# AG Mathematische Physik, Luca Giorgetti (Rom): Crossing symmetry and Fourier transforms

**Luca Giorgetti (Rom)**

**Crossing symmetry and Fourier transforms**

Abstract:

Crossing symmetry is an invariance property of scattering amplitudes involving (pairs of) particles and their TCP conjugates. Recently, see arXiv:2212.02298,crossing symmetry has been reformulated as an analytic extension property ofcertain functions defined using Tomita-Takesaki modular theory. In the talk, I will introduce a linear map on bounded operators on the tensor square of a given complex Hilbert space, called the “crossing map”, whose fixed points describe crossing symmetry. I describe its basic properties and, among its many surprising relations with other mathematical objects, I will describe how it turns out to be a special case of the subfactor theoretical Fourier transform (which generalizes, e.g., the ordinary Fourier transform for finite abelian groups). Joint work with R. Correa da Silva and G. Lechner, arXiv: 2402.15763