O(N)-symmetric vector models for N large

2021 ◽  
pp. 436-457
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Field Theory ◽  
Critical Systems ◽  
Fixed Dimension ◽  
Non Linear ◽  
Universal Properties ◽  
Limiting Case ◽  
Remarkable Relation ◽  
Insight Into

So far, universal properties of O(N) symmetric critical systems have been derived within the framework of the formal ϵ = 4−d expansion. Therefore, it is reassuring to verify that the results obtained in this way remain valid, at least in some limiting case, even when ϵ is not infinitesimal. Here it is shown in the example of the O(N) symmetric (ϕ2)2 field theory, the same universal properties can also be derived at fixed dimension in the large N limit and, more generally, order by order in an 1/N expansion. Moreover, large N techniques are also useful, because they provide an insight into other non-perturbative questions, including issues relevant to four-dimensional physics like renormalons and triviality. Using a large N expansion, we exhibit a remarkable relation between the (ϕ2)2 field theory and the non-linear σ-model, valid to all orders.

Stability of renormalization group fixed points and decay of correlations

2019 ◽  
pp. 101-110
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Field Theory ◽  
Renormalization Group ◽  
Fixed Points ◽  
Gradient Flow ◽  
Scalar Fields ◽  
Effective Field ◽  
Coupling Constants ◽  
Critical Systems ◽  
Universal Properties ◽  
General Physical

Chapter 7 is devoted to a discussion of the renormalization group (RG) flow when the effective field theory that describes universal properties of critical phenomena depends on several coupling constants. The universal properties of a large class of macroscopic phase transitions with short range interactions can be described by statistical field theories involving scalar fields with quartic interactions. The simplest critical systems have an O(N) orthogonal symmetry and, therefore, the corresponding field theory has only one quartic interaction. However, in more general physical systems, the flow of quartic interactions is more complicated. This chapter examines these systems from the RG viewpoint. RG beta functions are shown to generate a gradient flow. Some examples illustrate the notion of emergent symmetry. The local stability of fixed points is related to the value of the scaling field dimension.

A Review of Conformal Field Theory in 2D

2014 ◽  
Vol 6 (2) ◽  
pp. 1079-1105
Author(s):  
Rahul Nigam
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Conformal Field Theory ◽  
Statistical Mechanics ◽  
Critical Behavior ◽  
Verma Module ◽  
Virasoro Algebra ◽  
Quantum Field ◽  
Conformal Field ◽  
Insight Into

In this review we study the elementary structure of Conformal Field Theory in which is a recipe for further studies of critical behavior of various systems in statistical mechanics and quantum field theory. We briefly review CFT in dimensions which plays a prominent role for example in the well-known duality AdS/CFT in string theory where the CFT lives on the AdS boundary. We also describe the mapping of the theory from the cylinder to a complex plane which will help us gain an insight into the process of radial quantization and radial ordering. Finally we will develop the representation of the Virasoro algebra which is the well-known "Verma module".  

scholarly journals A master exceptional field theory

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Guillaume Bossard ◽  
Axel Kleinschmidt ◽  
Ergin Sezgin
Keyword(s):  
Field Theory ◽  
Gauge Invariance ◽  
Sigma Model ◽  
Linear Equations ◽  
Field Theories ◽  
Gauge Invariant ◽  
Non Linear ◽  
Non Linear Equations

Abstract We construct a pseudo-Lagrangian that is invariant under rigid E11 and transforms as a density under E11 generalised diffeomorphisms. The gauge-invariance requires the use of a section condition studied in previous work on E11 exceptional field theory and the inclusion of constrained fields that transform in an indecomposable E11-representation together with the E11 coset fields. We show that, in combination with gauge-invariant and E11-invariant duality equations, this pseudo-Lagrangian reduces to the bosonic sector of non-linear eleven-dimensional supergravity for one choice of solution to the section condi- tion. For another choice, we reobtain the E8 exceptional field theory and conjecture that our pseudo-Lagrangian and duality equations produce all exceptional field theories with maximal supersymmetry in any dimension. We also describe how the theory entails non-linear equations for higher dual fields, including the dual graviton in eleven dimensions. Furthermore, we speculate on the relation to the E10 sigma model.

Vortex-Induced Vibrations of a Rigid Cylinder on Non-Linear Elastic Supports

2006 ◽  
Author(s):  
S. Bourdier ◽  
J. R. Chaplin
Keyword(s):  
Circular Cylinder ◽  
Phase Flow ◽  
Linear Vibration ◽  
Chaotic Motions ◽  
Frequency Spectra ◽  
Elastic Supports ◽  
Linear Elastic ◽  
Vortex Induced Vibrations ◽  
Non Linear ◽  
Insight Into

The dynamics of vortex-induced vibrations of a rigid circular cylinder with structural non-linearities, introduced by means of discontinuities in the support system, are studied experimentally. The analysis of the measurements is carried out using non-linear vibration tools, i.e phase-flow portraits, frequency spectra, Lyapunov exponents and correlation dimensions, to provide an insight into the dynamical changes in the system brought about by restricting the motion. We show that chaotic motions can occur due to the structural non-linearities.

scholarly journals Applications of a working framework for the measurement of representative learning design in Australian football

PLoS ONE ◽  
2020 ◽  
Vol 15 (11) ◽  
pp. e0242336
Author(s):  
Peter R. Browne ◽  
Carl T. Woods ◽  
Alice J. Sweeting ◽  
Sam Robertson
Keyword(s):  
Association Rule ◽  
Analytical Techniques ◽  
Competitive Environment ◽  
Skill Transfer ◽  
Learning Design ◽  
Training Task ◽  
Linear Modelling ◽  
Non Linear ◽  
Australian Football ◽  
Insight Into

Representative learning design proposes that a training task should represent informational constraints present within a competitive environment. To assess the level of representativeness of a training task, the frequency and interaction of constraints should be measured. This study compared constraint interactions and their frequencies in training (match simulations and small sided games) with competition environments in elite Australian football. The extent to which constraints influenced kick and handball effectiveness between competition matches, match simulations and small sided games was determined. The constraints of pressure and time in possession were assessed, alongside disposal effectiveness, through an association rule algorithm. These rules were then expanded to determine whether a disposal was influenced by the preceding disposal. Disposal type differed between training and competition environments, with match simulations yielding greater representativeness compared to small sided games. The subsequent disposal was generally more effective in small sided games compared to the match simulations and competition matches. These findings offer insight into the measurement of representative learning designs through the non-linear modelling of constraint interactions. The analytical techniques utilised may assist other practitioners with the design and monitoring of training tasks intended to facilitate skill transfer from preparation to competition.

scholarly journals Kinetic field theory: Non-linear cosmic power spectra in the mean-field approximation

2021 ◽  
Vol 10 (6) ◽  
Author(s):  
Matthias Bartelmann ◽  
Johannes Dombrowski ◽  
Sara Konrad ◽  
Elena Kozlikin ◽  
Robert Lilow ◽  
...  
Keyword(s):  
Field Theory ◽  
Power Spectrum ◽  
Mean Field ◽  
Power Spectra ◽  
Field Approximation ◽  
Density Fluctuations ◽  
Mean Field Approximation ◽  
Non Linear ◽  
Two Parameters ◽  
Kinetic Field

We use the recently developed Kinetic Field Theory (KFT) for cosmic structure formation to show how non-linear power spectra for cosmic density fluctuations can be calculated in a mean-field approximation to the particle interactions. Our main result is a simple, closed and analytic, approximate expression for this power spectrum. This expression has two parameters characterising non-linear structure growth which can be calibrated within KFT itself. Using this self-calibration, the non-linear power spectrum agrees with results obtained from numerical simulations to within typically \lesssim10\,\%≲10% up to wave numbers k\lesssim10\,h\,\mathrm{Mpc}^{-1}k≲10hMpc−1 at redshift z = 0z=0. Adjusting the two parameters to optimise agreement with numerical simulations, the relative difference to numerical results shrinks to typically \lesssim 5\,\%≲5%. As part of the derivation of our mean-field approximation, we show that the effective interaction potential between dark-matter particles relative to Zel’dovich trajectories is sourced by non-linear cosmic density fluctuations only, and is approximately of Yukawa rather than Newtonian shape.

The Basics of Non-Linear Optics

2014 ◽  
pp. 214-250
Keyword(s):  
Field Theory ◽  
Nonlinear Optical ◽  
Linear Optics ◽  
Classical Level ◽  
Non Linear Optics ◽  
Light Coupling ◽  
Electromagnetic Field Theory ◽  
Non Linear ◽  
Linear Effects ◽  
Optical Phenomena

At a semi-classical level, the main analytical electromagnetic field theory tools were first used to describe the non-linear effects of light-light coupling as a basic cause of nonlinear optical phenomena and applications.

New insight into thend→3Hγprocess at thermal energy with pionless effective field theory

2014 ◽  
Vol 89 (6) ◽  
Author(s):  
M. Moeini Arani ◽  
H. Nematollahi ◽  
N. Mahboubi ◽  
S. Bayegan
Keyword(s):  
Field Theory ◽  
Thermal Energy ◽  
Effective Field Theory ◽  
Effective Field ◽  
Insight Into

CONFORMALLY COVARIANT COUPLED NON-LINEAR FIELD THEORY ON THE HYPERCONE: VACUUM SOLUTIONS AND QUANTIZATION OF NORMAL MODES

1988 ◽  
Vol 03 (02) ◽  
pp. 161-165
Author(s):  
T. AÇIKTEPE ◽  
K.G. AKDENIZ ◽  
A.O. BARUT ◽  
J. KALAYCI
Keyword(s):  
Field Theory ◽  
Normal Modes ◽  
Scalar Fields ◽  
Particle Spectrum ◽  
Geometric Meaning ◽  
Model Type ◽  
Quantized Fields ◽  
Non Linear ◽  
Vacuum Solutions ◽  
Linear Field

For the conformally covariant coupled non-linear spinor-scalar fields of the σ -model type we show that the non-trivial vacuum instanton solutions have a geometric meaning as constant spinors on the five-dimensional hypercone. The quantized fields around these solutions correspond to the normal modes of the hypercone. A connection is thus established between field theory, particle spectrum of the fields and quantized excitations of a geometry (the hypercone).

IMPROVED COSMOLOGICAL PARAMETERS AND TREATMENT OF THE 5-DIMENSIONAL KLEIN–GORDON FIELD AND DIRAC-FIELD ON THE BASIS OF THE PROJECTIVE UNIFIED FIELD THEORY

2009 ◽  
Vol 18 (09) ◽  
pp. 1903-1916 ◽  
Author(s):  
ERNST SCHMUTZER
Keyword(s):  
Field Theory ◽  
Continuum Mechanics ◽  
Cosmological Parameters ◽  
Dirac Field ◽  
Unified Field Theory ◽  
Unified Field ◽  
Klein Gordon ◽  
Insight Into

The 5-dimensional Projective Unified Field Theory (PUFT) elaborated and further developed by the author since 1957 is a geometrical semi-unified field theory restricting to gravitation, electromagnetism and scalarism. Up till now the substrate (matter) was described on a 5-dimensional phenomenological continuum mechanics. But it proved rather important to investigate the Klein–Gordon field and the Dirac field within this 5-dimensional concept of PUFT in order to get a better insight into some relationships of continuum mechanics mentioned, particularly with respect to cosmology.

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