Elementary particle physics has adapted the idea of phases and phase transitions from condensed matter physics. In particle physics, these ideas are applied to the `vacuum' itself. It is believed that, at very high temperatures, there should be transitions to a phase where quarks and gluons are unconfined and to a phase in which the symmetry between photons and W and Z particles is restored. For this reason, we study particle physics in thermal equilibrium at high temperatures. The work is applicable to experiments which seek to cause a phase transition by the collision between heavy atomic ions, producing a plasma of free quarks and gluons.

The universe may have undergone phase transitions as it cooled from its early, hot state. Possible relics of this process include `cosmic strings', filaments with regions in the symmetric phase trapped in their core. The influence of these on galaxy formation and on the imbalance between the numbers of protons and antiprotons is being investigated.

The traditional picture of an elementary particle is of a point-like structure, but extended objects are being investigated. One promising approach to building a unified theory of all interactions is to assume that the fundamental constituents of matter are string-like. The strings, in their different modes of internal motion, could give all the particles of nature, including the quantum, the gravitational force. The physical relevance of string theory has yet to be established at this fundamental level, but work is in progress both in building more realistic versions of the theory and in trying to understand its mathematical structure, a subject of great interest in its own right.

There are some three-dimensional extended objects which are made stable in a topological way and which appear very naturally as twists which cannot be untied. The study of the interactions of such objects involves fascinating new mathematical techniques. Protons and neutrons may be modelled in this way, providing a wholly new approach to nuclear physics.

The theory of quarks and gluons is a strongly nonlinear one and calculations are difficult. One approach is to approximate spacetime by a discrete lattice of points and then to approximate the required multi-dimensional integrals by `Monte Carlo' methods. The techniques used can be applied also to statistical physics and to problems in fluid flow. A similar discretised numerical approach is being made to the equations of general relativity.

The connections between elementary particle physics and condensed matter physics go far. Any system in statistical mechanics at a second-order phase transition has a field theory corresponding to it. This correspondence has been exploited in both directions. Two-dimensional statistical systems correspond to two-dimensional field theories, which have a very rich structure. They can be classified by algebraic methods, and for each class characteristic behaviour near the phase transition may be predicted. There are also many examples of two-dimensional statistical models which are exactly soluble. The method of solution involves new algebraic structures called `quantum groups' which are of considerable mathematical interest.

Also under study is the general problem of the relation between the classical and quantum descriptions of a dynamical system. Except in the simple `text-book' cases of point particles moving in flat space, this problem is far from solved. The whole question of the physical interpretation of quantum mechanics is open, and is being addressed.

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