Emergence and Reduction Combined in Phase Transitions

In another paper (Butterfield 2011), one of us argued that emergence and reduction are compatible, and presented four examples illustrating both. The main purpose of this paper is to develop this position for the example of phase transitions. We take it that emergence involves behaviour that is novel compared with what is expected: often, what is expected from a theory of the system's microscopic constituents. We take reduction as deduction, aided by appropriate definitions. Then the main idea of our reconciliation of emergence and reduction is that one makes the deduction after taking a limit of an appropriate parameter $N$. Thus our first main claim will be that in some situations, one can deduce a novel behaviour, by taking a limit $N\to\infty$. Our main illustration of this will be Lee-Yang theory. But on the other hand, this does not show that the $N=\infty$ limit is physically real. For our second main claim will be that in such situations, there is a logically weaker, yet still vivid, novel behaviour that occurs before the limit, i.e. for finite $N$. And it is this weaker behaviour which is physically real. Our main illustration of this will be the renormalization group description of cross-over phenomena.Comment: 24 pp, v2: one minor change. Contribution to the Frontiers of Fundamental Physics (FFP 11) Conference Proceeding

On emergence in gauge theories at the 't Hooft limit

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 (or charges) goes to infinity. The simplifications that can happen in this limit, and that we will consider, are: (i) the theory's Feynman diagrams can be drawn on a plane without lines intersecting (called `planarity'); and (ii) 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 thus with gravity. Previous philosophical literature about how one theory (or a sector, or regime, of a 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 (i) and (ii) 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.Comment: 44 pp. arXiv admin note: text overlap with arXiv:1012.3983, arXiv:hep-ph/9802419, arXiv:1012.3997 by other author

Towards open-closed string duality: Closed Strings as Open String Fields

We establish a translation dictionary between open and closed strings, starting from open string field theory. Under this correspondence, (off-shell) level-matched closed string states are represented by star algebra projectors in open string field theory. Particular attention is paid to the zero mode sector, which is indispensable in order to generate closed string states with momentum. As an outcome of our identification, we show that boundary states, which in closed string theory represent D-branes, correspond to the identity string field in the open string side. It is to be remarked that closed string theory D-branes are thus given by an infinite superposition of star algebra projectors.Comment: 29 page

Preheating in Dirac-Born-Infeld inflation

We study how the universe reheats following an era of chaotic Dirac-Born-Infeld inflation, and compare the rate of particle production with that in models based on canonical kinetic terms. Particle production occurs through non-perturbative resonances whose structure is modified by the nonlinearities of the Dirac-Born-Infeld action. We investigate these modifications and show that the reheating process may be efficient. We estimate the initial temperature of the subsequent hot, radiation-dominated phase.Comment: 23 page

Duality invariance of all free bosonic and fermionic gauge fields

We give a simple general extension to all free bosonic and fermionic massless gauge fields of a recent proof that spin 2 is duality invariant in flat space. We also discuss its validity in (A)dS backgrounds and the relevance of supersymmetry.Comment: 3 page

Parent form for higher spin fields on anti-de Sitter space

We construct a first order parent field theory for free higher spin gauge fields on constant curvature spaces. As in the previously considered flat case, both Fronsdal's and Vasiliev's unfolded formulations can be reached by two different straightforward reductions. The parent theory itself is formulated using a higher dimensional embedding space and turns out to be geometrically extremely transparent and free of the intricacies of both of its reductions.Comment: 39 pages, LaTeX; misprints corrected, references adde

Transverse Invariant Higher Spin Fields

It is shown that a symmetric massless bosonic higher-spin field can be described by a traceless tensor field with reduced (transverse) gauge invariance. The Hamiltonian analysis of the transverse gauge invariant higher-spin models is used to control a number of degrees of freedom.Comment: 12 pages, no figures. The general proof and the example of a spin-3 adde

The asymptotic solution of a singularly perturbed Cauchy problem for Fokker-Planck equation

The asymptotic method is a very attractive area of applied mathematics. There are many modern research directions which use a small parameter such as statistical mechanics, chemical reaction theory and so on. The application of the Fokker-Planck equation (FPE) with a small parameter is the most popular because this equation is the parabolic partial differential equations and the solutions of FPE give the probability density function. In this paper we investigate the singularly perturbed Cauchy problem for symmetric linear system of parabolic partial differential equations with a small parameter. We assume that this system is the Tikhonov non-homogeneous system with constant coefficients. The paper aims to consider this Cauchy problem, apply the asymptotic method and construct expansions of the solutions in the form of two-type decomposition. This decomposition has regular and border-layer parts. The main result of this paper is a justification of an asymptotic expansion for the solutions of this Cauchy problem. Our method can be applied in a wide variety of cases for singularly perturbed Cauchy problems of Fokker-Planck equations.Асимптотические методы - очень важная область прикладной математики. Существует множество современных направлений исследований, в которых используется малый параметр, например статистическая механика, теория химических реакций и др. Использование уравнения Фоккера-Планка с малым параметром очень востребовано, поскольку это уравнение является параболическим дифференциальным уравнением в частных производных, а решения этого уравнения дают функцию плотности вероятности. В работе исследуется сингулярно возмущённая задача Коши для симметричной линейной системы параболических дифференциальных уравнений в частных производных с малым параметром. Мы предполагаем, что эта система является неоднородной системой тихоновского типа с постоянными коэффициентами. Цель исследования - рассмотреть эту задачу Коши, применить асимптотический метод и построить асимптотические разложения решений в виде двухкомпонентного ряда. Таким образом, это разложение имеет регулярную и погранслойную части. Основным результатом данной работы является обоснование асимптотического разложения для решений этой задачи Коши. Наш метод может быть применён для широкого круга сингулярно возмущённых задач Коши для уравнений Фоккера-Планка

A Note on Chern-Simons Solitons - a type III vortex from the wall vortex

We study some properties of topological Chern-Simons vortices in 2 + 1 dimensions. As has already been understood in the past, in the large magnetic flux limit, they are well described by a Chern-Simons domain wall, which has been compactified on a circle with the symmetric phase inside and the asymmetric phase on the outside. Our goal is two-fold. First we want to explore how the tension depends on the magnetic flux discretized by the integer n. The BPS case is already known, but not much has been explored about the non-BPS potentials. A generic renormalizable potential has two dimensionless parameters that can be varied. Variation of only one of them lead to a type I and type II vortex, very similar to the Abrikosov-Nielsen-Olesen (ANO) case. Variation of both the parameters leads to a much richer structure. In particular we have found a new type of vortex, which is type I-like for small flux and then turns type II-like for larger flux. We could tentatively denote it a type III vortex. This results in a stable vortex with number of fluxes which can be greater than one. Our second objective is to study the Maxwell-Chern-Simons theory and and understand how the large n limit of the CS vortex is smoothly connected with the large n limit of the ANO vortex.Comment: 27 pages, 17 figures; v2.: references added, subsection 3.2 added, explanation added in section 2.