Current Research

Physical theories today are based on Lie algebras. Taken modulo isomorphism these form a (moduli) space of rich structure, a rugged landscape. Its peaks and ridges are singular groups without useful finite representations. They are flanked by regular Lie algebras with useful finite-dimensional representations, including simple Lie algebras. Early theorists built their castles on the peaks for ideological reasons. As a result these theories proved to be unstable and riddled with infinities. Heisenberg's quantization moved theories from the peaks to adjacent ridges, leaving them still singular and unstable but less so. General quantization moves theories to the valley, leaving singularity behind; specifically to full matrix algebras. Then the theory is finite.

Self-organization to account for the lesser symmetry of the singular theory. It also requires specifying a representation of the new algebras by fixing new quantum numbers, the eigenvalues of central invariants. The theorist need not fix these quantum numbers; they may be left to experiment.

Practice [quant-ph\0601002]. General relativity too has functional Lie algebras whose products obeys the laws of Lie algebra when they are defined, but which are infinite dimensional because their elements depend on arbitrary functions. General quantization must replace them by Lie algebras of high dimension.

The resulting space-time is composed of a finite number of elementary dynamical processes, "chronons," typically represented by units of a Clifford algebra. This implies a finite time-quantum: chronons are only finitely local, coupling nearest neighbors. Einstein's infinitesimal locality only appears in the singular continuum limit, not in a regular theory.