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Journal of Mathematical Physics

Geometry and uncertainty

Francis J. Narcowich

Published Feb 1, 1990• 43 citations
Publisher

Abstract

In the context of the Wigner–Weyl phase space formulation of quantum mechanics, a version of the uncertainty relations invariant under affine canonical transformations is derived. For a fixed Wigner distribution function possessing a finite covariance, ‘‘directions’’ of minimal uncertainty are found. The geometry of the Wigner ellipsoid and its Legendre transform, the dual Wigner ellipsoid, both of which are associated with the same covariance, is discussed. The results obtained are generalizations of the well-known fact that, for one degree of freedom, the area of the Wigner ellipse must be of order ℏ or larger. Instead of area, which is an invariant only when n=1, these results involve Poincaré invariants of certain curves and surfaces.

DOI
10.1063/1.528922

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In the context of the Wigner–Weyl phase space formulation of quantum mechanics, a version of the uncertainty relations invariant under affine canonical transformations is derived. For a fixed Wigner distribution function possessing a finite covariance, ‘‘directions’’ of minimal uncertainty are found. The geometry of the Wigner ellipsoid and its Legendre transform, the dual Wigner ellipsoid, both of which are associated with the same covariance, is discussed. The results obtained are generalizations of the well-known fact that, for one degree of freedom, the area of the Wigner ellipse must be of order ℏ or larger. Instead of area, which is an invariant only when n=1, these results involve Poincaré invariants of certain curves and surfaces.
We extend the Wigner-Weyl-Moyal phase-space formulation of quantum mechanics to general curved configuration spaces. The underlying phase space is based on the chosen coordinates of the manifold and their canonically conjugate momenta. The resulting Wigner function displays the axioms of a quasiprobability distribution, and any Weyl-ordered operator gets associated with the corresponding phase-space function, even in the absence of continuous symmetries. The corresponding quantum Liouville equation reduces to the classical curved space Liouville equation in the semiclassical limit. We demonstrate the formalism for a point particle moving on two-dimensional manifolds, such as a paraboloid or the surface of a sphere. The latter clarifies the treatment of compact coordinate spaces, as well as the relation of the presented phase-space representation to symmetry groups of the configuration space.
We derive Heisenberg uncertainty principles for pairs of Linear Canonical Transforms of a given function, by resorting to the fact that these transforms are just metaplectic operators associated with free symplectic matrices. The results obtained synthesize and generalize previous results found in the literature, because they apply to all signals, in arbitrary dimension and any metaplectic operator (which includes Linear Canonical Transforms as particular cases). Moreover, we also obtain a generalization of the Robertson-Schrödinger uncertainty principle for Linear Canonical Transforms. We also propose a new quadratic phase-space distribution, which represents a signal along two intermediate directions in the time-frequency plane. The marginal distributions are always non-negative and permit a simple interpretation in terms of the Radon transform. We also give a geometric interpretation of this quadratic phase-space representation as a Wigner distribution obtained upon Weyl quantization on a non-standard symplectic vector space. Finally, we derive the multidimensional version of the Hardy uncertainty principle for metaplectic operators and the Paley-Wiener theorem for Linear Canonical Transforms.
The quest to reconcile quantum mechanics with gravitational theory motivates the exploration of frameworks that treat quantum uncertainty and spacetime geometry under a unified approach. A promising candidate that emerges from this pursuit is the relativistic quantum phase space (QPS) formalism, which extends classical phase space by incorporating both mean values and variance-covariance matrices of quantum states, providing a unified setting where the uncertainty principle and relativistic covariance coexist. For the signature (1,4)(1,4)(1,4), we construct a scalar from the mean values and the inverse variance-covariance matrix and prove its invariance under linear canonical transformations (LCTs). Motivated by the form of the variance-covariance matrix in a particular reference frame, we identify this invariant as Γ=L2/ℓ2Γ= L^2/\ell^2Γ=L2/ℓ2 for states that saturate the uncertainty relations, where LLL and ℓ\ellℓ are two fundamental length scales that can be identified with the de Sitter radius and the Planck length, respectively. From this invariant, we obtain a geometric equation that unifies mean values and quantum fluctuations. In the limit ℓ→0\ell \to 0ℓ→0, the equation reduces to the de Sitter spacetime equation; in the limit L→∞L \to \inftyL→∞, it yields a curved momentum space reminiscent of Born reciprocity. In the Minkowski limit (both ℓ→0\ell \to 0ℓ→0 and L→∞L \to \inftyL→∞), the familiar relativistic relations for rest mass and proper time emerge. These limiting cases show how the Planck length and the cosmological constant can be unified within a single geometric constraint, establishing the QPS geometry as a promising framework for exploring the interplay between quantum mechanics and gravity.
It is shown that, in the phase-space representation of quantum mechanics, the uncertainty relation can be stated in terms of the integral invariant of Poincar\'e. The uncertainty relation for spreading free wave packets is discussed as an illustrative example. This phase-space approach can be extended to the relativistic regime. It is shown that Lorentz boosts are area-preserving canonical transformations in the phase space of the light-cone variables. The harmonic oscillator is discussed in detail as an illustrative example for the covariant realization of the uncertainty principle.
Uncertainty relations are usually stated as bounds on selected combinations of variances, but the full covariance matrix contains substantially richer information about the geometry of quantum state space and about the operational capabilities of quantum systems. Here we characterize the covariance matrices attainable by a finite-dimensional canonical pair of observables related by the discrete Fourier transform, the natural analogue of position and momentum in a finite Hilbert space. We combine analytic arguments with convex-geometric and semidefinite-programming methods based on joint numerical ranges to describe the admissible region through unitary invariants, in particular the trace and determinant of the covariance matrix. This provides a systematic way to identify extremal states, generalizing the notion of minimum-uncertainty states, and to quantify how the discrete uncertainty geometry approaches its continuous counterpart with increasing dimension. We further show that the resulting covariance-matrix characterization has direct consequences for applications: it yields accuracy bounds for multi parameter estimation protocols and separability criteria for finite-dimensional bipartite systems, including discrete analogues of continuous-variable EPR-type witnesses. Our results establish a systematic and versatile platform for connecting uncertainty relations, convex quantum geometry, metrology, and entanglement detection in finite-dimensional systems.