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Journal of Mathematical Physics / Volume 27 / Issue 1

J. Math. Phys. 27, 29 (1986); http://dx.doi.org/10.1063/1.527333 (8 pages)

The metaplectic group within the Heisenberg–Weyl ring

Mauricio García‐Bullé1, Wolfgang Lassner2, and Kurt Bernardo Wolf3

1Centro Internacional de Física y Matemáticas Orientadas (CIFMO), Mexico and Facultad de Ciencias, Universidad Nacional Autónoma de México, 01000 México D.F., Mexico
2Centro Internacional de Física y Matemáticas Orientadas (CIFMO), Mexico and Naturwissenschaftlich‐Theoretisches Zentrum, Sektion Mathematik, Karl‐Marx‐Universität, Leipzig, German Democratic Republic
3Centro Internacional de Física y Mathemáticas Orientadas (CIFMO), Mexico and Departamento de Matemáticas, Universidad Autónoma Metropolitana, Iztapalapa 09340 México DF, Mexico

(Received 18 February 1985; accepted 14 June 1985)

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.

KEYWORDS and PACS

Keywords

UNITARITY, CANONICAL TRANSFORMATIONS, INTEGRAL TRANSFORMATIONS, QUANTIZATION, GROUP THEORY, INVARIANCE PRINCIPLES, WAVE PROPAGATION, ABERRATIONS, HILBERT SPACE, SYMMETRY GROUPS, GEOMETRICAL OPTICS, LIE GROUPS

PACS

  • 02.20.Rt

    Discrete subgroups of Lie groups

  • 41.20.Jb

    Electromagnetic wave propagation; radiowave propagation

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.527333

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

For access to fully linked references, you need to log in.
    M. Moshinsky and C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772 (1971JMAPAQ000012000008001772000001)
    “Canonical transformations and matrix elements,” ibid. 12, 1780 (1971JMAPAQ000012000008001780000001).

    D. C. Rivier, “On the one-to-one correspondence between infinitesimal canonical transformations and infinitesimal unitary transformations,” Phys. Rev. 83, 674 (1951).

    L. Cohen, “Generalized phase-space distribution functions,” J. Math. Phys. 7, 781 (1966JMAPAQ000007000005000781000001).

    D. Basu and K. B. Wolf, “The unitary irreducible representations of SL(2,R) in all subgroup reductions,” J. Math. Phys. 23, 189 (1982JMAPAQ000023000002000189000001).

    A. Royer, “Wigner function as the expectation value of a parity operator,” Phys. Rev. A 15, 449 (1977).



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