Algebraic combinatorics and QFT

2021 ◽  
pp. 76-94
Author(s):  
Adrian Tanasa
Keyword(s):  
Hopf Algebra ◽  
Hochschild Cohomology ◽  
Algebraic Combinatorics ◽  
Fourth Section ◽  
Analytic Combinatorics ◽  
Multi Scale ◽  
Starting Point ◽  
Feynman Graphs ◽  
Perturbative Renormalization

We have seen in the previous chapter some of the non-trivial interplay between analytic combinatorics and QFT. In this chapter, we illustrate how yet another branch of combinatorics, algebraic combinatorics, interferes with QFT. In this chapter, after a brief algebraic reminder in the first section, we introduce in the second section the Connes–Kreimer Hopf algebra of Feynman graphs and we show its relation with the combinatorics of QFT perturbative renormalization. We then study the algebra's Hochschild cohomology in relation with the combinatorial Dyson–Schwinger equation in QFT. In the fourth section we present a Hopf algebraic description of the so-called multi-scale renormalization (the multi-scale approach to the perturbative renormalization being the starting point for the constructive renormalization programme).

scholarly journals Aristotle’s Refutation of the Eleatic Argument in Physics I.8

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Takashi Oki
Keyword(s):  
Starting Point ◽  
Composite Nature

In this paper, I show that Aristotle’s refutation of the Eleatic argument in Physics I.8 is based on the idea that a thing at the starting point of coming to be is composite and is made up of what underlies and a priva­tion. In doing so, I clarify how the concept of accidentality as used in his solution should be understood in relation to the composite nature of what comes to be. I also suggest an explanation of why Aristotle’s discus­sion of the Eleatic dilemma in Physics I.8, unlike his discussion in the previous chapter, is not clear.

The Lyman-α‎ Line as a Probe of the Early Universe

2013 ◽  
Author(s):  
Abraham Loeb ◽  
Steven R. Furlanetto
Keyword(s):  
Simple Model ◽  
High Redshift ◽  
Young Star ◽  
H Ii Regions ◽  
Host Galaxies ◽  
Star Forming ◽  
Starting Point ◽  
Good Starting Point ◽  
High Redshift Universe

This chapter investigates a number of specific observational probes of the high-redshift Universe. It examines the Lyman-α‎ line, an extraordinarily rich and useful—albeit complex—probe of both galaxies and the intergalactic medium (IGM). As established in the previous chapter, young star-forming galaxies can produce very bright Lyman-α‎ emissions. Although the radiative transfer of these photons through their host galaxies is typically very complex, a good starting point is a simple model in which a fraction of stellar ionizing photons are absorbed within their source galaxy, forming embedded H II regions. The resulting protons and electrons then recombine, producing Lyman-α‎ photons. Assuming ionization equilibrium, the rate of these recombinations must equal the rate at which ionizing photons are produced.

Institutional Theory and E-Government Research

2008 ◽  
pp. 349-360 ◽  
Author(s):  
Shahidul Hassan ◽  
J. Ramon Gil-Garcia
Keyword(s):  
Information Systems ◽  
Institutional Theory ◽  
Information Technologies ◽  
Future Research ◽  
Fourth Section ◽  
Current State ◽  
Starting Point ◽  
Recent Developments ◽  
Good Starting Point ◽  
Government Research

Recent developments in institutional theory are highly promising for the study of e-government. Scholars in various disciplines, such as economics (North, 1999; Rutherford, 1999), sociology (Brinton & Nee, 1998), and political science (March & Olsen, 1989; Peters, 2001), have used institutional approaches to understand diverse social and organizational phenomena. Insights gained from these studies can be valuable for guiding research in e-government. In fact, there are some initial efforts in information systems and e-government research that have applied institutional theory and proved useful in generating new insights about how information technologies are adopted (Teo, Wei, & Benbasat, 2003; Tingling & Parent, 2002), designed and developed (Butler, 2003; Klein, 2000; Laudon, 1985), implemented (Robey & Holmstrom, 2001), and used (Fountain, 2001) in organizations. In this chapter, we provide a brief overview of some of these initial studies to highlight the usefulness of institutional theory in e-government research. We also suggest some opportunities for future research in e-government using institutional theory. This chapter does not capture all the essential theoretical and empirical issues related to using institutional theory in information systems and e-government research. Instead, it is a brief review and a good starting point to explore the potential of institutional theory. We hope that e-government scholars find it interesting and useful. The chapter is organized in five sections, including this introduction. The second section provides a brief overview of institutional theory in various disciplinary traditions, with an emphasis on institutional theory in sociology. Then the chapter identifies various patterns of the use of institutional theory in information systems and e-government research. Based on our analysis of the current state of the art, the fourth section suggests some opportunities for future research. Finally, the fifth section provides some final comments.

Legal Form

2020 ◽  
pp. 43-56
Author(s):  
Zoe Adams
Keyword(s):  
Capitalist Society ◽  
Labour Law ◽  
Legal Form ◽  
Fourth Section ◽  
The Third ◽  
The Relationship ◽  
Shed Light

The chapter builds on the analysis in Chapter 1 with a view to exploring the nature of law and its relationship with capitalist society in more detail. The previous chapter used an analysis of capitalism’s deep structures to explore the nature of law’s role(s) in capitalism, engaging with the various legal ‘functions’ that capitalism presupposes. The purpose of this chapter is to explore the implications of this understanding of law’s role (or function) when it comes to understanding law’s form. The first section begins by developing a theory of the legal form by engaging with the work of Evgeny Pashukanis. The second section teases out the implications of this analysis for our understanding of the relationship between the legal form and capitalism’s contradictions. The third section draws on this analysis to shed light on the relationship between legal form and content. The fourth section makes some tentative conclusions about the implications of this analysis for our understanding of labour law.

scholarly journals THE DIRICHLET HOPF ALGEBRA OF ARITHMETICS

2007 ◽  
Vol 16 (04) ◽  
pp. 379-438 ◽  
Author(s):  
BERTFRIED FAUSER ◽  
P. D. JARVIS
Keyword(s):  
Hopf Algebra ◽  
Arithmetic Functions ◽  
Algebraic Structures ◽  
Axiom A ◽  
Matrix Mechanics ◽  
Starting Point ◽  
Index Sets ◽  
Coprime Integers ◽  
Free Development ◽  
Quantum Matrix

Many constructs in mathematical physics entail notational complexities, deriving from the manipulation of various types of index sets which often can be reduced to labelling by various multisets of integers. In this work, we develop systematically the "Dirichlet Hopf algebra of arithmetics" by dualizing the addition and multiplication maps. Then we study the additive and multiplicative antipodal convolutions which fail to give rise to Hopf algebra structures, but form only a weaker Hopf gebra obeying a weakened homomorphism axiom. A careful identification of the algebraic structures involved is done featuring subtraction, division and derivations derived from coproducts and chochains using branching operators. The consequences of the weakened structure of a Hopf gebra on cohomology are explored, showing this has major impact on number theory. This features multiplicativity versus complete multiplicativity of number theoretic arithmetic functions. The deficiency of not being a Hopf algebra is then cured by introducing an "unrenormalized" coproduct and an "unrenormalized" pairing. It is then argued that exactly the failure of the homomorphism property (complete multiplicativity) for non-coprime integers is a blueprint for the problems in quantum field theory (QFT) leading to the need for renormalization. Renormalization turns out to be the morphism from the algebraically sound Hopf algebra to the physical and number theoretically meaningful Hopf gebra (literally: antipodal convolution). This can be modelled alternatively by employing Rota–Baxter operators. We stress the need for a characteristic-free development where possible, to have a sound starting point for generalizations of the algebraic structures. The last section provides three key applications: symmetric function theory, quantum (matrix) mechanics, and the combinatorics of renormalization in QFT which can be discerned as functorially inherited from the development at the number-theoretic level as outlined here. Hence the occurrence of number theoretic functions in QFT becomes natural.

scholarly journals Repensando las políticas de privatización en educación: El cercamiento de la escuela

2016 ◽  
Vol 24 ◽  
pp. 123 ◽  
Author(s):  
Noelia Fernández-González
Keyword(s):  
Land Rights ◽  
Reform Process ◽  
State Reform ◽  
Fourth Section ◽  
Autonomous Region ◽  
The Public ◽  
Starting Point ◽  
Communal Land ◽  
The Commons ◽  
Current Context

This article aims to introduce the concept of enclosure as a category to think about privatization policies in education. The concept of enclosure refers to the process by which communal land rights and uses were removed between the 14th and 18th centuries, making possible the passage from feudalism to capitalism. Nowadays, a discourse named as a “commons paradigm” (Bollier, 2007) exposes privatization dynamics as a contemporary movement of enclosure. This paradigm stablishes an analogy between the enclosures that made possible the primitive accumulation and the contemporary dynamics of privatization. In this text, privatization policies in education are analyzed as a movement of enclosure in the school. The text is divided into four sections. Firstly, it analyzes the state-reform process in the current context of globalization and the blurring of boundaries between the public and the private. Secondly, it focuses in the “commons paradigm”, followed by its critics in the next section. The fourth section reflects on the enclosure of the school taking as starting point previous research about privatization policies introduced in Spain and particularly in the Autonomous Region of Madrid.

Against the Yellowwashing of Israel: The BDS Movement and Liberatory Solidarities across Settler States

2017 ◽  
Author(s):  
Candace Fujikane
Keyword(s):  
World War Ii ◽  
Asian American ◽  
Pacific Islanders ◽  
Japanese American ◽  
World War ◽  
Alternative Evaluation ◽  
Starting Point

Following the focus on Palestine in the previous chapter, Chapter 9 takes as a critical starting point the complicit relationship Asian American politicians such as Senator Daniel Inouye shared with Israel. Such complicit “yellowwashing,” which involves a strategic remembrance of World War II–era Japanese American incarceration, presages Fujikane’s alternative evaluation of “liberatory solidarities” between Pacific Islanders and Palestinians.

Tree weights and renormalization in QFT

2021 ◽  
pp. 39-49
Author(s):  
Adrian Tanasa
Keyword(s):  
Partition Function ◽  
Fundamental Theorem ◽  
Functional Integral ◽  
Perturbative Expansion ◽  
Functional Integrals ◽  
Perturbative Series ◽  
Fundamental Step ◽  
Feynman Graphs ◽  
Perturbative Renormalization ◽  
Feynman Amplitudes

In this chapter we define specific tree weights which appear natural when considering a certain approach to non-perturbative renormalization in QFT, namely the constructive renormalization. Several examples of such tree weights are explicitly given in Appendix A. A fundamental step in QFT is to compute the logarithm of functional integrals used to define the partition function of a given model This comes from a fundamental theorem of enumerative combinatorics, stating the logarithm counts the connected objects. The main advantage of the perturbative expansion of a QFT into a sum of Feynman amplitudes is to perform this computation explicitly: the logarithm of the functional integral is the sum of Feynman amplitudes restricted to connected graphs. The main disadvantage is that the perturbative series indexed by Feynman graphs typically diverges.

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