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Journal of Mathematical Physics

The metaplectic group within the Heisenberg–Weyl ring

Mauricio García-Bullé, Wolfgang Lassner, Kurt Bernardo Wolf

Published Jan 1, 1986• 27 citations
Publisher

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.

DOI
10.1063/1.527333

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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.
The symplectic group Sp(n) acts on phase space while the unitary representation of its double cover, Mp(n), the metaplectic group, acts on functions defined on configuration space. We will construct an extension Mp(n) of Mp(n) acting on square integrable functions on phase space. This is performed using previous results of ours involving explicit expressions of the twisted Weyl symbols of metaplectic operators and Bopp pseudodifferential operators, which are phase space extensions of the usual Weyl operators.
The Heisenberg-Weyl group H W ( d ) related to a d -dimensional Hilbert space H ( d ) , is enlarged into the Heisenberg-Weyl-parity group H W P ( d ) that incorporates parity transformations. It consists of 2 d 3 elements, of which d 3 elements belong to the H W ( d ) subgroup, and extra d 3 elements which are related through a Fourier transform with the former ones. It is shown that H W P ( d ) is a generalised version of the dihedral group. The properties of operators that combine displacements and parity, are discussed. H W P ( d ) is shown to be a solvable group, and commutators of its elements perform displacement and parity transformations of quantum states, along loops in the discrete phase space. 2 d 2 coherent states related to the H W P ( d ) group are introduced, which consist of d 2 coherent states related to the H W ( d ) subgroup, and extra d 2 coherent states which are related through a Fourier transform with the former ones. In noisy cases, expansion of an arbitrary state in terms of the 2 d 2 coherent states with Bargmann coefficients, is advantageous in comparison to expansion in terms of the d 2 coherent states related to H W ( d ) . One of the consequences of the H W P ( d ) group, is a natural unification of the Wigner and Weyl functions. The properties of the unified Wigner–Weyl function are discussed.
The Heisenberg-Weyl group HW(d)HW(d)HW(d) related to a ddd-dimensional Hilbert space H(d)H(d)H(d), is enlarged into the Heisenberg-Weyl-parity group HWP(d)HWP(d)HWP(d) that incorporates parity transformations. It consists of 2d32d^32d3 elements, of which d3d^3d3 elements belong to the HW(d)HW(d)HW(d) subgroup, and extra d3d^3d3 elements which are related through a Fourier transform with the former ones. It is shown that HWP(d)HWP(d)HWP(d) is a generalised version of the dihedral group. The properties of operators that combine displacements and parity, are discussed. HWP(d)HWP(d)HWP(d) is shown to be a solvable group, and commutators of its elements perform displacement and parity transformations of quantum states, along loops in the discrete phase space.2d22d^22d2 coherent states related to the HWP(d)HWP(d)HWP(d) group are introduced, which consist of d2d^2d2 coherent states related to the HW(d)HW(d)HW(d) subgroup, and extra d2d^2d2 coherent states which are related through a Fourier transform with the former ones. In noisy cases, expansion of an arbitrary state in terms of the 2d22d^22d2 coherent states with Bargmann coefficients, is advantageous in comparison to expansion in terms of the d2d^2d2 coherent states related to HW(d)HW(d)HW(d). One of the consequences of the HWP(d)HWP(d)HWP(d) group, is a natural unification of the Wigner and Weyl functions. The properties of the unified Wigner-Weyl function are discussed.
It is shown that the representation matrices of the quantum mechanical group in the coherent state basis can be used as bases for the expansion of arbitrary square integrable functions on the group. Although this basis is nonorthogonal the expansion can be inverted in a manner similar to an orthogonal basis. Next a coherent state ‘‘wave function’’ is defined in analogy with the Schrödinger wave function and the above matrix elements are used to determine the action of the group in the space of the coherent state wave functions. The function space in this realization is Bargmann’s Hilbert space of analytic functions. It is shown that the mixed basis matrix element between the coordinate and coherent state bases is essentially the integral kernel of Bargmann. The transpose of this matrix element yields the kernel for the inversion of the transform. In this construction the Bargmann transform appears as the unitary transformation connecting the Schrödinger and coherent state wave functions. This considerably simplifies Bargmann’s original arguments and yields the integral transform as well as its inversion formula in a simple way.
Any quantization maps linearly functions on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary transformations) symmetries. Integral means that we use all ressources of integral calculus, in order to implement the method when we apply it to singular functions, or distributions, for which the integral calculus is an essential ingredient. In this paper we emphasize the deep connection between Fourier transform and covariant integral quantization when the Weyl-Heisenberg and affine groups are involved. We show with our generalisations of the Wigner-Weyl transform that many properties of the Weyl integral quantization, commonly viewed as optimal, are actually shared by a large family of integral quantizations.