Subsystems and independence in relativistic microscopic physics☆
Introduction
Without the possibility of analyzing the universe into subsystems, it is hardly conceivable how the sciences could be carried out. Common experience certainly supports this possibility; however, common experience is neither quantum nor relativistic, so it is far from obvious whether one can sensibly speak of microscopic subsystems, despite the fact that much of science is carried out as if one could do so. Indeed, what appears to be straightforward in a classical, nonrelativistic setting turns out to be highly nontrivial and, to some degree, impossible in a relativistic quantum setting. However, it is not our purpose here to rehearse the well-known controversies concerning the various kinds of nonlocality and interdependence of subsystems manifest in relativistic quantum theory (cf. Butterfield, 2007; Clifton & Halvorson, 2001; Peres & Terno, 2004; Summers, 2008a for recent discussions and reviews). Instead, our intent is to indicate how, in spite of these, one can still speak meaningfully of subsystems in relativistic quantum theory. Although no new theorems will be proven in this paper, we shall draw together results scattered through many highly technical papers to make a coherent case for this claim. The technicalities will be minimized as much as possible, however.
What is a subsystem of a relativistic quantum system? We shall not answer this question here. Indeed, as explained in Section 7, even after the analysis carried out below, there are other subtle matters to deal with before such a definition can be attempted. But whatever a subsystem is, it is not merely a spatially distinguished portion of the full system. To be conceptually most useful, a subsystem should be an identifiable component of the system which can subsist independently of the other subsystems comprising the system, e.g. it can be suitably screened off from the other subsystems and studied experimentally without their influence. The analyzability of the universe into subsystems therefore requires a concept of the “independence” of the subsystems, of which the relativistic quantum world supports many distinct notions which coincide or are trivial in the classical setting. The complex relation between these notions will only be adumbrated here; the emphasis will be placed upon the warrant for and the consequences of a particular notion of subsystem independence, which, it is proposed, should be viewed as primary and which, it is argued, provides a reasonable framework within which to sensibly speak of relativistic quantum subsystems.
In order to formulate in a mathematically rigorous manner the notion of independent subsystems and to understand its consequences, it is necessary to choose a mathematical framework which is sufficiently general to subsume large classes of relativistic quantum models, is powerful enough to facilitate the proof of nontrivial assertions of physical interest, and yet is conceptually simple enough to have a direct, if idealized, interpretation in terms of operationally meaningful physical quantities. Such a framework is provided by algebraic quantum field theory (AQFT) and algebraic quantum statistical mechanics (Araki, 1999; Bratteli and Robinson, 1979, Bratteli and Robinson, 1981; Haag, 1992), also called collectively local quantum physics, which is based on operator algebra theory, itself initially developed by von Neumann, 1932, von Neumann, 1962 for the express purpose of providing quantum theory with a rigorous and flexible foundation. This framework is briefly described in the next section.
In Section 3 we discuss three of the many notions of independence which have been examined in the literature, indicating briefly their operational meaning and their logical interrelations. But what we regard as the operationally primary notion of independence—the split property—is initially discussed in Section 4. This property is strictly stronger than all those treated in Section 3. After the somewhat abstract discussion in Section 4, we present in Section 5 a number of equivalent characterizations of the split property which all have operational meaning. Further physically significant consequences of the split property are reviewed in Section 6 to buttress our contention that the split property should be viewed as the primary independence notion. Various aspects of the warrant for the split property are considered in 4 The split property, 5 Physical characterizations of the split property, 6 Further consequences of the split property. Finally, in Section 7 we draw our conclusions and indicate why the analysis of the notion of independent subsystems in relativistic quantum theory is far from complete.
Section snippets
Mathematical framework
The operationally fundamental objects in a laboratory are the preparation apparata—devices which prepare in a repeatable manner the individual quantum systems which are to be examined—and the measuring apparata—devices which are applied to the prepared systems and which measure the “value” of some observable property of the system. The physical notion of a “state” can be viewed as a certain equivalence class of such preparation devices, and the physical notion of an “observable” (or “effect”)
Some formulations of subsystem independence
There are various technical conditions used in algebraic quantum theory to formulate the notion of the independence of subsystems. This is only to be expected, since there are clearly different quantitative and qualitative aspects of such independence. The study of these formulations and their logical relations is therefore of some conceptual interest. In the context of classical mechanics or classical field theory, these notions are either trivial or mutually equivalent. However, in the
The split property
We turn now to the split property, an important structure property of inclusions of von Neumann algebras, which has been intensively studied for the purposes of both abstract operator algebra theory and local quantum physics. We shall see that it provides a particularly useful formalization of subsystem independence and propose this as primary among notions of independence. In the following denotes the (unique ) tensor product of two von Neumann algebras and , which can be thought of
Physical characterizations of the split property
In light of the above, one may tentatively conclude that the split property obtains in some generality in physically relevant quantum field models. We further examine the warrant for this property by explaining some physically meaningful characterizations of the split property. We begin with one of the first found, which generalized a characterization proven in Buchholz, Doplicher, and Longo (1986). We present it in a form given in Summers (1990), since the original (Werner, 1987) requires the
Further consequences of the split property
In general, in relativistic quantum field theory one has global energy, momentum and charge observables (say ) which have meaning for the full quantum system (Araki, 1999, Haag, 1992). These cannot be localized in any region with nonempty causal complement and cannot directly refer to any subsystem. But to any subsystem worth the name one must be able to attribute such quantities. This is a highly nontrivial matter, but if the funnel property holds,7
Concluding remarks
We conclude that it is meaningful to speak of independent subsystems in relativistic quantum theory, if they can be localized in spacetime regions , resp. , such that and satisfy the split property. For then their observables are mutually commensurable, they can be independently and locally prepared in arbitrary states, they are “operationally independent,” and they possess mutually compatible localized energy, momentum and charge observables, to mention just a few desirable
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This is an expanded version of an invited talk given at the biennial meeting of the Philosophy of Science Association, held in Pittsburgh, PA, on November 6–9, 2008.