Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:hep-th/0105226 · Modern Physics Letters A

NEGATIVE PROBABILITY AND UNCERTAINTY RELATIONS

THOMAS CURTRIGHT, COSMAS ZACHOS

Published Dec 7, 2001• 26 citations
arXivHTMLPDF

Abstract

A concise derivation of all uncertainty relations is given entirely within the context of phase-space quantization, without recourse to operator methods, to the direct use of Weyl's correspondence, or to marginal distributions of x and p.
DOI
10.1142/s021773230100576x

Related papers

The Generalized “Position–Momentum” Uncertainty Relation in Quantum Mechanics and in the Theory of Brownian Motion

Alexander D. Sukhanov

Theoretical and Mathematical Physics•Apr 1, 2004•2 citations

Negative Probability and Uncertainty Relations

Thomas Curtright, COSMAS K. ZACHOS

A concise derivation of all uncertainty relations is given entirely within the context of phase-space quantization, without recourse to operator methods, to the direct use of Weyl’s correspondence, or to marginal distributions of x and p. Phase-space quantization is the third autonomous and logically complete formulation of quantum mechanics beyond the conventional ones based on operators in Hilbert space or path integrals [1, 2, 3]. It is free of operators and wavefunctions: observables and matrix elements are computed through phasespace integrals of c-number functions (“classical kernels”) weighted by a Wigner function (WF) [4, 3]. This is a phase-space distribution function which is not positive semi-definite, and constitutes the Weyl correspondent [5] of the density matrix in the conventional formulation, fmn(x,p) = 1 2π dye −iyp ψ ∗ m(x − � 2 y)ψn(x + � 2 y) = f ∗ nm(x,p). (1)

•Jan 1, 2001•15 citations

Generalization of uncertainty relation for quantum and stochastic systems

Tomoi Koide C, Takeshi Kodama

arXiv:1208.0258•Physics Letters A•Apr 6, 2018•22 citations

A Metaplectic Perspective of Uncertainty Principles in the Linear Canonical Transform Domain

Nuno Costa Dias, Maurice A. de Gosson, João Nuno Prata

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.

arXiv:2405.10651•Journal of Functional Analysis•Jan 1, 2022•15 citations

Operator-ordering identities for mutual transformation of power of coordinate-momentum operators obtained by a new concise method

Fan Hong-Yi, Lou Sen‐Yue, Zhang Peng-Fei

Since the foundation of quantum mechanics, operator-ordering identities for mutual transformation of power of coordinate-momentum operators have been a fundamental and tough topic. To the best of our knowledge, this topic has not been tackled smoothly because there is no elegant and direct way to investigate it. In this paper we report a very concise and novel method to handle this topic, i.e., we employ the generating function of two-variable Hermite polynomial and the characteristics of ordered operators to derive a series of operator-ordering identities for mutual transformation of power of coordinate-momentum operators: they surly possess potential applications. The essence of our method lies in the fact that coordinate-momentum operators can be permutable within ordered product of operators, just as the scenarios in P-Q ordering, Q-P ordering and Weyl ordering. We also derive the integration transformation formula about two-variable Hermite polynomial in phase space. The correspondence relation between operator ordering and quantization recipe is established. The beauty of theoretical physics is embodied extensively in the paper.

Acta Physica Sinica•Jan 1, 2015•2 citations

A quantum mechanical derivation of the general uncertainty relation for real signals in communication theory

Emilio Borchi, G. Pelosi

Signal Processing•Jul 1, 1980•8 citations
A concise derivation of all uncertainty relations is given entirely within the context of phase-space quantization, without recourse to operator methods, to the direct use of Weyl’s correspondence, or to marginal distributions of x and p. Phase-space quantization is the third autonomous and logically complete formulation of quantum mechanics beyond the conventional ones based on operators in Hilbert space or path integrals [1, 2, 3]. It is free of operators and wavefunctions: observables and matrix elements are computed through phasespace integrals of c-number functions (“classical kernels”) weighted by a Wigner function (WF) [4, 3]. This is a phase-space distribution function which is not positive semi-definite, and constitutes the Weyl correspondent [5] of the density matrix in the conventional formulation, fmn(x,p) = 1 2π dye −iyp ψ ∗ m(x − � 2 y)ψn(x + � 2 y) = f ∗ nm(x,p). (1)
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.
Since the foundation of quantum mechanics, operator-ordering identities for mutual transformation of power of coordinate-momentum operators have been a fundamental and tough topic. To the best of our knowledge, this topic has not been tackled smoothly because there is no elegant and direct way to investigate it. In this paper we report a very concise and novel method to handle this topic, i.e., we employ the generating function of two-variable Hermite polynomial and the characteristics of ordered operators to derive a series of operator-ordering identities for mutual transformation of power of coordinate-momentum operators: they surly possess potential applications. The essence of our method lies in the fact that coordinate-momentum operators can be permutable within ordered product of operators, just as the scenarios in P-Q ordering, Q-P ordering and Weyl ordering. We also derive the integration transformation formula about two-variable Hermite polynomial in phase space. The correspondence relation between operator ordering and quantization recipe is established. The beauty of theoretical physics is embodied extensively in the paper.