Mirrors to Transforms: The Fourier Act

We present the Fourier transform as the canonical intertwiner between Schrodinger models of the Heisenberg representation associated to different Lagrangian polarizations. This "Fourier = change of polarization" principle explains convolution, the Weyl relations, and the privileged role of Gaussians, and it naturally introduces the Wigner transform as the phase-space incarnation of a wavefunction.

We then apply the formalism to fields on a plane-wave background. In Brinkmann-type coordinates, the scalar wave operator admits a separation in which Fourier transform in the transverse/ignorable directions reduces the PDE to a one-parameter family of evolution equations with a quadratic generator (a time-dependent harmonic-oscillator Hamiltonian on phase space). Consequently, the propagator is metaplectic: it is determined by a linear symplectic flow, giving explicit integral kernels and a clean description of focusing/caustics. This metaplectic "propagation calculus" is the bridge between classical Fourier analysis and the geometric constructions that appear in later talks.