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arXiv:1802.09350

Inter-Theory Relations in Physics: Case Studies from Quantum Mechanics and Quantum Field Theory

Joshua Rosaler

Published Jan 1, 2013• 9 citations
arXivHTMLPDF

Abstract

I defend three general claims concerning inter-theoretic reduction in physics. First, the popular notion that a superseded theory in physics is generally a simple limit of the theory that supersedes it paints an oversimplified picture of reductive relations in physics. Second, where reduction specifically between two dynamical systems models of a single system is concerned, reduction requires the existence of a particular sort of function from the state space of the low-level (purportedly more accurate and encompassing) model to that of the high-level (purportedly less accurate and encompassing) model that approximately commutes, in a specific sense, with the rules of dynamical evolution prescribed by the models. The third point addresses a tension between, on the one hand, the frequent need to take into account system-specific details in providing a full derivation of the high-level theory’s success in a particular context, and, on the other hand, a desire to understand the general mechanisms and results that under- write reduction between two theories across a wide and disparate range of different systems; I suggest a reconciliation based on the use of partial proofs of reduction, designed to reveal these general mechanisms of reduction at work across a range of systems, while leaving certain gaps to be filled in on the basis of system-specific details. After discussing these points of general methodology, I go on to demonstrate their application to a number of particular inter-theory reductions in physics involving quantum theory. I consider three reductions: first, connecting classical mechanics and non-relativistic quantum mechanics; second,connecting classical electrodynamics and quantum electrodynamics; and third, connecting non-relativistic quantum mechanics and quantum electrodynamics. I approach these reductions from a realist perspective, and for this reason consider two realist interpretations of quantum theory - the Everett and Bohm theories - as potential bases for these reductions. Nevertheless, many of the technical results concerning these reductions pertain also more generally to the bare, uninterpreted formalism of quantum theory. Throughout my analysis, I make the application of the general methodological claims of the thesis explicit, so as to provide concrete illustration of their validity.

DOI
10.48550/arxiv.1802.09350

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I defend three general claims concerning inter-theoretic reduction in physics. First, the popular notion that a superseded theory in physics is generally a simple limit of the theory that supersedes it paints an oversimplified picture of reductive relations in physics. Second, where reduction specifically between two dynamical systems models of a single system is concerned, reduction requires the existence of a particular sort of function from the state space of the low-level (purportedly more accurate and encompassing) model to that of the high-level (purportedly less accurate and encompassing) model that approximately commutes, in a specific sense, with the rules of dynamical evolution prescribed by the models. The third point addresses a tension between, on the one hand, the frequent need to take into account system-specific details in providing a full derivation of the high-level theory’s success in a particular context, and, on the other hand, a desire to understand the general mechanisms and results that under- write reduction between two theories across a wide and disparate range of different systems; I suggest a reconciliation based on the use of partial proofs of reduction, designed to reveal these general mechanisms of reduction at work across a range of systems, while leaving certain gaps to be filled in on the basis of system-specific details. After discussing these points of general methodology, I go on to demonstrate their application to a number of particular inter-theory reductions in physics involving quantum theory. I consider three reductions: first, connecting classical mechanics and non-relativistic quantum mechanics; second,connecting classical electrodynamics and quantum electrodynamics; and third, connecting non-relativistic quantum mechanics and quantum electrodynamics. I approach these reductions from a realist perspective, and for this reason consider two realist interpretations of quantum theory - the Everett and Bohm theories - as potential bases for these reductions. Nevertheless, many of the technical results concerning these reductions pertain also more generally to the bare, uninterpreted formalism of quantum theory. Throughout my analysis, I make the application of the general methodological claims of the thesis explicit, so as to provide concrete illustration of their validity.
I distinguish two types of reduction within the context of quantum-classical relations, which I designate "formal" and "empirical". Formal reduction holds or fails to hold solely by virtue of the mathematical relationship between two theories; it is therefore a two-place, a priori relation between theories. Empirical reduction requires one theory to encompass the range of physical behaviors that are well-modeled in another theory; in a certain sense, it is a three-place, a posteriori relation connecting the theories and the domain of physical reality that both serve to describe. Focusing on the relationship between classical and quantum mechanics, I argue that while certain formal results concerning singular limits as Planck's constant goes to zero have been taken to preclude the possibility of reduction between these theories, such results at most block reduction in the formal sense; little if any reason has been given for thinking that they block reduction in the empirical sense. I then briefly outline a strategy for empirical reduction that is suggested by work on decoherence theory, arguing that this sort of account remains a fully viable route to the empirical reduction of classical to quantum mechanics and is unaffected by such singular limits.
I show explicitly how concerns about wave function collapse and ontology can be decoupled from the bulk of technical analysis necessary to recover localized, approximately Newtonian trajectories from quantum theory. In doing so, I demonstrate that the account of classical behavior provided by decoherence theory can be straightforwardly tailored to give accounts of classical behavior on multiple interpretations of quantum theory, including the Everett, de Broglie-Bohm and GRW interpretations. I further show that this interpretation-neutral, decoherence-based account conforms to a general view of inter-theoretic reduction in physics that I have elaborated elsewhere, which differs from the oversimplified picture that treats reduction as a matter of simply taking limits. This interpretation-neutral account rests on a general three-pronged strategy for reduction between quantum and classical theories that combines decoherence, an appropriate form of Ehrenfest's Theorem, and a decoherence-compatible mechanism for collapse. It also incorporates a novel argument as to why branch-relative trajectories should be approximately Newtonian, which is based on a little-discussed extension of Ehrenfest's Theorem to open systems, rather than on the more commonly cited but less germane closed-systems version. In the Conclusion, I briefly suggest how the strategy for quantum-classical reduction described here might be extended to reduction between other classical and quantum theories, including classical and quantum field theory and classical and quantum gravity.
Today there is no agreement on which, if any, of the several known realist interpretations of quantum mechanics is the correct one, and disagreement on this matter is not merely verbal, but substantial. However, I will show that the interpretations share a common aim: to recover the classical world given that it is quantum. This aim responds to implicit and well-entrenched philosophical intuitions that can be phrased in terms of intertheory relations, traditionally involving theory reduction, in philosophy, and the quantum-classical limit, in the foundations of physics. However, not one of those notions is free from controversies, and many contest that there is a smooth transition from the quantum to the classical. Hence, the account of the relationship between quantum mechanics and classical mechanics is an unresolved problem, and the philosophical character of the underlying framework can be contested. This thesis will offer a critical analysis of current well-known realist interpretations and will also put forward an alternative framework. I will critically examine traditional views on intertheory relations and the recent view of interstructuralism. My main claim will be that the role of intertheory relations is overrated, because a more basic question has to be answered first: 'what is a quantum system?'. That will motivate my novel view. I will critically evaluate and defend an alternative view based on the philosophy of Werner Heisenberg. The view proposes that physical theories should be regarded as "closed" systems. This has immediate implications for how we should understand intertheory relations in general, as well as scientific realism in particular. My view will appear radical in comparison to traditional views of quantum mechanics. Yet, I will examine fruitful comparison with forms of realism such as perspectivism and metaphysical pluralism. I shall conclude with indications for future work.
“the view advanced by H. W. Everett (1957)—that there simply is no reduction—quite fails to account for the actual phenomena of physics, which we may summarize as the appearance of reduction: namely, our experience of definite observations.” (Stein 1983, p. 573.) In an interesting recent paper, Geroch (1984) has sought to state the content of, “and to some extent to advocate” a view toward quantum mechanics which is at least closely akin to the view advanced by Everett, known popularly (but, arguably, misleadingly) as the many-worlds interpretation. While sensitive to objections along the lines of that by Stein quoted above, Geroch believes that these can adequately be answered. The key to answering such objections is to argue that physics is not responsible for accounting for our experience of physical phenomena—it is enough that it account for the physical phenomena themselves. It is this last argument that I wish to examine in this paper, both in general, and specifically in its application to Everett's interpretation of quantum mechanics.
In 1964, John Stewart Bell famously demonstrated that the laws of standard quantum mechanics demand a physical world that cannot be described entirely according to local laws. The present article argues that this nonlocality must be gravitationally related, as it comes about only with quantum state reduction, this being claimed to be a gravitational effect. A new formalism for curved space–times, palatial twistor theory, is outlined, which appears now to be able to accommodate gravitation fully, providing a nonlocal description of the physical world. Nonlocality and Quantum State Reduction Although quantum entanglement is a normal consequence of the unitary (Schrödinger) evolution U of a multiparticle quantum system, this evolution is nevertheless local in the sense that it is described as the continuous (local) evolution in the relevant configuration space. What the EPR situations considered by John Stewart Bell [1] demonstrated was that when widely separated quantum measurements , of appropriate kinds, are performed on such entangled states (a current record for distance separation being 143 km, for entangled photon pairs [2]), the results of such measurements (which may be probabilistic or of a yes/no character) cannot be mimicked by any local realistic model. Thus, in trying to propose a mathematical model of what is going on realistically in the physical world, one would need to face up to the actual physical process involved in quantum state reduction R, which is an essential feature of quantum measurement. For many years, my own position on the state-reduction issue has been that R would have to involve a fundamental mathematical extension of current quantum theory, giving a description of something objectively taking place ‘ out there’ in the physical world (OR: objective reduction), rather than R being the effect of some kind of ‘interpretation’ of the standard unitary quantum formalism. Moreover, the extended theory should be able to describe the one physical world that we all experience, rather than some sort of coexisting superposition of vast numbers of alternatives. To be more specific about my own viewpoint, it has long been my position that such necessary deviation from unitarity U in this OR would result from a correct melding of the quantum formalism with that of general relativity .