Physics in the 20th century is founded on the twin pillars of quantum mechanics and the theory of relativity. However, in spite of the enormous successes of each theory individually, the two appear to be incompatible. This embarrassing contradiction at the very heart of theoretical physics remains one of the great outstanding challenges in science.
The general theory of relativity gives a superbly accurate account of gravitation. Applying quantum mechanics to the gravitational field is the subject of quantum gravity. At first sight, constructing a theory of quantum gravity looks no more problematic than the theory of quantum electrodynamics (QED), which was satisfactorily worked out half a century ago.
Central to QED is the description of electromagnetic forces in terms of the exchange of so-called virtual photons. These are photons that are emitted and then quickly reabsorbed; the Heisenberg uncertainty principle means that they do not have to conserve energy and momentum. Thus the electrostatic repulsion between two electrons can be thought of as due to virtual photons being emitted by one electron and absorbed by the other.
Similarly, one could envisage the gravitational attraction between two bodies as being due to the exchange of virtual “gravitons” – the quanta of the gravitational field. The fact that nobody has ever detected a graviton is no surprise, because gravity is so much weaker than electromagnetism. It turns out that an exchange of single gravitons between point masses does indeed yield an expression for the gravitational field with its familiar inverse-square law.
Problems set in, however, when more complicated processes involving many gravitons are examined. Gravity differs crucially from electromagnetism inasmuch as the gravitational field is nonlinear. This nonlinearity arises because the gravitational field possesses energy, and energy has mass, which gravitates. In quantum language, this implies that gravitons will interact with other gravitons, unlike photons, which interact only with electric charges and currents and not with other photons. Because gravitons respond to each other, particles of matter are surrounded by complex networks of virtual gravitons that form “closed loops”, as well as “branching trees”.
In quantum field theory, closed loops are a sign of trouble; they normally produce infinite answers in calculations of physical processes. In QED, such loops occur when an electron emits and reabsorbs its own photon. The resulting infinities are removed by a mathematical procedure that is known as renormalization. If this is done properly, sensible answers are obtained. QED is termed a renormalizable theory because all such infinities can be removed by a systematic procedure; in effect, a single set of mathematical operations is sufficient to remove the infinities.
Unfortunately, no such systematic procedure exists when quantum mechanics is applied to general relativity; the theory is therefore “non-renormalizable”. Each process involving progressively more closed loops of gravitons introduces new varieties of infinite terms. This renders the theory useless for investigating almost all phenomena of interest, and suggests that there is something basically wrong with general relativity, quantum mechanics, or both.
Over the past few decades, several attempts have been made to evade the problem of non-renormalizability in quantum gravity. The most promising is superstring theory. This is based on the hypothesis that the fundamental entities from which the physical world is built are not particles but loops of string 1020 times smaller than an atomic nucleus. Roughly speaking, different vibrational modes of the strings correspond to different particles – electrons, quarks, neutrinos, photons, gravitons and so on. Interactions take place between strings as they do between particles, but, when processes involving closed loops are examined, the answers are found to be finite.
The natural scale that enters string theory is determined by the Planck energy, which is around 1019 GeV. This is 1017 times higher than the sorts of energies probed by the largest particle accelerators, so it is impossible to observe the stringy nature of matter directly. Theorists hope that, at everyday energy scales, familiar physics – general relativity, electromagnetism, the weak and strong nuclear forces, the familiar subatomic particles – will emerge from string theory as an approximation. Thus, superstring theory is not just a putative description of quantum gravity, but also an attempt to unify the forces and particles of nature.
Unfortunately, there is no unique low-energy limit to superstring theory and no unique superstring theory either. For a time this looked like an insuperable obstacle, but in recent years a more abstract mathematical scheme known as “M theory” has been constructed, and it appears to contain the various superstring theories within it.
It is too soon to say whether M theory will finally reconcile gravitation and quantum mechanics, but, if such a theory is to live up to expectations, it should explain some basic facts about the physical world. For example, four-dimensional space-time would have to emerge from the theory, rather than be put into it. The forces and particles of nature also ought to be described, preferably including their key properties, such as interaction strengths and masses. However, unless M theory, or a future variant, can be “projected down” into the relatively low-energy realm of laboratory physics to yield such information, it will remain little more than an elegant exercise in mathematics.
