Geometry and Mathematical Physics Group

Quantum Gravity and Quantum Fields Theory : Physical, Mathematical and Philosophical perspectives

Phys Math

The relation between mathematics and physics is one with a long tradition. For Galileo the book of nature is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures.

A giant step forwards in using the language of nature to describe physical phenomena was made by Isaac Newton, who developed and applied the calculus to the study of dynamics and whose universal law of gravitation explained everything from the fall of an apple to the orbits of the planets.

The nineteenth century witnessed the greater sophistication of Maxwell’s equations to include the behaviour of electromagnetism, and the twentieth century saw this process take a major step forwards with Einstein’s theory of special relativity and then of general relativity. At this stage both gravitation and electromagnetism were formulated as field theories in four-dimensional space–time, and this fusion of geometry and classical physics provided a strong stimulus to mathematicians in the field of differential geometry.

However, by this time, it had already been realized that atomic physics required an entirely new mathematical framework in the form of quantum mechanics, using radically new concepts, such as the linear superposition of states and the uncertainty principle, that no longer allowed the determination of both the position and momentum of a particle. Here the mathematical links were not with geometry, but with the analysis of linear operators and spectral theory. As experimental physics probed deep into the subatomic region, the quantum theories became increasingly complex and physics appeared to be diverging from classical mathematics.

The picture began to change around 1955, with the advent of the Yang–Mills equations, which showed that particle physics could be treated by the same kind of geometry as Maxwell’s theory, but with quantum mechanics playing a dominant role. However, it took to the beginning of the 1970s before it became clear that these non-Abelian gauge theories are indeed at the heart of the standard model of particle physics, which describes the known particles and their interactions within the context of quantum field theory. It is a remarkable achievement that all the building blocks of this theory can be formulated in terms of geometrical concepts such as vector bundles, connections, curvatures, covariant derivatives and spinors.

This combination of geometrical field theory with quantum mechanics worked well for the structure of matter but seemed to face a brick wall when confronted with general relativity and gravitation.
But over the past 30 years a new type of interaction has taken place, probably unique, in which physicists, exploring their new and still speculative theories, have stumbled across a whole range of mathematical discoveries. These are derived by physical intuition and heuristic arguments, which are beyond the reach, as yet, of mathematical rigour, but which have withstood the tests of time and alternative methods. There is great intellectual excitement in these mutual exchanges.

The impact of these discoveries on mathematics has been profound and widespread. Areas of mathematics such as topology and algebraic geometry, which lie at the heart of pure mathematics and appear very distant from the physics frontier, have been dramatically affected. This development has led to many hybrid subjects, such as topological quantum field theory, quantum cohomology or quantum groups, which are now central to current research in both mathematics and physics. The meaning of all this is still unclear.

Finally, the interactions between mathematics and quantum physics played an important role in recent developments in mathematics. Areas of mathematics which have been inspired very much from physics are Conformal Field Theory, Mirror Symmetry, and Noncommutative Geometry.