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

Galilean Covariance Almost Fixes the Quantum Probability Current

Nidhal Mghirbi

Published Oct 30, 2026• 0 citations
Publisher

Abstract

The continuity equation fixes only the divergence of the quantum probability current, leaving the flow of probability underdetermined. We show that Galilean boost covariance, imposed pointwise at fixed time, nearly resolves this ambiguity. For any one-body current, local or nonlocal, pure, mixed, or many-body, the boost law holds exactly when the current equals the Schrödinger current plus a linear functional of the density. Equivalently, the current is covariant under local phase changes, or its density and momentum density satisfy the canonical commutator of the nonrelativistic current algebra. Among translation-invariant currents, rotational covariance together with conservation removes the remaining density term in three or more dimensions, while conservation together with the momentum–translation relation removes it in every dimension. The same framework classifies known alternatives: Nelson and Doebner–Goldin currents obey the algebra but are not conserved; chiral currents are conserved but violate the algebra; convolution kernels violate the boost law; and a conserved boost-covariant current exists exactly when the \(f\)-sum rule holds.

DOI
10.5281/zenodo.23815694

Related papers

Covariance of the Schroedinger Equation under Low Velocity Boosts

van Oosten

It is well-known that Schrodinger wave functions are not covariant under Galilean boosts. To obtain the correct result the boost transformation, t′ = t and ~x′ = ~x − ~vt, must be followed by the phase shift δφ = 12mv 2t + m~v · ~r. A generally accepted approach is to absorb the phase shift into the Galilean boost, construct the Schrodinger group and claim Galilean invariance of the Schrodinger wave function. Here I address the physical meaning of the phase shift. It is not a coordinate transformation since it depends on the mass of the Schrodinger particle. Consequently, one needs as many Schrodinger groups as there are distinct masses. The phase shift does not follow from Lorentz boost per se in the low velocity limit. Covariance of the non-relativistic quantum mechanical kinetic energy and momentum under pure coordinate transformations can be satisfied only by the boost t′ = t ( 1 + 12 v2 c2 + ~v·~r c2 )

Apeiron•Oct 1, 2006•3 citations

Galilean boost invariance does not survive the trace: symmetry breaking in open quantum systems

Leonardo F. Calderón, Esteban Marulanda, Santiago Morales +1 more

Tracing out a Galilean-invariant Caldeira-Leggett environment breaks Galilean boost covariance of the reduced dynamics, while spatial translations and rotations survive intact. An operator-level analysis of the exact Hu-Paz-Zhang master equation localizes the violation entirely in the dissipative anticommutator term, scaling with the damping coefficient $Γ(t)f(t)$. The fluctuation-dissipation theorem ties this coefficient to the absorptive bath response that drives equilibrium momentum diffusion, so for any non-trivial bath spectral density bilinear-coupled Galilean invariance, the fluctuation-dissipation theorem, and reduced boost covariance cannot hold simultaneously. The stochastic decomposition of the influence functional extends the mechanism beyond the quadratic regime. The dimensionless ratio $\hbarγ/k_\mathrm{B} T$ delineates the crossover: cold atoms in dissipative optical lattices and ultracold molecules sit at its edge. Parametric driving offers a one-directional escape: the squeezing rate that protects nonequilibrium entanglement above the standard quantum limit also suppresses boost-breaking over a driving cycle.

quant-pharXiv:2604.27459•May 1, 2026•0 citations

The Hamiltonian and generating functional for a nonrelativistic local current algebra

Ralph Menikoff

The nonrelativistic current algebra with conserved current consisting of ρ(x), the particle number density, and J(x), the flux density of particles, is studied. The Hamiltonian for any time reversal invariant system of spinless particles, interacting via a two-body interaction potential, is expressed as a Hermitian form in the currents. This leads to a functional equation for the generating functional, which is the ground state expectation value of exp[i ∫d xρ(x)f(x)]. In the N / V limit an expression for the generating functional in terms of correlation functions is given. Representations of the exponentiated current algebra which are translation invariant satisfy the cluster decomposition property, and those which have different Hamiltonians are shown to be unitarily inequivalent.

Journal of Mathematical Physics•Jul 1, 1974•28 citations

Algebraic quantum kinematics: Galilean covariance with positive energy confines unequal-time commutation to null sets

Leonardo A. Pachon

We ask which kinematical groups allow local algebras of canonical quantum fields to commute at unequal times. The Galilei group acts transitively on pairs of non-simultaneous events with a given time separation, so a Galilean-invariant commutation relation between sharp-time algebras at unequal times also holds at coincident positions; at arbitrarily short separations it contradicts the canonical commutation relations, with no spectral assumption. Under unitary Bargmann covariance and positivity of the energy modulo the central mass, the vacuum is annihilated by the annihilation fields, hence not separating for local algebras, and the vacuum two-point function of a canonical field factorises into the free Schrödinger propagator and the characteristic function of an internal-energy distribution. Sharp-time algebras at unequal times then commute only at a closed null set of time separations, bounded below by the Mandelstam-Tamm and Margolus-Levitin times and empty for sharp or generic internal energy; any finite set of separations occurs. Algebras of open spacetime regions containing time-smeared fields never commute. Positivity cannot be dropped: without it, commutation can hold at all separations beyond any given one. Among spatially isotropic homogeneous spacetimes with absolute time, only the galilean ones are transitive at fixed time separation, up to discrete exceptions in the two oscillating cases, realised by a harmonic trap in the Newton-Hooke one. Non-commutation transfers from the vacuum to every representation of a simple C*-algebra on which the dynamics acts by automorphisms, including free Bose and Fermi gases at all temperatures and the Galilei-covariant cutoff dynamics of interacting fermions of Narnhofer and Thirring; without such a vacuum representation it can fail even in thermal equilibrium.

quant-pharXiv:2604.26267•Apr 30, 2026•0 citations

Modified quantum current density under minimal uncertainty in momentum and nonlocal potential within noncommutative Framework

Ilyas Haouam

In this paper, we investigate the Schrödinger and continuity equations in the presence of both local and nonlocal potentials, the latter arising from electron–electron interactions, within the framework of minimal momentum uncertainty in both commutative and noncommutative phase–spaces. In particular, a Frahn–Lemmer-type nonlocal potential is employed. The combined effects of phase–space noncommutativity and potential nonlocality on the quantum density and current density are thoroughly analyzed. It is found that the conventional definition of current density fails to satisfy the current conservation condition under these generalized settings. Therefore, a modified definition of current density that properly accounts for these contributions is proposed. We subsequently demonstrate that the current obtained from this new formulation satisfies the continuity equation. Additionally, numerical results are presented to extensively explore the behavior of the system under various conditions. Phase–space noncommutativity is introduced via both the Moyal [Formula: see text]product and the linear Bopp-shift transformation.

Modern Physics Letters A•Feb 11, 2026•0 citations

The Converse Madelung Question

Jonathan R Dunkley

We pose the converse Madelung question: not whether Fisher information can reproduce quantum mechanics, but whether it is necessary. We work with minimal, physically motivated axioms on density and phase: locality, probability conservation, Euclidean invariance with a global phase symmetry, reversibility, and convex regularity. Within the resulting class of first order local Hamiltonian field theories, these axioms single out the canonical Poisson bracket on density and phase under the Dubrovin and Novikov assumptions for local hydrodynamic brackets. Using a pointwise, gauge covariant complex change of variables that maps density and phase to a single complex field, we show that the only convex, rotationally invariant, first derivative local functional of the density whose Euler Lagrange term yields a reversible completion that is exactly projectively linear is the Fisher functional. When its coefficient equals Planck constant squared divided by twice the mass, the dynamics reduce to the linear Schrodinger equation. For many body systems, a single local complex structure across sectors enforces the same relation species by species, fixing a single Planck constant. Galilean covariance appears through the Bargmann central extension, with the usual superselection consequences. Comparison with the Doebner and Goldin family identifies the reversible zero diffusion corner with linear Schrodinger dynamics. We provide operational falsifiers via residual diagnostics for the continuity and Hamilton Jacobi equations and report numerical minima at the Fisher scale that are invariant under Galilean boosts. In this setting, quantum mechanics emerges as a reversible fixed point of Fisher regularised information hydrodynamics. A code archive enables direct numerical checks, including a superposition stress test that preserves exact projective linearity within our axioms.

quant-pharXiv:2511.03552•Nov 6, 2025•0 citations
The continuity equation fixes only the divergence of the quantum probability current, leaving the flow of probability underdetermined. We show that Galilean boost covariance, imposed pointwise at fixed time, nearly resolves this ambiguity. For any one-body current, local or nonlocal, pure, mixed, or many-body, the boost law holds exactly when the current equals the Schrödinger current plus a linear functional of the density. Equivalently, the current is covariant under local phase changes, or its density and momentum density satisfy the canonical commutator of the nonrelativistic current algebra. Among translation-invariant currents, rotational covariance together with conservation removes the remaining density term in three or more dimensions, while conservation together with the momentum–translation relation removes it in every dimension. The same framework classifies known alternatives: Nelson and Doebner–Goldin currents obey the algebra but are not conserved; chiral currents are conserved but violate the algebra; convolution kernels violate the boost law; and a conserved boost-covariant current exists exactly when the (f)-sum rule holds.
It is well-known that Schrodinger wave functions are not covariant under Galilean boosts. To obtain the correct result the boost transformation, t′ = t and ~x′ = ~x − ~vt, must be followed by the phase shift δφ = 12mv 2t + m~v · ~r. A generally accepted approach is to absorb the phase shift into the Galilean boost, construct the Schrodinger group and claim Galilean invariance of the Schrodinger wave function. Here I address the physical meaning of the phase shift. It is not a coordinate transformation since it depends on the mass of the Schrodinger particle. Consequently, one needs as many Schrodinger groups as there are distinct masses. The phase shift does not follow from Lorentz boost per se in the low velocity limit. Covariance of the non-relativistic quantum mechanical kinetic energy and momentum under pure coordinate transformations can be satisfied only by the boost t′ = t ( 1 + 12 v2 c2 + ~v·~r c2 )
Tracing out a Galilean-invariant Caldeira-Leggett environment breaks Galilean boost covariance of the reduced dynamics, while spatial translations and rotations survive intact. An operator-level analysis of the exact Hu-Paz-Zhang master equation localizes the violation entirely in the dissipative anticommutator term, scaling with the damping coefficient Γ(t)f(t)Γ(t)f(t)Γ(t)f(t). The fluctuation-dissipation theorem ties this coefficient to the absorptive bath response that drives equilibrium momentum diffusion, so for any non-trivial bath spectral density bilinear-coupled Galilean invariance, the fluctuation-dissipation theorem, and reduced boost covariance cannot hold simultaneously. The stochastic decomposition of the influence functional extends the mechanism beyond the quadratic regime. The dimensionless ratio ℏγ/kBT\hbarγ/k_\mathrm{B} Tℏγ/kB​T delineates the crossover: cold atoms in dissipative optical lattices and ultracold molecules sit at its edge. Parametric driving offers a one-directional escape: the squeezing rate that protects nonequilibrium entanglement above the standard quantum limit also suppresses boost-breaking over a driving cycle.
The nonrelativistic current algebra with conserved current consisting of ρ(x), the particle number density, and J(x), the flux density of particles, is studied. The Hamiltonian for any time reversal invariant system of spinless particles, interacting via a two-body interaction potential, is expressed as a Hermitian form in the currents. This leads to a functional equation for the generating functional, which is the ground state expectation value of exp[i ∫d xρ(x)f(x)]. In the N / V limit an expression for the generating functional in terms of correlation functions is given. Representations of the exponentiated current algebra which are translation invariant satisfy the cluster decomposition property, and those which have different Hamiltonians are shown to be unitarily inequivalent.
We ask which kinematical groups allow local algebras of canonical quantum fields to commute at unequal times. The Galilei group acts transitively on pairs of non-simultaneous events with a given time separation, so a Galilean-invariant commutation relation between sharp-time algebras at unequal times also holds at coincident positions; at arbitrarily short separations it contradicts the canonical commutation relations, with no spectral assumption. Under unitary Bargmann covariance and positivity of the energy modulo the central mass, the vacuum is annihilated by the annihilation fields, hence not separating for local algebras, and the vacuum two-point function of a canonical field factorises into the free Schrödinger propagator and the characteristic function of an internal-energy distribution. Sharp-time algebras at unequal times then commute only at a closed null set of time separations, bounded below by the Mandelstam-Tamm and Margolus-Levitin times and empty for sharp or generic internal energy; any finite set of separations occurs. Algebras of open spacetime regions containing time-smeared fields never commute. Positivity cannot be dropped: without it, commutation can hold at all separations beyond any given one. Among spatially isotropic homogeneous spacetimes with absolute time, only the galilean ones are transitive at fixed time separation, up to discrete exceptions in the two oscillating cases, realised by a harmonic trap in the Newton-Hooke one. Non-commutation transfers from the vacuum to every representation of a simple C*-algebra on which the dynamics acts by automorphisms, including free Bose and Fermi gases at all temperatures and the Galilei-covariant cutoff dynamics of interacting fermions of Narnhofer and Thirring; without such a vacuum representation it can fail even in thermal equilibrium.
In this paper, we investigate the Schrödinger and continuity equations in the presence of both local and nonlocal potentials, the latter arising from electron–electron interactions, within the framework of minimal momentum uncertainty in both commutative and noncommutative phase–spaces. In particular, a Frahn–Lemmer-type nonlocal potential is employed. The combined effects of phase–space noncommutativity and potential nonlocality on the quantum density and current density are thoroughly analyzed. It is found that the conventional definition of current density fails to satisfy the current conservation condition under these generalized settings. Therefore, a modified definition of current density that properly accounts for these contributions is proposed. We subsequently demonstrate that the current obtained from this new formulation satisfies the continuity equation. Additionally, numerical results are presented to extensively explore the behavior of the system under various conditions. Phase–space noncommutativity is introduced via both the Moyal [Formula: see text]product and the linear Bopp-shift transformation.
We pose the converse Madelung question: not whether Fisher information can reproduce quantum mechanics, but whether it is necessary. We work with minimal, physically motivated axioms on density and phase: locality, probability conservation, Euclidean invariance with a global phase symmetry, reversibility, and convex regularity. Within the resulting class of first order local Hamiltonian field theories, these axioms single out the canonical Poisson bracket on density and phase under the Dubrovin and Novikov assumptions for local hydrodynamic brackets. Using a pointwise, gauge covariant complex change of variables that maps density and phase to a single complex field, we show that the only convex, rotationally invariant, first derivative local functional of the density whose Euler Lagrange term yields a reversible completion that is exactly projectively linear is the Fisher functional. When its coefficient equals Planck constant squared divided by twice the mass, the dynamics reduce to the linear Schrodinger equation. For many body systems, a single local complex structure across sectors enforces the same relation species by species, fixing a single Planck constant. Galilean covariance appears through the Bargmann central extension, with the usual superselection consequences. Comparison with the Doebner and Goldin family identifies the reversible zero diffusion corner with linear Schrodinger dynamics. We provide operational falsifiers via residual diagnostics for the continuity and Hamilton Jacobi equations and report numerical minima at the Fisher scale that are invariant under Galilean boosts. In this setting, quantum mechanics emerges as a reversible fixed point of Fisher regularised information hydrodynamics. A code archive enables direct numerical checks, including a superposition stress test that preserves exact projective linearity within our axioms.