Introduction to the special issue Hermann Weyl and the philosophy of the ‘New Physics’

This Special Issue Hermann Weyl and the Philosophy of the ‘New Physics’ has two main objectives: first, to shed fresh light on the relevance of Weyl's work for modern physics and, second, to evaluate the importance of Weyl's work and ideas for contemporary philosophy of physics. Regarding the first objective, this Special Issue emphasizes aspects of Weyl's work (e.g. his work on spinors in n dimensions) whose importance has recently been emerging in research fields across both mathematical and experimental physics, as well as in the history and philosophy of physics. Regarding the second objective, this Special Issue addresses the relevance of Weyl's ideas regarding important open problems in the philosophy of physics, such as the problem of characterizing scientific objectivity and the problem of providing a satisfactory interpretation of fundamental symmetries in gauge theories and quantum mechanics. In this Introduction, we sketch the state of the art in Weyl studies and we summarize the content of the contributions to the present volume.     

Why be regular? Part II

We provide a novel perspective on “regularity” as a property of representations of the Weyl algebra. In Part I, we critiqued a proposal by Halvorson [2004, “Complementarity of representations in quantum mechanics”, Studies in History and Philosophy of Modern Physics 35 (1), pp. 45–56], who advocates for the use of the non-regular “position” and “momentum” representations of the Weyl algebra. Halvorson argues that the existence of these non-regular representations demonstrates that a quantum mechanical particle can have definite values for position or momentum, contrary to a widespread view. In this sequel, we propose a justification for focusing on regular representations, pace Halvorson, by drawing on algebraic methods.     

Why Be regular?, part I

We provide a novel perspective on “regularity” as a property of representations of the Weyl algebra. We first critique a proposal by Halvorson [2004, “Complementarity of representations in quantum mechanics”, Studies in History and Philosophy of Modern Physics 35 (1), pp. 45–56], who argues that the non-regular “position” and “momentum” representations of the Weyl algebra demonstrate that a quantum mechanical particle can have definite values for position or momentum, contrary to a widespread view. We show that there are obstacles to such an intepretation of non-regular representations. In Part II, we propose a justification for focusing on regular representations, pace Halvorson, by drawing on algebraic methods.     

How Weyl stumbled across electricity while pursuing mathematical justice

It is argued that Weyl's theory of gravitation and electricity came out of ‘mathematical justice’: out of the equal rights of direction and length. Such justice was manifestly at work in the context of discovery, and is enough to derive all of source-free electromagnetism. Weyl's repeated references to coordinates and gauge are taken to express equal treatment of direction and length.     

Theoretical construction in physics – The role of Leibniz for Weyl's ‘Philosophie der Mathematik und Naturwissenschaft’

This paper aims at closing a gap in recent Weyl research by investigating the role played by Leibniz for the development and consolidation of Weyl's notion of theoretical (symbolic) construction. For Weyl, just as for Leibniz, mathematics was not simply an accompanying tool when doing physics—for him it meant the ability to engage in well-guided speculations about a general framework of reality and experience. The paper first introduces some of the background of Weyl's notion of theoretical construction and then discusses particular Leibnizian inheritances in Weyl's ‘Philosophie der Mathematik und Naturwissenschaft’, such as the general appreciation of the principles of sufficient reason and of continuity. Afterwards the paper focuses on three themes: first, Leibniz's primary quality phenomenalism, which according to Weyl marked the decisive step in realizing that physical qualities are never apprehended directly; second, the conceptual relation between continuity and freedom; and third, Leibniz's notion of ‘expression’, which allows for a certain type of (surrogative) reasoning by structural analogy and which gave rise to Weyl's optimism regarding the scope of theoretical construction.     

From canonical transformations to transformation theory, 1926–1927: The road to Jordan's Neue Begründung

We sketch the development from matrix mechanics as formulated in the Dreimännerarbeit of Born, Heisenberg, and Jordan, completed in late 1925, to transformation theory developed independently by Jordan and Dirac in late 1926. Focusing on Jordan, we distinguish three strands in this development: the implementation of canonical transformations in matrix mechanics (the main focus of our paper), the clarification of the relation between the different forms of the new quantum theory (matrix mechanics, wave mechanics, q-numbers, and operator calculus), and the generalization of Born's probability interpretation of the Schrödinger wave function. These three strands come together in a two-part paper by Jordan published in 1927, “On a new foundation [neue Begründung] of quantum mechanics.”     

Zweideutigkeit about “Zweideutigkeit”: Sommerfeld,Pauli, and the methodological origins of quantum mechanics

In early 1925, Wolfgang Pauli (1900–1958) published the paper for which he is now most famous and for which he received the Nobel Prize in 1945. The paper detailed what we now know as his “exclusion principle.” This essay situates the work leading up to Pauli's principle within the traditions of the “Sommerfeld School,” led by Munich University's renowned theorist and teacher, Arnold Sommerfeld (1868–1951). Offering a substantial corrective to previous accounts of the birth of quantum mechanics, which have tended to sideline Sommerfeld's work, it is suggested here that both the method and the content of Pauli's paper drew substantially on the work of the Sommerfeld School in the early 1920s. Part One describes Sommerfeld's turn away from a faith in the power of model-based (modellmässig) methods in his early career towards the use of a more phenomenological emphasis on empirical regularities (Gesetzmässigkeiten) during precisely the period that both Pauli and Werner Heisenberg (1901–1976), among others, were his students. Part two delineates the importance of Sommerfeld's phenomenology to Pauli's methods in the exclusion principle paper, a paper that also eschewed modellmässig approaches in favour of a stress on Gesetzmässigkeiten. In terms of content, a focus on Sommerfeld's work reveals the roots of Pauli's understanding of the fundamental Zweideutigkeit (ambiguity) involving the quantum number of electrons within the atom. The conclusion points to the significance of these results to an improved historical understanding of the origin of aspects of Heisenberg's 1925 paper on the “Quantum-theoretical Reformulation (Umdeutung) of Kinematical and Mechanical Relations.”     

The case for black hole thermodynamics part II: Statistical mechanics

I present in detail the case for regarding black hole thermodynamics as having a statistical-mechanical explanation in exact parallel with the statistical-mechanical explanation believed to underlie the thermodynamics of other systems. (Here I presume that black holes are indeed thermodynamic systems in the fullest sense; I review the evidence for that conclusion in the prequel to this paper.) I focus on three lines of argument: (i) zero-loop and one-loop calculations in quantum general relativity understood as a quantum field theory, using the path-integral formalism; (ii) calculations in string theory of the leading-order terms, higher-derivative corrections, and quantum corrections, in the black hole entropy formula for extremal and near-extremal black holes; (iii) recovery of the qualitative and (in some cases) quantitative structure of black hole statistical mechanics via the AdS/CFT correspondence. In each case I briefly review the content of, and arguments for, the form of quantum gravity being used (effective field theory; string theory; AdS/CFT) at a (relatively) introductory level: the paper is aimed at readers with some familiarity with thermodynamics, quantum mechanics and general relativity but does not presume advanced knowledge of quantum gravity. My conclusion is that the evidence for black hole statistical mechanics is as solid as we could reasonably expect it to be in the absence of a directly-empirically-verified theory of quantum gravity.     

Quantum Mechanics as a Principle Theory

I show how quantum mechanics, like the theory of relativity, can be understood as a ‘principle theory’ in Einstein's sense, and I use this notion to explore the approach to the problem of interpretation developed in my book Interpreting the Quantum World.     

Gauge theories and holisms

Those looking for holism in contemporary physics have focused their attention primarily on quantum entanglement. But some gauge theories arguably also manifest the related phenomenon of nonseparability. While the argument is strong for the classical gauge theory describing electromagnetic interactions with quantum “particles”, it fails in the case of general relativity even though that theory may also be formulated in terms of a connection on a principal fiber bundle. Anandan has highlighted the key difference in his analysis of a supposed gravitational analog to the Aharonov–Bohm effect. By contrast with electromagnetism in the original Aharonov–Bohm effect, gravitation is separable and exhibits no novel holism in this case. Whether the nonseparability of classical gauge theories of nongravitational interactions is associated with holism depends on what counts as the relevant part–whole relation. Loop representations of quantized gauge theories of nongravitational interactions suggest that these conclusions about holism and nonseparability may extend also to quantum theories of the associated fields.     

Why history matters to philosophy of physics

Naturalized metaphysics remains the default presupposition of much contemporary philosophy of physics. As metaphysics is supposed to concern the general structure of reality, so scientific naturalism draws upon our best physical theories to attempt to answer the foundational question “par excellence ” viz., “how could the world possibly be the way this theory says it is?” A particular case study, Hilbert's attempt to analyze and explain a seeming “pre-established harmony” between mind and nature, is offered as a salutary reminder that naturalism's ready inference from physical theory to ontology may be too quick.     

Foundations of quantum gravity: The role of principles grounded in empirical reality

When attempting to assess the strengths and weaknesses of various principles in their potential role of guiding the formulation of a theory of quantum gravity, it is crucial to distinguish between principles which are strongly supported by empirical data – either directly or indirectly – and principles which instead (merely) rely heavily on theoretical arguments for their justification. Principles in the latter category are not necessarily invalid, but their a priori foundational significance should be regarded with due caution. These remarks are illustrated in terms of the current standard models of cosmology and particle physics, as well as their respective underlying theories, i.e., essentially general relativity and quantum (field) theory. For instance, it is clear that both standard models are severely constrained by symmetry principles: an effective homogeneity and isotropy of the known universe on the largest scales in the case of cosmology and an underlying exact gauge symmetry of nuclear and electromagnetic interactions in the case of particle physics. However, in sharp contrast to the cosmological situation, where the relevant symmetry structure is more or less established directly on observational grounds, all known, nontrivial arguments for the “gauge principle” are purely theoretical (and far less conclusive than usually advocated). Similar remarks apply to the larger theoretical structures represented by general relativity and quantum (field) theory, where – actual or potential – empirical principles, such as the (Einstein) equivalence principle or EPR-type nonlocality, should be clearly differentiated from theoretical ones, such as general covariance or renormalizability. It is argued that if history is to be of any guidance, the best chance to obtain the key structural features of a putative quantum gravity theory is by deducing them, in some form, from the appropriate empirical principles (analogous to the manner in which, say, the idea that gravitation is a curved spacetime phenomenon is arguably implied by the equivalence principle). Theoretical principles may still be useful however in formulating a concrete theory (analogous to the manner in which, say, a suitable form of general covariance can still act as a sieve for separating theories of gravity from one another). It is subsequently argued that the appropriate empirical principles for deducing the key structural features of quantum gravity should at least include (i) quantum nonlocality, (ii) irreducible indeterminacy (or, essentially equivalently, given (i), relativistic causality), (iii) the thermodynamic arrow of time, (iv) homogeneity and isotropy of the observable universe on the largest scales. In each case, it is explained – when appropriate – how the principle in question could be implemented mathematically in a theory of quantum gravity, why it is considered to be of fundamental significance and also why contemporary accounts of it are insufficient. For instance, the high degree of uniformity observed in the Cosmic Microwave Background is usually regarded as theoretically problematic because of the existence of particle horizons, whereas the currently popular attempts to resolve this situation in terms of inflationary models are, for a number of reasons, less than satisfactory. However, rather than trying to account for the required empirical features dynamically, an arguably much more fruitful approach consists in attempting to account for these features directly, in the form of a lawlike initial condition within a theory of quantum gravity.     

“Local–Global”: the first twenty years

This paper investigates how and when pairs of terms such as “local–global” and “im Kleinen–im Grossen” began to be used by mathematicians as explicit reflexive categories. A first phase of automatic search led to the delineation of the relevant corpus, and to the identification of the period from 1898 to 1918 as that of emergence. The emergence appears to have been, from the very start, both transdisciplinary (function theory, calculus of variations, differential geometry) and international, although the AMS-Göttingen connection played a specific part. First used as an expository and didactic tool (e.g. by Osgood), it soon played a crucial part in the creation of new mathematical concepts (e.g. in Hahn’s work), in the shaping of research agendas (e.g. Blaschke’s global differential geometry), and in Weyl’s axiomatic foundation of the manifold concept. We finally turn to France, where in the 1910s, in the wake of Poincaré’s work, Hadamard began to promote a research agenda in terms of “passage du local au general.”     

Formal and physical equivalence in two cases in contemporary quantum physics

The application of analytic continuation in quantum field theory (QFT) is juxtaposed to T-duality and mirror symmetry in string theory. Analytic continuation—a mathematical transformation that takes the time variable t to negative imaginary time—it—was initially used as a mathematical technique for solving perturbative Feynman diagrams, and was subsequently the basis for the Euclidean approaches within mainstream QFT (e.g., Wilsonian renormalization group methods, lattice gauge theories) and the Euclidean field theory program for rigorously constructing non-perturbative models of interacting QFTs. A crucial difference between theories related by duality transformations and those related by analytic continuation is that the former are judged to be physically equivalent while the latter are regarded as physically inequivalent. There are other similarities between the two cases that make comparing and contrasting them a useful exercise for clarifying the type of argument that is needed to support the conclusion that dual theories are physically equivalent. In particular, T-duality and analytic continuation in QFT share the criterion for predictive equivalence that two theories agree on the complete set of expectation values and the mass spectra and the criterion for formal equivalence that there is a “translation manual” between the physically significant algebras of observables and sets of states in the two theories. The analytic continuation case study illustrates how predictive and formal equivalence are compatible with physical inequivalence, but not in the manner of standard underdetermination cases. Arguments for the physical equivalence of dual theories must cite considerations beyond predictive and formal equivalence. The analytic continuation case study is an instance of the strategy of developing a physical theory by extending the formal or mathematical equivalence with another physical theory as far as possible. That this strategy has resulted in developments in pure mathematics as well as theoretical physics is another feature that this case study has in common with dualities in string theory.     

Have we lost spacetime on the way? Narrowing the gap between general relativity and quantum gravity

Important features of space and time are taken to be missing in quantum gravity, allegedly requiring an explanation of the emergence of spacetime from non-spatio-temporal theories. In this paper, we argue that the explanatory gap between general relativity and non-spatio-temporal quantum gravity theories might significantly be reduced with two moves. First, we point out that spacetime is already partially missing in the context of general relativity when understood from a dynamical perspective. Second, we argue that most approaches to quantum gravity already start with an in-built distinction between structures to which the asymmetry between space and time can be traced back.     

The physics and metaphysics of Tychistic Bohmian Mechanics

The paper takes up Bell's (1987) “Everett (?) theory” and develops it further. The resulting theory is about the system of all particles in the universe, each located in ordinary, 3-dimensional space. This many-particle system as a whole performs random jumps through 3N-dimensional configuration space – hence “Tychistic Bohmian Mechanics” (TBM). The distribution of its spontaneous localisations in configuration space is given by the Born Rule probability measure for the universal wavefunction. Contra Bell, the theory is argued to satisfy the minimal desiderata for a Bohmian theory within the Primitive Ontology framework (for which we offer a metaphysically more perspicuous formulation than is customary). TBM's formalism is that of ordinary Bohmian Mechanics (BM), without the postulate of continuous particle trajectories and their deterministic dynamics. This “rump formalism” receives, however, a different interpretation. We defend TBM as an empirically adequate and coherent quantum theory. Objections voiced by Bell and Maudlin are rebutted. The “for all practical purposes”-classical, Everettian worlds (i.e. quasi-classical histories) exist sequentially in TBM (rather than simultaneously, as in the Everett interpretation). In a temporally coarse-grained sense, they quasi-persist. By contrast, the individual particles themselves cease to persist.     

Open or closed? Dirac,Heisenberg, and the relation between classical and quantum mechanics

This paper describes a long-standing, though little known, debate between Dirac and Heisenberg over the nature of scientific methodology, theory change, and intertheoretic relations. Following Heisenberg's terminology, their disagreements can be summarized as a debate over whether the classical and quantum theories are “open” or “closed.” A close examination of this debate sheds new light on the philosophical views of two of the great founders of quantum theory.     

Space–time philosophy reconstructed via massive Nordström scalar gravities? Laws vs. geometry,conventionality, and underdetermination

What if gravity satisfied the Klein–Gordon equation? Both particle physics from the 1920–30s and the 1890s Neumann–Seeliger modification of Newtonian gravity with exponential decay suggest considering a “graviton mass term” for gravity, which is algebraic in the potential. Unlike Nordström׳s “massless” theory, massive scalar gravity is strictly special relativistic in the sense of being invariant under the Poincaré group but not the 15-parameter Bateman–Cunningham conformal group. It therefore exhibits the whole of Minkowski space–time structure, albeit only indirectly concerning volumes. Massive scalar gravity is plausible in terms of relativistic field theory, while violating most interesting versions of Einstein׳s principles of general covariance, general relativity, equivalence, and Mach. Geometry is a poor guide to understanding massive scalar gravity(s): matter sees a conformally flat metric due to universal coupling, but gravity also sees the rest of the flat metric (barely or on long distances) in the mass term. What is the ‘true’ geometry, one might wonder, in line with Poincaré׳s modal conventionality argument? Infinitely many theories exhibit this bimetric ‘geometry,’ all with the total stress–energy׳s trace as source; thus geometry does not explain the field equations. The irrelevance of the Ehlers–Pirani–Schild construction to a critique of conventionalism becomes evident when multi-geometry theories are contemplated. Much as Seeliger envisaged, the smooth massless limit indicates underdetermination of theories by data between massless and massive scalar gravities—indeed an unconceived alternative. At least one version easily could have been developed before General Relativity; it then would have motivated thinking of Einstein׳s equations along the lines of Einstein׳s newly re-appreciated “physical strategy” and particle physics and would have suggested a rivalry from massive spin 2 variants of General Relativity (massless spin 2, Pauli and Fierz found in 1939). The Putnam–Grünbaum debate on conventionality is revisited with an emphasis on the broad modal scope of conventionalist views. Massive scalar gravity thus contributes to a historically plausible rational reconstruction of much of 20th–21st century space–time philosophy in the light of particle physics. An appendix reconsiders the Malament–Weatherall–Manchak conformal restriction of conventionality and constructs the ‘universal force’ influencing the causal structure.Subsequent works will discuss how massive gravity could have provided a template for a more Kant-friendly space–time theory that would have blocked Moritz Schlick׳s supposed refutation of synthetic a priori knowledge, and how Einstein׳s false analogy between the Neumann–Seeliger–Einstein modification of Newtonian gravity and the cosmological constant Λ generated lasting confusion that obscured massive gravity as a conceptual possibility.     

Subjective probability and quantum certainty

In the Bayesian approach to quantum mechanics, probabilities—and thus quantum states—represent an agent's degrees of belief, rather than corresponding to objective properties of physical systems. In this paper we investigate the concept of certainty in quantum mechanics. Particularly, we show how the probability-1 predictions derived from pure quantum states highlight a fundamental difference between our Bayesian approach, on the one hand, and Copenhagen and similar interpretations on the other. We first review the main arguments for the general claim that probabilities always represent degrees of belief. We then argue that a quantum state prepared by some physical device always depends on an agent's prior beliefs, implying that the probability-1 predictions derived from that state also depend on the agent's prior beliefs. Quantum certainty is therefore always some agent's certainty. Conversely, if facts about an experimental setup could imply agent-independent certainty for a measurement outcome, as in many Copenhagen-like interpretations, that outcome would effectively correspond to a preexisting system property. The idea that measurement outcomes occurring with certainty correspond to preexisting system properties is, however, in conflict with locality. We emphasize this by giving a version of an argument of Stairs [(1983). Quantum logic, realism, and value-definiteness. Philosophy of Science, 50, 578], which applies the Kochen–Specker theorem to an entangled bipartite system.     

Mesh and measure in early general relativity

In the early days of general relativity, several of Einstein׳s readers misunderstood the role of coordinates or “mesh-system” in ways that threatened the basic predictions of the theory. This confusion largely derived from intrinsic defects of Einstein׳s first systematic exposition of his theory. A few of Einstein׳s followers, including Arthur Eddington, Hermann Weyl, and Max von Laue, identified the interpretive difficulties and solved them by combining a deeply geometrical understanding of the theory with detailed attention to the concrete conditions of measurement.