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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.

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