The Metaplectic and Anaplectic Representations

Abstract

In this chapter, we briefly review some basic aspects of the metaplectic representation, especially in the one-dimensional and two-dimensional cases. Then, we shall introduce the new anaplectic analysis on the real line, in which the spectrum of the harmonic oscillator is ℤ rather than $\frac{1}{2} + \mathbb{N}$. The basic space $\mathfrak{A}$ substituting for L ²(ℝ) consists of functions on the line extending as entire functions, typically increasing like “bad” Gaussian functions at infinity. Nevertheless, there is on $\mathfrak{A}$ a well-defined translation-invariant concept of integral, and (in place of the scalar product of L ²(ℝ)) a pseudoscalar product reminiscent of indefinite forms occurring in Physics. All symmetries of usual analysis expressing themselves by means of such objects as the Heisenberg representation, the Fourier transformation, and, more generally, the metaplectic representation, have counterparts in anaplectic analysis. Note that in Sect. 4.1, we shall have to consider the parameter-dependent ? -anaplectic analysis. The one considered in the present chapter (in Sect. 2.2) corresponds to $\nu = - \frac{1}{2}$: it will also be shown in Sect. 4.2 that the case when $\nu = 0$ yields an analysis containing the usual one.