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arXiv:1405.3254

Causality and chance in relativistic quantum field theories

Richard Healey

Published May 13, 2014• 0 citations
arXivHTMLPDF

Abstract

Bell appealed to the theory of relativity in formulating his principle of local causality. But he maintained that quantum field theories do not conform to that principle, even when their field equations are relativistically covariant and their observable algebras satisfy a relativistically motivated microcausality condition. A pragmatist view of quantum theory and an interventionist approach to causation prompt the reevaluation of local causality and microcausality. Local causality cannot be understood as a reasonable requirement on relativistic quantum field theories: it is unmotivated even if applicable to them. But microcausality emerges as a sufficient condition for the consistent application of a relativistic quantum field theory.

DOI
10.48550/arxiv.1405.3254

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Bell appealed to the theory of relativity in formulating his principle of local causality. But he maintained that quantum field theories do not conform to that principle, even when their field equations are relativistically covariant and their observable algebras satisfy a relativistically motivated microcausality condition. A pragmatist view of quantum theory and an interventionist approach to causation prompt the reevaluation of local causality and microcausality. Local causality cannot be understood as a reasonable requirement on relativistic quantum field theories: it is unmotivated even if applicable to them. But microcausality emerges as a sufficient condition for the consistent application of a relativistic quantum field theory.
By moving to the context of relativistic space-time structure, this chapter completes the argument of Chapter 4 that we can use quantum theory locally to explain correlations that violate Bell inequalities with no instantaneous action at a distance. Chance here must be relativized not just to time but to a space-time point, so that an event may have more than one chance at the same time—it may even be certain relative to one space-time point but ‘at the same time’ completely uncertain relative to another. This renders Bell’s principle of Local Causality either inapplicable or intuitively unmotivated. Counterfactual dependence between the outcomes of measurements on systems assigned an entangled state is not causal since neither outcome is subject to intervention: but it may still be appealed to in a non-causal explanation of one in terms of the other.
This paper surveys the issue of relativistic causality within the framework of algebraic quantum field theory (AQFT). In doing so, we distinguish various notions of causality formulated in the literature and study their relationships, and thereby we offer what we hope to be a useful taxonomy. We propose that the most direct expression of relativistic causality in AQFT is captured not (as has sometimes been claimed) by the spectrum condition but rather by the axiom of local primitive causality, in that it entails a form of local determinism for quantum fields which generalizes the constraint of no superluminal propagation of classical field theories to relativistic quantum field theory. We discuss the status of the axiom of micro-causality by locating its place within a large family of separability/independence/locality conditions developed for AQFT and also by relating it to so-called no-signalling theorems. And we also provide a critical survey of attempts to understand the implications for relativistic causality of the distant correlations endemic to the states in models of AQFT satisfying the standard axioms, and we provide an assessment of attempts to employ Reichenbach's common cause principle in AQFT to defuse worries that these distant correlations implicate direct causal connections between relatively spacelike events.
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.
J.S. Bell's work has convinced many that correlations in violation of CHSH inequalities show that the world itself is non-local, and that there is an apparently essential conflict between any sharp formulation of quantum theory and relativity. Against this consensus, this paper argues that there is no conflict between quantum theory and relativity. Quantum theory itself helps us explain such (otherwise) puzzling correlations in a way that contradicts neither Bell's intuitive locality principle nor his local causality condition. The argument depends on understanding quantum theory along pragmatist lines, and on a more general view of how that theory helps us explain. Quantum theory is compatible with Bell's intuitive locality principle and with his local causality condition not because it conforms to them, but because they are simply inapplicable to quantum theory, as so understood.
Axiomatic quantum field theory is one approach to the project of merging the special theory of relativity with that of ordinary quantum mechanics. The project begins with the postulation of a set of axioms. Axioms should be motivated by reasonable physical principles in a way that illustrates how a given axiom is true. Motivations are often grounded in the principles of the parent theories: ordinary quantum mechanics or the theory of special relativity. Amongst the set of axioms first proposed by Haag and Kastler in 1963 is the axiom of microcausality. Microcausality requires the observables of regions at space-like separation to commute. This thesis seeks to answer the question ‘What principles from the special theory of relativity or ordinary quantum mechanics motivate, or justify, accepting microcausality as an axiom?’ The first chapter will provide the necessary background to investigate this question and the second chapter will undertake that investigation. In conclusion, microcausality cannot be well-motivated by individual principles rooted in the special theory of relativity or ordinary quantum mechanics.
Locally covariant algebraic quantum field theory (LCQFT) satisfies Einstein causality through microcausality and operational no-signalling, yet Bell-type correlations persist in entangled field states across spacelike regions. We demonstrate that this apparent tension reflects a fundamental covariance obstruction: no assignment of classical past variables can simultaneously be covariant under spacetime embeddings, screen off quantum correlations, and reproduce AQFT statistics. This obstruction is distinct from dynamical nonlocality or signalling violations. We formalize this as a no-go theorem in the category-theoretic framework of LCQFT, showing that Bell's notion of local causality -- requiring factorization conditioned on a common past -- is structurally incompatible with diffeomorphism covariance. The failure of Bell locality thus reflects not a breakdown of relativistic causality but the impossibility of embedding quantum correlations into a classical causal framework without introducing preferred foliations or non-covariant beables. This clarifies the conceptual status of nonlocality in relativistic quantum theory.