Exploring the Space of Linear Canonical Transforms

Synopsis

The Fourier transform is one of the most powerful tools in a physicist's arsenal, offering a bridge between real-space and reciprocal-space representations of functions. By considering the Fourier transform as a 90-degree rotation of the time-frequency plane, we can naturally rediscover the fractional Fourier transform and, more generally, a family of integral transforms called linear canonical transforms. Parameterized by 2×2 matrices with unitary determinant, these transforms describe area-preserving operations on phase space, such as rotation, dilation, and shearing. Linear canonical transforms arise naturally in many different areas, providing a connection between classical wave optics and quantum dynamics. In this talk, we will forgo derivations and develop a visual intuition of these transforms by performing them on an arbitrary test function.