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My Encounters -- As a Physicist -- With Mathematics Date 2004-7-3 0:19 |
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[Editor's Note: The following essay first appeared in "Physics Today",
vol. 49, no. 2 (February 1996). It is reprinted here by the kind permission
of the author and the editor of "Physics Today". Please notice the copyright
and the "Physics Today" web-access information at the end of this essay.]
My Encounters -- As a Physicist -- With Mathematics
___________________________________________________
by Roman Jackiw
Laboratory for Nuclear Science and Department of Physics
Massachusetts Institute of Technology
(first printed - Vol. 2, No. 8 - May 1996)
Descriptions of the physical universe and proofs of abstract
mathematical theorems are sometimes intimately related.
---------------------
Mathematical Physics is not a discipline with its own identity. Rather
there is mathematics and there is physics, and their cyclical relationship
enjoys periods of cooperation interspersed with periods of mutual
indifference. New initiatives leading to rapid development in physics
frequently are accompanied by mathematical innovation. Examples from the past
are the construction of particle mechanics and differential calculus by Isaac
Newton and Gottfried Leibniz, developments in general relativity and
differential geometry at the time of Albert Einstein and Hermann Minkowski,
and advances in group theory and analysis following the invention of quantum
mechanics and quantum field theory.
But the links remain sporadic and, as in any two societies that are
separated, the languages develop differently, thereby hindering communication.
Also, attitudes about achievement acquire distinct emphases: Physicists prize
empirical problem solving and model building, in contrast to the
mathematicians' appreciation of rigorously proven theorems. Part of the fun of
being a mathematical physicist is discovering how the different terminologies
describe the same things and how the distinct goals benefit each other. (See
figure 1.)
___________________________________________________________________________
| |
| Volume 67B, number 2 PHYSICS LETTERS 28 March 1977 |
| |
| ON REGULAR SOLUTIONS OF EUCLIDEAN YANG-MILLS EQUATIONS |
| |
| A. S. SCHWARZ |
| Moscow Institute of Physical Engineering, Moscow M-409, USSR |
| |
| Received 28 January 1977 |
| |
| The number of instantons and the number of zero fermion modes in the |
| field of instanton are calculated |
|_________________________________________________________________________|
| |
| Volume 67B, number 2 PHYSICS LETTERS 28 March 1977 |
| |
| DEGREES OF FREEDOM IN PSEUDOPARTICLE SYSTEMS* |
| |
| R. JACKWI and C. REBBI |
| Laboratory for Nuclear Science and Department of Physics, |
| Massachusetts Institute of Technology, |
| Cambridge, Massachusetts 02139, USA |
| |
| Received 1 February 1977 |
| |
| We show that at least 8n-3 parameters are required to specify an |
| n-pseudoparticle solution in Eclidean SU(2) Yang-Mills theory |
|_________________________________________________________________________|
Figure 1. Frontispieces of two papers that prove the same result,
once in mathematical style by use of the Atiyah-Singer index theorem,
and once in physical style by explicit solution of differential
equations.
Early in my career, my attitude about the relation between mathematics
and physics and about the utility of the former for the pursuit of the latter
was informed by a talk I heard given by Paul Dirac, in the Harvard University
Loeb lecture series on physics history. Although I did not take notes, later I
found a printed version in which he reiterated his forceful position in favor
of mathematics in physics:
The most powerful method of advance [in physics] ... is to
employ all the resources of pure mathematics in attempts to
perfect and generalize the mathematical formalism that forms
the existing basis of theoretical physics, and ... to try to
interpret the new mathematical features in terms of physical
entities.
Today there is intense cross-fertilization between mathematics and
physics, specifically between geometry and field theory. The contact,
initially established through Einstein's theory of general relativity tin
which the gravitational field is described in terms of the geometry of
space-time), surged again about two decades ago. Some of my own research took
place during that new beginning, so I present here a reminiscence in the form
of a case history of a physics-mathematics encounter, with comments on how
actual experience has modified my preconceived opinion.
Kinks and zero modes
____________________
By the early 1970s, quantum field theory was very much in favor with
theoretical physicists, but the quantized equations continued to resist
solution. It then occurred to many people that it would be worthwhile to
ignore the quantal nature of the fields and solve the equations as if they
described nonlinear, classical dynamical systems. Interesting, localized and
nondissipative solutions were found very quickly. They were the kinks in one
dimension, relevant to physics on a line; vortices in two dimensions, in
planar physics; skyrmions and magnetic monopoles in three dimensions--our
physical world., Collectively, such solutions were called "solitons," the name
being taken from applied mathematics. Another class of solutions comprised the
"instantons" in four-dimensional space-time.
With colleagues at MIT, I addressed the question of how to extract
information on the quantum theory from these classical results--that is, we
wanted to determine the quantum meaning of classical fields. We made progress
on this problem, and at a certain stage Claudio Rebbi and I decided it was
important to study the linear small fluctuations about the nonlinear field
profiles of solitons and instantons, and also the coupling of other linear
systems, such as fermions, to solitons and instantons.
Thus we were led to linear eigenvalue equations, and we realized that
the zero modes, which correspond to vanishing eigenvalues, contain especially
important information about the quantum physics. The zero modes of the
small-fluctuation equations measure the allowed deformations of the soliton or
instanton, and the number of these modes gives the dimension of the moduli
space for the solution of the nonlinear equation. (The moduli space consists
of the set of all possible distinct solutions. For example, if a soliton
solution is described by two parameters--say, its location a and its size
c--then the moduli space will be some region in the space of {a, c} values.)
In the fermionic Dirac equation, the eigenvalues measure energy:
Positive-energy modes describe quantum particles, negative-energy modes
correspond to antiparticles and zero modes -- when they exist -- signal a
degenercy that gives rise to unexpected quantum numbers, such as fractional
fermion charge.
Rebbi and I were delighted to find precisely such zero modes in the
equations we studied, and also to establish their physical consequences. But
we were surprised that the existence of these special solutions did not depend
on the details of the localized profiles of the background solitons and
instantons; instead, only their large-distance behavior mattered. The
long-range features characterize the topological properties of solitons and
instantons. Such properties are not tied to the detailed shape of a field
configuration but characterize the configuration as a whole -for example, by
specifying its critical points or its nontrivial behavior at infinity.
With this idea in mind, we began to suspect that; the occurrence of
zero modes was not an accident of our analysis, but a consequence of having
nontrivial topological backgrounds. Such backgrounds have nontrivial,
nonvanishing large-distance properties, in contrast to the usual case where
behavior at; infinity is assumed t;o be trivial and irrelevant.
The corridors of MIT
____________________
Rebbi and I wanted to find out what mathematicians knew about these
ideas. All buildings at MIT are connected, and the math department is in the
same structure as my work space. However, locked doors as well as the
chemistry department intervene, so communication is obstructed. Nevertheless,
we walked the corridors of the math department, looking for anyone willing to
spend time understanding our questions and answering them in a way
comprehensible to nonspecialists, to us physicists. Initially we had no luck,
but then we met Barry Simon, who did not have specific information on our
problem but suggested that the work of Michael Atiyah and Isadore Singer might
be relevant.
Singer had temporarily moved from MIT to University of California,
Berkeley, but it so happened that my colleague and collaborator Jeffrey
Goldstone knew Atiyah from their student days in Cambridge, England, and had
information that Atiyah was coming to visit his mathematics friends in
Cambridge, Massachusetts.
So Rebbi and I arranged a meeting in my office. We invited physicists
who were working on soliton-instanton questions, and we listened to Atiyah
explain how his index theorem with Singer counts instanton zero modes, and how
their spectral-flow theorem with V. Patodi is relevant to Fractional charge.
Here the "index" is the number of zero eigenvalues of a relevant differential
operator, and "spectral-flow" describes the passage of eigenvalues across
zero, as parameters in the differential operator are varied.
It was especially thrilling for us to learn that the four-dimensional
index is given by an,integral over the gauge curvature-form F, specifically by
/
| F ^ F .
/
That was because the integrand, F ^ F, had also arisen in physics papers by
John Bell and me, as well as by Stephen Adler, as the anomalous divergence of
the chiral fermion current, thereby controlling neutral pion decay. (See
figure 2.) ("Gauge curvature-form F" is mathematical terminology and notation
for what physicists call F_{mu, nu} the electromagnetic field-strength tensor
or its generalizations. Similarly, the wedge "^" denotes a generalized cross
product.)
___________________________________________________________________________
| |
| | |
| | pi_o : |
| \|/ : |
| | : |
| | : |
| / \ : _________________ |
| / \ Physics Mathematics | Atyah-Singer | |
| |/ |\ <------------ F^F ------------> | index theorem | |
| / \ |_______________| |
| /____\____\ : |
| \ / \ : |
| / / : |
| \ \ : |
| / / : |
| gamma-1 gamma-2 : |
| |
|_________________________________________________________________________|
Figure 2. The observed Decay of the neutral pion into teo
photons is physically allowed only because of a chiral anomaly
that arises in renormalization. Mathematically, the anomaly is
given by the wedge product of the electromagnetic field strength,
F, with itself. The same quantity is deeply related to index
theorems in mathematics.
Evidently the chiral anomaly and the index theorem are related; they
had been elaborated in the late 1960s at different ends of the same MIT
building, by people working in ignorance of each other!
We appreciated very much Atiyah's efforts to make his presentation
understandable to us; still, exchanging information was not easy. One young
member of the audience impressed Atiyah, who encouraged the fellow to speak
because he seemed to understand, better than anyone else, what Atiyah was
saying. That person was Edward Witten; as is now well known, he has continued
to impress Atiyah and other mathematicians.
Soon thereafter, I was asked to review these exciting new results
about quantum field theory at a meeting of the American Physical Society.
Since Singer was present, I yielded some of my time to him, with the
suggestion that he describe the mathematical connection. But a detailed
presentation could not be fit in, so he merely eulogized collaboration between
mathematics and physics with an ode:
In this day and age
The physicist sage
Writes page after page
On the current rage
The gauge
Mathematicians so blind
Follow slowly behind
With their clever minds
A theorem they'll find
Duly written and signed
But gauges have flaws
God hems and haws
As the curtain He draws
O'er His physical laws
It may be a lost cause
Index theory also received a contribution from physics. The
Atiyah-Singer theorem applies to even-dimensional spaces on which a connection
is defined. ("Connection" is a mathematical name for the physicist's gauge
potential.) However, physicists are also interested in odd-dimensional
spaces--where one-dimensional kinks or three-dimensional skyrmions and
monopoles reside. These configurations can lead to zero modes, even in the
absence of a gauge connection. So we asked the mathematicians about
odd-dimensional index theorems, but apparently they knew nothing about them.
At that time I had a mathematically minded student, Costas Callias, and I
asked him to prove such a theorem. He achieved success, which then prompted
Raul Bott and Robert Seeley to publish a mathematical exegesis of the result,
immediately following Callias's paper in *Communications in Mathematical
Physics*. Since then I have been happy to see the "Callias index theorem" used
and cited.
Shakespeare and Churchill
_________________________
The two approaches to solving problems--the explicit, goal-oriented
methods of the physicists and the general theorems of the mathematicians--are
well illustrated by the determination of the dimensionality for instanton
moduli space: the n-instanton SU(2) solution depends on 8n-3 parameters. This
result appeared twice in the same issue of *Physics Letters*, once in a paper
by Albert Schwarz, who used the Atiyah-Singer theorem, and a few pages later
in a paper by Rebbi and me--we solved differential equations to find
explicitly 8n - 3 zero modes. (See figure 1.)
Gauge theories in general and instantons in particular continued to
interest mathematicians. They established the topological properties of the
instanton moduli space and produced a construction--but not an explicit
formula--for the general solution. The most general explicit expression was
given bv Craig Nohl, Rebbi and me, but it does not exhaust all The parameters.
Further developments on four-dimensional gauge fields led
mathematicians to Donaldson theory. In three dimensions, the Chern-Simons
term--a gauge structure that Steven Templeton and I, as well as Jonathan
Schonfeld, introduced to physics--has been related by mathematicians to knot
theory on manifolds with various topologies, while physicists have applied
this term to experimental phenomena on the plane, such as the Hall effect.
(See PHYSICS TODAY, March 1995, page 17, for developments involving Donaldson
theory and physics.)
These parallel investigations by physicists and mathematicians also
highlight our differences: We physicists use mathematics as a language for
recording observations about physical systems, and this limits our interest in
what fascinates mathematicians -- the full range of mathematical possibility.
For example, the general instanton solution does not appear to be physically
relevant; only the original one-instanton and the explicit, but limited,
multi-instanton solutions have illuminated physical theory. Even the
physicist's language need not always be mathematical. The fractional charge
phenomenon, which Rebbi and I inferred from zero modes or from spectral flow,
was also independently established by Wu-Pei Su, J. Robert Schrieffer and Alan
Heeger, who found a physical Realization in linear polyacetylene chains. One
of their derivations uses the pictorial language of chemical bonds, in which
the only mathematics consists of counting! (See PHYSICS TODAY, July 1981, page
19, for a discussion of fractional charge.)
An analogy comes here to mind. The English language contains over 200
000 words, and all of them interest the lexicographer. For William Shakespeare
20 000 diferent words sufficed to express his ideas in plays and sonnets,
whereas Winston Churchill used fewer than 2000 different words in his
historically decisive speeches. Physicists, like Churchill. can achieve their
goals by making effective use of a limited vocabulary
Nature's chosen subset
______________________
The process of gaining knowledge goes through the same steps in both
physics and mathematics. First there is the intuition/guess, then follows the
proposal/conjecture and finally comes the verification. For the mathematician,
verification consists of constructing a proof, establishing a theorem
according to rules whose legitimacy evolves slowly under the direction of the
entire mathematics community. But the physicist verifies his ideas by finding
a physical correlative: Neutral pion decay validates chiral anomalies (see
figure 2), properties of solitons in polyacetylene establish fractional charge
(see figure 3). Within physics the rules for giving a proof are constantly and
rapidly changing--presuppositions can become modified, experimental facts can
evolve.
Because the terms "proof" and "theorem" carry intellectual prestige
and pleasure, occasionally there are attempts to employ them in physics. To my
mind such efforts are mostly futile and sterile. For example, physicists
wanted very much to combine internal and spacetime symmetries in a nontrivial
fashion and were not daunted by a proven "impossibility theorem." Rather the
"theorem" was circumvented by the simple device of replacing commutators with
anticommutators and by grading the algebra. Thus was born supersymmetry, which
now is also influencing mathematics. Similarly, when field-theory
constructivists "proved" the existence of quantum {lambda phi^4} theory in 1+1
dimensions, they were correct but missed the entire quantum soliton
phenomenon, which is the only physically interesting feature of that model.
(See figures 3 and 4.)
___________________________________________________________________________
| |
| * V ^ * |
| * | * |
| * | * |
| * | * |
| * | * |
| * | * |
| * | * |
| * | * |
| * *** * |
| * ** | ** * |
| * * | * * |
| * * | * * |
| ----------***-----------***--------------> phi |
| -a +a |
| |
|_________________________________________________________________________|
Figure 3. The potential in {lambda phi^4} theory. The theory
consists of a single scalar field {phi} with a {phi^4}
interaction term and a negative mass-squared term multiplying
{phi^2}. The resulting potential has two minima or vacuum
states, at {phi = +/- a}. This model has long been studied by
mathematical physicists and has a physical realization in
linear polyacetylene chains, which exhibit fractional charge
phenomena. More complicated models involving gauge fields
relate to knot theory and the classification of manifolds
in mathematics.
___________________________________________________________________________
| |
| phi ^ |
| | |
| +a| |
| ***************************************** |
| ######## | @@@@@@@@@@ |
| ##### | @@@@@ |
| #### | @@@@ |
| ###|@@@ |
| -------------------0--------------------> x |
| @@@|### |
| @@@@ | #### |
| @@@@@ | ##### |
| @@@@@@@@ | ########## |
| ***************************************** |
| -a| |
| | |
|_________________________________________________________________________|
Figure 4. Classical solutions of the {lambda phi^4} theory in
1 space and 1 time dimension: Each of the two vacua(**** curve,
offset slightly for clarity) has trival topology, being a
constant value throughout space. Soliton solutions such as the
kink (@@@@ curve) and antikink (#### curve) also exist. These
are topologically nontrivial, as they approach different values
of {phi} at spatial {+infinity} and {-infinity}; a kink cannot
be smoothly transformed into an antikink or either of the
vacua by a finite process.
These days I believe that a statement by C. N. Yang accurately
describes physicists' historical use of mathematics:
Physics is not mathematics, just as mathematics is not physics.
Somehow nature chooses only a subset of the very beautiful and
complex and intricate mathematics that mathematicians develop,
and that precise subset is what the the theoretical physicist
is trying to look for.
This conservative view of mathematics differs from Dirac's radical advice
(cited on page 1) that physicists should "try to interpret ... mathematical
features in terms of physical entities."
Nowadays, however, when we are facing the absence of new experimental
data about fundamental phenomena, particle physics theory--as realized in the
string program-is driven by mathematics in the manner advocated by Dirac. That
was not the way things worked in the past, not even for Dirac. When first
confronting his negative-energy solutions, he identified them with the
proton---then the only known positively charged particle. As a physicist he
did not at first trust his mathematics enough to postulate the existence of
the positron!
Thus a question about the future development of physics remains: Shall
we succeed in obtaining further and deeper understanding of nature by
theoretical, mathematical reasoning alone, or must we wait for new
experimental, physical discoveries? New ideas of physics that occupy many
people today are built entirely on mathematics, indeed they help create new
mathematics. I am immensely curious to learn the ultimate fate of this
activity and to see whether it succeeds.
-----------------
This article is adapted from my Heineman Prize acceptance speech at the April
1995 meeting of the American Physical Society in Washington, DC. Now as then,
I acknowledge the colleagues who contributed to the research recognized by the
prize: John Bell on quantum anomalies, Steven Weinberg on the thermal field
theory, Claudio Rebbi on quantal topological effects, and Stanley Deser,
Gerard 't Hooft and So-Young Pi on planar field theories.
-----------------
Copyright 1996, the American Institute of Physics. All rights reserved.
Individual articles in Physics Today are copyrighted by
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"Physics Today" maintains a world-wide-web site at:
URL: http://aip.org/pt/phystoday.html
"Physics Today - February 1996" and
"My Encounters -- As a Physicist -- With Mathematics" are at:
URL: http://aip.org/pt/cont9602.html
The editor of "Metaphysical Review" would like to thank Roman Jackiw
at MIT, and Graham Collins and Ann Perlman at AIP/Physics Today for
their help in this project.
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