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Journal of Physics A Mathematical and General

Spectrum condition, analyticity, Reeh-Schlieder and cluster properties in non-relativistic Galilei-invariant quantum theory

Manfred Requardt

Published Dec 1, 1982• 9 citations
Publisher

Abstract

The author shows that properties like spectrum condition, analyticity of n-point functions, space-like clustering of correlation functions stronger than any inverse power, the Reeh-Schlieder property do hold in Galilei invariant quantum theory. Furthermore, the range of validity of Haag's theorem is briefly discussed. The results seems to be of relevance both in the non-relativistic regime proper and as a hint that many of the properties typically attributed to relativistic field theories are actually a common feature of every theory with a zero mean-particle density and translation-invariant Hamiltonian.

DOI
10.1088/0305-4470/15/12/023

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quant-pharXiv:2604.26271•Apr 30, 2026
The author shows that properties like spectrum condition, analyticity of n-point functions, space-like clustering of correlation functions stronger than any inverse power, the Reeh-Schlieder property do hold in Galilei invariant quantum theory. Furthermore, the range of validity of Haag's theorem is briefly discussed. The results seems to be of relevance both in the non-relativistic regime proper and as a hint that many of the properties typically attributed to relativistic field theories are actually a common feature of every theory with a zero mean-particle density and translation-invariant Hamiltonian.
Many of the "counterintuitive" features of relativistic quantum field theory have their formal root in the Reeh-Schlieder theorem, which in particular entails that local operations applied to the vacuum state can produce any state of the entire field.It is of great interest then that I.E.Segal and, more recently, G. Fleming (in a paper entitled "Reeh-Schlieder meets Newton-Wigner") have proposed an alternative "Newton-Wigner" localization scheme that avoids the Reeh-Schlieder theorem.In this paper, I reconstruct the Newton-Wigner localization scheme and clarify the limited extent to which it avoids the counterintuitive consequences of the Reeh-Schlieder theorem.I also argue that there is no coherent interpretation of the Newton-Wigner localization scheme that renders it free from act-outcome correlations at spacelike separation.
Abstract Although it is usual to read that non-relativistic quantum mechanics is covariant, and even invariant, under the Galilei transformations, this topic has scarcely been addressed in the standard literature on the theory. This fact has, at least, two undesirable consequences. On the one hand, the meanings of the concepts of covariance and invariance are not precisely elucidated. On the other hand, one is left without a clear idea about in what sense the covariance and/or invariance of quantum mechanics can be asserted. This situation has its counterpart in the field of the interpretation of quantum mechanics: the relevance of the Galilei group is rarely discussed in the impressive amount of literature on the subject. The general premise underlying the present chapter is that the relationship between interpretation and Galilei transformations deserves to be seriously analyzed: the fact that the theory is covariant under the Galilei group does not guarantee the same property for the interpretation because, in general, interpretations add interpretative postulates to the formal structure of the theory. On this basis, and after clarifying the concepts of covariance and invariance, this chapter considers in what sense quantum mechanics is covariant and under what conditions it is invariant under the transformations of the Galilei group. This conceptual framework will provide the tools to study the result of the application of the Galilei transformations to the Modal-Hamiltonian interpretative postulates.
In this paper, I provide a formal set of assumptions and give a natural criterion for a quantum field theory to admit particles. I construct a naïve approach to localization for a free bosonic quantum field theory and show how this localization scheme, as a consequence of the Reeh-Schlieder theorem, fails to satisfy this criterion. I then examine the Newton-Wigner concept of localization and show that it fails to obey strong microcausality and thus is subject to a more general version of the Reeh-Schlieder theorem. I review approaches to quantum field theoretic explanations of particle detection events and explain how particles can be regarded as emergent phenomena of a relativistic field theory. In particular, I show that effective localization of Hilbert space vectors is equivalent to an approximate locality of observable algebras.
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​.