6.1 General Relativity and Quantum Field Theory

    A mathematician represents the motion of the planets of the solar system by a flow line of an incompressible fluid in a 54-dimensional phase space, whose volume is given by the Liouville measure; while a famous physicist is said to have made the statement that the whole purpose of physics is to find a number, with decimal points, etc.!, otherwise you have not done anything (Manin, 1980). Despite such extreme points of view it can be safely said that the relations between mathematics and physics (and astronomy) have been very productive in past epochs of the evolution of these disciplines (Barrow and Tipler, 1988).

    Mathematical structures entered the development of physics, and problems emanating from physics influenced developments in mathematics. Examples are the role of differential geometry in general relativity, the dynamical theory of space and time, and the influence of quantum mechanics in the development of functional analysis and built on the understanding of Hilbert spaces. A prospective similar development occurred only recently when non-Abelian gauge theories emerged as the quantum field theories for describing fundamental particle interactions. Yang-Mills theory found its mathematical formulation in the theory of principal fiber bundles. The understanding of anomalies in gauge theories involved the theory of families of elliptic operators and representation theory of infinite-dimensional Lie algebras and their cohomologies. Based on these developments, there are two fundamental theories in modern physics: General relativity and quantum field theory. General relativity describes gravitational forces on an astronomical scale. Quantum field theory describes the interaction of fundamental particles, electromagnetism, weak, and strong forces. The formal quantization of general relativity is leading to infinities and the current understanding is that the mathematical machinery is missing to accomplish the unification of general relativity and quantum field theory. This is particularly due to the nonlinear character of general relativity which is incompatible with the linearity needed for the superposition principle in quantum theory. More recently attention has turned to the exploration of the mathematical structure of non-Abelian gauge theories and to the more ambitious attempts to construct unified theories of all the fundamental interactions of matter together with gravity. A success of such attempts may reveal new insights for structural elements of the microcosmos and macrocosmos (Schmid, 1992).

6.2 Macroscopic Dimension of Space

    Of all the fundamental constants the most familiar one is the dimension of physical space, N = 3. Variation of such a fundamental characteristic as the dimension N may lead to unpredictable changes of physical laws. It was Paul Ehrenfest (1880-1933) who, in 1917, was trying to answer the question of why physical space is three-dimensional. From physics, we are familiar with the analogy between Coulomb's law and Newton's law [Equations (1) and (2)]. In both cases, the force is F proportional_alpha r-2. In physics these laws are treated separately. This lack of coherency obscures a profound relationship of the electromagnetic and the gravitational forces with the properties of space, in particular, with its dimension. Two properties are common to the gravitational and the electromagnetic interactions: Both are weak and long-range. In modern language, this means that the mass of the gauge boson is zero in both cases, implying that the interaction radius is infinite and that the interaction constants are small, alphag, alphae much smaller 1 [Equations (3) and (4)]. In the language of physics these properties mean, that the lines of forces, originating at the point where their source is located, run to infinity, not intersecting with each other, provided that no other source is present. The fact that the lines of force extend to infinity reflects the long-range character of the gravitational and electromagnetic forces; the absence of intercepts signifies that there is no reciprocal action between the lines of force, i.e., that the interactions under consideration are weak. The combination of both properties, the weakness and the long-range action, is not characteristic for the other interactions. The force F exerted by one particle on another particle, a distance r apart, is proportional to the density ni of the lines of force. Accordingly,

F proportional to n_sub_i = f / 4 Pi r^2 . (30)

    The proportionality constant f in (30) is by definition equal to the product of the charges of both particles (Coulomb's law) or of their masses (Newton's law). The denominator in (30) gives the surface area S of a sphere of radius r. For the three-dimensional space, this quantity equals S3 = 4pir2 = a3r3 - 1. These considerations can be repeated for the more general case of an N-dimensional space. The surface area of a sphere in such a space is SNrN - 1. Therefore, the force FN, acting in an N-dimensional space, is

F_sub_N = b_sub_N / r^(N-1) . (31)

    Accordingly, the potential energy has the form

U_sub_N = -b_sub_N / [(N - 2) * r^(N - 2)] , (32)

    where N not equal 2; for N = 2, the dependence is logarithmic. It should be stressed that these expressions apply only for integer N, and only for long-range forces in the quasistatic approximation, i.e., for motion in a central force field. The existence of stable orbits in a central force field in an N-dimensional space is determined by (31) and consequently by the dimension N. From mechanics, it is known that the existence of stable orbits depends on the form of the r-dependence of the effective potential

U_sub_Ne = U_sub_N + M^2 / (2 m r^2) , (33)

    where M2/2mr2 is the centrifugal energy, M the angular momentum, and m the mass of the body moving at a distance r. A stable state is possible if the dependence UNe(r) has a minimum at a value of r different from zero or infinity. Henceforth the attractive forces are considered for which UN < 0. The results of an analysis of the function UNe(r) with regard to the existence of an extremum, are the following:

    The existence of a minimum in the dependence UNe(r) is a necessary condition for the stability of motion. For this reason, the existence and the properties of closed orbits are determined by the dimension of space, N. For N > 4, there is no minimum at r not equal 0, infinity; consequently there are no stable closed orbits. Any motion, caused by long-range forces would be of one of the following two types: Either it is infinite (the body escapes to infinity) or otherwise, the moving body falls on a massive central body. At N = 2 or N = 3, all types of motion are possible: Infinite motion, fall onto a central body and, notably, motion in stable, closed orbits. For N = 1, only finite motion is possible; a body cannot escape to infinity. In the one-dimensional case, no orbital motion occurs, and the centrifugal potential is zero (M = 0). The effective potential, Equation (32), is then U1 = b1r, and the force F1 = const. This corresponds to an infinitely deep potential well. To remove the body from this well, an infinitely large force has to be applied; this means it would be impossible for the body to escape to infinity. Hence, the degree of stability grows as the dimension N decreases. For N > 4, there are no analogues to planetary systems. Similar considerations in the framework of quantum mechanics have demonstrated that for N > 4, stable atomic systems do not exist, either. It appears that the absence of analogues of planets and atoms for N > 4 is a clue to the understanding of the significance of the space dimension N = 3, with regard to the existence of structural elements in the universe.

6.3 Microscopic Dimension of Space

    In classical physics, particles have definite locations and follow exact trajectories in spacetime. In quantum mechanics, wavepackets propagate through spacetime, their positions and velocities uncertain according to Heisenberg's uncertainty principle. In string theory, point particles are replaced by tiny loops with the result that the concept of spacetime becomes "fuzzy" at scales comparable to the square root of a new fundamental constant, alpha' round (10-32 cm)2, introduced as string tension in string theory. Employing both string tension (alpha' not equal 0) and quantum effects (h not equal 0) leads to results that may change the conventional notion of spacetime (Donaldson, 1996). The situation of introducing alpha' in string theory is similar to passing from classical to quantum mechanics by the introduction of Planck's constant h. In string theory the one-dimensional trajectory of a particle in spacetime is replaced by a two-dimensional orbit of a string. According to Heisenberg's uncertainty principle, at a momentum p one can probe a distance x round h/p. However, the introduction of (alpha', acts as if Heisenberg's uncertainty principle would have two terms, deltax greater equal h/deltap + alpha'(deltap/h), where the second term reflects the fuzziness due to string theory. Then, the constant alpha' would be the absolute minimum uncertainty in length in any physical experiment. The consequences of this type of quantum field theory for fundamental particles and the structure of the universe are still to be discovered.