The metaplectic group within the Heisenberg–Weyl ring

Abstract

The Heisenberg–Weyl ring contains the metaplectic group of canonical transforms acting unitarily on ℒ 2(ℛ). These ring elements are characterized through (i) the integral transform kernels, (ii) coset distributions, and (iii) classical functions under any quantization scheme. The isomorphism under group composition leads to several new relations involving twisted products and quantization of Gaussian classical functions. The Wigner inversion operator is a special central group element. It is shown that the only quantization scheme invariant under metaplectic transformations is the Weyl scheme. The structure studied here appears to be relevant to the study of wave optics with aberration.

Quantum Harmonic Analysis of the Density Matrix

“…For a complete study of the metaplectic group in quantum mechanics see our book [40]; on a slightly more general and technical level see [35,36]. An excellent introduction to the symplectic group is given in Arvind et al [2], also see García-Bullé et al [30]. It is an easy exercise to show that if S is symplectic then S −1 and S T are symplectic as well, hence this defining relation is equivalent to SJS T = J.…”

“…Finally, || A|| HS = 0 is equivalent to Aψ j = 0 for every index j that is to A = 0. The space L 2 (H) is complete for that norm, and hence a Banach space (it is actually even a Hilbert space when equipped with the scalar product (30). In addition L 2 (H) is closed under multiplication (and hence an algebra).…”

“…For a complete study of the metaplectic group in quantum mechanics see our book [40]; on a slightly more general and technical level see [35,36]. An excellent introduction to the symplectic group is given in Arvind et al [2], also see García-Bullé et al [30].…”

Linear transformations and aberrations in continuous and finite systems

“…The contours in the density plot of the Wigner function are chosen both to highlight the shape of its main peak, and to display the fine structure in the regions where it is less than 5% of the top value π . (26). The real, imaginary, and absolute values of the signals are indicated by dots of increasing size, joined by dashed, dotted, and continuous lines, respectively.…”

New two-fold integration transformation for the Wigner operator in phase space quantum mechanics and its relation to operator ordering

“…The quantizations of q m p n and q-p exponentials in all orderings, for example, can be found in Wolf's book. [11,12] In the present work we shall adopt another different approach for tackling three ordering schemes: Weyl, P -Q and Q-P , and will present a well-ordered compendium of formulae of this type. The work is arranged as follows: In Section 2 we briefly review the Weyl ordered form of the Wigner operator.…”