On emergence in gauge theories at the ’t Hooft limit‘

Abstract

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. The idea of the limit is that the number N of colours goes to infinity. The simplifications that can happen in this limit, and that we will consider, are: the theory’s Feynman diagrams can be drawn on a plane without lines intersecting ; and the theory, or a sector of it, becomes integrable, and indeed corresponds to a well-studied system, viz. a spin chain. Planarity is important because it shows how a quantum field theory can exhibit extended, in particular string-like, structures; in some cases, this gives a connection with string theory, and so with its representation of gravity. Previous philosophical literature about how one theory might be emergent from, and-or reduced to, another one has tended to emphasize cases, such as occur in statistical mechanics, where the system before the limit has finitely many degrees of freedom. But here, our quantum field theories, including those on the way to the ’t Hooft limit, will have infinitely many degrees of freedom. Nevertheless, we will show how a recent schema by Butterfield and taxonomy by Norton apply to the quantum field theories we consider; and we will classify three physical properties of our theories in these terms. These properties are planarity and integrability, as in and above; and the behaviour of the beta-function reflecting, for example, asymptotic freedom. Our discussion of these properties, especially the beta-function, will also relate to recent philosophical debate about the propriety of assessing quantum field theories, whose rigorous existence is not yet proven

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Jeremy Butterfield
Cambridge University

References found in this work

Taking Particle Physics Seriously: A Critique of the Algebraic Approach to Quantum Field Theory.David Wallace - 2010 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 42 (2):116-125.
Quantum Mechanics: An Empiricist View.Paul Teller & Bas C. van Fraassen - 1995 - Philosophical Review 104 (3):457.

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Citations of this work

Dualities and Emergent Gravity: Gauge/Gravity Duality.Sebastian de Haro - 2017 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 59:109-125.
Emergence in Holographic Scenarios for Gravity.Dennis Dieks, Jeroen van Dongen & Sebastian de Haro - 2015 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 52 (Part B):203-216.
Comparing Dualities and Gauge Symmetries.Sebastian De Haro, Nicholas Teh & Jeremy N. Butterfield - 2017 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 59:68-80.
Interpreting Theories Without a Spacetime.Sebastian De Haro & Henk W. de Regt - 2018 - European Journal for Philosophy of Science 8 (3):631-670.

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