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arXiv:2604.26267

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

Leonardo A. Pachon

quant-phgr-qcmath-phSubmitted Apr 29, 2026• Announced Apr 30, 2026• 0 citations• Revised Sep 27, 2026 (v2)
v1 · Apr 29, 2026v2 · Sep 27, 2026
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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.

Comments
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

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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.
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×IB\times IB×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_11​ 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_11​.
The Galilean symmetry of a free particle in one-dimensional space is examined under the scope of non-Abelian quantum kinematics. Within the Hilbert space that carries the regular ray representation of the Galilei group the Schrödinger operator appears as one of the three fundamental invariants of the extended kinematic algebra. By means of a superselection rule the physical Hilbert subspaces of the system are identified, in which a complementary ray representation of the Galilean transformation produces the time-dependent Schrödinger equation, and the Feynman space-time propagator. The quantization approach used in this paper is purely group theoretic and relativistic.
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This paper is the second and final part of a two-part study. We construct positive-energy relativistic spatial localization observables in Minkowski spacetime within standard quantum field theory, using the stress--energy--momentum tensor smeared with suitable test functions. For each fixed timelike direction, the construction gives positive operator-valued measures (POVMs) on spacelike hypersurfaces, well defined on every nnn-particle sector and satisfying a relativistic causality condition excluding superluminal propagation of detection probabilities. The observables are built from local or quasi-local field-theoretic quantities, thus providing a rigorous version of earlier heuristic proposals. In the one-particle sector, the construction reduces to the observable previously introduced by the author, and its first moment gives the Newton--Wigner position operator under appropriate normalization and centering assumptions. Because the Reeh--Schlieder theorem prevents the normally ordered stress--energy--momentum tensor from being positive on the full Fock space, we use quantum energy inequalities to obtain lower bounds controlling deviations from positivity. This leads to regularized operator families, bounded from below, which approximate the localization effects. Finally, we define conditional localization observables for finite laboratories through modified local energy operators. By Haag duality, the corresponding conditional POVMs belong to local von Neumann algebras and commute for causally separated regions, in accordance with the Araki--Haag--Kastler framework. The results show how commutativity of localization observables is recovered for conditional measurements in finite spacetime regions.
A general theoretical framework is developed for the treatment of a class of quantum field theories that are explicitly exactly solvable, but require the use of C*-algebraic techniques because time-dependent scattering theory cannot be constructed in any one natural representation of the observable algebra. The purpose is to exhibit mechanisms by which inequivalent representations of the observable algebra can arise in quantum field theory, in a setting free of other complications commonly associated with the specification of dynamics. One of two major results is the development of necessary and sufficient conditions for the concurrent unitary implementation of two automorphism groups in a class of quasifree representations of the algebra of the canonical commutation relations (CCR). The automorphism groups considered are induced by one-parameter groups of symplectic transformations on the classical phase space over which the Weyl algebra of the CCR is built; each symplectic group is conjugate by a fixed symplectic transformation to a one-parameter unitary group. The second result, an analog to the Birman–Belopol’skii theorem in two-Hilbert-space scattering theory, gives sufficient conditions for the existence of Mo/ller wave morphisms in theories with time-development automorphism groups of the above type. In a paper which follows, this framework is used to analyze a particular model system for which wave operators fail to exist in any natural representation of the observable algebra, but for which wave morphisms and an associated S matrix are easily constructed.
As a way of studying the consequences of a possible breakdown of Lorentz invariance, the canonical commutation relations are enlarged to include time and energy as additional conjugate variables, and with this enlargement, they are shown to be invariant under the Lorentz transformation under the assumption that commutators of non-conjugate operators vanish. Although time does not commute with the energy operator, it does commute with the Hamiltonian. It is also shown that the invariance of the enlarged commutation relations holds for arbitrary non-singular linear transformations that describe a velocity boost to another uniformly moving frame, under the same assumptions about non-conjugate commutators. A covariant enlargement of the Poisson brackets, together with Dirac's interpretation, that would predict this invariance of the commutators is presented. There is a discussion of improper coordinate transformations, and a canonical time-reversal operator is introduced, and compared with the standard time-reversal operator T. It is concluded that even if the Lorentz transformation were to fail, and were to be replaced by another linear transformation, the enlarged canonical commutation relations could still hold.