飞儿钩编杯垫图解:THE GREENING OF QUANTUM FIELD THEORY GEORGE AND I*
THE GREENING OF QUANTUM FIELD THEORY
GEORGE AND I*
Julian Schwinger
University of California, Los Angeles, CA 90024-1547
The young theoretical physicists of a generation or two earlier subscribed to the belief
that: If you haven’t done something important by age 30, you never will. Obviously, they
were unfamiliar with the history of George Green, the miller of Nottingham.
Born, as we all know, exactly two centuries ago, he received, from the age 8, only a
few terms of formal education. Thus, he was self-educated in mathematics and physics,
when in 1828, at age 35, he published, by subscription, his first and most important work:
An Essay on the Applications of Mathematical Analysis to the Theory of Electricity and
Magnetism. The Essay was dedicated to a noble patron of the “Sciences and Literature”,
the Duke of Newcastle. Green sent his own copy to the Duke. I do not know if it was
acknowledged. Indeed, as Albert Einstein is cited as effectively saying, during his 1930
visit to Nottingham, Green, in writing the Essay, was years ahead of his time.
There are those who cannot accept that someone, of modest social status and limited
formal education, could produce formidable feats of intellect. There is the familiar example
of William Shakespeare of Stratford on Avon. It took almost a century and a half to
surface, and yet another century to strongly promote, the idea that Will of Stratford could
not possibly be the source of the plays and the sonnets which had to have been written
by Francis Bacon. Or was it the earl of Rutland? Or perhaps it was William, the sixth
earl of Derby? The most recent pretender is Edward deVir, Seventeenth earl of Oxford,
notwithstanding the fact that he had been dead for 12 years when Will was put to rest.
I have always been surprised that no one has suggested an analogous conspiracy to
explain the remarkable mathematical feats of the Miller of Nottingham. So I invented one.
* Lecture of July 14, 1993, at Nottingham.
Descended from one of the lines of the earl of Nottingham was the branch of the earls
of Effindham, which was separated from the Howards in 1731. The fourth holder of the
title died in 1816, with apparently no claimant. In that year, George Green, age 23, could
well have reached the maturity that led, 12 years later, to the publication of the Essay.
And what of the remarkable fact that, in the same year that the earldom was revived,
1837, George Green graduated fourth wrangler at Cambridge University?
The conspiracy at which I hint darkly is one in which I believe quite as much as I
think Edward deVir is the real Shakespeare.
I consider myself to be largely self-educated. A major source of information came
from my family’s possession of the Encyclopedia Brittanica Eleventh Edition. I recently
became curious to know what I might have, and probably did, learn about George Green,
some 65 years before.
There is no article detailing the life of George Green. There are, however, 4 brief
references that indicate the wide range of Green’s interests.
First, in the article Electricity, as a footnote to the description of Lord Kelvin’s work,
is this:
In this connexion the work of George Green (1793-1841) must not be forgotten.
Green’s Essay on the application of mathematical analysis to the theories of electricity
and magnetism, published in 1828, contains the first exposition of the theory of potential.
An important theorem contained in it is known as Green’s theorem, and is of great value.
It was, of course, Lord Kelvin, or rather William Thomson, who rescued Green’s work
from total obscurity.
Then, in the article Hydromechanics, after several applications of Green’s transformation,
which is to say, the theorem, there appears, under the heading The Motion of a
Solid through a Liquid:
The ellipsoid was the shape first worked out, by George Green, in his Research on the
vibration of a pendulum in a fluid medium (1833).
On to the article Light under the heading Mechanical Models of the Electromagnetic
Medium. After some negative remarks about Fresnel, one reads:
Thus, George Green, who was the first to apply the theory of elasticity in an unobjectional
manner ...
This is the content of On the Laws of Reflexion and Refraction of Light (1837).
Finally, the paper On the Propagation of Light in Crystallized Media (1839) appears
in the Brittanica article Wave as follows:
The theory of waves diverging from a center in an unlimited crystaline medium has
been investigated with a view to optical theory by G. Green.
The word “propagation” is a signal to us that, in little more than 10 years, George
Green had significantly widened his physical framework. From the static three-dimensional
Green function that appears in potential theory, he had arrived at the concept of a dynamical,
four-dimensional Green function. It would be invaluable a century later.
To continue the saga of George Green and me—my next step was to trace the influences
of George Green on my own works. Here I spent no time over ancient documents. I went
directly to a known source: THE WAR.
I presume that in Britain, unlike the United States, the war has a unique connotation.
Apart from a brief sojourn in Chicago, to see if I wanted to help develop The Bomb—I
didn’t—I spent the war years helping to develop microwave radar. In the earlier hands of
the British, that activity, famous for its role in winning the Battle of Britain, had begun
with electromagnetic radio waves of high frequency, to be followed by very high frequency,
which led to very high frequency, indeed.
Through those years in Cambridge (Massachusetts, that is), I gave a series of lectures
on microwave propagation. A small percentage of them is preserved in a slim volume
entitled Discontinuities in Waveguides. The word propagation will have alerted you to
the presence of George Green. Indeed, on pages 10 and 18 of an introduction there are
applications of two different forms of Green’s identity.
Then, on the first page of Chapter 1, there is Green’s function, symbolized by G. In
the subsequent 138 pages the references to Green in name or symbol are more than 200 in
number.
As the war in Europe was winding down, the experts in high power microwaves began
to think of those electric fields as potential electron accelerators. I took a hand in that and
devised the microtron which relies on the properties of relativistic energy. I have never
seen one, but I have been told that it works. More important and more familiar is the
synchrotron.
Here I was mainly interested in the properties of the radiation emitted by an accelerated
relativistic electron. I used the four-dimensionally invariant proper time formulation
of action. It included the electromagnetic self-action of the charge, which is to say that
it employed a four-dimensionally covariant Green’s function. I was only interested in the
resistive part, describing the flow of energy from the mechanical system into radiation, but
I could not help noticing that the mechanical mass had an invariant electromagnetic mass
added to it, thereby producing the physcial mass of an electron. I had always been told
that such a union was not possible. The simple lesson? To arrive at covariant results, use
a covariant formulation, and maintain covariance throughout.
Quantum field theory, or more precisely, quantum electrodynammics, was forced from
childhood into adolescence by the experimental results announced at Shelter Island early
in June, 1947. The relativistic theory of the electron created by Dirac in 1928 was wrong.
Not very wrong, but measurably so.
A few days later, I left on a honeymoon tour across the United States. Not until
September did I begin to work on the obvious hypothesis that electrodynamic effects were
responsible for the experimental deviations, one on the magnetic moment of the electron,
the other on the energy spectrum of the hydrogen atom.
Although a covariant method was in order, I felt I could make up time with the then
more familiar non-covariant methods of the day. By the end of November I had the results.
The predicted shift in magnetic moment agreed with experiment. As for the energy shift
in hydrogen, one ran into an expected problem.
Consider the electromagnetic momentum associated with a charge moving at constant
speed. The ratio of that momentum to the speed is a mass–an electromagnetic mass. It
differs from the electromagnetic mass inferred from the electromagnetic energy. Analogously,
the magnetic dipole moment inferred for an electron moving in an electric field is
wrong. Replacing it by the correct dipole moment leads to an energy level displacement
that was correct in 1947, and remains correct today at that level of accuracy as governed
by the fine structure constant.
I described all this at the January 1948 meeting of the American Physical Society,
after which Richard Feynman stood up and announced that he had a relativistic method.
Well, so did I, but I also had the numbers. Indeed, several months later, at the opening
of the Pocono Conference, he ran over to me, shook my hand, and said “Congratulations,
Professor! You got it right,” which left me somewhat bewildered. It turned out he had
completed his own calculation of the additional magnetic moment. Later we compared
notes and found much in common.
Unfortunately, one of the things we shared was an incorrect treatment of low energy
photons. Nothing fundamental was involved; it was a matter of technique in making a
transition between two different gauges. But, as in American politics these days, the less
important the subject, the louder the noise. When that lapse was set right, the result of
1947 was regained. Incidentally, even Lord Rayleigh once made a mistake. That’s one
reason for its being called the Rayleigh-Jeans law.
To keep to the main thrust of the talk—the evolution of Green’s function in the
quantum mechanical realm—I move on to 1950, and a paper entitled On Gauge Invariance
and Vacuum Polarization.
This paper makes extensive use of Green’s functions, in a proper-time context, to deal
with a variety of problems: non-linearities of the electromagnetic field, the photon decay
of a neutral meson, and a short, but not the shortest derivation of the additional electron
magnetic moment. The latter ends with the remark that “The concepts employed here
will be discussed at length in later publications.” I cannot believe I wrote that.
The first, rather brief, discussion of those concepts appeared in a pair of 1951 papers,
entitled On the Green’s Functions of Quantized Fields. One would not be wrong to trace
the origin of today’s lecture back 42 years to these brief notes. This is how paper I begins:
The temporal development of quantized fields, in its particle aspect, is described
by propagation functions, or Green’s functions. The construction of these functions for
coupled fields is usually considered from the viewpoint of perturbation theory. Although
the latter may be resorted to for detailed calculations, it is desirable to avoid founding the
formal theory of the Green’s functions on the restricted basis provided by the assumption of
expandability in powers of the coupling constants. These notes are a preliminary account
of a general theory of Green’s functions, in which the defining property is taken to be the
representation of the fields of prescribed sources.
We employ a quantum dynammical principle for fields which has been described in
the 1951 paper entitled The Theory of Quantized Fields. This (action) principle is a differential
characterization of the function that produces a transformation from eigenvalues of
a complete set of commuting operators on one space-like surface to eigenvalues of another
set on a different surface.
In one example of a rigorous formulation, Green’s function, for an electron-positron,
obeys an inhomogeneous Dirac differential equation for an electromagnetic vector potential
that is supplemented by a functional derivative with respect to the photon source; and, the
vector potential obeys a differential equation in which the photon source is supplemented
by a vectorial part of the electron-positron Green’s function. (It looks better than it
sounds.) It is remarked that, in addition to such one-particle Green’s functions, one can
also have multiparticle Green’s functions.
The second note begins with:
In all the work of the preceding note there has been no explicit reference to the particular
states on (the space-like surfaces) that enter the definitions of the Green’s functions.
This information must be contained in boundary conditions that supplement the differential
equations. We shall determine these boundary conditions for the Green’s functiions
associated with vacuum states on both (surfaces).
And then:
We thus encounter Green’s functions that obey the temporal analog of the boundary
condition characteristic of a source radiating into space. In keeping with this analogy,
such Green’s functions can be derived from a retarded proper time Green’s function by a
Fourier decomposition with respect to the mass.
The text continues with the introduction of auxiliary quantities: the mass operator
M that gives a non-local extension to the electron mass; a somewhat analogous photon
polarization operator P; and ?, the non-local extension of the coupling between the electromagnetic
field and the fields of the charged particles. Then, in the context of two-particle
Green’s functions, there is the interaction operator I.
The various operators that enter in the Green’s function equations M, P, ?, I, can be
constructed by successive approximation. Perturbation theory, as applied in this manner,
must not be confused with the expansion of the Green’s functions in powers of the charge.
The latter procedure is restricted to the treatment of scattering problems.
Then one reads:
It is necessary to recognize, however, that the mass operator, for example, can be
largely represented in its effect by an alteration in the mass constant and by a scale change
of the Green’s functiion. Similarly, the major effect of the polarization operator is to
multiply the photon Green’s function by a factor, which everywhere appears associated
with the charge. It is only after these renormalizations have been performed that we deal
with wave equations that involve the empirical mass and charge, and are thus of immediate
physcial applicability.
In the period 1951-1952, two colleagues of mine at Harvard, and I, wrote a series of
papers under the title Electrodynamic Displacements of Atommic Energy Levels. The third
paper, which does not carry my name, is subtitled The Hyperfine Structure of Positronium.
I quote a few lines:
The discussion of the bound states of the electron-positron system is based upon a
rigorous functional differential equation for the Green’s function of that system.
And,
Theory and experiment are in agreement.
As for the rest of the 50’s, I focus on two highlights. First: although it could have
appeared any time after 1951, it was 1958 when I published The Euclidean Structure of
Relativistic Field Theory. Here is how it begins:
The nature of physcial experience is largely conditioned by the topology of spacetime,
with its indefinite Lorentz metric. It is somewhat remarkable, then, to find that
a detailed correspondence can be established between relativistic quantum field theory
and a mathematical image based on a four-dimensional Euclidean manifold. The objects
that convey this correspondence are the Green’s functions of quantum field theory, which
contain all possible physcial information. The Green’s functions can be defined as vacuumstate
expectation values of time-ordered field products.
I well recall the reception this received, running the gamut from “It’s wrong” to “It’s
trivial.” It is neither.
Second (high light):
Another Harvard colleague and I had spent quite some time evolving the techniques
before we published a 1959 paper entitled Theory of Many-Particle Systems. It was intended
to bring the full power of quantum field theory to bear on the problems encountered
in solid state physics, for example. That required the extension of vacuum Green’s functions,
which refer to absolute zero temperature, into those for finite temperature. This is
accomplished by a change of boundary conditions, which become statements of periodicity,
or anti-periodicity, for the respective BE or FD statistics, in response to an imaginary time
displacement.
As an off shoot of this paper, I published in 1960, Field Theory of Unstable Particles.
Here is how it begins:
Some attention has been directed recently to the field theoretic description of unstable
particles. Since this question is conceived as a basic problem for field theory, the
responses have been some special device or definition, which need not do justice to the
physical situation. If, however, one regards the descriptiion of unstable particles to be fully
contained in the framework of the general theory of Green’s function, it is only necessary
to emphasize the relevant structure of these functions. That is the purpose of this note.
What is essentially the same question, the propagation of excitations in many-particle
systems where stable or long-lived “particles” can occur under exceptional circumstances,
has already been discussed along thse lines.
One might be forgiven for assuming that this saga of George and me efectively ended
with this paper. But that was 1/3 century ago!
To set the stage for what actually happened, I remind you that operator field theory
is an extrapolatiion of ordinary quantum mechanics, with its finite number of degrees
of freedom, to a continuum labeled by the spatial coordinates. The use of such spacetime
dependent variables presumes the availability, in principle, of unlimited amounts of
momentum and energy. It is, therefore, a hypothesis about all possible phenomena of that
type, the vast majority of which lies far outside the realm of accessible physics. In honor
of a failed economic policy, I call such procedures: trickle-down theory.
In the real world of physics, progress comes from tentative excursions beyond the
established framework of experiment and theory–the grass roots–indeed, the Green grass
roots. What is sought here, in contrast with the speculative approach of trickle-down
theory, is a phenomenological theory–a coherent account of the phenomena that is anabatic
(from anabassis: going up).
The challenge was to reconstruct quantum field theory, without operator fields. The
source concept was introduced in 1951 as a mathematical device–it was a source of fields.
It took 15 years to appreciate that, with a finite, rather than an unlimited, supply of energy
available, it made better sense to use the more physical–if idealized–concept of a particle
source. Indeed, during that time period one had become accustomed to the fact that to
study a particle of high energy physics, one had to create it. And, the act of detection
involved the annihilation of that particle.
This idea first appeared in an article, entitled Particles and Sources, which recorded
a lecture of the 1966 Tokyo Summer Lectures in Theoretical Physics. The preface begins
with:
It is proposed that the phenomenological theory of particles be based on the source
concept, which is abstracted from the physical possibility of creating or annihilating any
particle in a suitable collision. The source representation displays both the momentum
(energy) and the space-time characteristics of particle behavior.
Then, in the introduction, one reads:
Any particle can be created in a collision, given suitable partners, before and after the
impact to supply the appropriate values of the spin and other quantum numbers, together
with enough energy to exceed the mass threshold. In identifying new particles it is basic
experimental principle that the specific reaction is not otherwise relevant. Then, let us
abstract from the physical presence of the additional particles involved in creating a given
one (this is the vacuum) and consider them simply as the source of the physcial properties
that are carried by the created particle. The ability to give some localization in space and
time to a creation act may be represented by a corresponding coordinate dependence of a
mathematical source function, S(x). The effectiveness of the source in supplying energy
and momentum may be described by another mathematical source function, S(p). The
complementarity of these source aspects can be given its customary quantum interpretation:
S(p) is the 4-dimensional Fourier transformation of S(x).
The basic physical act begins with the creation of a particle by a source, followed by the
propagation (aha!) of that particle between the neighborhoods of emission and detection,
and is closed by the source annihilation of the particle. Relativistic requirements largely
constrain the structure of the propagation function—Green’s function.
We now have a situation in which Green’s function is not a secondary quantity, implied
by a more fundamental aspect of the theory, but rather, is a primary part of the foundation
of that theory. Of course fields, initially inferred as derivative concepts, are of great
importance, as witnessed by the title I gave to the set of books I began to write in 1968:
Particles, Sources, and Fields.
The quantum electrodynamics that began to emerge in 1947 still bothers some people
because of the divergences that appear prior to renormalization. That objection is
removed in the phenomenological source theory where there are no divergences, and no
renormalization.
As another example of such clarification I cite a 1975 paper entitled Casimir Effect
in Source Theory. The abstract reads:
The theory of the Casimir effect, including its temperature dependence is rederived by
source theory methods, which do not employ the concept of (divergent) zero point energy.
What source theory does have is a photon Green’s function, which changes in response to
the change of boundary conditions, as one conducting sheet is pushed into the proximity
of another one.
A few years later, I, and two colleagues at the University of California (UCLA),
who had joined me from Harvard with their new doctorates, extended this treatment to
dielectric bodies where forces of attraction also appear.
Having said this, I can move up to the present day, and the fascinating phenomenon
of coherent sonoluminescense.
It has only recently been discovered that a single air bubble in water can be stabilized
by an acoustical field. And, that the bubble emmits pulses of light, including ultra violet
light, in synchronism with the sonic frequency.
During the phase of negative acoustical pressure the bubble expands. That is followed
by a contraction which, as Lord Rayleigh already recognized in his 1917 study of cavitation,
turns into run away collapse. The recent measurements find speeds in excess of Mach 1 in
air.
Then the collapse abruptly slows, and a blast of photons is emitted. In due time, the
expansion slowly begins, and it all repeats, and repeats.
When confronted with a new phenomena, everyone tends to see in it something that is
already familiar. So, when told about this new aspect of sonoluminescence, I immediately
said “It’s the Casimir effect!” Not the static Casimir effect, of course, but the dynamical
one of accelerated dielectric bodies. I have had no occasion to change my mind.
I can imagine a member of this audience thinking: “That’s nice, but what is the role
of George Green in this?”
Looking in at the center of the water container, one sees a steady blue light. A photomultiplier
tube registers the succession of pulses, each containing a substantial number of
photons, which can be an incomplete count because, deep in the ultraviolet, water becomes
opaque.
A quantum mechanical description seeks the probabilities of emmitting various numbers
of photons, all of which probabilities are referred to the basic probability, that for
emitting no photons. The latter probability dips below one—in some analogy with synchrotron
radiation—because of the self-action carried by the electromagnetic field, as described
by Green’s functiion. And that function must obey the requirements imposed by
an accelerated surface discontinuity, with water, the dielectric material, on one side, and a
dielectric vacuum, air, on the other side. Carrying out that program is—as one television
advertiser puts it–job one. Very fascinating, indeed.
So ends our rapid journey through 200 years. What, finally, shall we say about George
Green? Why, that he is, in a manner of speaking, alive, well, and living among us.