Department of Physics

Superstring Theory and Supersymmetric Quantum Field Theory

Of the four known fundamental interactions between elementary particles, only the electromagnetic interaction and the gravitational interaction operate over macroscopic distances.

On these length scales, there are well-known classical descriptions of the interactions in terms of electrodynamics and general relativity respectively. Classical electrodynamics fails, however, on atomic length scales. It must be replaced by quantum electrodynamics.

On large distance scales, this quantum description yields the same results as the classical description, but it is fundamentally different on short distance scales. These differences are measurable and quantum electrodynamics has been verified experimentally to very high accuracy.

The quantum description of electromagnetism is in some sense much simpler than the classical description, in that it only involves particles. An interaction between a pair of charged particles, which is really an exchange of energy and momentum between them, occurs because they exchange particles (virtual photons, which carry energy and momentum from one charged particle to the other).

Two of the other fundamental interactions, namely the strong and the weak interactions, also have a quantum description in terms of the exchange of particles. The strong interaction, responsible for binding quarks together to form protons and neutrons (and, indirectly, for binding protons and neutrons together into atomic nuclei) is mediated by the exchange of virtual particles called "gluons".

The weak interaction is responsible, amongst other things, for radioactive decay, and is mediated by the exchange of virtual W and Z bosons. Since both these interactions have extremely short ranges, we do not experience them on everyday length scales, and so a classical description is neither required nor appropriate. The quantum descriptions of the strong and weak interactions have been tested to very high precision in particle accelerators.

The theories which describe the strong, weak and electromagnetic interactions in terms of virtual particle exchange are called quantum field theories, and more specifically, Yang-Mills theories.

The quantum description of electromagnetism was developed in the 1950s, while the quantum descriptions of strong and weak interactions were developed in the late 1970s and the early 1980s.

It was only natural that the attention of theoretical physics should turn to obtaining a quantum description of the remaining fundamental interaction, gravitation. This is still an unsolved problem, but an immensely important one in terms of achieving a full description of nature.

As mentioned, gravitational interactions are well described on large distance scales by classical general relativity, but at sufficiently small length scales, quantum considerations will become important and this classical description will fail. The length scale involved is the Planck length, about 10-35m. Using the biggest particle accelerators, it is currently only possible to probe down to length scales of 10-20m, so that quantum gravitational effects have not been observed yet. This lack of experimental input will be a hindrance to the testing of a quantum theory of gravitation, but indirect tests involving the early Universe and cosmological processes will be possible.

On the basis of our experience with the other three interactions, there is an expectation among theoretical physicists that the quantum description of the gravitational interaction will be in terms of the exchange of a virtual particle, dubbed the "graviton". However, all attempts to use the mathematical formalism (quantum field theory) that so successfully describes the other three interactions have failed spectacularly when applied to gravitation.

The resulting theory gives infinite results for most measurable quantities! This leads to the conclusion that we must either give up the possibility of explaining the gravitational interaction in terms of virtual particle exchange, or that the mathematical formalism which worked so well for the description of virtual particle exchange in the case of the other three interactions needs extension to cope with the gravitational interaction.

No satisfactory progress has been made with theories which abandon the idea of gravitational interactions mediated by virtual graviton exchange. However, considerable progress has been made recently in attempts to extend the type of mathematical formalism that works for the other three interactions in such a way that it incorporates virtual gravitons.

These extensions of quantum field theory are called superstring theories. Even more exciting is the fact that they naturally incorporate the known description of the strong, weak and electromagnetic interactions via Yang-Mills theories, and as such have the potential to provide a unified description of all four of the fundamental interactions.

As a result, superstring theory is currently the subject of enormous scrutiny and excitement in the theoretical physics community.

Conventional quantum field theories describe the propagation and interaction of a finite set of particles. Superstring theories may be crudely characterised as quantum field theories containing a finite set of massless particles together with an infinite tower of particles of increasing mass, all interacting with each other in a very tightly constrained way.

It is this infinite tower of massive states which "cures" the problems that arise in attempts to describe the gravitational interaction in terms of conventional quantum field theories. The process of spontaneous symmetry breaking can give rise to small masses for some of the massless states, which are then candidates for the known elementary particles.

Mathematically, the infinite spectrum of states in superstring theory arises from the quantised oscillations of a one-dimensional spatially extended object (a "superstring"). The interactions of the infinite spectrum of particles are coded mathematically into the splitting and joining of the superstrings.

The superstrings themselves are not observable as fundamental degrees of freedom because their length scale is about 10-35m, far below the length scales that can be probed in particle accelerators. Crudely, a superstring looks like a point particle when viewed from afar.

The massive states in superstring theory have energies of the order of the Planck energy MP = 1.22 × 1019 GeV, which means that they cannot be produced as physical states in current (or foreseeable future!) particle accelerators1. Nevertheless, they can still occur as virtual intermediate states2 in quantum interactions between the low energy (massless) states of superstring theory, which can be produced by current and future accelerators.

The interactions of the massless states are modified by the presence of these virtual intermediate states, and the modified interactions can be encoded in a "low energy effective action" involving quantum fields for only the finite set of massless states.

In other words, the low energy effective action is a conventional quantum field theory that describes the indirect effects of an underlying superstring theory on processes that occur at energies that are currently experimentally accessible.

1. To get a feeling of the actual value of MP, it is worth pointing out that the energies currently achieved in the world's largest particles accelerators are about 1000 GeV.

2. Quantum field theory allows an amount of energy DE to be "borrowed" for a short time Dt provided DEDt £ h/2p. This allows for particles with energies much higher than that set by the experimental scale to be produced for very short times as intermediate states; these are the "virtual states". Quantum field theory describes fundamental interactions in terms of the exchange of virtual states.

For a good introduction to string theory try visiting any of the following sites:

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Last updated:
Tuesday, 16 February, 2010 12:38 PM