Affine connections in special relativity

Abstract

Affine geometry, implemented by a ''Lorentz connection'' for accelerated frames of reference in pseudo-Euclidean space-times, a ''Fourier connection'' for abstract Hilbert spaces associated with classical Fourier analysis, and a ''quantum connection'' for quantum-mechanical Hilbert spaces, give snew perspectives on special relativity, where affine connections are usually interpreted phenomenologically, in a manner which obscures theri geometric significance. The connections are determined by ''absolute constants,'' whose covariant derivatives, expressed in terms of the connection coefficients, vanish identically. In the deterministic theory of prequantum physics, the space-time connection is expressed in terms of ''local Lorentz transformations,'' which represent the motion of accelerated frames relative to inertial frames. In quantum theory, relative motion is not well defined, but local Lorentz transformations, represented by locally isomorphic symmetry groups, remain well defined, and become the basis of quantum field theory.