Relativistic Causality in Algebraic Quantum Field Theory

Abstract

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.

Acknowledgements

We are grateful to Jeremy Butterfield, Geoffrey Hellmann, Laura Ruetsche, Gárbor Hofer-Szabó, and Miklós Rédei for helpful comments on an earlier draft of this paper. We also wish to thank the anonymous referees for suggestions that led to substantive improvements.

Notes

[1] Some discussions of locality also associate non-locality with holism. We will have nothing to say about the vexed notion of holism, except that the additivity axioms of AQFT may be thought of as expression of an anti-holism for quantum fields (see section 2).

[2] Readers who want to go beyond the superficial recounting of highlights given here may consult Haag (1992) and Horuzhy (1990). For those interested in foundations issues, the review article by Halvorson (2007) and the book by Ruetsche (2011) are recommended.

[3] The concrete version is referred to in the literature as Haag–Araki theory while the abstract version is called Haag–Klaster theory.

[4] Also called Einstein locality or simply locality.

[5] While ‘time sharp’ observables are possible for non-interacting fields, this is almost certainly not true for interacting fields.

[6] This result follows from the fact that a vector is cyclic for a von Neumann algebra if and only if it is separating for the commutant . That a vector is separating for means that , , implies that .

[7] , the von Neumann algebra generated by . For Type I factors, .

[8] Notice that the Buchholz–Summers condition remains meaningful even if the algebras do not commute. In fact, Buchholz and Summers went on to employ it to define a measure of the degree of non-commutativity of two arbitrary algebras.

[9] Let be an inertial coordinate system for Minkowski spacetime. Then the right Rindler wedge with vertex at the origin consists of those points , . Reflecting about the origin gives the left Rindler wedge.

[10] A diamond (aka double cone) region is the interior of the intersection of the forward light cone of a spacetime point p with the backward light cone of a point q where q lies to the chronological future of p.

[11] Lüders selective conditionalization representing the upshot of a selective measurement of A results in the change of initial state to the new state where is some particular spectral projector of A, the denominator being required for to be normalized. The significance of the fraught selective vs. non-selective distinction will be discussed in section 4.4.

[12] Thus, for present purposes we are not concerned about no-go results for conditional expectations for continuous observables; see Davis (1976).

[13] The tensor product structure follows via the split property if it is assumed that and are mutually commuting Type I factors.

[14] The operation is normal just in case, if is a sequence of positive bounded operators converging to A, then the sequence converges to . The dual of an normal operation maps normal states onto normal states.

[15] This worry is undercut if the quantum state is given an epistemic interpretation in the manner quantum Bayesians propose. We do not believe that quantum Bayesianism is a viable view, but will not try to argue that here.

[16] Note that if our condition for no superluminal propagation of the field holds then it follows that changing the initial data on a portion of the initial value hypersurface, while holding the data fixed elsewhere, may result in a change to the solution to the future side of only in the region belonging to causal future of S (i.e. the set of all spacetime points p that can be reached from S by a future-directed causal curve). Of course, by time symmetry changing the initial data on may also necessitate a change in the solution of the region belonging to the causal past of S (i.e. the set of all spacetime points p that can be reached from S by a past-directed directed causal curve). The latter change is often ignored perhaps because it is thought that causal influences only operate in the past-to-future direction. This issue will be encountered again in the discussion of the common cause principle (see section 8.3).

[17] For example, the DEC ‘can be interpreted as saying that the speed of energy flow of matter is always less than [or equal to] the speed of light’ (Wald 1984, 219).

[18] Additionally, Hegerfeldt stresses that Buchholz and Yngvason's argument applies to a kind of causality which he refers to as Weak Causality, but it does not apply to what he calls Strong Causality, that has to do with individual processes rather than expectation values only. We shall not dwell further on such a distinction here.

[19] Notice that in the orthodox quantization scheme, algebras of local observables can also be associated with spatial regions S of a time slice. But in the orthodox scheme scheme .

[20] See Hamhalter (2003) for the needed technical restriction on the algebra.

[21] Hellman (1982) thought that the formulation of stochastic Einstein locality needed to come with a proviso in order to ward off potential counterexamples where relatively spacelike events are subject to lawlike constraints, such as conservation laws. We do not think that any proviso is needed. As as analogy we note that Maxwell's field equations for classical electromagnetic fields place constraints on relatively spacelike data (see section 7.2); but such constraints are demonstrably compatible with local determinism for the Maxwell field. It might be thought, however, that this analogy is not apposite because in the quantum case ‘state’ has a probabilistic interpretation and because the probability of events in cannot be determined by the state on some appropriate region of when a constraint entails that, say, for any physically possible global state there are correlations between events in and those in with relatively spacelike to . AQFT shows that this intuition is incorrect.

[22] Note in passing the trade off regarding non-locality in Segal's (1964) heterodox quantization of the Klein–Gordon field. The vacuum state is a product state for the local Segal algebras and for disjoint regions and lying on some spacelike hyperplane. On the other hand, and may fail to commute if and lie on different hyperplanes even though and are relatively spacelike (see Halvorson 2001). And LPC and SEL also fail in Segal's scheme.

[23] Allowance can be made for cases where there is not a unique nearest F (or ); but this complication does not affect the main points to be made here.

[24] In addition Elga (2001) has argued that the appeal to ‘small miracles’ does not succeed in grounding the directionality of causation.

[25] By ‘rolling up’ Minkowski spacetime along the space axis it is possible to produce a flat cylindrical spacetime in which is the entirety of the spacetime.

[26] We will not broach the issue of how the common cause principle can be extended to more than two events. On this matter, see Uffink (1999).

[27] The terminology is due to Hofer-Szabó, Rédei, and Szabó (2002) and Rédei and Summers (2002).

[28] Moreover, Hofer-Szabó, Rédei, and Szabó (2002) provide examples where no extension of the probability space will produce a common common-cause.

[29] For a general perspective on the status of the CCP, see Arntzenius (2010).

[30] The particle horizon for an observer is a null surface that forms the boundary between those spacetime points from which the observer can receive a light signal and those from which she cannot receive a light signal.

[31] For more discussion of the significance of the difference between (to use the Budapest school terminology) common causes and common common causes, see Butterfield (2007a).

[32] Perhaps the failure of CCP is due to the ‘coarseness’ of the lattice model. This suspicion would have to be justified by showing that CCP holds in some appropriate continuum limit.

[33] For results on non-commutative common causes, see Hofer-Szabó and Vecsernyés (2012b).

[34] More precisely, the only positive result for commutative common causes; non-commutative common causes are another matter.

[35] Here we are indebted to an anonymous referee and to Erik Curiel who made a similar point in at Workshop on Relativistic Causality, held at the University of Pittsburgh Center for Philosophy of Science in March 2013.