scholarly journals AN ALTERNATIVE DIMENSIONAL REDUCTION PRESCRIPTION

1996 ◽  
Vol 11 (13) ◽  
pp. 1037-1045 ◽  
Author(s):  
J.D. EDELSTEIN ◽  
C. NÚÑEZ ◽  
F.A. SCHAPOSNIK ◽  
J.J. GIAMBIAGI
Keyword(s):  
Dimensional Reduction ◽  
Dimensional Space ◽  
Dimensional Case ◽  
Green Functions ◽  
Two Dimensional ◽  
Spatial Coordinate ◽  
Higher Dimensional

We propose an alternative dimensional reduction prescription which in respect with Green functions corresponds to dropping the extra spatial coordinate. From this, we construct the dimensionally reduced Lagrangians both for scalars and fermions, discussing bosonization and supersymmetry in the particular two-dimensional case. We argue that our proposal is in some situations more physical in the sense that it maintains the form of the interactions between particles thus preserving the dynamics corresponding to the higher-dimensional space.

A Graphical Exposition of the Ising Problem

1971 ◽  
Vol 12 (3) ◽  
pp. 365-377 ◽  
Author(s):  
Frank Harary
Keyword(s):  
Dimensional Case ◽  
Two Dimensional ◽  
One Dimensional ◽  
Higher Dimensional ◽  
The One

Ising [1] proposed the problem which now bears his name and solved it for the one-dimensional case only, leaving the higher dimensional cases as unsolved problems. The first solution to the two dimensional Ising problem was obtained by Onsager [6]. Onsager's method was subsequently explained more clearly by Kaufman [3]. More recently, Kac and Ward [2] discovered a simpler procedure involving determinants which is not logically complete.

Some Helical Structures Generated by Reflexions

1985 ◽  
Vol 38 (3) ◽  
pp. 299 ◽  
Author(s):  
AC Hurley
Keyword(s):  
Dimensional Space ◽  
Helical Structure ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Helical Structures ◽  
Higher Dimensional ◽  
Translation Component ◽  
Explicit Formulae ◽  
Different Dimensions ◽  
Early Discussion

There has recently been a revival of interest in the helical structure built up as a column of face-sharing tetrahedra, because of possible applications in structural crystallography (Nelson 1983). This structure and its analogues in spaces of different dimensions are investigated here. It is shown that the only crystallographic cases are the structures in one- and two-dimensional space. For three and higher dimensional space the structures are all non-crystallographic. For the physically important case of three dimensions, this result is implicit in an early discussion by Coxeter (1969). Results obtained here include explicit formulae for the positions of all vertices of the simplexes for dimensions n = 1-4 and a demonstration that, for arbitrary n, the ratio of the translation component of the screw to the edge of the simplex is {6/ n(n+ I)(n+ 2)}1/2

scholarly journals Geometric and Differential Features of Scators as Induced by Fundamental Embedding

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1880
Author(s):  
Artur Kobus ◽  
Jan L. Cieśliński
Keyword(s):  
Building Block ◽  
Differential Calculus ◽  
Dimensional Space ◽  
Dimensional Case ◽  
Main Building ◽  
Higher Dimensional ◽  
Algebraic Operations ◽  
Low Dimensional

The scator space, introduced by Fernández-Guasti and Zaldívar, is endowed with a product related to the Lorentz rule of addition of velocities. The scator structure abounds with definitions calculationally inconvenient for algebraic operations, like lack of the distributivity. It occurs that situation may be partially rectified introducing an embedding of the scator space into a higher-dimensonal space, that behaves in a much more tractable way. We use this opportunity to comment on the geometry of automorphisms of this higher dimensional space in generic setting. In parallel, we develop commutative-hypercomplex analogue of differential calculus in a certain, specific low-dimensional case, as also leaned upon the notion of fundamental embedding, therefore treating the map as the main building block in completing the theory of scators.

Shadowing property of geodesics in Hedlund's metric

1997 ◽  
Vol 17 (1) ◽  
pp. 187-203 ◽  
Author(s):  
MARK LEVI
Keyword(s):  
Geodesic Flow ◽  
Rotation Vector ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Shadowing Property ◽  
Mather Theory ◽  
Higher Dimensional

In this paper we show that the geodesic flow in a Hedlund-type metric on the 3-torus possesses the shadowing property. This implies, in particular, that any rotation vector is represented by a geodesic, a fact that in the two-dimensional case is given by the Aubry–Mather theory, while in the higher-dimensional case is still unknown.

scholarly journals VANISHING DIMENSIONS: A REVIEW

2013 ◽  
Vol 28 (37) ◽  
pp. 1330034 ◽  
Author(s):  
DEJAN STOJKOVIC
Keyword(s):  
Dimensional Reduction ◽  
Dimensional Space ◽  
Planck Scale ◽  
Cosmic Ray ◽  
High Energy ◽  
Convincing Evidence ◽  
Quantum Theories ◽  
Higher Dimensional ◽  
Dynamical Triangulation ◽  
Lower Dimensional Space

We review a growing theoretical motivation and evidence that the number of dimensions actually reduces at high energies. This reduction can happen near the Planck scale, or much before, the dimensions that are reduced can be effective, spectral, topological or the usual dimensions, but many things point toward the fact that the high energy theories appear to propagate in a lower-dimensional space, rather than a higher-dimensional one. We will concentrate on a particular scenario of "vanishing" or "evolving dimensions" where the dimensions open up as we increase the length scale that we are probing, but will also mention related models that point to the same direction, i.e. the causal dynamical triangulation, asymptotic safety, as well as evidence coming from a noncommutative quantum theories, the Wheeler–DeWitt equation and phenomenon of "asymptotic silence". It is intriguing that experimental evidence for the high energy dimensional reduction may already exist — a statistically significant planar alignment of events with energies higher than TeV has been observed in high altitude cosmic ray experiments. A convincing evidence for dimensional reduction may be found in future in collider experiments and gravity waves observatories.

METHOD OF SOLVING CONFORMAL MODELS IN D-DIMENSIONAL SPACE III SECONDARY FIELDS IN D>2 AND THE SOLUTION OF TWO-DIMENSIONAL MODELS

1998 ◽  
Vol 13 (28) ◽  
pp. 4837-4888
Author(s):  
E. S. FRADKIN ◽  
M. YA. PALCHIK
Keyword(s):  
Hilbert Space ◽  
Dimensional Space ◽  
Conformal Symmetry ◽  
Solvable Model ◽  
Virasoro Algebra ◽  
Green Functions ◽  
Two Dimensional ◽  
Ward Identities ◽  
Fundamental Field ◽  
Conformal Models

We proceed with the study (started in Refs. 1 and 2) of the Hilbert space of conformal field theory in D di mensions. We discuss an infinite family of secondary fields [Formula: see text] generated by the action of the components of energy–momentum tensor Tμν on the fundamental (primary) field. It is shown that the states of these fields form a specific sector of the Hilbert space H which is determined by the Ward identities and [Formula: see text]-dimensional conformal symmetry. We demonstrate that for D = 2 the subspace H coincides with the space of representation of the Virasoro algebra. Each exactly solvable model in the case of D ≥ 2 is defined by the requirement of vanishing of a certain state Qs(x)|0> ⊂ H analogous to the null vector of two-dimensional theory. The Green functions of the fields [Formula: see text] are calculated in terms of the Green functions of the fundamental field. It is shown that all the Green functions of the type [Formula: see text] satisfy the anomalous Ward identities. The anomalous contributions are given by the fields [Formula: see text], where s′≤s-1. The fields Qs are constructed as superpositions of secondary fields with the anomalous contribution equal to zero, An approach developed is based on a finite-dimensional conformal symmetry for any D ≥ 2. Nevertheless the resulting models have the structure analogous to that of two-dimensional conformal theories. This analogy is discussed in detail. It is shown that for D = 2 the family of models coincides with the well-known family of conformal models based on infinite-dimensional conformal symmetry. The analysis of this phenomenon indicates the existence of the D-dimensional analog of the Virasoro algebra.

scholarly journals Kaluza–Klein reduction and Bergmann–Wagoner bi-scalar general action of scalar–tensor gravity

2015 ◽  
Vol 12 (10) ◽  
pp. 1550106 ◽  
Author(s):  
Kazuharu Bamba ◽  
Davood Momeni ◽  
Ratbay Myrzakulov
Keyword(s):  
Dimensional Reduction ◽  
Dimensional Space ◽  
Space Time ◽  
De Sitter ◽  
Higher Dimensional Space Time ◽  
Higher Dimensional ◽  
Dimensional Space Time ◽  
General Action ◽  
The Stability ◽  
Kaluza Klein

We examine the Kaluza–Klein (KK) dimensional reduction from higher dimensional space-time and the properties of the resultant Bergmann–Wagoner general action of scalar–tensor theories. With the analysis of the perturbations, we also investigate the stability of the anti-de Sitter (AdS) space-time in the (D ∈ 𝒩)-dimensional Einstein gravity with the negative cosmological constant. Furthermore, we derive the conditions for the dimensional reduction to successfully be executed and present the KK compactification mechanism.

scholarly journals Higher-Dimensional SUSY Quantum Mechanics

1997 ◽  
Vol 12 (08) ◽  
pp. 581-588 ◽  
Author(s):  
A. Das ◽  
S. Okubo ◽  
S. A. Pernice
Keyword(s):  
Quantum Mechanics ◽  
Spin Structure ◽  
Supersymmetric Quantum Mechanics ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Supersymmetry Algebra ◽  
General Properties ◽  
Higher Dimensional ◽  
Spatial Dimensions

Higher-dimensional supersymmetric quantum mechanics is studied. General properties of the two-dimensional case are presented. For three spatial dimensions or higher, a spin structure is shown to arise naturally from the nonrelativistic supersymmetry algebra.

Two-dimensional shear Alfvén-wave turbulence in a plasma with arbitrary β

1996 ◽  
Vol 55 (1) ◽  
pp. 121-130 ◽  
Author(s):  
G. Brodin
Keyword(s):  
Dimensional Space ◽  
Alfven Wave ◽  
Dimensional Case ◽  
Wave Turbulence ◽  
Two Dimensional ◽  
One Dimensional ◽  
Coupled Equations ◽  
Parallel Propagation ◽  
Derivative Nonlinear Schrödinger Equation ◽  
Space Dependence

Two-dimensional shear Alfvén-wave turbulence is investigated for a plasma with arbitrary β value. A set of three equations, which reduces to the derivative nonlinear Schrödinger equation in the limit of parallel propagation, is derived. The genuinely two-dimensional case is governed by two coupled equations. For a non-parallel one-dimensional space dependence, the nonlinearity is shown to vanish to all orders.

scholarly journals The Velocity Tensor and the Momentum Tensor

10.14311/1356 ◽  
2011 ◽  
Vol 51 (2) ◽  
Author(s):  
T. Lanczewski
Keyword(s):  
Momentum Tensor ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Higher Dimensional ◽  
Velocity Tensor

This paper introduces a new object called the momentum tensor. Together with the velocity tensorit forms a basis for establishing the tensorial picture of classical and relativistic mechanics. Some properties of the momentum tensor are derived as well as its relation with the velocity tensor. For the sake of clarity only two-dimensional case is investigated. However, general conclusions are also valid for higher dimensional spacetimes.

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