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This article investigates the Reeh-Schlieder property within Galilean quantum field theory, analyzing how local algebras behave in vacuum versus thermal states for Bose and Fermi gases. The study determines that while thermal states exhibit specific cyclic and separating properties leading to type III₁ factor algebras, Galilean vacua fail to be separating for local spatial regions, highlighting fundamental differences from relativistic locality axioms.
Galilean Reeh-Schlieder obstruction in the vacuum and the thermal Reeh-Schlieder property
arXiv:2604.26271v2 Announce Type: replace Abstract: 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 $B\times I$ with temporal extent whenever the spatial algebras at all times generate the global algebra, as for free gases. For the free Schr\"odinger 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 III$_1$ 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 III$_1$.
Abstract: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 $B\times I$ 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 III$_1$ 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 III$_1$.
17 pages, 2 figures, 1 table. v2: rewritten, new title. The locality axiom (G2) of v1 required commutation at unequal times, which even the free field violates; v2 uses nets generated by time-zero fields. The non-separating vacuum is kept. Withdrawn: the claims of v1 that Galilean theories cannot have the Reeh-Schlieder property or modular flow, and its Corollary 3 on five published models
arXiv:2604.26271 [quant-ph]
(or arXiv:2604.26271v2 [quant-ph] for this version)
Full text under the publisher's open licence · 375 words
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