Magic, Mystery, and Matrix Edward Witten

In the twentieth century, the quest for deeper
understanding of the laws of nature has
largely revolved around the development of
two great theories: namely, general relativity
and quantum mechanics.
General relativity is, of course, Einstein’s theory
according to which gravitation results from the curvature
of space and time; the mathematical framework
is that of Riemannian geometry. While previously
spacetime was understood as a fixed arena,
given ab initio, in which physics unfolds, in general
relativity spacetime evolves dynamically, according
to the Einstein equations. Part of the problem
of physics, according to this theory, is to
determine, given the initial conditions as input, how
spacetime will develop in the future.
The influence of general relativity in twentiethcentury
mathematics has been clear enough. Learning
that Riemannian geometry is so central in
physics gave a big boost to its growth as a mathematical
subject; it developed into one of the most
fruitful branches of mathematics, with applications
in many other areas.
While in physics general relativity is used to
understand the behavior of astronomical bodies
and the universe as a whole, quantum mechanics
is used primarily to understand atoms, molecules,
and subatomic particles. Quantum theory has had
a much more complex history than general relativity,
and in some sense most of its influence on
mathematics belongs to the twenty-first century.
The quantum theory of particles—which is more
commonly called nonrelativistic quantum mechanics—
was put in its modern form by 1925 and
has greatly influenced the development of functional
analysis, and other areas.
But the deeper part of quantum theory is the
quantum theory of fields, which arises when one
tries to combine quantum mechanics with special
relativity (the precursor of general relativity, in
which the speed of light is the same in every inertial
frame but spacetime is still flat and given ab
initio). This much more difficult theory, developed
from the late 1920s to the present, encompasses
most of what we know of the laws of physics, except
gravity. In its seventy years there have been
many milestones, ranging from the theory of “antimatter”,
which emerged around 1930, to a more
precise description of atoms, which quantum field
theory provided by 1950, to the “standard model
of particle physics” (governing the strong, weak,
and electromagnetic interactions), which emerged
by the early 1970s, to new predictions in our own
time that one hopes to test in present and future
accelerators.
Quantum field theory is a very rich subject for
mathematics as well as physics. But its development
in the last seventy years has been mainly by
physicists, and it is still largely out of reach as a
rigorous mathematical theory despite important efforts
in constructive field theory. So most of its impact
on mathematics has not yet been felt. Yet in
many active areas of mathematics, problems are
Edward Witten is professor of physics at the Institute for
Advanced Study. His e-mail address is witten@ias.edu.
This article is based on the Josiah Willard Gibbs Lecture
given at the Joint Meetings in Baltimore in January 1998.
The author’s work is supported in part by NSF Grant PHY