which
continue
Unfortunately,
despite
there still has not been found a satisfactory (e.g., renormalizable)
quantum field
of
It
is
that
ironic
which
theory
gravitation.
provided the first classi
gravitation,
cal field theory, has so far resisted incorporation
into the general framework of
quantum field theory.
this history I have put great emphasis on the condition of renor
Throughout
that it should be possible to eliminate all infinities
the
malizability,
requirement
in a quantum field theory by a redefinition of a small number of physical param
eters. Many physicists would
disagree with this emphasis, and indeed, itmay
or not, are
be
found
that
all
quantum field theories, renormalizable
eventually
underlying
There
equally
satisfactory.
However,
it has
always
seemed
to me
that
the
requirement
of renormalizability
has just the kind of restrictiveness
that we need in a funda
mental physical theory. There are very few renormalizable
quantum field theo
ries. For instance,
to construct
it is possible
field theories of
quantum
we like, but
in
moment
which
the
electron
has
any magnetic
electromagnetism
one
a
to
of
these
moment
of
theories,
only
corresponding
magnetic
. . . times the Dirac value, is renormalizable.
we
1.0011596523
as
have
Also,
seen, it took a long time before it was found that there are any renormalizable
theories at all of the weak interactions. We very much need a
guiding principle
to help us to pick the quantum field
like renormalizability
theory of the real
world out of the infinite variety of conceivable quantum field theories. Thus,
if
to
is
some
be
with
other
I
condition,
renormalizability
ultimately
replaced
STEVEN
34
WEINBERG
be one that is equally or even more restrictive. After all,
to describe the world as we find it, but to explain to the
extent why it has to be the way it is.
would hope that itwill
we do not want
merely
greatest
possible
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advice from Felix Bloch, Freeman Dyson,
Francis Everitt,
grateful for valuable
and Victor Weisskopf.
^his
often quoted summary of Newton's
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theory of gravitation
revised and annotated
p. 546, in the translation of A. Motte,
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by
F. Cajori
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on
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article
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26C. D. Anderson,
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QUANTUM
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35
35W. Heisenberg,
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(1945); ibid., (19) (1946). (Heisenberg
actually
a fundamental
an energy,
introduced
but the two are equivalent
notions:
any
length rather than
an energy,
the energy of a photon with
that wavelength).
length determines
38P. A. M. Dirac, Proc. Roy. Soc. (London), A180
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and R. P. Feynman,
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17 (1945): 157; ibid., 21 (1949): 425.
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Medd.,
in QED.
reprinted
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report of Kramers' work that I can find is in the Rapports du VIH Conseil de
Kramers
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Physique Salvay 1948 (ed. R. Stoops, Brussels,
reported on his work at the
Shelter
Island Conference.
44W. E. Lamb,
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Jr. and R. C. Retherford,
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inQED.
ibid., 3 (1948): 290. Some of these articles are reprinted
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Tomonaga,
Tomonaga,
664, 914; 91
in
reprinted
QED.
50R. P. Feynman,/?^.
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749,
367;Phys. Rev. 74(1948):
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in QED.
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E. B. Nelson,
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and H. M. Foley, Phys. Rev.,
72 (1947):
Phys. Rev.,
1256.
53For early calculations
of the Lamb shift, see J. B. French and V. F.
Weisskopf,
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ibid. (1949): 388; R. P. Feynman,
(1949): 1240; N. M. Kroll and W. E. Lamb,
Phys. Rev., 74 (1948):
and S. Tomonaga,
1430; H. Fukuda, Y. Miyamoto,
Prog. Theor. Phys. (Japan), 4 (1948): 47, 121.
The article by Kroll and Lamb
is reprinted
in QED.
54F. J. Dyson,
75 (1949): 486,
in QED.
1736, reprinted
Phys. Rev.,
to complete Dyson's
needed
55Elements
later supplied by A. Salam, Phys. Rev., 82
proof were
118 (1960): 838.
(1951): 217; 84 (1951): 426; and S. Weinberg,
Phys. Rev.
56S. Pasternack,
1113. Also see W. V. Houston,
51 (1937): 446;
P/ryj. ?w.,
Phys. Rev., 54(1938):
R. C. Williams,
results were reported by J. W. Drinkwater,
O.
Phys. Rev., 54 (1938): 558. Contrary
and W. E. Williams,
Proc. Roy. Soc,
174 (1940): 164.
Richardson,
57I have recently discussed
this point in a quite different
context
in The First Three Minutes?A
Modern View of the Origin of the Universe
(New York: Basic Books,
1977), Ch. 6.
58For a review of this subject on a technical
to the
see
level, with references
literature,
original
Rev. Mod. Phys., 46 (1974): 255. I have also
accounts
in
e.g. S. Weinberg,
given nonmathematical
231 (July 1974): 50; and in Bull. Am. Acad. Arts and Sciences, 24 (4) (1976): 13;
Scientific American,
in Plural,
of
(in Spanish)
(1976); and in American Scientist, 65 (March 1977): 171. Aspects
reprinted
this subject are treated in two recent books: J. C.
(Cam
Gauge Theories of Weak Interactions
Taylor,
bridge, England: Cambridge
University
Achievements
of Twentieth Century Physics
Press,
1976); G. Feinberg, What Is theWorld Made Of? The
Anchor
(Garden City, N.Y.:
1977).
Press/Doubleday,