Mathematical analysis as required by a tachyonic observer

Abstract

We rely on an analysis of initial states of the wave function associated with the simplest relativistic particle—the Weyl neutrino—to construct two series of representations of \documentclass[12pt]{minimal}\begin{document}$SL(2,\mathbb {R})$\end{document}SL(2,R), or of the twofold cover of this group, by operators acting on scalar functions defined on the real line. The first one, which depends on a parameter p = 0, 1, …, contains the usual one-dimensional metaplectic, or oscillator, representation. The second one, a series of representations no longer unitary but, for certain values of the parameter, pseudo-unitary with respect to some non-degenerate indefinite pseudo-scalar product, is built in a comparable way, only exchanging the time coordinate with one of the spatial ones. The first series of representations was originally introduced in connection with automorphic pseudodifferential analysis; the second one is new, except for one value of the (continuous, in this case) parameter ρ, in which case it coincides with the recently introduced anaplectic representation. For each value of the parameter λ (= p or ρ), the basic operators Q and P from the usual Heisenberg pair give way to a new pair (Q, Pλ) and to another analysis of functions on the real line, with as rich a collection of (most of the time totally new) symmetries as the usual one. The unusual set of coordinates in Minkowski's space under consideration can be regarded as corresponding to a “tachyonic observer,” a notion which does not require venturing into debatable questions of Physics. On the other hand, the paper indicates in a special example the way one may recover well-posedness (in a non-classical sense) for an initial-value problem with data on a timelike surface.