revealing an astounding degree of agreement.
The gravitational force, described classically by Einstein’s general relativity theory, does not
seem to lend itself to a QFT description. String theory appears to provide a more appropriate
description of the quantum theory of gravity. String theoryis an extension of QFT, whose very
formulation is built squarely on QFT and which reduces to QFTin the low energy limit.
The devlopment of quantum field theory has gone hand in hand with developments in Condensed
Matter theory and Statistical Mechanics, especially critical phenomena and phase transitions.
A final remark is in order on QFT and mathematics. Contrarily to the situation with general
relativity and quantum mechanics, there is no good “axiomatic formulation” of QFT, i.e. one is very
hard pressed to lay down a set of simple postulates from whichQFT may then be constructed in
a deductive manner. For many years, physicists and mathematicians have attempted to formulate
such a set of axioms, but the theories that could be fit into this framework almost always seem
to miss the physically most relevant ones, such as Yang-Mills theory. Thus, to date, there is no
satsifactory mathematical “definition” of a QFT.
Conversely, however, QFT has had a remarkably strong influence on mathematics over the past
25 years, with the development of Yang-Mills theory, instantons, monopoles, conformal field theory,
Chern-Simons theory, topological field theory and superstring theory. Some developments is QFT
have led to revolutions in mathematics, such as Seiberg-Witten theory. It is suspected by some
that this is only the tip of the iceberg, and that we are only beginning to have a glimpse at the
powerful applications of quantum field theory to mathematics.