Many Dimensions in Gravity Theory

Albert Einstein didn't like quantum mechanics. Spending some time with the theorists who are trying to quantize his theory of general relativity can lead to the impression that the antipathy was mutual. Although Einstein made some important early contributions to quantum physics, he became dismayed as the philosphical implications of quantum mechanics revealed themselves. Classical physics is rigidly deterministic: a given cause leads inexorably to a given result, always and everywhere. In quantum mechanics a given cause can lead to any of several results. Individual cases cannot be determined. The only laws are statistical ones for large numbers of instances. Growling his famous remark about God not throwing dice, Einstein withdrew mostly to the contemplation of classical field theory. In order to save classical field theory Einstein destroyed it. The general relativity theory that he developed wrought a revolution in our concepts of space, time and force, giving them a radically different form from what had gone before. And Einstein destroyed the absolutism that had characterized the physics of the past: the notion that there is an absolute evenly flowing time, which is the same for all observers, and that there is a special spatial frame of reference that is absolutely at rest, against which motions can be measured absolutely. Now time differs for different observers; all spatial reference frames are of equal rank; there is no absolute rest, and forces and motions are relative, artifacts of geometry. Nevertheless, general relativity is still a classical field theory in the sense that it is deterministic. The uncertainties and statistical analyses that are fundamental to quantum mechanics do not play a role in it. But, somehow, eventually it must be mated with them. To do this, to quantize general relativity, has long been seen as a necessary step in the completion of theoretical physics. In past decades it did not seem too urgent. Quantum mechanics deals with behavior on the atomic and subatomic level. relativity is primarily a theory of gravitation. Practically, the effects of gravity in ordinary particle-physics and atomic-physics experiments are so minuscule as virtually to vanish. There was no clamoring market for a theory of gravity compatible with particle physics. Now there is. The theory of particle physics, at least, has now moved into realms where gravity is important. These are domains of very high energy, which may actually have existed in the early moments of the history of the universe. Particle physicists have begun to theorize like cosmologists and to act like astronomers, scrutinizing such relics of the early universe as are available for some evidence of the things they have theorized. They need a theory of gravity compatible with particle physics to aid them in what they are doing. The quest was evident recently in New Orleans at the Second New Orleans Conference on Quantum Theory and Gravitation. It seems clear from the discussion at the meeting that to get such a theory will require some radical changes in some of our present basic ideas about space, time and matter and possibly some spectacular violations of common sense. (Much in modern physics violates common sense.) It could require the quantization of space and time, a shift from the continuous space and time, used by geometers from Euclid to Einstein, in which one location or instant shades imperceptibly and indivisibly into the next, to some conception that is bumpy and jumpy like the processes of quantum physics. It could mean a further geometrization of the properties of matter. It will probably mean something of both. One thing it surely needs is more dimensions. In the words of John William Moffatt of the University of Toronto, General relativity is based,on the algebra of numbers in four-dimensional space. Real numbers are the ordinary ones we count with, and four-dimensional space is the space-time of ordinary perception real spacetime it is often called. It has the three space dimensions in which we see ourselves free to move in any direction we like (assuming there are no physical restraints on us) and time, in which we can move in only one direction. This difference between space and time is by no means scanted in general relativity, but as the theory is formulated, it is nevertheless possible to treat time as a dimension. The only forces in the theory are gravitational ones. (Einstein tried for 40 years to work electromagnetic forces in but could not). Gravity is seen as an effect of the curvature of space-time, and that curvature is determined by a quantity that represents the amount of matter and energy in a given neighborhood. Thus force and matter are in a sense geometrized. When subatomic particles come into the picture, geometrizing their properties requires more dimensions. The four dimensions of ordinary space-time represent the external of of a body, its ability to move in space and the changes in time that come over it. Subatomic particles are more than simple point masses exerting gravitational forces. They have what are known as internal degrees of freedom, intrinsic properties that change by quantum jumps. As these changes occur they alter the physical state and often the identity of the particles to which they occur. In cases where there are more degrees of freedom, more quantities that can change than the four dimensions of ordinary space-time, physicists have long found it useful to retain geometric imagery by using, as calculational devices, spaces with enough dimensions to accommodate all the degrees of freedom relevant to the problem. Leopold Halpern of-Florida State University in Tallahassee quotes an 1837 statement of Bernhard Riemann: Physics and geometry are complementary in the description of nature. Various spaces are chosen as approaches to the problem according to criteria that seem important to a particular theorist. Moffatt chooses an eight-dimensional space because, he says, it gives him the connection to spin that he wants. Spin is an important property of subatomic particles. It seems also to play a crucial role in the amalgamation of quantum physics and general relativity. Halpern is led to a 10-dimensional space by considerations of the geometry of general relativity theory. Motions in general relativity are described with the aid of particular mathematical groups called de Sitter groups. These groups are 10-dimensional. Other theorists opt for 12 dimensions. The point is that for these people it is easier to calculate out the physics in these multidimensional spaces than to try to make sense of it in four-dimensional space-time. The work is done with equations, not by trying to draw 12 dimensions. Much of the mathematics is done