Giannelli and Chiribella derived an observable-generator duality for energy from a collision model of informational nonequilibrium. We study a continuous-variable version aimed at the Galilean one-particle sector. A smooth family of reference states around an isotropic equilibrium supplies time, translation, rotation, and boost directions. The local observable-generator correspondence is obtained by differentiating a smooth extension of the single-state duality map, and the norm-one property of localization is obtained from a fiducial focusing assumption together with covariance. Combined with the standard smearing form of covariant localization observables, this yields sharp localization. With local inertial composition, the spin-cover action of rotations, and a central boost-translation holonomy, every irreducible sector is unitarily equivalent to the Hilbert space L2(R3) tensored with a (2s+1)-dimensional spin space. In that representation translations are generated by the canonical momentum, the holonomy is a scalar mass m > 0, boosts at t = 0 are generated by m times the position observable, the Hamiltonian is the free-particle kinetic term plus a constant E0, and the total angular momentum is orbital plus spin.
Giannelli and Chiribella derived an observable-generator duality for energy from a collision model of informational nonequilibrium. We study a continuous-variable version aimed at the Galilean one-particle sector. A smooth family of reference states around an isotropic equilibrium supplies time, translation, rotation, and boost directions. The local observable-generator correspondence is obtained by differentiating a smooth extension of the single-state duality map, and the norm-one property of localization is obtained from a fiducial focusing assumption together with covariance. Combined with the standard smearing form of covariant localization observables, this yields sharp localization. With local inertial composition, the spin-cover action of rotations, and a central boost-translation holonomy, every irreducible sector is unitarily equivalent to the Hilbert space L2(R3) tensored with a (2s+1)-dimensional spin space. In that representation translations are generated by the canonical momentum, the holonomy is a scalar mass m > 0, boosts at t = 0 are generated by m times the position observable, the Hamiltonian is the free-particle kinetic term plus a constant E0, and the total angular momentum is orbital plus spin.
The definition of the total momentum, position, and spin for Galilean- or Lorentz-invariant two-free-particle systems is discussed by using the requirement that the generators of the respective invariance groups should have the same form expressed in terms of them as for a single particle. Internal c.m. dynamical variables are introduced by applying the singular transformation due to Gartenhaus and Schwartz on the basic single-particle dynamical variables, the transformation mapping the whole Hilbert space onto the c.m. subspace spanned by states of zero total momentum. The form of the internal c.m. position operator is given for particles with spin, for what appears to be the first time. Using these dynamical variables, it is shown how an interaction can be introduced while maintaining Galilean or Lorentz invariance and satisfying the asymptotic condition of freely propagating particles for large separations.
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.
This note isolates the free fixed-background sector of the capacity-geometric Schrodinger action and proves that it already carries the full standard nonrelativistic free-particle package. In density-phase variables the action is Galilei covariant with the exact boost law for rho and S; the constant-density plane-wave sector reproduces the standard dispersion E = |p|^2/(2m); and the free evolution operator has the usual oscillatory kernel together with the correct composition law. The stationary-phase analysis of that kernel recovers the classical straight-line ray with momentum p* = m(x-x0)/t. The novelty is not free-particle quantum mechanics itself. The genuine contribution is a hard consistency check: once the SGOC geometric inputs are fixed, the free sector lands on the standard boost law, standard dispersion, and standard propagator without further tuning, extra correction terms, or a hidden preferred frame. For a derivation programme, that is a real rigidity test rather than a decorative benchmark. The claim boundary is narrow and explicit. This is a fixed-background, force-free note. It does not derive new physics, does not add interactions, and does not try to reinterpret the free kernel beyond showing that the SGOC action reproduces the ordinary nonrelativistic structure exactly where it should. Within SGOC, this is the baseline sanity-check paper for the quantum lane. It confirms that the fixed-background forced-action theorem is not only formally equivalent to Schrodinger evolution on paper, but also reproduces the operational free-sector structures that any credible downstream extension must preserve. For mathematical physicists, this matters because derivation programmes often fail at the simplest covariance or kernel tests. Here the basic free package survives intact. That makes the later electromagnetic, spin, and relativistic extensions scientifically worth reading rather than suspect from the outset. Keywords: Schrodinger equation, Galilean covariance, free propagator, plane waves, capacity geometry, SGOC, free sector. DOI this paper: 10.5281/zenodo.19593108. Related SGOC papers: - The Schrodinger Equation from Capacity Geometry: General Forced-Action Theorem on a Fixed Capacity Background - Local U(1) Gauge Covariance, Minimal Electromagnetic Coupling, and Gauge-Covariant Spectral Recovery on a Capacity Background - A Pauli-Type Spinor Completion of the Capacity-Geometric Schrodinger Action Programme summary files - before and beyond Volume Three SGOC Programme Compendium - Pedagogical 10.5281/zenodo.19614652 SGOC Programme Compendium - Technical 10.5281/zenodo.19614677
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.
We ask in which states, and for which local algebras, Galilean quantum fields have the Reeh-Schlieder property. The nets are generated by time-zero canonical fields on spatial regions and by the dynamics. The Bargmann mass over the charge of the field is a number operator for the time-zero fields, bounded from below by positivity of the energy and boost covariance, and Chaiken's theorem gives Galilean Fock rigidity: these fields form a direct sum of Fock representations, and the time-zero annihilation fields annihilate every translation-invariant vector. If the vacuum is the only vector invariant under spatial translations (not implied by the usual uniqueness axiom), the vacuum sector of the observable net is one-dimensional. No vector of finite mass is cyclic or separating for an algebra of a spatial region at fixed time. In the vacuum of Bose gases with bounded or Coulomb-type pair potentials (with Buchholz's renormalised dynamics if not H-stable), the fields of any region over any time interval generate all bounded operators. By contrast, a KMS state is cyclic and separating for the algebra of every region B×I with temporal extent whenever the spatial algebras at all times generate the global algebra, as for free gases. For the free Schrödinger field in the vacuum, the thermal states of Bose and Fermi gases and the Fermi sea, fixed-time field algebras are type I factors, for which thermal vectors and the Fermi sea are separating but not cyclic, and every field algebra with temporal extent is the global one: type I in pure states, type III1 in thermal states, with the grand-canonical dynamics as modular group. Galilean vacua are thus never separating for local algebras. For the free field the type of the algebras of regions with temporal extent depends on the state, and what is specifically relativistic is the locality of type III1.
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.