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Cambridge University Press eBooks

Causal closedness of quantum field theory

Gábor Hofer‐Szabó, Miklós Rédei, László E. Szabó

Published May 16, 2013• 0 citations
Publisher

Abstract

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 .

DOI
10.1017/cbo9781139094344.009

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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 .
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.
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.
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.
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.
Reichenbach's Common Cause Principle is the claim that if two events are correlated, then either there is a causal connection between the correlated events that is responsible for the correlation or there is a third event, a so called (Reichenbachian) common cause, which brings about the correlation. The paper reviews some results concerning Reichenbach's notion of common cause, results that are directly relevant to the problem of how one can falsifY Reichenbach's Common Cause Principle. Special emphasis will be put on the question of whether EPR-type correlations can have an explanation in terms of Reichenbachian common causes. Most of the results to be recalled indicate that falsifYing Reichenbach's Common Cause Principle is much more tricky than one may have thought and that, contrary to some claims in the literature, there is no conclusive proof yet that the EPR correlations predicted by ordinary, non-relativistic quantum mechanics cannot have a common cause; furthermore, recent results about possible Reichenbachian common causes of correlations predicted by quantum field states between spacelike separated local observable algebras in algebraic quantum field theory strongly indicate that there may very well exist Reichenbachian common causes of superluminal correlations predicted by quantum field theory.
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.