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This paper establishes a rigorous mathematical constraint within non-relativistic quantum mechanics, proving that for Galilean-invariant systems with positive energy, local field algebras at unequal times commute only on a null set of time separations. The result relies fundamentally on the positivity of energy and Bargmann covariance, demonstrating that dropping these assumptions allows commutation relations to hold universally beyond specific bounds. This work advances foundational quantum kinematics by clarifying the precise conditions under which canonical commutation relations persist in non-relativistic spacetimes.
Algebraic quantum kinematics: Galilean covariance with positive energy confines unequal-time commutation to null sets
arXiv:2604.26267v2 Announce Type: replace Abstract: 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 Schr\"odinger 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.
Abstract: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.
27 pages, 3 figures, 1 table. v2: completely rewritten, new title (v1: "Algebraic quantum kinematics and SR-selection"). The SR-selection conjecture of v1 is replaced by a theorem excluding Galilean unequal-time locality and by results on commutation sets; the photon-sector and modular material of v1 is not included
arXiv:2604.26267 [quant-ph]
(or arXiv:2604.26267v2 [quant-ph] for this version)
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