Sachs Equations and Plane Waves, V: Ward, Fourier, and Heisenberg Symmetry on Plane Waves

TL;DR

This paper explores wave equations on plane wave spacetimes, linking geometric, Fourier, and Schrödinger perspectives, and connecting to theta functions and the Weil representation.

Contribution

It develops an integrated framework connecting Ward, Fourier, and Schrödinger representations on plane waves, highlighting the role of the conformal tensor and Maslov phase.

Findings

Describes the Ward progressing-wave representation of solutions.

Analyzes the Fourier structure of the associated Heisenberg group.

Connects the Schrödinger propagator with theta functions and the Weil representation.

Abstract

This article studies wave equations and their solutions on plane wave spacetimes of arbitrary dimension, developing the interplay among three structural layers: the Ward progressing-wave representation of solutions to the scalar wave equation, the Fourier analysis of the Heisenberg group naturally associated to the plane wave, and the Schr\"odinger propagator governing the evolution of initial data. The central geometric object is a positive curve in the Lagrangian Grassmannian determined by the plane wave metric, previously studied in the authors' series. The conformal tensor $H(u)$ that parametrises this curve plays a dual role: it encodes the null-cone geometry of the spacetime and simultaneously appears as the time-dependent parameter in the Schr\"odinger representation of the Heisenberg group acting by isometries on the plane wave. Parallel to the classical Fourier inversion theorem, convolution by Lagrangian delta distributions on the Heisenberg group furnishes an intrinsic description of the Schr\"odinger propagator, and the intertwining of different polarisations by this propagator is captured by a diagram that commutes up to a Maslov phase. The theta functions and Bargmann transforms that arise from imaginary polarisations complete the analytic picture, connecting the present work to the theory of the Weil representation as developed by Lion--Vergne and to Mumford's systematic treatment of theta functions.