Open Questions: Mathematics and Physics

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See also: Symmetry -- Superstring theory -- M-theory -- Quantum field theory -- Geometry and topology -- Noncommutative geometry -- Quantum geometry -- Mathematical analysis and differential equations -- Chaos theory and dynamical systems

Since gauge fields, including in particular the electromagnetic field, are fiber bundles, all gauge fields are thus based on geometry. To us it is remarkable that a geometrical concept formulated without reference to physics should turn out to be exactly the basis of one, and indeed maybe all, of the fundamental interactions of the physical world.

T. T. Wu and C. N. Yang



Introduction

Hilbert's sixth problem

Yang-Mills theory

Solitons and instantons

Supersymmetry

Seiberg-Witten theory

Quantum geometry

Knot theory

Complex manifolds

Superstrings and M-theory

Chaos and renormalizaton theory

Ramanujan's modular function

Zeta functions


Recommended references: Web sites

Recommended references: Magazine/journal articles

Recommended references: Books

Introduction

The distinguished physicist Eugene Wigner wrote a well-known essay entitled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". (From which this quote is taken.) Many other scientists have, like Wigner, gone beyond agreeing with Galileo that mathematics is the language of science to noting that there is something a bit eerie and uncanny about this situation, especially in the physical (and increasingly, the biological) sciences.

How does it happen, for example, that the abstract geometry developed by the mathematician G. F. B. Riemann in the 1850s turned out to be exactly the tool Einstein needed in 1915 to formulate his general theory of relativity? This was not, after all, a case of building a tool to suit the problem (which is what Newton did when he invented calculus). It was, instead, a matter of Einstein's finding an existing, quite sophisticated mathematical theory which, when pressed into service for describing gravitation, turns out to make some astonishing physical predictions -- from the precession of Mercury's perihelion to black holes -- new examples of which are still being confirmed today.

At present, we have no satisfactory answers to this question. That is, it remains in the province of philosophy -- it is still an "open question". Looking at some of the current ways in which mathematics and physics interact will emphasize the remarkable interconnection -- and deepen the mystery of the relationship.

It should not be thought, however, that all is sweetness and light between mathematicians and physicists. The chances that the two disciplines will merge anytime in the foreseeable future is quite remote, because the characteristic modes of reasoning employed by mathematicians and physicists are quite different.

Although this is a large oversimplification, it is still not unfair to say that the ultimate goal of physicists is to discover basic laws of nature. There are no hard-and-fast rules about how such discovery may be done. Pure guessing, for instance, is allowed. All that matters, ultimately, is that the resulting "law" stand up to all attempts to find experimental or observational evidence which might falsify some logical prediction of the supposed law. In other words, if there are experimental or observational tests which can be made, and if the theory passes all of them, it is accepted -- until it either fails a test or some other equally successful theory comes along with is manifestly simpler.

Therefore, it does not fundamentally matter to physicists (in general) if steps taken in deriving the theory or law from starting premises and known facts are lacking in mathematical rigor. If the steps are "intuitively" correct and the answer is right, that's enough. Quantum field theory, for instance, is a prime example. The relativistic theory of electromagnetism, known as quantum electrodynamics (QED) produces answers which agree with experiment to more than 10 decimal places. Yet the techniques used to derive the theory (such as "path integrals" and "renormalization") have yet be put on a rigourous mathematical foundation.

Physicists are content to leave the development of such rigorous foundations to the mathematicians. Indeed, there are a number of important examples where work of this sort provides important open questions facing mathematics.

From the mathematicians' point of view (again oversimplifying), rigor in formulating a theory and making deductions is all important. This attitude goes right back to Euclid, over 2500 years ago, whose science is so durable that it alone among its contemporaries is still taught almost unchanged in schools today. As far as mathematicians are concerned, it is only such formalization and rigor that guarantees the correctness of results. Whether or not the results can be applied or put to experimental tests is all but irrelevant. (And yet, Euclidean geometry is anything but useless or irrelevant, 2500 years later.)

But in fact, mathematicians and physicists usually get along well enough when they have occasion to interact. Physicists are sensitive to considerations of mathematical elegance and aesthetics. From Maxwell's equations, through special and general relativity, to contemporary superstring theory there are numerous examples of this. If given the chance, physicists are often guided in their thinking by mathematical aesthetics.

Likewise, mathematicians cannot fail to be impressed by physical laws which seem to meet all experimental challenges -- such as quantum field theory and the standard model of particle physics -- in spite of less than rock-solid mathematical foundations. Cases like this provide the inspiration to attempt a more rigorous formulation of the theory, and a seemingly endless supply of difficult open questions.

It's worth noting that there certainly have been many cases when physicists and mathematicians have resisted or resented encroachments from the other side on "their" territory. For example, a leading American physicist, John Slater, referred contemptuously to the application of group theory in quantum mechanics as the "Gruppenpest". He wrote, as late as 1975,

As soon as [my] paper became known, it was obvious that a great many other physicists were as disgusted as I had been with the group-theoretical approach to the problem. As I heard later, there were remarks made such as "Slater has slain the 'Gruppenpest'". I believe that no other piece of work I have done was so universally popular.
Poor Slater. From today's perspective, it's clear that the ideas of symmetry and group theory are absolutely pervasive in physics.

Mathmematicians have also been guilty of chauvinism. The number theorist G. H. Hardy, for instance, liked to boast that none of his work had any practical applications at all. But number theory (primes and factorization) and the related theory of elliptic curves are now of fundamental importance in cryptography. And many number theorists now suspect -- with perhaps justified pride -- that deep results yet to come in their field may underlie physics at the most fundamental level.


Here are some of the more esoteric mathematical topics, developed initially for their own intrinsic interest which have been found essential in formulating important parts of modern physics:


Hilbert's sixth problem


Seiberg-Witten theory

The mathematician Simon Donaldson had been doing some extremely important work on the geometry of 4-manifolds. This work involved a many concepts from physics, such as "instantons" and Yang-Mills equations. Among other things, it led to the very surprising conclusion that 4-dimensional Euclidean space (R4) had an infinite number of distinct "differentiable structures". That is, R4 could be regarded as a differentiable manifold in infinitely many different ways. What's so surprising is that Rn, for every other n, has a unique differentiable structure. This aroused the interest of physicists, because they need to know which is the right structure for doing physics.

Donaldson's work also led to the construction of new "invariants" of 4-manifolds, which are numeric or algebraic objects that help classify manifolds. (The Euler number is an example.) The physicist Nathan Seiberg, who had been working with supersymmetry, became interested in this research. In 1993 he collaborated with Witten on an investigation of supersymmetric gauge theory related to Donaldson's theory. What they found were new equations that greatly simplified what Donaldson had discovered.


Recommended references: Web sites

Site indexes

Math Forum Internet Mathematics Library: Physics
Alphabetized list of links with extensive annotations.
Open Directory Project: Mathematical Physics
Categorized and annotated mathematical physics links. A version of this list is at Google, with entries sorted in "page rank" order.
The Net Advance of Physics: Geometry and Topology
An index of tutorial and research articles located at the physics preprint archive. Areas covered include differential geometry, differential forms, geometric probability, noncommutative and quantum geometry, topology, topological field theory, Seiberg-Witten theory.
The Net Advance of Physics: Group Theory and Algebra
An index of tutorial and research articles located at the physics preprint archive. Topics include Clifford algebras, quaternions, Grassmann algebra, Hopf algebras, Lie algebras, quantum groups, noncommutative geometry, group theory.
Galaxy: Mathematical Physics
Categorized site directory. Entries usually include descriptive annotations.


Sites with general resources

This Week's Finds in Mathematical Physics
John Baez writes an excellent and useful column on recent news, papers, and books about mathematical physics that he's come across. The column appears roughly twice a month. It can be searched by issue number, keyword, or through a large table of contents.
String Theory Mathematics
Excellent (but brief) summaries of the mathematical topics that are essential in string theory, from the undergraduate level to advanced research. Part of the Official String Theory Web Site.
Annotated Bibliography for Seiberg-Witten Theory
Good bibliography of books and papers, by Jonathan Poritz.
Calabi-Yau Home Page
Resource for information about Calabi-Yau manifolds.
Mathematical Physics Electronic Journal
Free, refereed online journal.


Surveys, overviews, tutorials

Mathematical physics
Article from Wikipedia. See also Gauge theory, Gauge field theory
Yang-Mills Theory
Brief description of the problem at the Clay Mathematics Institute site, by Arthur Jaffe and Edward Witten. (A more complete description is available as a PDF file.)
Twists and Supersymmetry
Slide presentation given by Arthur Jaffe at the 2001: A Spacetime Odyssey conference.
Preparation for Gauge Theory
"Class lecture notes at a beginning graduate level on the mathematical background needed to understand classical gauge theory." By George Svetlichny.


Recommended references: Magazine/journal articles

Geometry of Solitons
Chuu-Lian Terng; Karen Uhlenbeck
Notices of the AMS, January 2000, pp. 17-25
Solitons are solitary wave solutions of certain nonlinear partial differential equations, like the Korteweg-deVries and sine-Gordon equations. Although solutions of nonlinear equations can't be added to produce additional solutions, sometimes solutions will be asymptotic to linear combinations of solutions.
[Article in PDF format]
Magic, Mystery, and Matrix
Edward Witten
Notices of the AMS, October 1998, pp. 1124-1129
A fairly nontechnical survey article on M-theory by the person responsible for the "second superstring revolution". The main historical developments and concepts are outlined, including quantum field theory, quantum gravity, string theory, and M-theory.
[Article in PDF format]
The Symmetries of Solitons
Richard S. Palais
Bulletin of the AMS, October 1997, pp. 339-403
The history of the investigation of solitons is presented in this long technical survey article. In particular, the evolution of equations of solitons have many conserved quantities, which suggests the existence of unobvious symmetries in their solutions.
[Abstract, references, downloadable text]
Solitary Waves
Russell Herman
American Scientist, July-August 1992, pp. 350-361
Solitary waves, or solitons, were first observed in 1834 and long thought rare. However, they have turned out to be common in nature. The absence of dispersion in solitons is a result of nonlinearities in the differential equations that govern them.
Fiber Bundles and Quantum Theory
Herbert J. Bernstein; Anthony V. Phillips
Scientific American, July 1981, pp. 123-137
Fiber bundles are a mathematical construct that can be applied in the differential geometry of manifolds to generalize the notion of curvature. In physics they may be applied to model the spin rotation of neutrons in a magnetic field.
Solitons
Claudio Rebbi
Scientific American, February 1979, pp. 92-116
Solitons are wave-like solutions of certain differential equations, which propagate but do not dissipate or disperse. They may represent predicted phenomena such as magnetic monopoles in elementary particle theory. When a soliton solution is limited in temporal as well as spatial extent it is called an instanton and can represent quantum tunneling.


Recommended references: Books

M. Waldschmidt; P. Moussa; J.-M. LUck; C. Itzykson, eds. -- From Number Theory to Physics
Normally we wouldn't list this sort of book here, as it's at a very advanced level (and expensive). It is a collection of expository papers intended for professionals in mathematics and physics. But its subject matter includes many topics which indicate the striking applicability of some very "deep" classical mathematical ideas to modern physics -- such as zeta functions, abelian varieties, elliptic curves, modular forms, algebraic number theory, and p-adic numbers.
Michael Monastyrsky -- Riemann, Topology, and Physics
Birkhäuser, 1987
This is really two (short) books in one: a biography and overview of the work of G. F. B. Riemann, one of the outstanding mathematicians of the 19th century, and a monograph on "topological themes in contemporary physics". The second part, as the mathematician/physicist Freeman Dyson says in the Foreword, illustrates "one of the central themes of science, the mysterious power of mathematical concepts to prepare the ground for physical discoveries which could not have been foreseen or even imagined by the mathematicians who gave the concepts birth."
Yu. I. Manin -- Mathematics and Physics
Birkhäuser, 1981
The author offers an extended essay (but not a long book) on the relation between mathematics and physics. Some of the concepts involved are advanced ones, but the reader isn't asked to follow detailed proofs. In the author's own words, "This book describes how mathematics associates to some important physical abstractions (models) its own mental constructions, which are far removed from the direct impressions of experience and physical experiment."

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