The expansion of physical quantities in terms of the irreducible representations of the scale-euclidean group and applications to the construction of scale-invariant correlation functions

1973 ◽  
Vol 50 (3) ◽  
pp. 194-221 ◽  
Author(s):  
H. E. Moses ◽  
A. F. Quesada
Keyword(s):  
Correlation Functions ◽  
Irreducible Representations ◽  
Euclidean Group ◽  
Scale Invariant ◽  
Physical Quantities

Best Matching: Technical Details

2018 ◽  
Author(s):  
Flavio Mercati
Keyword(s):  
Point Particle ◽  
Euclidean Group ◽  
Scale Invariant ◽  
Similarity Group ◽  
N Body Problem ◽  
Technical Details ◽  
Matching Procedure ◽  
Principal Fibre ◽  
Principal Fibre Bundle ◽  
Gauge Connection

The best matching procedure described in Chapter 4 is equivalent to the introduction of a principal fibre bundle in configuration space. Essentially one introduces a one-dimensional gauge connection on the time axis, which is a representation of the Euclidean group of rotations and translations (or, possibly, the similarity group which includes dilatations). To accommodate temporal relationalism, the variational principle needs to be invariant under reparametrizations. The simplest way to realize this in point–particle mechanics is to use Jacobi’s reformulation of Mapertuis’ principle. The chapter concludes with the relational reformulation of the Newtonian N-body problem (and its scale-invariant variant).

BIRTH OF ANOMALOUS SCALING IN A MODEL OF HYDRODYNAMIC TURBULENCE WITH A TUNABLE PARAMETER

Fractals ◽  
2002 ◽  
Vol 10 (03) ◽  
pp. 291-296
Author(s):  
D. PIEROTTI ◽  
V. S. L'VOV ◽  
A. POMYALOV ◽  
I. PROCACCIA
Keyword(s):  
Numerical Simulations ◽  
Correlation Functions ◽  
Order Structure ◽  
Anomalous Scaling ◽  
Scale Invariant ◽  
Scaling Exponents ◽  
Closed Set ◽  
Hydrodynamic Turbulence ◽  
Turbulent Field ◽  
Random Components

We introduce a model of hydrodynamic turbulence with a tunable parameter ε, which represents the ratio between deterministic and random components in the coupling between N identical copies of the turbulent field. To compute the anomalous scaling exponents ζn (of the nth order structure functions) for chosen values of ε, we consider a systematic closure procedure for the hierarchy of equations for the n-order correlation functions, in the limit N → ∞. The parameter ε regularizes the closure procedure, in the sense that discarded terms are of higher order in ε compared to those retained. It turns out that after the terms of O(1), the first nonzero terms are O(ε4). Within this ε-controlled procedure, we have a finite and closed set of scale-invariant equations for the 2nd and 3rd order statistical objects of the theory. This set of equations retains all terms of O(1) and O(ε4) and neglects terms of O(ε6). On this basis, we expect anomalous corrections δ ζn in the scaling exponents ζn to increase with εn. This expectation is confirmed by extensive numerical simulations using up to 25 copies and 28 shells for various values of εn. The simulations demonstrate that in the limit N → ∞, the scaling is normal for ε < ε cr with ε cr ≈ 0.6. We observed the birth of anomalous scaling at ε = ε cr with [Formula: see text] according to our expectation.

MODY: a program for calculation of symmetry-adapted functions for ordered structures in crystals

2004 ◽  
Vol 37 (6) ◽  
pp. 1015-1019 ◽  
Author(s):  
Wiesława Sikora ◽  
Franciszek Białas ◽  
Lucjan Pytlik
Keyword(s):  
Phase Transition ◽  
Magnetic Moments ◽  
Numerical Data ◽  
Irreducible Representations ◽  
High Symmetry ◽  
Ordered Structure ◽  
Ordered Structures ◽  
Symmetry Adapted Functions ◽  
Model Structures ◽  
Physical Quantities

This paper describes a computer program, based on the theory of groups and representations, which calculates symmetry-adapted functions used for the description of various ordered structures in crystals. It is assumed that the ordered structure, which is formed by a configuration of occupational probabilities, ion displacements, magnetic moments, quadrupolar moments or other local physical quantities, is obtained from a high-symmetry crystal structure with a given space groupG, as a result of a symmetry-lowering phase transition. The detailed characteristics of the phase transition are given by the specification of the irreducible representations of groupG, active in the transition. The symmetry-adapted functions obtained from the calculation are perfect tools for the construction of model structures, which can be used for comparison with experimental (e.g.neutron diffraction) data, and can be a great help in numerical data elaboration by reducing the number of adjustable parameters describing the structure of a given symmetry.

Canonical representations for massless particles and zero-mass limits of the helicity representation

1965 ◽  
Vol 285 (1401) ◽  
pp. 287-296 ◽  
Keyword(s):  
Lorentz Group ◽  
Irreducible Representations ◽  
Two Dimensional ◽  
Euclidean Group ◽  
Positive Mass ◽  
Continuous Spin ◽  
Massless Particles ◽  
Zero Mass ◽  
Canonical Representations ◽  
Inhomogeneous Lorentz Group

The global forms of the unitary irreducible representations of the inhomogeneous Lorentz group corresponding to zero mass and finite or continuous spin are constructed by means of the little-group technique from those of the two-dimensional Euclidean group, and it is shown that these representations may be derived from the helicity representation for positive mass by taking suitable limits.

scholarly journals INDECOMPOSABLE REPRESENTATIONS OF THE EUCLIDEAN ALGEBRA 𝔢(3) FROM IRREDUCIBLE REPRESENTATIONS OF

2011 ◽  
Vol 83 (3) ◽  
pp. 439-449 ◽  
Author(s):  
ANDREW DOUGLAS ◽  
JOE REPKA
Keyword(s):  
Euclidean Space ◽  
Lie Algebra ◽  
Semidirect Product ◽  
Lie Group ◽  
Three Dimensional ◽  
Irreducible Representations ◽  
Dimensional Euclidean Space ◽  
Euclidean Group ◽  
New Class ◽  
Euclidean Algebra

AbstractThe Euclidean group E(3) is the noncompact, semidirect product group E(3)≅ℝ3⋊SO(3). It is the Lie group of orientation-preserving isometries of three-dimensional Euclidean space. The Euclidean algebra 𝔢(3) is the complexification of the Lie algebra of E(3). We embed the Euclidean algebra 𝔢(3) into the simple Lie algebra $\mathfrak {sl}(4,\mathbb {C})$ and show that the irreducible representations V (m,0,0) and V (0,0,m) of $\mathfrak {sl}(4,\mathbb {C})$ are 𝔢(3)-indecomposable, thus creating a new class of indecomposable 𝔢(3) -modules. We then show that V (0,m,0) may decompose.

LOOP AVERAGES AND PARTITION FUNCTIONS IN U(N) GAUGE THEORY ON TWO-DIMENSIONAL MANIFOLDS

1990 ◽  
Vol 05 (09) ◽  
pp. 693-703 ◽  
Author(s):  
B. YE. RUSAKOV
Keyword(s):  
Gauge Theory ◽  
Gauge Group ◽  
Point Of View ◽  
Irreducible Representations ◽  
Partition Functions ◽  
Two Dimensional ◽  
Geometrical Characteristics ◽  
Physical Quantities

Loop averages and partition functions in the U(N) gauge theory are calculated for loops without intersections on arbitrary two-dimensional manifolds including nonorientable ones. The physical quantities are directly expressed through geometrical characteristics of a manifold (areas enclosed by loops and the genus) and gauge group parameters (Casimir eigenvalues and dimensions of the irreducible representations). It is shown that, from the physical quantities’ point of view, non-orientability of the manifold is equivalent to its non-compactness.

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