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| QUANTUM GRAVITY |
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| Quantum Gravity is reputed to be one of the most difficult puzzles of science. In practical terms it is probably of no direct relevance and may even be impossible to verify by experiment.But for physicists it is the holy grail which may enable them to complete the unification of all fundamental laws of physics. The problem is to put together general relativity and quantum mechanics into one selfconsistent theory. The difficulty is that the two parts seem to be incompatible, both in concept and in practice. Conceptually, it is the nature of space and time which present fundamental differences. A direct approach, attempting to combine general relativity and quantum mechanics, while ignoring conceptual differences, leads to a meaningless quantum field theory with unmanageable divergences. There have, in fact, been many attempts to create a theory of quantum gravity. In this article I will first outline the nature of general relativity and quantum mechanics with emphasis on their similarities and differences. Then I will briefly review some of the main stream approaches to quantum gravity. Finally I will talk about some ways in which these ideas now seem to be converging. Attempts to do Quantum Gravity The most direct way to try to quantise quantum gravity is to use perturbative quantum field theory. This is a procedure which has been applied with great success to electrodynamics. To do the same thing for gravity it is necessary to first construct a system of non-interacting gravitons which represent a zero order approximation to quantised gravitational waves in flat space-time. These hypothetical gravitons must be spin two massless particles because of the form of the metric field in general relativity. The next step is to describe the interactions of these gravitons using the perturbation theory of quantum mechanics, which are defined by a set of Feynman diagrams derived from Einstein's gravitational field equations. For electrodynamics this can be made to work, but only after conveniently cancelling divergent anomalies which appear in the calculations. For gravity this simply cannot be done. The resulting quantum field theory is said to be unrenormalisable and is incapable of giving any useful result. Because quantum gravity is an attempt to combine two different fields of physics, there are two distinct groups of physicists involved. These two groups form a different interpretation of the failure of the direct attack. The relativists say that it is because gravity cannot be treated perturbatively. To try to do so destroys the basic principles on which relativity was founded. It is, for them, no surprise that this should not work. Particle physicists say that if a field theory is non-renormalisable then it is because it is incomplete. The theory must be modified and new fields must be added to cancel divergences. Canonical Quantum Gravity While particle physicists were making a lot of noise about superstring theory, relativists have been quietly trying to do things differently. Many of them take the view that to do quantum gravity properly you must respect its diffeomorphism symmetry. The Wheeler-DeWitt equation together with a Hamiltonian constraint equation,describe the way in which the quantum state vector should evolve according to this canonical approach.For a long time there seemed little hope of finding any solutions to the Wheeler-DeWitt equation. Then in 1986 Ashtekar found a way to reformulate Einstein's equations of gravity in terms of new variables. Soon afterwards a way was discovered to find solutions to the equations. This is now known as the loop representation of quantum gravity. Mathematicians were surprised to learn that knot theory was an important part of the concept.The results from the canonical approach seem very different from those of string theory. There is no need for higher dimensions or extra fields to cancel divergences. Relativists point to the fact that a number of field theories which appear to be unrenormalisable have now been quantised exactly. There is no need to insist on a renormalisable theory of quantum gravity. On the other hand, the canonical approach still has some technical problems to resolve. It could yet turn out that the theory can only be made fully consistent by including supersymmetry As well as their differences, the two approaches have some striking similarities. In both cases they are trying to be understood in terms of symmetries based on loop like structures. It seems quite plausible that they are both aspects of one underlying theory.Other mathematical fields are common features of both, such as knot theory and topology. Indeed there is now a successful formulation of quantum gravity in three dimensional space-time which can be regarded as either a loop representation or a string theory. A number of physicists such as Lee Smolin are looking for a more general common theory uniting the two approaches. |
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