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Journal of Mathematical Physics

Local States

A. L. Licht

Published Sep 1, 1966• 20 citations
Publisher

Abstract

A heuristic discussion is given of the preparation of states in finite space-time regions. Some axioms concerning such states are derived from this discussion. It is shown that states requiring for their preparation a selection of events according to the outcome of a measurement will not be ``strictly local.'' Nonselective states will, however, be ``strictly local.'' The mathematical structure of such states is investigated.

DOI
10.1063/1.1705079

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A heuristic discussion is given of the preparation of states in finite space-time regions. Some axioms concerning such states are derived from this discussion. It is shown that states requiring for their preparation a selection of events according to the outcome of a measurement will not be strictly local.'' Nonselective states will, however, be strictly local.'' The mathematical structure of such states is investigated.
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The axiomatic formulation of an Algebraic Relativistic Quantum Field Theory leads naturally -via the Haag-Kastler axioms- to the fact that operators representing observables generate a von Neumann algebra localized in a space-time region in which said observations are made. I present and explain the axioms, as well as the mathematical preliminaries needed to understand them and apply them to the so-called causal diamond and Fermi's Two-Atom System. The latter shows the importance of formulating axioms concerning causality that try to make compatible the requirement from Special Relativity according to which no causal influence can travel faster than the speed of light and the quantum mechanical theory of measurement and preparability of states. For a relativistic field-theoretic setting, these concepts must be re-formulated, and this reasoning suggests that the operator algebra must be a type III von Neumann factor. I also introduce some philosophical problems concerning our usual notion of causality from Hume to contemporary theories of counterfactual causation and show how the algebraic structure of the theory is deeply related to the causal one and how this theory can be extended if we change the way we think about causality, measurements, and local preparability of states.
This paper presents the first component of the Unified Space-Time and Scale-Dependent Field Theory (USSFT), a framework in which quantum behavior arises from the statistical structure of an underlying U-field. We derive entanglement, the Most Probable Outcome (MPO) principle, and the Born Rule directly from multiplicity distributions rather than assuming them as axioms. Measurement is modeled through the Probabilistic Bias Functional, which selects outcomes according to local multiplicity density, eliminating the need for non-physical wavefunction collapse. A new entanglement analogy is introduced—an extended mathematical sequence whose endpoints shift instantaneously when any part of the expression is changed. This captures the nonlocal correlations observed in quantum systems while preserving the locality of the underlying U-field dynamics. Paper I establishes the mathematical and conceptual foundation for Papers II–V, forming the quantum mechanical pillar of USSFT.
We review what it means for a relativistic particle to be localized and argue that the most useful approach is through local measurement. This perspective ties localization directly to relativistic causality and measurable outcomes in spacetime. Strict localization in quantum field theory is formalized by states that give the same expectation value as vacuum for all local observables outside some spacetime region. With this formalism, single photons cannot be strictly localized, but coherent states can. The structure and interpretation of local observables and local operator algebras are discussed in connection with the Reeh-Schlieder theorem. Since probabilities of local measurement outcomes cannot vanish in vacuum, particle detection events must be phrased carefully. Key characteristics of strictly localized states are surveyed alongside concrete strategies for constructing them. The relationship between localization, causality, and on-demand sources is analyzed, finding that strictly localized states can represent localized, propagating information in quantum field theory. Explicit examples of localized photonic states are presented with an emphasis on quantum optics, including states that approximate single photons and (entangled) Bell states.