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) associated with spacelike separated spacetime regions
V1,V2 have a (Reichenbachian) common cause located in the union of the backward light cones of
V1 and
V2. Further comments on causality and independence in quantum field theory are made.
Abstract We employ reduced density matrices to investigate the localization of relativistic particles within the framework of
(1+1) ( 1 + 1 ) -dimensional quantum field theory. In the case of bosons, we prove that scalar particles cannot be localized within any compact region, although they can be made arbitrarily close to localized states. For massless fermions, by exploiting properties of the reduced density matrices of coherent states, we show that the fermionic single-particle state which, under constructive bosonization, corresponds to a bosonic coherent state is strictly localized at a single spatial point; however, the energy of this state is infinite. Moreover, since the notion of spatial localization based on reduced density matrices is consistent with relativistic causality, even a particle with infinite energy that is localized at a single spatial point will neither spread outside the light cone nor propagate superluminally. Indeed, the explicit expression for the localized fermionic state derived in our work shows that a massless fermion can preserve its sharply localized profile while propagating at the speed of light, in full agreement with causality.
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.