The main problem of contemporary theoretical physics is how to combine the theory of gravity with theories of other fundamental forces into a single fully unified theory. In Einstein's theory of gravity, the geometry of space-time is a dynamic variable. But in theories of the strong, weak and electromagnetic forces, background geometry is a predominant factor.
The theories of these three forces are the so-called gauge theories, characterised by space-time symmetries with a fixed geometrical structure. There exist clear schemes allowing us to change classical gauge theories into corresponding quantum theories. General relativity can also be regarded as a gauge theory if all sufficiently smooth space and time transformations (all space-time diffeomorphisms) are taken as its characteristic symmetry. However, this symmetry, being very broad, permutes various mathematical structures in a much more radical way than is the case in other gauge theories: because of this, standard quantisation schemes cannot be directly applied to general relativity.
Geometries of both gauge theories and general relativity become more similar if they are done not on the ordinary space-time, but rather on a much more abstract (and rich) space, which mathematicians call the "connection space modulo the symmetry group". Given the set of all possible loops in this space, the similarity becomes yet more pronounced. Another surprise was that there is a connection between this approach and the mathematical theory of knots. On the one hand, mathematicians hope that ideas from physics can be used to generalise some concepts of the theory of knots and manifolds; on the other hand knot theoretical, even some diagrammatic techniques can serve as tools to solve some algebraic problems related to physics.
This highly interesting book contains papers from a workshop on knots and quantum gravity at the University of California at Riverside in 1994 and others from physicists and mathematicians in the field. The last paper in the book, "Knotted surfaces, braid movies, and beyond", by J. Scott Carter and Masahico Saito is the only one belonging to pure mathematics. According to the authors, "the motion picture method for studying knotted surfaces is one of the most well known". To some aristocratic class of mathematicians, perhaps. Mathematics is an abstract and purely formal discipline which can deal with everything, even knots, braids and movies. But it was a surprise when Ed Witten discovered a connection between aspects of knot theory and quantum field theory (in the Chern-Simons approach). The universe, at its deepest level, can have a "knotted structure".
Louis Crane, another contributor, writes that to work with beautiful mathematics in the hope of uniting general relativity with quantum mechanics for the entire universe is very much "in the spirit of the drunkard who looks for his keys near a street light". Physics in its history has offered us a lot of surprises. It might just turn out that the keys are indeed near a street light.
Michael Heller is a physicist with the Pontifical Academy of Theology, Krakow, Poland.
Knots and Quantum Gravity
Editor - John C. Baez
ISBN - 0 19 853490 6
Publisher - Clarendon Press, Oxford
Price - £.50
Pages - 226