Mapping the axial current onto the hypersphere

1990 ◽  
Vol 68 (1) ◽  
pp. 54-57 ◽  
Author(s):  
D. G. C. McKeon
Keyword(s):  
Euclidean Space ◽  
Stereographic Projection ◽  
Axial Current ◽  
Dimensional Euclidean Space ◽  
Gauge Invariant ◽  
Five Dimensions ◽  
Conformally Invariant

A stereographic projection has been used to map conformally invariant spinor electrodynamics from four-dimensional Euclidean space onto the surface of a sphere in five dimensions. We extend electrodynamics so that the spinors also couple in a gauge invariant way to the current on the surface of the hypersphere that is the image of the axial current [Formula: see text] in Euclidean space under this mapping.

scholarly journals A relation between the right triangle and circular tori with constant mean curvature in the unit 3-sphere

2004 ◽  
Vol 76 (4) ◽  
pp. 639-643 ◽  
Author(s):  
Abdênago Barros
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Stereographic Projection ◽  
Inverse Image ◽  
Dimensional Euclidean Space ◽  
3 Dimensional ◽  
The Right

In this note we will show that the inverse image under the stereographic projection of a circular torus of revolution in the 3-dimensional euclidean space has constant mean curvature in the unit 3-sphere if and only if their radii are the catet and the hypotenuse of an appropriate right triangle.

Operator regularization on the hypersphere

1989 ◽  
Vol 67 (7) ◽  
pp. 669-677 ◽  
Author(s):  
D. G. C. McKeon
Keyword(s):  
Euclidean Space ◽  
Sigma Model ◽  
Gauge Theories ◽  
Stereographic Projection ◽  
Mass Parameter ◽  
Field Theories ◽  
Dimensional Euclidean Space ◽  
Two Dimensional ◽  
Yang Mills ◽  
Infrared Cutoff

Operator regularization has proved to be a viable way of computing radiative corrections that avoids both the insertion of a regulating parameter into the initial Lagrangian and the occurrence of explicit infinities at any stage of the calculation. We show how this regulating technique can be used in conjunction with field theories defined on an n + 1-dimensional hypersphere, which is the stereographic projection of n-dimensional Euclidean space. The radius of the hypersphere acts as an infrared cutoff, thus eliminating the need to insert a mass parameter to serve as an infrared regulator. This has the advantage of leaving conformai symmetry present in massless theories, intact. We illustrate our approach by considering [Formula: see text], massless Yang–Mills gauge theories and the two-dimensional nonlinear bosonic sigma model with torsion. In the last model, the lowest mode is used as an infrared cutoff.

scholarly journals GAUGE MODEL WITH EXTENDED FIELD TRANSFORMATIONS IN EUCLIDEAN SPACE

2000 ◽  
Vol 15 (02) ◽  
pp. 227-250 ◽  
Author(s):  
D. G. C. McKEON ◽  
T. N. SHERRY
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Dirac Field ◽  
Fundamental Representation ◽  
Gauge Invariant ◽  
Invariant Model ◽  
Gauge Transformations ◽  
Gauge Models ◽  
Extended Field ◽  
Matter Fields

An SO(4) gauge-invariant model with extended field transformations is examined in four-dimensional Euclidean space. The gauge field is (Aμ)αβ=½tμνλ(Mνλ)αβ where Mνλ are the SO(4) generators in the fundamental representation. The SO(4) gauge indices also participate in the Euclidean space SO(4) transformations giving the extended field transformations. We provide the decomposition of the reducible field tμνλ in terms of fields irreducible under SO(4). The SO(4) gauge transformations for the irreducible fields mix fields of different spin. Reducible matter fields are introduced in the form of a Dirac field in the fundamental representation of the gauge group and its decomposition in terms of irreducible fields is also provided. The approach is shown to be applicable also to SO(5) gauge models in five-dimensional Euclidean space.

On Vitali's Theorem for Groups of Motions of Euclidean Space

1999 ◽  
Vol 6 (4) ◽  
pp. 323-334
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Finite Dimensional

Abstract We give a characterization of all those groups of isometric transformations of a finite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem [Sul problema della misura dei gruppi di punti di una retta, 1905] holds true. This characterization is formulated in purely geometrical terms.

Some properties of Wigner coefficients and hyperspherical harmonics

1956 ◽  
Vol 52 (3) ◽  
pp. 424-430 ◽  
Author(s):  
A. P. Stone
Keyword(s):  
Angular Momentum ◽  
Euclidean Space ◽  
Rotation Group ◽  
Dimensional Euclidean Space ◽  
Hyperspherical Harmonics ◽  
Shift Operators ◽  
Analytical Relations

ABSTRACTGeneral shift operators for angular momentum are obtained and applied to find closed expressions for some Wigner coefficients occurring in a transformation between two equivalent representations of the four-dimensional rotation group. The transformation gives rise to analytical relations between hyperspherical harmonics in a four-dimensional Euclidean space.

On inequalities of the Tchebychev type

1963 ◽  
Vol 59 (1) ◽  
pp. 135-146 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Probability Theory ◽  
Euclidean Space ◽  
Random Variable ◽  
Dimensional Euclidean Space ◽  
Real Random Variable ◽  
Finite Dimensional

1. A type of problem which frequently occurs in probability theory and statistics can be formulated in the following way. We are given real-valued functions f(x), gi(x) (i = 1, 2, …, k) on a space (typically finite-dimensional Euclidean space). Then the problem is to set bounds for Ef(X), where X is a random variable taking values in , about which all we know is the values of Egi(X). For example, we might wish to set bounds for P(X > a), where X is a real random variable with some of its moments given.

Curve following in n—dimensional Euclidean space

SIMULATION ◽  
1973 ◽  
Vol 21 (5) ◽  
pp. 145-149 ◽  
Author(s):  
John Rees Jones
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space

scholarly journals On the Volume of the Zero Cell of a Class of Isotropic Poisson Hyperplane Tessellations

2014 ◽  
Vol 46 (3) ◽  
pp. 622-642 ◽  
Author(s):  
Julia Hörrmann ◽  
Daniel Hug
Keyword(s):  
Euclidean Space ◽  
Asymptotic Behaviour ◽  
Explicit Formula ◽  
Dimensional Euclidean Space ◽  
Arbitrary Dimensions ◽  
Zero Cell ◽  
Parametric Class

We study a parametric class of isotropic but not necessarily stationary Poisson hyperplane tessellations in n-dimensional Euclidean space. Our focus is on the volume of the zero cell, i.e. the cell containing the origin. As a main result, we obtain an explicit formula for the variance of the volume of the zero cell in arbitrary dimensions. From this formula we deduce the asymptotic behaviour of the volume of the zero cell as the dimension goes to ∞.

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