Holography and emergence

In this paper, I discuss one form of the idea that spacetime and gravity might ‘emerge’ from quantum theory, i.e. via a holographic duality, and in particular via AdS/CFT duality. I begin by giving a survey of the general notion of duality, as well as its connection to emergence. I then review the AdS/CFT duality and proceed to discuss emergence in this context. We will see that it is difficult to find compelling arguments for the emergence of full quantum gravity from gauge theory via AdS/CFT, i.e. for the boundary theory's being metaphysically more fundamental than the bulk theory.     

Holographic space and time: Emergent in what sense?

This paper proposes a metaphysics for holographic duality. In addition to the AdS/CFT correspondence I also consider the dS/CFT conjecture of duality. Both involve non-perturbative string theory and both are exact dualities. But while the AdS/CFT keeps time at the margins of the story, the dS/CFT conjecture gives to time the “space” it deserves by presenting an interesting holographic model of it. My goals in this paper can be summarized in the following way. First, I argue that the formal structure and physical content of the duality do not support the standard philosophical reading of the relation in terms of grounding. Second, I put forward a philosophical scheme mainly extrapolated from the double aspect monism theory. I read holographic duality in this framework as it seems to fit the mathematical and physical structure of the duality smoothly. Inside this framework I propose a notion of spacetime emergence alternative to those ones commonly debated in the AdS/CFT physics and philosophy circles.     

Comparing dualities and gauge symmetries

We discuss some aspects of the relation between dualities and gauge symmetries. Both of these ideas are of course multi-faceted, and we confine ourselves to making two points. Both points are about dualities in string theory, and both have the ‘flavour’ that two dual theories are ‘closer in content’ than you might think. For both points, we adopt a simple conception of a duality as an ‘isomorphism’ between theories: more precisely, as appropriate bijections between the two theories’ sets of states and sets of quantities.The first point (Section 3) is that this conception of duality meshes with two dual theories being ‘gauge related’ in the general philosophical sense of being physically equivalent. For a string duality, such as T-duality and gauge/gravity duality, this means taking such features as the radius of a compact dimension, and the dimensionality of spacetime, to be ‘gauge’.The second point (4 Gauge/gravity duality, 5 Some complications for gauge invariance, 6 Galileo׳s ship, (Local)) is much more specific. We give a result about gauge/gravity duality that shows its relation to gauge symmetries (in the physical sense of symmetry transformations that are spacetime-dependent) to be subtler than you might expect. For gauge theories, you might expect that the duality bijections relate only gauge-invariant quantities and states, in the sense that gauge symmetries in one theory will be unrelated to any symmetries in the other theory. This may be so in general; and indeed, it is suggested by discussions of Polchinski and Horowitz. But we show that in gauge/gravity duality, each of a certain class of gauge symmetries in the gravity/bulk theory, viz. diffeomorphisms, is related by the duality to a position-dependent symmetry of the gauge/boundary 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.     

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.     

Suppressing spacetime emergence

One of the primary tasks in building a quantum theory of gravity is discovering how to save spatiotemporal phenomena using a theory which, putatively, does not include spacetime. Some have taken this task a step further and argue for the actual emergence of spacetime from a non-spatiotemporal ontology in the low-energy regime. In this paper, it is argued that the account of spacetime emergence presented in Huggett and Wüthrich (2013) and then assumed in Baron (2019), Crowther (2016), Wüthrich (2017), and Wüthrich and Lam (2018) fails to accomplish the task to which it is set. There is a prima facie contradiction between the scale-independent ontology of spacetime in GR and the scale-dependent account of emergence proposed by this literature. One can avoid this contradiction but only at the cost of changing the target of emergence and by endorsing a perspectival theory of ontology – a view I call “ontic-perspectivism”. Though this paper explicitly addresses spacetime emergence, many of the following arguments are applicable to other accounts where objects of ontology, or their properties, are claimed to emerge in the low-energy regime.     

Einstein׳s Equations for Spin 2 Mass 0 from Noether׳s Converse Hilbertian Assertion

An overlap between the general relativist and particle physicist views of Einstein gravity is uncovered. Noether׳s 1918 paper developed Hilbert׳s and Klein׳s reflections on the conservation laws. Energy-momentum is just a term proportional to the field equations and a ‘curl’ term with identically zero divergence. Noether proved a converse “Hilbertian assertion”: such “improper” conservation laws imply a generally covariant action.Later and independently, particle physicists derived the nonlinear Einstein equations assuming the absence of negative-energy degrees of freedom (“ghosts”) for stability, along with universal coupling: all energy-momentum including gravity׳s serves as a source for gravity. Those assumptions (all but) imply (for 0 graviton mass) that the energy-momentum is only a term proportional to the field equations and a symmetric “curl,” which implies the coalescence of the flat background geometry and the gravitational potential into an effective curved geometry. The flat metric, though useful in Rosenfeld׳s stress-energy definition, disappears from the field equations. Thus the particle physics derivation uses a reinvented Noetherian converse Hilbertian assertion in Rosenfeld-tinged form.The Rosenfeld stress-energy is identically the canonical stress-energy plus a Belinfante curl and terms proportional to the field equations, so the flat metric is only a convenient mathematical trick without ontological commitment. Neither generalized relativity of motion, nor the identity of gravity and inertia, nor substantive general covariance is assumed. The more compelling criterion of lacking ghosts yields substantive general covariance as an output. Hence the particle physics derivation, though logically impressive, is neither as novel nor as ontologically laden as it has seemed.     

The role of symmetry in the interpretation of physical theories

The symmetries of a physical theory are often associated with two things: conservation laws (via e.g. Noether׳s and Schur׳s theorems) and representational redundancies (“gauge symmetry”). But how can a physical theory׳s symmetries give rise to interesting (in the sense of non-trivial) conservation laws, if symmetries are transformations that correspond to no genuine physical difference? In this paper, I argue for a disambiguation in the notion of symmetry. The central distinction is between what I call “analytic” and “synthetic“ symmetries, so called because of an analogy with analytic and synthetic propositions. “Analytic“ symmetries are the turning of idle wheels in a theory׳s formalism, and correspond to no physical change; “synthetic“ symmetries cover all the rest. I argue that analytic symmetries are distinguished because they act as fixed points or constraints in any interpretation of a theory, and as such are akin to Poincaré׳s conventions or Reichenbach׳s ‘axioms of co-ordination’, or ‘relativized constitutive a priori principles’.     

On the relation between the probabilistic characterization of the common cause and Bell׳s notion of local causality

In this paper the relation between the standard probabilistic characterization of the common cause (used for the derivation of the Bell inequalities) and Bell׳s notion of local causality will be investigated in the isotone net framework borrowed from algebraic quantum field theory. The logical role of two components in Bell׳s definition will be scrutinized; namely that the common cause is localized in the intersection of the past of the correlated events; and that it provides a complete specification of the ‘beables’ of this intersection.     

Linear structures,causal sets and topology

Causal set theory and the theory of linear structures (which has recently been developed by Tim Maudlin as an alternative to standard topology) share some of their main motivations. In view of that, I raise and answer the question how these two theories are related to each other and to standard topology. I show that causal set theory can be embedded into Maudlin׳s more general framework and I characterise what Maudlin׳s topological concepts boil down to when applied to discrete linear structures that correspond to causal sets. Moreover, I show that all topological aspects of causal sets that can be described in Maudlin׳s theory can also be described in the framework of standard topology. Finally, I discuss why these results are relevant for evaluating Maudlin׳s theory. The value of this theory depends crucially on whether it is true that (a) its conceptual framework is as expressive as that of standard topology when it comes to describing well-known continuous as well as discrete models of spacetime and (b) it is even more expressive or fruitful when it comes to analysing topological aspects of discrete structures that are intended as models of spacetime. On one hand, my theorems support (a). The theory is rich enough to incorporate causal set theory and its definitions of topological notions yield a plausible outcome in the case of causal sets. On the other hand, the results undermine (b). Standard topology, too, has the conceptual resources to capture those topological aspects of causal sets that are analysable within Maudlin׳s framework. This fact poses a challenge for the proponents of Maudlin׳s theory to prove it fruitful.     

The curvature argument

Dasgupta (2015) has recently put forward a novel argument, which he calls the ‘curvature argument’, that aims to show that Galilean spacetime is not an ideal setting for our classical theory of motion. This paper examines the curvature argument and argues that it is not sound. The discussion yields a remark about the conditions under which a ‘symmetry argument’ demonstrates that a particular spacetime is a non-ideal setting for our theory of motion.     

Friedman׳s thesis

This essay examines Friedman׳s recent approach to the analysis of physical theories. Friedman argues against Quine that the identification of certain principles as ‘constitutive’ is essential to a satisfactory methodological analysis of physics. I explicate Friedman׳s characterization of a constitutive principle, and I evaluate his account of the constitutive principles that Newtonian and Einsteinian gravitation presuppose for their formulation. I argue that something close to Friedman׳s thesis is defensible.     

Taking times out: Tense logic as a theory of time

Ulrich Meyer’s book The Nature of Time uses tense logic to argue for a ‘modal’ view of time, which replaces substantial times (as in Newton’s Absolute Time) with ‘ersatz times’ constructed using conceptually basic tense operators. He also argues against Bertrand Russell’s relationist theory, in which times are classes of events, and against the idea that relativity compels the integration of time and space (called by Meyer the Inseparability Argument). I find fault with each of these negative arguments, as well as with Meyer’s purported reconstruction of empty spacetime from tense operators and substantial spatial points. I suggest that Meyer’s positive project is best conceived as an elimination of time in the mode of Julian Barbour's The End of Time.     

The logic of experimental tests,particularly of Everettian quantum theory

Claims that the standard procedure for testing scientific theories is inapplicable to Everettian quantum theory, and hence that the theory is untestable, are due to misconceptions about probability and about the logic of experimental testing. Refuting those claims by correcting those misconceptions leads to an improved theory of scientific methodology (based on Popper׳s) and testing, which allows various simplifications, notably the elimination of everything probabilistic from the methodology (‘Bayesian’ credences) and from fundamental physics (stochastic processes).     

Theories of Newtonian gravity and empirical indistinguishability

In this essay, I examine the curved spacetime formulation of Newtonian gravity known as Newton–Cartan gravity and compare it with flat spacetime formulations. Two versions of Newton–Cartan gravity can be identified in the physics literature—a “weak” version and a “strong” version. The strong version has a constrained Hamiltonian formulation and consequently a well-defined gauge structure, whereas the weak version does not (with some qualifications). Moreover, the strong version is best compared with the structure of what Earman (World enough and spacetime. Cambridge: MIT Press) has dubbed Maxwellian spacetime. This suggests that there are also two versions of Newtonian gravity in flat spacetime—a “weak” version in Maxwellian spacetime, and a “strong” version in Neo-Newtonian spacetime. I conclude by indicating how these alternative formulations of Newtonian gravity impact the notion of empirical indistinguishability and the debate over scientific realism.     

Minkowski spacetime and Lorentz invariance: The cart and the horse or two sides of a single coin?

Michel Janssen and Harvey Brown have driven a prominent recent debate concerning the direction of an alleged arrow of explanation between Minkowski spacetime and Lorentz invariance of dynamical laws in special relativity. In this article, I critically assess this controversy with the aim of clarifying the explanatory foundations of the theory. First, I show that two assumptions shared by the parties—that the dispute is independent of issues concerning spacetime ontology, and that there is an urgent need for a constructive interpretation of special relativity—are problematic and negatively affect the debate. Second, I argue that the whole discussion relies on a misleading conception of the link between Minkowski spacetime structure and Lorentz invariance, a misconception that in turn sheds more shadows than light on our understanding of the explanatory nature and power of Einstein׳s theory. I state that the arrow connecting Lorentz invariance and Minkowski spacetime is not explanatory and unidirectional, but analytic and bidirectional, and that this analytic arrow grounds the chronogeometric explanations of physical phenomena that special relativity offers.     

Multiplicity in Everett׳s interpretation of quantum mechanics

Everett׳s interpretation of quantum mechanics was proposed to avoid problems inherent in the prevailing interpretational frame. It assumes that quantum mechanics can be applied to any system and that the state vector always evolves unitarily. It then claims that whenever an observable is measured, all possible results of the measurement exist. This notion of multiplicity has been understood in different ways by proponents of Everett׳s theory. In fact the spectrum of opinions on various ontological questions raised by Everett׳s approach is rather large, as we attempt to document in this critical review. We conclude that much remains to be done to clarify and specify Everett׳s approach.     

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.     

Newton–Cartan theory and teleparallel gravity: The force of a formulation

It is well-known that Newtonian gravity, commonly held to describe a gravitational force, can be recast in a form that incorporates gravity into the geometry of the theory: Newton–Cartan theory. It is less well-known that general relativity, a geometrical theory of gravity, can be reformulated in such a way that it resembles a force theory of gravity; teleparallel gravity does just this. This raises questions. One of these concerns theoretical underdetermination. I argue that these theories do not, in fact, represent cases of worrying underdetermination. On close examination, the alternative formulations are best interpreted as postulating the same spacetime ontology. In accepting this, we see that the ontological commitments of these theories cannot be directly deduced from their mathematical form. The spacetime geometry involved in a gravitational theory is not a straightforward consequence of anything internal to that theory as a theory of gravity. Rather, it essentially relies on the rest of nature (the non-gravitational interactions) conspiring to choose the appropriate set of inertial frames.