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The horizontal lines represent the operators and the ordered vertical lines represent the contractions between the individual fields inside the trace. (a) and (b) are planar; while (c) is nonplanar, since lines must cross.
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Quantum field theories are notoriously difficult to understand, physically as well as philosophically. The aim of this paper is to contribute to a better conceptual understanding of gauge quantum field theories, such as quantum chromodynamics, by discussing a famous physical limit, the 't Hooft limit, in which the theory concerned often simplifies....
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Citations
... So, the 'ordinary' spacetime of the theory has disappeared, and has given rise to another, emergent spacetime, whose properties are different from the original one. For a philosophical discussion of this kind of emergence, see Butterfield and Bouatta (2015). ...
While the relation between visualization and scientific understanding has been a topic of long-standing discussion, recent developments in physics have pushed the boundaries of this debate to new and still unexplored realms. For it is claimed that, in certain theories of quantum gravity, spacetime ‘disappears’: and this suggests that one may have sensible physical theories in which spacetime is completely absent. This makes the philosophical question whether such theories are intelligible, even more pressing. And if such theories are intelligible, the question then is how they manage to do so. In this paper, we adapt the contextual theory of scientific understanding, developed by one of us, to fit the novel challenges posed by physical theories without spacetime. We construe understanding as a matter of skill rather than just knowledge. The appeal is thus to understanding, rather than explanation, because we will be concerned with the tools that scientists have at their disposal for understanding these theories. Our central thesis is that such physical theories can provide scientific understanding, and that such understanding does not require spacetimes of any sort. Our argument consists of four consecutive steps: (a) We argue, from the general theory of scientific understanding, that although visualization is an oft-used tool for understanding, it is not a necessary condition for it; (b) we criticise certain metaphysical preconceptions which can stand in the way of recognising how intelligibility without spacetime can be had; (c) we catalogue tools for rendering theories without a spacetime intelligible; and (d) we give examples of cases in which understanding is attained without a spacetime, and explain what kind of understanding these examples provide.
... I will defend this claim by showing, in detail, how the steps underlying the perturbative construction of an effective QFT are physically justified and how the resulting parts of the theory are physically meaningful, unambiguously characterized and coherently related to one another-and this independently of the particular local QFT considered. And I will show how a careful comparison between the two approaches: (i) helps to dispel the mystery surrounding the success of the renormalization procedure discussed early on (e.g., Teller, 1988Teller, , 1989Weingard, 1995, 1996) but never fully dispelled in my sense, not even in the most recent literature (e.g., Butterfield and Bouatta, 2015;Crowther and Linnemann, 2017;Fraser J., 2017;; (ii) helps to clarify the various notions of renormalizability; and (iii) gives reasons to temper Butterfield and Bouatta's claim that some continuum QFTs are ripe for metaphysical inquiry (Butterfield and Bouatta, 2014;Butterfield, 2014). ...
... 6-9) has rightly argued, I believe, that effective Lagrangian QFTs are as well-defined as any of the past theories that we usually take to be mathematically well-defined, and therefore should be considered fit for foundational and philosophical scrutiny. Butterfield and Bouatta (2014) recently extended this claim to continuum QFTs (see also Butterfield, 2014, pp. 8-9, sec. ...
In this paper, I propose a general framework for understanding renormalization by drawing on the distinction between effective and continuum Quantum Field Theories (QFTs), and offer a comprehensive account of perturbative renormalization on this basis. My central claim is that the effective approach to renormalization provides a more physically perspicuous, conceptually coherent and widely applicable framework to construct perturbative QFTs than the continuum approach. I also show how a careful comparison between the two approaches: (i) helps to dispel the mystery surrounding the success of the renormalization procedure; (ii) clarifies the various notions of renormalizability; and (iii) gives reasons to temper Butterfield and Bouatta's claim that some continuum QFTs are ripe for metaphysical inquiry (Butterfield and Bouatta, 2014).
... Making sense of the approximations and idealisations of these kind of infinite limits has been the subject of a considerable philosophical literature of the last decade. 13 In fact, (Bouatta and Butterfield 2015) make a direct comparison between the 't Hooft limit and the thermodynamic limit. The important point for our purposes is that in the context of a 1/N expansion, the difference between 3 and ∞ at first order is small enough that subsisting one expansion with the other is a valid approximation provided one is only interested in order of magnitude estimates. ...
We provide an analysis of the empirical consequences of the AdS/CFT duality with reference to the application of the duality in a fundamental theory, effective theory and instrumental context. Analysis of the first two contexts is intended to serve as a guide to the potential empirical and ontological status of gauge/gravity dualities as descriptions of actual physics at the Planck scale. The third context is directly connected to the use of AdS/CFT to describe real quark-gluon plasmas. In the latter context, we find that neither of the two duals are confirmed by the empirical data.
... Making sense of the approximations and idealisations of these kind of infinite limits has been the subject of a considerable philosophical literature of the last decade. 13 In fact, (Bouatta and Butterfield 2015) make a direct comparison between the 't Hooft limit and the thermodynamic limit. The important point for our purposes is that in the context of a 1/N expansion, the difference between 3 ...
We provide an analysis of the empirical consequences of the AdS/CFT duality with reference to the application of the duality in a fundamental theory, effective theory and instrumental context. Analysis of the first two contexts is intended to serve as a guide to the potential empirical and ontological status of gauge/gravity dualities as descriptions of actual physics at the Planck scale. The third context is directly connected to the use of AdS/CFT to describe real quark-gluon plasmas. In the latter context, we find that neither of the two duals are confirmed by the empirical data.
... So, the 'ordinary' spacetime of the theory has disappeared, and has given rise to another, emergent spacetime, whose properties are different from the original one. For a philosophical discussion of this kind of emergence, see Butterfield and Bouatta (2015). ...
While the relation between visualization and scientific understanding has been a topic of long-standing discussion, recent developments in physics have pushed the boundaries of this debate to new and still unexplored realms. For it is claimed that, in certain theories of quantum gravity, spacetime 'disappears': and this suggests that one may have sensible physical theories in which spacetime is completely absent. This makes the philosophical question whether such theories are intelligible, even more pressing. And if such theories are intelligible, the question then is how they manage to do so. In this paper, we adapt the contextual theory of scientific understanding, developed by one of us, to fit the novel challenges posed by physical theories without spacetime. We construe understanding as a matter of skill rather than just knowledge. The appeal is thus to understanding, rather than explanation, because we will be concerned with the tools that scientists have at their disposal for understanding these theories. Our central thesis is that such physical theories can provide scientific understanding, and that such understanding does not require spacetimes of any sort. Our argument consists of four consecutive steps: (a) We argue, from the general theory of scientific understanding, that although visualization is an oft-used tool for understanding, it is not a necessary condition for it; (b) we criticise certain metaphysical preconceptions which can stand in the way of recognising how intelligibility without spacetime can be had; (c) we catalogue tools for rendering theories without a spacetime intelligible; and (d) we give examples of cases in which understanding is attained without a spacetime, and explain what kind of understanding these examples provide.
... 15 But we will not need such details here: for which we refer to e.g. Butterfield et al. (2015), of which we will now recall some notions:-An approximation is a particular kind of relation between theories, as follows. Consider a theory T b (where the subscript stands for 'better, bottom or basic') and a theory T t (for 'tainted, tangible or top' theory). ...
... Butterfield et al. (2015) relate their schema to a trichotomy ofNorton's (2011), allowing for both cases of idealisations and (good or bad) approximations. They relate this schema to their discussion of emergence, but this will not concern us here.16 ...
In this paper we have two aims: first, to draw attention to the close connexion between interpretation and scientific understanding; second, to give a detailed account of how theories without a spacetime can be interpreted, and so of how they can be understood. In order to do so, we of course need an account of what is meant by a theory `without a spacetime': which we also provide in this paper. We describe three tools, used by physicists, aimed at constructing interpretations which are adequate for the goal of understanding. We analyse examples from high-energy physics illustrating how physicists use these tools to construct interpretations and thereby attain understanding. The examples are: the 't Hooft approximation of gauge theories, random matrix models, causal sets, loop quantum gravity, and group field theory.
... 16 But we will not need such details here: for which we refer to e.g. Butterfield et al. (2015), of which we will now recall some notions:-An approximation is a particular kind of relation between theories, as follows. Consider a theory T b (where the subscript stands for 'better, bottom or basic') and a theory T t (for 'tainted, tangible or top' theory). ...
... Butterfield et al. (2015) relate their schema to a trichotomy ofNorton's (2011), allowing for both cases of idealisations and (good or bad) approximations. They relate this schema to their discussion of emergence, but this will not concern us here.17 ...
In this paper we have two aims: first, to draw attention to the close connexion between interpretation and scientific understanding; second, to give a detailed account of how theories without a spacetime can be interpreted, and so of how they can be understood. In order to do so, we of course need an account of what is meant by a theory `without a spacetime': which we also provide in this paper. We describe three tools, used by physicists, aimed at constructing interpretations which are adequate for the goal of understanding. We analyse examples from high-energy physics illustrating how physicists use these tools to construct interpretations and thereby attain understanding. The examples are: the 't Hooft approximation of gauge theories, random matrix models, causal sets, loop quantum gravity, and group field theory.
... While I am sympathetic to demanding that an interesting novel (i.e., emergent) property be robust sensu Butterfield and Bouatta, I find it helpful to distinguish the representational issues from the inferential ones. (Indeed, one finds in their later expositions that robustness enters only nominally (Bouatta and Butterfield, 2015) or not at all (Butterfield, 2014).) ...
I review and amplify on some of the many uses of representing a scientific theory in a particular context as a collection of models endowed with a similarity structure, which encodes the ways in which those models are similar to one another. This structure, which is related to topological structure, proves fruitful in the analysis of a variety of issues central to the philosophy of science. These include intertheoretic reduction, emergent properties, the epistemic connections between modeling and inference, the semantics of counterfactual conditionals, and laws of nature. The morals are twofold: first, the further adoption of formal methods for describing similarity (and related topological) structure has the potential to aid in decisive progress in philosophy of science; and second, the selection and justification of such structure is not a matter of technical convenience, but rather often involves great conceptual and philosophical subtlety. I conclude with various directions for future research.
... 3, this is the extension of the Poincaré group with the two additional generators D and K μ ; cf. (11)(12) and (16)(17). On the other hand, the SO(6) symmetry generators of S 5 correspond to the R-symmetry (see Sect. 3) transformations of the field theory (see e.g. ...
... These two principles will be explicated in (i)-(ii) below, which is called the minimalist conception of background-independence. Furthermore, there is an extended conception of background-independence, which will in addition add (iii) below: roughly, the condition that: (3) the boundary conditions also be background-independent. 17 Before proceeding, let us cash out the difference between the minimalist and the extended conceptions of background-independence, for gauge/gravity dualities, in a simple way as follows. The minimalist conception is the requirement of the background-independence of the bulk theory, i.e. the gravity theory in the (d + 1)-dimensional spacetime: so it is the sense in which general relativity is itself background-independent. ...
... Our understanding of the minimalist and the extended conceptions is thus that the former is the more essential principle to be preserved, while the latter can easily yield to other principles. 17 An earlier version of this paper only mentioned the minimalist conception of background-independence. In order to clarify the notion, and also in the light of [35] and [77], we have added an exposition of the extended conception of background-independence, from [33]. ...
We give an introductory review of gauge/gravity duality, and associated ideas
of holography, emphasising the conceptual aspects. The opening Sections gather
the ingredients, viz. anti-de Sitter spacetime, conformal field theory and
string theory, that we need for presenting, in Section 5, the central and
original example: Maldacena's AdS/CFT correspondence. Sections 6 and 7 develop
the ideas of this example, also in applications to condensed matter systems,
QCD, and hydrodynamics. Sections 8 and 9 discuss the possible extensions of
holographic ideas to de Sitter spacetime and to black holes. Section 10
discusses the bearing of gauge/gravity duality on two philosophical topics: the
equivalence of physical theories, and the idea that spacetime, or some features
of it, are emergent.
... For an introduction of the 't Hooft limit, see section 1.7.5 ofAmmon et al. (2015).Bouatta and Butterfield (2015) is a conceptual discussion of the 't Hooft limit. 8 There are numerous books that introduce string theory at various levels. Canonical and fairly comprehensive volumes introducing string theory at the graduate level includeGreen, Schwarz, Witten (1987) andPolchinski (1998). A volume also suitable for advanced undergraduates due to its m ...
We give an introductory review of gauge/gravity duality, and associated ideas of holography, emphasising the conceptual aspects. The opening Sections gather the ingredients, viz. anti-de Sitter spacetime, conformal field theory and string theory, that we need for presenting, in Section 5, the central and original example: Maldacena's AdS/CFT correspondence. Sections 6 and 7 develop the ideas of this example, also in applications to condensed matter systems, QCD, and hydrodynamics. Sections 8 and 9 discuss the possible extensions of holographic ideas to de Sitter spacetime and to black holes. Section 10 discusses the bearing of gauge/gravity duality on two philosophical topics: the equivalence of physical theories, and the idea that spacetime, or some features of it, are emergent.
