The Geometry of Reduction: Model Embedding, Compound Reduction, and Overlapping State Space Domains

The relationship according to which one physical theory encompasses the domain of empirical validity of another is widely known as "reduction." Here it is argued that one popular methodology for showing that one theory reduces to another, associated with the so-called "Bronstein cube" of physical theories, rests on an over-simplified characterization of the type of mathematical relationship between theories that typically underpins reduction. An alternative methodology, based on a certain simple geometrical relationship between dis- tinct state space models of the same physical system, is then described and illustrated with examples. Within this approach, it is shown how and under what conditions inter-model reductions involving distinct model pairs can be composed or chained together to yield a direct reduction between theoretically remote descriptions of the same system. Building on this analysis, we consider cases in which a single reduction between two models may be effected via distinct composite reductions differing in their intermediate layer of description, and motivate a set of formal consistency requirements on the mappings between model state spaces and on the subsets of the model state spaces that characterize such reductions. These constraints are explicitly shown to hold in the reduction of a non-relativistic classical model to a model of relativistic quantum mechanics, which may be effected via distinct composite reductions in which the intermediate layer of description is either a model of non-relativistic quantum mechanics or of relativistic classical mechanics. Some brief speculations are offered as to whether and how this sort of consistency requirement between distinct composite reductions might serve to constrain the relationship that any unification of the Standard Model with general relativity must bear to these theories

Inter-Theory Relations in Physics: Case Studies from Quantum Mechanics and Quantum Field Theory (Doctoral Dissertation - University of Oxford, 2013)

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 First, the popular notion that a superseded theory in physics is generally a simple limit of the theory that supersedes it paints an oversimplified picture of reductive relations in physics.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

Ehrenfest Theorems, Deformation Quantization, and the Geometry of Inter-Model Reduction

This study attempts to spell out more explicitly than has been done previously the connection between two types of formal correspondence that arise in the study of quantum–classical relations: one the one hand, deformation quantization and the associated continuity between quantum and classical algebras of observables in the limit ℏ→0, and, on the other, a certain generalization of Ehrenfest’s Theorem and the result that expectation values of position and momentum evolve approximately classically for narrow wave packet states. While deformation quantization establishes a direct continuity between the abstract algebras of quantum and classical observables, the latter result makes in-eliminable reference to the quantum and classical state spaces on which these structures act — specifically, via restriction to narrow wave packet states. Here, we describe a certain geometrical re-formulation and extension of the result that expectation values evolve approximately classically for narrow wave packet states, which relies essentially on the postulates of deformation quantization, but describes a relationship between the actions of quantum and classical algebras and groups over their respective state spaces that is non-trivially distinct from deformation quantization. The goals of the discussion are partly pedagogical in that it aims to provide a clear, explicit synthesis of known results; however, the particular synthesis offered aspires to some novelty in its emphasis on a certain general type of mathematical and physical relationship between the state spaces of different models that represent the same physical system, and in the explicitness with which it details the above-mentioned connection between quantum and classical models

Naturalness, Wilsonian Renormalization, and 'Fundamental Parameters' in Quantum Field Theory

The Higgs naturalness principle served as the basis for the so far failed prediction that signatures of physics beyond the Standard Model (SM) would be discovered at the LHC. One influential formulation of the principle, which prohibits fine tuning of bare Standard Model (SM) parameters, rests on the assumption that a particular set of values for these parameters constitute the ``fundamental parameters" of the theory, and serve to mathematically define the theory. On the other hand, an old argument by Wetterich suggests that fine tuning of bare parameters merely reflects an arbitrary, inconvenient choice of expansion parameters and that the exact choice of parameters in an EFT is therefore a matter of convention. We argue that these two interpretations of Higgs fine tuning reflect distinct ways of formulating and interpreting effective field theories (EFTs) within the Wilsonian framework: the first takes an EFT to be defined by a single set of physical, fundamental bare parameters, while the second takes a Wilsonian EFT to be defined instead by a whole Wilsonian renormalization group (RG) trajectory, associated with a one-parameter class of physically equivalent parametrizations. From this latter perspective, no single parametrization constitutes the physically correct, fundamental parametrization of the theory, and the delicate cancellation between bare Higgs mass and quantum corrections appears as an eliminable artifact of the arbitrary, unphysical reference scale with respect to which the physical amplitudes of the theory are parametrized. While the notion of fundamental parameters is well motivated in the context of condensed matter field theory, we explain why it may be superfluous in the context of high energy physics

Towards Enhanced Local Explainability of Random Forests: a Proximity-Based Approach

We initiate a novel approach to explain the out of sample performance of random forest (RF) models by exploiting the fact that any RF can be formulated as an adaptive weighted K nearest-neighbors model. Specifically, we use the proximity between points in the feature space learned by the RF to re-write random forest predictions exactly as a weighted average of the target labels of training data points. This linearity facilitates a local notion of explainability of RF predictions that generates attributions for any model prediction across observations in the training set, and thereby complements established methods like SHAP, which instead generates attributions for a model prediction across dimensions of the feature space. We demonstrate this approach in the context of a bond pricing model trained on US corporate bond trades, and compare our approach to various existing approaches to model explainability.Comment: 5 pages, 6 figure

Theory Reduction in Physics: A Model-Based, Dynamical Systems Approach

In 1973, Nickles identified two senses in which the term `reduction' is used to describe the relationship between physical theories: namely, the sense based on Nagel's seminal account of reduction in the sciences, and the sense that seeks to extract one physical theory as a mathematical limit of another. These two approaches have since been the focus of most literature on the subject, as evidenced by recent work of Batterman and Butterfield, among others. In this paper, I discuss a third sense in which one physical theory may be said to reduce to another. This approach, which I call `dynamical systems (DS) reduction,' concerns the reduction of individual models of physical theories rather than the wholesale reduction of entire theories, and specifically reduction between models that can be formulated as dynamical systems. DS reduction is based on the requirement that there exist a function from the state space of the low-level (more encompassing) model to that of the high-level (less encompassing) model that satisfies certain general constraints and thereby serves to identify quantities in the low-level model that mimic the behavior of those in the high-level model - but typically only when restricted to a certain domain of parameters and states within the low-level model. I discuss the relationship of this account of reduction to the Nagelian and limit-based accounts, arguing that it is distinct from both but exhibits strong parallels with a particular version of Nagelian reduction, and that the domain restrictions employed by the DS approach may, but need not, be specified in a manner characteristic of the limit-based approach. Finally, I consider some limitations of the account of reduction that I propose and suggest ways in which it might be generalised. I offer a simple, idealised example to illustrate application of this approach; a series of more realistic case studies of DS reduction is presented in another paper

"Formal" vs. "Empirical" Approaches to Quantum-Classical Reduction

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

Interpretation Neutrality in the Classical Domain of Quantum Theory

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 and often ambiguous picture that treats reduction simply as a matter of 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

Reduction as an A Posteriori Relation

Reduction between theories in physics is often approached as an a priori relation in the sense that reduction is often taken to depend only on a comparison of the mathematical structures of two theories. I argue that such approaches fail to capture one crucial sense of “reduction,” whereby one theory encompasses the set of real behaviors that are well-modeled by the other. Reduction in this sense depends not only on the mathematical structures of the theories, but also on empirical facts about where our theories succeed at describing real systems, and is therefore an a posteriori relation