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Topic: Quantum Field Theory
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Evelyn Sander

Posts: 187
Registered: 12/3/04
Quantum Field Theory
Posted: Oct 15, 1993 11:23 AM
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"This is a period in which mathematics and physics are closely
linked," says Geometry Center visiting professor Dan Freed. "This has
happened quite frequently in the past. For example, the Greeks
developed trigonometry to try to describe the stars' placements in the
sky. Newton invented calculus while trying explain Kepler's work about
astronomy. Gauss studied the geometry of surfaces because he was
trying to understand and describe the curvature of the earth.
Einstein's theory of general relativity gave importance to the subject
of Riemannian geometry. Now it is the work of physicists in quantum
field theory which is giving mathematics a new push." Dan Freed is
currently working in the mathematical discipline of topological
quantum field theory which relates closely to the physical results.

Using quantum field theory, physicists have brought new insight to
unsolved problems in mathematics by using a physical intuition for the
problem not available to mathematicians. For example, here is a
mathematical situation which became more clear by physical intuition:
Four-dimensional objects are have some problems not seen in any other
dimension. There are some four-dimensional manifolds which are
indistinguishable by traditional means, such as classical homotopy
theory. They are homeomorphic but not diffeomorphic. Therefore, the
mathematician Donaldson came up with a series of computable quantities
which are invariant for a given manifold. These quantities are called
Donaldson invariants. They enable one to distinguish between the
four-dimensional manifolds which were previously indistinguishable by
any computable means. Physicist Ed Witten fit these Donaldson
invariants into a general framework using quantum field theory. Just
this week Freed heard that Witten used quantum field theory techniques
to "prove" (in the physicist's sense) an interesting result about the
Donaldson invariants, generalizing recent work of Kronheimer and

In another case, mathematicians were trying to compute the number of a
certain kind of rational curves of a given degree. They had computed
some of the numbers of curves, but it was quite difficult and
cumbersome. Then the physicist Candelas and his collaborators came up
with a generating function for the number of curves. The numbers did
not agree with those that the mathematicians had calculated in one
case. This resulted in a great deal of debate. However, it turned out
that the mathematicians had made a mistake, and Candelas' function has
been successful in predicting the answers.

After the results on the Donaldson invariants, Witten did some further
work in which he linked quantum field theory to the Jones polynomials,
an important tool for knot theorists. While creating this link, Witten
discovered a new invariant in three-dimensional manifolds. Though this
has not been earth-shattering for topology, it is a rare and thus
exciting discovery.

The physicists' work is not what mathematicians would call rigorous.
Many of the methods used are cannot be justified mathematically. Yet
the results seem to work mathematically. Freed says, "I would not say
that the physicists are doing things that are wrong, but certainly
they are using tools which do not belong to the toolkits of
mathematicians. Though to mathematicians certain steps do not make
sense, the physical framework of quantum field theory serves as an
excellent prediction and motivation for mathematical results. Over
the past decade we have seen many instances where these
predictions--often numerical and very concrete--have proved correct.
The challenge for mathematicians is to create the new mathematics
behind these physical insights."

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