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Remarks on Causality in Relativistic Quantum Field Theory

Miklós Rédei, Stephen J. Summers

Published Feb 14, 2003• 20 citations
OpenAlex

Abstract

It is shown that the correlations predicted by relativistic quantum field theory in locally normal states between projections in local von Neumann algebras $\\cA(V_1),\\cA(V_2)$ associated with spacelike separated spacetime regions $V_1,V_2$ have a (Reichenbachian) common cause located in the union of the backward light cones of $V_1$ and $V_2$. Further comments on causality and independence in quantum field theory are made.

Related papers

Local Primitive Causality and the Common Cause Principle in Quantum Field Theory

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The Common Cause Principle in algebraic relativistic quantum field theory Algebraic quantum field theory (AQFT) predicts correlations between projections A, B lying in von Neumann algebras A ( V 1 ), A ( V 2 ) associated with spacelike separated spacetime regions V 1 , V 2 in Minkowski spacetime. These spacelike correlations predicted by AQFT lead naturally to the question of the status of Reichenbach's Common Cause Principle within AQFT. The aim of this chapter is to investigate the problem of status of Reichenbach's Common Cause Principle within AQFT. Since the correlated projections belong to algebras associated with spacelike separated regions, a direct causal influence between them is excluded by the Special Theory of Relativity. Consequently, compliance of AQFT with Reichenbach's Common Cause Principle would mean that for every correlation between projections A and B lying in von Neumann algebras A ( V 1 ) and A ( V 2 ), respectively, associated with spacelike separated spacetime regions V 1 , V 2 , there must exist a projection C possessing the probabilistic properties that qualify it to be a Reichenbachian common cause of the correlation between A and B . However, since observables and hence also the projections in AQFT must be localized, one also has to specify the spacetime region V with which the von Neumann algebra A(V) is associated that contains the common cause C . Intuitively, the region V should be disjoint from both V 1 and V 2 but should not be causally disjoint from them in order to leave room for a causal effect of C on the events A and B so that C can account for the correlation between A and B .

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Reichenbach's Pre-Common Cause Principle and Logical Independence

Yuichiro Kitajima

In algebraic quantum field theory, some von Neumann algebra is associated with any finite open set in Minkowski space-time. Such a von Neumann algebra is called a local algebra. Let V1 and V2 be two spacelikely separated spacetime regions and let N(V1) and N(V2) be local algebras associated with V1 and V2 respectively. In the present paper it is shown that N(V1) and N(V2) are logically independent if and only if any correlation between projections of N(V1) and N(V2) has a Reichenbach's pre-common cause explanation.

Journal of the Japan Association for Philosophy of Science•Jan 1, 2009•0 citations
It is shown that the correlations predicted by relativistic quantum field theory in locally normal states between projections in local von Neumann algebras cA(V1),cA(V2)\\cA(V_1),\\cA(V_2)cA(V1​),cA(V2​) associated with spacelike separated spacetime regions V1,V2V_1,V_2V1​,V2​ have a (Reichenbachian) common cause located in the union of the backward light cones of V1V_1V1​ and V2V_2V2​. Further comments on causality and independence in quantum field theory are made.
If \{A(V)\} is a net of local von Neumann algebras satisfying standard axioms of algebraic relativistic quantum field theory and V_1 and V_2 are spacelike separated spacetime regions, then the system (A(V_1),A(V_2),ϕ) is said to satisfy the Weak Reichenbach's Common Cause Principle iff for every pair of projections A \in A(V_1), B \in A(V_2) correlated in the normal state ϕthere exists a projection C belonging to a von Neumann algebra associated with a spacetime region V contained in the union of the backward light cones of V_1 and V_2 and disjoint from both V_1 and V_2, a projection having the properties of a Reichenbachian common cause of the correlation between A and B. It is shown that if the net has the local primitive causality property then every local system (A(V_1),A(V_2),ϕ) with a locally normal and locally faithful state ϕand open bounded V_1 and V_2 satisfies the Weak Reichenbach's Common Cause Principle.
The Common Cause Principle in algebraic relativistic quantum field theory Algebraic quantum field theory (AQFT) predicts correlations between projections A, B lying in von Neumann algebras A ( V 1 ), A ( V 2 ) associated with spacelike separated spacetime regions V 1 , V 2 in Minkowski spacetime. These spacelike correlations predicted by AQFT lead naturally to the question of the status of Reichenbach's Common Cause Principle within AQFT. The aim of this chapter is to investigate the problem of status of Reichenbach's Common Cause Principle within AQFT. Since the correlated projections belong to algebras associated with spacelike separated regions, a direct causal influence between them is excluded by the Special Theory of Relativity. Consequently, compliance of AQFT with Reichenbach's Common Cause Principle would mean that for every correlation between projections A and B lying in von Neumann algebras A ( V 1 ) and A ( V 2 ), respectively, associated with spacelike separated spacetime regions V 1 , V 2 , there must exist a projection C possessing the probabilistic properties that qualify it to be a Reichenbachian common cause of the correlation between A and B . However, since observables and hence also the projections in AQFT must be localized, one also has to specify the spacetime region V with which the von Neumann algebra A(V) is associated that contains the common cause C . Intuitively, the region V should be disjoint from both V 1 and V 2 but should not be causally disjoint from them in order to leave room for a causal effect of C on the events A and B so that C can account for the correlation between A and B .
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.
States in algebraic quantum field theory ”typically” establish correlation between spacelike separated events. Reichenbach’s Common Cause Principle, generalized to the quantum field theoretical setting, offers an apt tool to causally account for these superluminal correlations. In the paper we motivate first why commutativity between the common cause and the correlating events should be abandoned in the definition of the common cause. Then we show that the Noncommutative Weak Common Cause Principle holds in algebraic quantum field theory with locally finite degrees of freedom. Namely, for any pair of projections A, B supported in spacelike separated regions VA and VB , respectively, there is a local projection C not necessarily commut- ing with A and B such that C is supported within the union of the backward light cones of VA and VB and the set {C, C⊥ } screens off the correlation between A and B.
States in algebraic quantum field theory “typically” establish correlation between spacelike separated events. Reichenbach's Common Cause Principle, generalized to the quantum field theoretical setting, offers an apt tool to causally account for these superluminal correlations. In the paper we motivate first why commutativity between the common cause and the correlating events should be abandoned in the definition of the common cause. Then we show that the Noncommutative Weak Common Cause Principle holds in algebraic quantum field theory with locally finite degrees of freedom. Namely, for any pair of projections A, B supported in spacelike separated regions VA and VB, respectively, there is a local projection C not necessarily commuting with A and B such that C is supported within the union of the backward light cones of VA and VB and the set {C, C⊥} screens off the correlation between A and B.
In algebraic quantum field theory, some von Neumann algebra is associated with any finite open set in Minkowski space-time. Such a von Neumann algebra is called a local algebra. Let V1 and V2 be two spacelikely separated spacetime regions and let N(V1) and N(V2) be local algebras associated with V1 and V2 respectively. In the present paper it is shown that N(V1) and N(V2) are logically independent if and only if any correlation between projections of N(V1) and N(V2) has a Reichenbach's pre-common cause explanation.