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Philosophy of Science

Is Algebraic Lorentz-Covariant Quantum Field Theory Stochastic Einstein Local?

Frederik Müller, Jeremy Butterfield

Published Sep 1, 1994• 7 citations
Publisher

Abstract

The general context of this paper is the locality problem in quantum theory. In a recent issue of this journal, Rédei (1991) offered a proof of the proposition that algebraic Lorentz-covariant quantum field theory is past stochastic Einstein local. We show that Rédei's proof is either spurious or circular, and that it contains two deductive fallacies. Furthermore, we prove that the mentioned theory meets the stronger condition of stochastic Haag locality.

DOI
10.1086/289814

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Butterfield's (1992a,b,c) claim of the equivalence of absence of Lewisian probabilistic counterfactual causality (LC) to Hellman's stochastic Einstein locality (SEL) is questioned. Butterfield's assumption on which the proof of his claim is based would suffice to prove that SEL implies absence of LC also for appropriately given versions of these notions in algebraic quantum field theory, but the assumption is not an admissible one. The conclusion must be that the relation of SEL and absence of LC is open, and that they may be independent.

Philosophy of Science•Dec 1, 1993•5 citations
The general context of this paper is the locality problem in quantum theory. In a recent issue of this journal, Rédei (1991) offered a proof of the proposition that algebraic Lorentz-covariant quantum field theory is past stochastic Einstein local. We show that Rédei's proof is either spurious or circular, and that it contains two deductive fallacies. Furthermore, we prove that the mentioned theory meets the stronger condition of stochastic Haag locality.
Based partly on proving that algebraic relativistic quantum field theory (ARQFT) is a stochastic Einstein local (SEL) theory in the sense of SEL which was introduced by Hellman (1982b) and which is adapted in this paper to ARQFT, the recently proved maximal and typical violation of Bell's inequalities in ARQFT (Summers and Werner 1987a–c) is interpreted in this paper as showing that Bell's inequalities are, in a sense, irrelevant for the problem of Einstein local stochastic hidden variables, especially if this problem is raised in connection with ARQFT. This leads to the question of how to formulate the problem of local hidden variables in ARQFT. By giving a precise definition of hidden-variable theory within the operator algebraic framework of quantum mechanics, it will be argued that the aim of hidden-variable investigations is to determine those classes of quantum theories whose elements represent a statistical content that cannot be reduced in a given way. In some particular way to be stated, a proposition will be stated which distinguishes quantum field theories whose statistical content cannot be reduced without violating some relativistic locality principle.
There has been some recent interest in applying the techniques of Algebraic Quantum Field Theory (AQFT) to entanglement problems in perturbative QFT. In particular, the Hilbert space independence of this formulation makes it particularly interesting in the context of curved spacetimes and the emphasis on the algebra of observables makes the treatment of Bell inequalities in QFT resemble such treatment in non-relativistic Quantum Mechanics. In this work, we present the mathematical structures needed for formulating AQFT in terms of the Haag-Araki-Kastler (HAK) axioms and discuss their implications. Moreover, we discuss the algebraic approach to quantum entanglement in the form of Bell inequalities. We provide an extension of this formulation to general globally hyperbolic spacetimes using the so-called Locally Covariant approach to QFT, which extends the HAK axioms to general spacetimes by means of the Category Theory language.
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.
I discuss various formulations of stochastic Einstein locality (SEL), which is a version of the idea of relativistic causality, that is, the idea that influences propagate at most as fast as light. SEL is similar to Reichenbach's Principle of the Common Cause (PCC), and Bell's Local Causality.My main aim is to discuss formulations of SEL for a fixed background spacetime. I previously argued that SEL is violated by the outcome dependence shown by Bell correlations, both in quantum mechanics and in quantum field theory. Here I reassess those verdicts in the light of some recent literature which argues that outcome dependence does not violate the PCC. I argue that the verdicts about SEL still stand.Finally, I briefly discuss how to formulate relativistic causality if there is no fixed background spacetime. 1. Introduction2. Formulating Stochastic Einstein Locality2.1. Events and regions2.2. The idea of SEL2.3. Three formulations of SEL2.3.1. The formulations2.3.2. Comparisons2.4. Implications between the formulations2.4.1. Conditions for the equivalence of SELD1 and SELD22.4.2. Conditions for the equivalence of SELS and SELD23. Relativistic Causality in the Bell Experiment3.1. The background3.1.1. The Bell experiment reviewed3.1.2. My previous position3.2. A common common cause? The Budapest school3.2.1. Resuscitating the PCC3.2.2. Known proofs of a Bell inequality need a strong PCC3.2.3. Two distinctions3.2.4. Szabó's model3.2.5. A common common cause is plausible3.2.6. Bell inequalities from a weak PCC: the Bern school3.3. SEL in the Bell experiment3.3.1. PCC and SEL are connected by PPSI3.3.2. The need for other judgments3.3.3. Weak vs. strong SELD4. SEL in Algebraic Quantum Field Theory4.1. The story so far4.2. Questions4.2.1. Our formulations4.2.2. The Budapest and Bern schools5. SEL in Dynamical Spacetimes5.1. SEL for metric structure?5.2. SEL for causal sets?5.2.1. The causal set approach5.2.2. Labelled causal sets; general covariance5.2.3. Deducing the dynamics5.2.4. The fate of SEL
Main Claim I argue in this chapter that Einstein and von Neumann meet in algebraic relativistic quantum field theory in the following metaphorical sense: algebraic quantum field theory was created in the late 1950s/early 1960s and was based on the theory of “rings of operators,” which von Neumann established in 1935–1940 (partly in collaboration with J. Murray). In the years 1936–1949, Einstein criticized standard, nonrelativistic quantum mechanics, arguing that it does not satisfy certain criteria that he regarded as necessary for any theory to be compatible with a field theoretical paradigm. I claim that algebraic quantum field theory (AQFT) does satisfy those criteria and hence that AQFT can be viewed as a theory in which the mathematical machinery created by von Neumann made it possible to express in a mathematically explicit manner the physical intuition about field theory formulated by Einstein. The argument in favor of this claim has two components: 1. Historical : An interpretation of Einstein's (semi)formal wordings of his critique of nonrelativistic quantum mechanics. This interpretation results in mathematically explicit operational independence definitions, which, I claim, express independence properties of systems that are localized in causally disjoint spacetime regions. Einstein regarded these as necessary for a theory to comply with field theoretical principles. 2. Systematic : The presentation of several propositions formulated in terms of AQFT that state that the operational independence conditions in question do in fact typically hold in AQFT.
Butterfield's (1992a,b,c) claim of the equivalence of absence of Lewisian probabilistic counterfactual causality (LC) to Hellman's stochastic Einstein locality (SEL) is questioned. Butterfield's assumption on which the proof of his claim is based would suffice to prove that SEL implies absence of LC also for appropriately given versions of these notions in algebraic quantum field theory, but the assumption is not an admissible one. The conclusion must be that the relation of SEL and absence of LC is open, and that they may be independent.