Clifford Space as a Generalization of Spacetime: Prospects for Unification in Physics
The geometric calculus based on Clifford algebra is a very useful tool for geometry and physics. It describes a geometric structure which is much richer than the ordinary geometry of spacetime. A Clifford manifold (C-space) consists not only of points, but also of 1-loops, 2-loops, etc.. They are associated with multivectors which are the wedge product of the basis vectors, the generators of Clifford algebra. We assume that C-space is the true space in which physics takes place and that physical quantities are Clifford algebra valued objects, namely, superpositions of multivectors, called Clifford aggregates or polyvectors. We explore some very promising features of physics in Clifford space. Instead of the usual theory of relativity in spacetime $M_N$ we have the relativity in C-space. The latter space has dimension $2^N$ and signature $+++ ... --- ...$, where the number of plus and minus signs is the same, namely $2^N/2$. This has consequences for string theory which can be formulated without central charges even when the dimension of the underlying spacetime is four, provided that the Jackiw definition of vacuum is employed. We do not need a higher dimensional target spacetime for a consistent formulation of (quantized) string theory. Instead of a higher dimensional space we have Clifford space which also provides a natural framework for description superstrings and supersymmetry, since spinors are just left or right minimal ideals of Clifford algebra. When considering field theory and assuming the Jackiw definition of vacuum state, the concept of C-space enables a formulation in which zero point energies belonging to positive and negative signature degrees of freedom cancel out whilst preserving the Casimir effect. This provides a resolution of the cosmological constant problem. Instead of flat C-space we may consider a curved C-space. As the passage from flat Minkowski spacetime to curved spacetime had provided us with a tremendous insight into the nature of one of the fundamental interactions, namely gravity, so the introduction of a curved C-space will presumably even further increase our understanding of the other fundamental interactions and their unificationwith gravity.