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

New Journal of Physics

Universal Heisenberg uncertainty relations for all massless particles

Iwo Białynicki‐Birula, Zofia Białynicka-Birula

Published Jan 1, 2026• 0 citations
Publisher

Abstract

Abstract We derive the uncertainty relations between position and momentum (or wavector) in a uniform way for all massless particles. As expected, the lower bound γ min = ( 1 + 1 / 4 + 2 h ) in the uncertainty relation grows with the particle’s helicity h .
DOI
10.1088/1367-2630/ae333b

Related papers

Heisenberg uncertainty relation for position and momentum beyond central potentials

Łukasz Rudnicki

Recently, S\'anchez-Moreno et al. [New J. Phys. 8, 330 (2006)] have shown that the three-dimensional Heisenberg uncertainty relation ${\ensuremath{\sigma}}_{r}{\ensuremath{\sigma}}_{p}\ensuremath{\ge}3\ensuremath{\hbar}/2$ might be significantly sharpened if the relevant quantum state describes the particle in a central potential. I extend that result to the case of states which are not the eigenstates of the square of the angular momentum operator. I derive a lower bound for ${\ensuremath{\sigma}}_{r}{\ensuremath{\sigma}}_{p}$ which involves the mean value and the variance of the square of the angular momentum operator.

arXiv:1112.5198•Physical Review A•Feb 8, 2012•11 citations

Heisenberg uncertainty relations for photons

Iwo Białynicki‐Birula, Zofia Białynicka-Birula

The idea to base the uncertainty relation for photons on the electromagnetic energy distribution in space enabled us to derive a sharp inequality that expresses the uncertainty relation [Phys. Rev. Lett. 108, 140401 (2012)]. An alternative version of the uncertainty relation derived in this paper is closer in spirit to the original Heisenberg relation because it employs the analog of the position operator for the photon---the center of the energy operator. The noncommutativity of the components of the center of the energy operator results in the increase of the bound $3\ensuremath{\hbar}/2$ in the standard Heisenberg uncertainty relation in three dimensions. This difference diminishes with the increase of the photon energy. In the infinite-momentum frame, the lower bound in the Heisenberg uncertainty relations for photons is the same as in nonrelativistic quantum mechanics. A similar uncertainty relation is also derived for coherent photon beams. This relation has direct experimental consequences since it gives a precise relationship between the spectral composition of the laser beam and the minimal focal volume.

arXiv:1205.0516•Physical Review A•Aug 30, 2012•28 citations

Localization of relativistic particles and uncertainty relations

Daniel R. Terno

Localization of relativistic particles and their position-momentum uncertainty relations are not yet fully understood. We discuss two schemes of photon localization that are based on the energy density. One scheme produces a positive operator-valued measure for localization. It coincides with the number density operator and reproduces an effective $3\ifmmode\times\else\texttimes\fi{}3$ polarization density matrix. Another scheme results in a probability distribution that is conditioned on the detection. In both schemes the uncertainty relations for transversal position and momentum approach the Heisenberg bound $\ensuremath{\Delta}p\ensuremath{\Delta}z=\frac{1}{2}\ensuremath{\hbar}$.

arXiv:1308.0479•Physical Review A•Apr 21, 2014•24 citations

Uncertainty Relations for Nonsimultaneous Measurements

Jean-Marc Lévy-Leblond

The usual Heisenberg uncertainty relations are generalized to the case of nonsimultaneous measurements of position and momentum. The product of the corresponding uncertainties is bounded from below by a quantity depending on the dynamics of the system (the potential), and which may be lower than ħ/2 and even vanish. The case of two successive measurements of momentum and position is also treated, and one recovers the standard phenomenon of spreading of the wave packet. Whereas the case of an arbitrary potential can only be treated to lowest orders in the lime delay, a complete solution is given for the harmonic oscillator.

American Journal of Physics•Jun 1, 1972•4 citations

A universal formulation of uncertainty relation for errors

Jaeha Lee, Izumi Tsutsui

We present a universal formulation of uncertainty relation valid for any conceivable quantum measurement of statistical nature. Owing to its simplicity and operational tangibility, our general relation is also experimentally verifiable. Our relation violates the traditional naïve bound ħ / 2 for the position-momentum measurement while respecting Heisenberg's original philosophy of the uncertainty principle. Our relation entails, among others, the Arthurs–Kelly–Goodman and Ozawa relations as corollaries to its special cases, and also seamlessly reduces itself to the standard Kennard–Robertson (Schrödinger) relation when the measurement is non-informative.

Physics Letters A•Oct 5, 2024•3 citations

On the relation between momentum uncertainty and thermal wavelength

Zi-Fan Zhu, Yao Wang

For quantum particles in a Boltzmann state, we derive an inequality between momentum uncertainty $Δp$ and thermal de Broglie wavelength $λ_{\rm th}$, expressed as $Δp \geq \sqrt{2π}\hbar/λ_{\rm th}$, as a corollary of the Boltzmann lower bound for the Heisenberg uncertainty product proposed in the previous work [EPL, 143, 20001 (2023)]

arXiv:2407.06715•Jul 9, 2024•0 citations
Recently, S\'anchez-Moreno et al. [New J. Phys. 8, 330 (2006)] have shown that the three-dimensional Heisenberg uncertainty relation ${\ensuremath{\sigma}}_{r}{\ensuremath{\sigma}}_{p}\ensuremath{\ge}3\ensuremath{\hbar}/2$ might be significantly sharpened if the relevant quantum state describes the particle in a central potential. I extend that result to the case of states which are not the eigenstates of the square of the angular momentum operator. I derive a lower bound for ${\ensuremath{\sigma}}_{r}{\ensuremath{\sigma}}_{p}$ which involves the mean value and the variance of the square of the angular momentum operator.
The idea to base the uncertainty relation for photons on the electromagnetic energy distribution in space enabled us to derive a sharp inequality that expresses the uncertainty relation [Phys. Rev. Lett. 108, 140401 (2012)]. An alternative version of the uncertainty relation derived in this paper is closer in spirit to the original Heisenberg relation because it employs the analog of the position operator for the photon---the center of the energy operator. The noncommutativity of the components of the center of the energy operator results in the increase of the bound $3\ensuremath{\hbar}/2$ in the standard Heisenberg uncertainty relation in three dimensions. This difference diminishes with the increase of the photon energy. In the infinite-momentum frame, the lower bound in the Heisenberg uncertainty relations for photons is the same as in nonrelativistic quantum mechanics. A similar uncertainty relation is also derived for coherent photon beams. This relation has direct experimental consequences since it gives a precise relationship between the spectral composition of the laser beam and the minimal focal volume.
Localization of relativistic particles and their position-momentum uncertainty relations are not yet fully understood. We discuss two schemes of photon localization that are based on the energy density. One scheme produces a positive operator-valued measure for localization. It coincides with the number density operator and reproduces an effective $3\ifmmode\times\else\texttimes\fi{}3$ polarization density matrix. Another scheme results in a probability distribution that is conditioned on the detection. In both schemes the uncertainty relations for transversal position and momentum approach the Heisenberg bound $\ensuremath{\Delta}p\ensuremath{\Delta}z=\frac{1}{2}\ensuremath{\hbar}$.
The usual Heisenberg uncertainty relations are generalized to the case of nonsimultaneous measurements of position and momentum. The product of the corresponding uncertainties is bounded from below by a quantity depending on the dynamics of the system (the potential), and which may be lower than ħ/2 and even vanish. The case of two successive measurements of momentum and position is also treated, and one recovers the standard phenomenon of spreading of the wave packet. Whereas the case of an arbitrary potential can only be treated to lowest orders in the lime delay, a complete solution is given for the harmonic oscillator.
We present a universal formulation of uncertainty relation valid for any conceivable quantum measurement of statistical nature. Owing to its simplicity and operational tangibility, our general relation is also experimentally verifiable. Our relation violates the traditional naïve bound ħ / 2 for the position-momentum measurement while respecting Heisenberg's original philosophy of the uncertainty principle. Our relation entails, among others, the Arthurs–Kelly–Goodman and Ozawa relations as corollaries to its special cases, and also seamlessly reduces itself to the standard Kennard–Robertson (Schrödinger) relation when the measurement is non-informative.
For quantum particles in a Boltzmann state, we derive an inequality between momentum uncertainty ΔpΔpΔp and thermal de Broglie wavelength λthλ_{\rm th}λth​, expressed as Δp≥2πℏ/λthΔp \geq \sqrt{2π}\hbar/λ_{\rm th}Δp≥2π​ℏ/λth​, as a corollary of the Boltzmann lower bound for the Heisenberg uncertainty product proposed in the previous work [EPL, 143, 20001 (2023)]