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arXiv:2411.08589 · Physical Review A

Uncertainty relations relative to phase-space quantum reference frames

Miguel Jorquera Riera, Leon Loveridge

Published Jun 9, 2025• 3 citations
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Abstract

The authors show that Heisenberg's uncertainty relation is modified when considered relative to a phase-space quantum reference frame, and that such a frame destroys the incompatibility of position and momentum. They recover the standard uncertainty bound by taking the classical limit of the frame, arguing that standard quantum theory relies on an implicit classical frame.

DOI
10.1103/physreva.111.l060201

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The authors show that Heisenberg's uncertainty relation is modified when considered relative to a phase-space quantum reference frame, and that such a frame destroys the incompatibility of position and momentum. They recover the standard uncertainty bound by taking the classical limit of the frame, arguing that standard quantum theory relies on an implicit classical frame.
Uncertainty relations and complementarity of canonically conjugate position and momentum observables in quantum theory are discussed with respect to some general coupling properties of a function and its Fourier transform. The question of joint localization of a particle on bounded position and momentum value sets and the relevance of this question to the interpretation of position-momentum uncertainty relations is surveyed. In particular, it is argued that the Heisenberg interpretation of the uncertainty relations can consistently be carried through in a natural extension of the usual Hilbert space frame of the quantum theory.
To investigate how quantum effects might modify special relativity, we will study a Lorentz transformation between classical and quantum reference frames and express it in terms of the four-dimensional (4D) momentum of the quantum reference frame. The transition from the classical expression of the Lorentz transformation to a quantum-mechanical one requires us to symmetrize the expression and replace all its dynamical variables with the corresponding operators, from which we can obtain the same conclusion as that from quantum field theory (given by Weinberg's formula): owing to the Heisenberg's uncertainty relation, a particle (as a quantum reference frame) can propagate over a spacelike interval.
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.
The Heisenberg uncertainty principle is a consequence of the postulate that coordinate and momentum representations are related to each other by the Fourier transform. This postulate has been accepted from the beginning of quantum theory by analogy with classical electrodynamics. We argue that the postulate is based neither on strong theoretical arguments nor on experimental data. A position operator proposed in our recent publication resolves inconsis-tencies of standard approach and sheds a new light on important problems of quantum theory. We do not assume that the reader is an expert in the given field and the content of the paper can be understood by a wide audience of physicists.
Viewing frames of reference as physical systems, subject to the same laws as the systems they describe, is central to the relational approach in physics. Under the assumption that quantum mechanics universally governs all physical entities, this perspective naturally leads to the concept of quantum reference frames (QRFs). We investigate the perspective-dependence of position and momentum uncertainties, correlations, covariance matrices, and entanglement within the QRF formalism. We show that the Robertson-Schrödinger uncertainty relations are frame-dependent, and so are correlations and variances, which satisfy various constraints described as inequalities. However, the determinant of the total covariance matrix, linked to the uncertainty volume in phase space, as well as variance-based entanglement criteria, remains invariant under changes of reference frame. Under specific conditions, the purities of subsystems are also invariant for different QRFs, but in general, they are perspective-dependent. These invariants suggest fundamental, robust measures of uncertainty and entanglement that persist despite changes in observational perspective, potentially inspiring dedicated quantum information protocols as well as further foundational studies.
Abstract The subject of quantum reference frame transformations gets popular lately with some interesting new theoretical development partly for the reason that the physics involved is becoming experimentally accessible. The position of a position eigenstate when observed from an object with ‘uncertainty’ in position would be seen with ‘uncertainty’. In fact, even the existence of entanglement is reference frame-dependent. We present an improved formulation of such a transformation and give a novel way to describe exactly by ‘how much’ the ‘value of the position’ has changed which fully encodes all information about the changes, including the ‘uncertainty’ and entanglement. That is an application of the notion of noncommutative values of physical quantities we introduced to understand the reality of quantum physics and beyond. Some implications on fundamental physics will also be discussed. In particular, we suggest thinking about quantum gravity as a theory of general quantum relativity, alleviating Penrose’s notion of incompatibility of qauntum mechanics with the relativity principle.