mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...

and

harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an e ...

, the quadratic Fourier transform is an

integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than i ...

that generalizes the

fractional Fourier transform In mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform. It can be thought of as the Fourier transform to the ''n''-th power, where ''n' ...

, which in turn generalizes the

Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...

. Roughly speaking, the Fourier transform corresponds to a change of variables from time to frequency (in the context of harmonic analysis) or from position to momentum (in the context of

quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, q ...

). In

phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usual ...

, this is a 90 degree rotation. The fractional Fourier transform generalizes this to any angle rotation, giving a smooth mixture of time and frequency, or of position and momentum. The quadratic Fourier transform extends this further to the group of all linear symplectic transformations in phase space (of which rotations are a subgroup). More specifically, for every member of the

metaplectic group In mathematics, the metaplectic group Mp2''n'' is a double cover of the symplectic group Sp2''n''. It can be defined over either real or ''p''-adic numbers. The construction covers more generally the case of an arbitrary local or finite field ...

(which is a double cover of the symplectic group) there is a corresponding quadratic Fourier transform.

References

Fourier analysis Integral transforms Time–frequency analysis {{Quantum-stub