Abstract The generating function for canonical transformations derived by Marinov has the important property of symplectic invariance (i. e. under linear canonical transformations). However, a more geometric approach to the rederivation of this function from the variational principle reveals that it is not free from caustic singularities after all. These singularities can be avoided without breaking the symplectic invariance by the definition of a complementary generating function bearing an analogous relation to the Woodward ambiguity function in telecommunications theory as that tying Marinov’s function to the Wigner function and the Weyl transform in quantum mechanics. Marinov’s function is specially apt to describe canonical transformations close to the identity, but breaks down for reflections through a point in phase space, easily described by the new generating function.Abstract The generating function for canonical transformations derived by Marinov has the important property of symplectic invariance (i. e. under linear canonical transformations). However, a more geometric approach to the rederivation of this function from the variational principle reveals that it is not free from caustic singularities after all. These singularities can be avoided without breaking the symplectic invariance by the definition of a complementary generating function bearing an analogous relation to the Woodward ambiguity function in telecommunications theory as that tying Marinov’s function to the Wigner function and the Weyl transform in quantum mechanics. Marinov’s function is specially apt to describe canonical transformations close to the identity, but breaks down for reflections through a point in phase space, easily described by the new generating function.

Proceedings of the Royal Society of London Series A Mathematical and Physical SciencesDec 8, 19908 citations

Symplectic structure and quantization

D. J. Simms, L. Castel

Jan 1, 19750 citations

Canonical transformations applied to the free Landau electron

M. H. Boon, T. H. Seligman

It is shown that the problem of a free eletron in a uniform magnetic field provides an interesting example for the use of recently developed canonical transformation methods in quantum mechanics. The Schrödinger equation is immediately solved by a unitary transformation corresponding to a finite linear symplectic transformation in phase space. The natural invariance and dynamical groups involve affine and not only linear symplectic transformations, and the Weyl group turns out to be an appropriate invariance group to account completely for the infinite degeneracies.It is shown that the problem of a free eletron in a uniform magnetic field provides an interesting example for the use of recently developed canonical transformation methods in quantum mechanics. The Schrödinger equation is immediately solved by a unitary transformation corresponding to a finite linear symplectic transformation in phase space. The natural invariance and dynamical groups involve affine and not only linear symplectic transformations, and the Weyl group turns out to be an appropriate invariance group to account completely for the infinite degeneracies.

Journal of Mathematical PhysicsSep 1, 197343 citations

Anisotropic uncertainty principles for metaplectic operators

Elena Cordero, Gianluca Giacchi, Edoardo Pucci

We establish anisotropic uncertainty principles (UPs) for general metaplectic operators acting on $L^2(\mathbb{R}^d)$ , including degenerate cases associated with symplectic matrices whose $B$ -block has nontrivial kernel. In this setting, uncertainty phenomena are shown to be intrinsically directional and confined to an effective phase-space dimension given by $\mathrm{rank}(B)$ . First, we prove sharp Heisenberg-Pauli-Weyl type inequalities involving only the directions corresponding to $\ker(B)^\perp$ , with explicit lower bounds expressed in terms of geometric quantities associated with the underlying symplectic transformation. We also provide a complete characterization of all extremizers, which turn out to be partially Gaussian functions with free behavior along the null directions of $B$ . Building on this framework, we extend the Beurling-Hörmander theorem to the metaplectic setting, obtaining a precise polynomial-Gaussian structure for functions satisfying suitable exponential integrability conditions involving both $f$ and its metaplectic transform. Finally, we prove a Morgan-type (or Gel'fand--Shilov type) uncertainty principle for metaplectic operators, identifying a sharp threshold separating triviality from density of admissible functions and showing that this threshold is invariant under metaplectic transformations. Our results recover the classical Fourier case and free metaplectic transformations as special instances, and reveal the geometric and anisotropic nature of uncertainty principles in the presence of symplectic degeneracies.We establish anisotropic uncertainty principles (UPs) for general metaplectic operators acting on $L^2(\mathbb{R}^d)$, including degenerate cases associated with symplectic matrices whose $B$-block has nontrivial kernel. In this setting, uncertainty phenomena are shown to be intrinsically directional and confined to an effective phase-space dimension given by $\mathrm{rank}(B)$. First, we prove sharp Heisenberg-Pauli-Weyl type inequalities involving only the directions corresponding to $\ker(B)^\perp$, with explicit lower bounds expressed in terms of geometric quantities associated with the underlying symplectic transformation. We also provide a complete characterization of all extremizers, which turn out to be partially Gaussian functions with free behavior along the null directions of $B$. Building on this framework, we extend the Beurling-Hörmander theorem to the metaplectic setting, obtaining a precise polynomial-Gaussian structure for functions satisfying suitable exponential integrability conditions involving both $f$ and its metaplectic transform. Finally, we prove a Morgan-type (or Gel'fand--Shilov type) uncertainty principle for metaplectic operators, identifying a sharp threshold separating triviality from density of admissible functions and showing that this threshold is invariant under metaplectic transformations. Our results recover the classical Fourier case and free metaplectic transformations as special instances, and reveal the geometric and anisotropic nature of uncertainty principles in the presence of symplectic degeneracies.

math.APCross-listedarXiv:2601.16279Jan 26, 2026

We establish anisotropic uncertainty principles (UPs) for general metaplectic operators acting on

L^2(\mathbb{R}^d)

, including degenerate cases associated with symplectic matrices whose

B

-block has nontrivial kernel. In this setting, uncertainty phenomena are shown to be intrinsically directional and confined to an effective phase-space dimension given by

\mathrm{rank}(B)

. First, we prove sharp Heisenberg-Pauli-Weyl type inequalities involving only the directions corresponding to

\ker(B)^\perp

, with explicit lower bounds expressed in terms of geometric quantities associated with the underlying symplectic transformation. We also provide a complete characterization of all extremizers, which turn out to be partially Gaussian functions with free behavior along the null directions of

B

. Building on this framework, we extend the Beurling-Hörmander theorem to the metaplectic setting, obtaining a precise polynomial-Gaussian structure for functions satisfying suitable exponential integrability conditions involving both

f

and its metaplectic transform. Finally, we prove a Morgan-type (or Gel'fand--Shilov type) uncertainty principle for metaplectic operators, identifying a sharp threshold separating triviality from density of admissible functions and showing that this threshold is invariant under metaplectic transformations. Our results recover the classical Fourier case and free metaplectic transformations as special instances, and reveal the geometric and anisotropic nature of uncertainty principles in the presence of symplectic degeneracies.