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Chemical Physics

Relativistic causality and quantum-mechanical states in the Fermi problem

G. Compagno, G. Massimo Palma, Roberto Passante, Franco Persico

Published Sep 1, 1995• 18 citations
Publisher
DOI
10.1016/0301-0104(95)00035-m

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The localization problem in relativistic quantum theory has persisted for more than seven decades, yet it is largely unknown and continues to perplex even those well-versed in the subject. At the heart of this problem lies a fundamental conflict between localizability and relativistic causality, which can also be construed as part of the broader dichotomy between measurement and unitary dynamics. This article provides a historical review of the localization problem in one-particle relativistic quantum mechanics, clarifying some persistent misconceptions in the literature, and underscoring the antinomy between causal dynamics and localized observables.
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.
We study the instantaneity of the state-reduction process in relativistic quantum mechanics. The conclusion of various authors that this instantaneity will restrict the set of relativistic observables to purely local ones (i.e., that the measurement of any nonlocal property of a system at a well-defined time would give rise to violations of relativistic causality) is found to be erroneous, and experiments (of a kind not encountered before in measurement theory) are described whereby certain nonlocal properties of some simple physical systems can be measured at a well-defined time without violating causality. The attempts of certain authors to reconcile the reduction process with the covariance of the relativistic quantum state are considered and found wanting, and it is argued that the covariance of relativistic quantum theories resides exclusively in the experimental probabilities, and not in the underlying quantum states. The problem of nonlocal measurement is considered in general: distinctions (which are not to be met with in the nonrelativistic case) arise in relativistic quantum mechanics between what can be measured for fermions and what can be measured for bosons, between what can be measured for individual systems and what can be measured for ensembles, and between what kinds of states can be verified by measurement and what kinds of states can be prepared by measurement; and these pose difficult questions about the nature of measurement itself.