The theory of Kairons (Q1027957)

The paper is motivated by the idea that space and time in a relativistic theory should be (up to a certain degree) dual. Therefore it should be possible to construct field equations which admit unique solutions for initial data on a timelike curve rather than a spacelike hypersurface. Since the author has general relativistic theories in mind he starts with a short but quite comprehensive review of some material from differential geometry and fibre bundle theory. This makes the presentation very self contained, however, a certain degree of familiarity with these topics is still needed to follow the paper. After that an infinite dimensional vector bundle \(\Upsilon\) over spacetime \(M\) called the ``Kairon bundle'' is defined. The fibre of \(\Upsilon\) at \(x \in M\) consists of all functions on the forward light cone at \(x\) which are homogeneous of order \(-m/2\) (with \(m = \dim(M)-1\)). For the sections of \(\Upsilon\) a first order field equation is introduced which admits a conserved current. In the special case for fields propagating in Minkowski space and a particular class \(\mathcal{T}\) of timelike curves it is shown that for each \(\gamma \in \mathcal{T}\) and arbitrary (smooth) initial data on \(\gamma\) this equation admits a unique solution. Finally a Hilbert space of solutions is introduced and shown to admit a unitary representation of the Poincaré group.