The Dirac spinor in six dimensions

1968 ◽  
Vol 64 (3) ◽  
pp. 765-778 ◽  
Author(s):  
E. A. Lord
Keyword(s):  
Lorentz Group ◽  
Dimensional Space ◽  
Rotation Group ◽  
Dimensional Subspace ◽  
Indefinite Metric ◽  
Dirac Spinor ◽  
Six Dimensions ◽  
Spinor Representations ◽  
Dirac Spinors

AbstractThe spinor representations of the rotation group in a six-dimensional space with indefinite metric are shown to be four-component spinors, which become the usual Dirac spinors when the formalism is restricted to a four-dimensional subspace. Eriksson's work on the five-dimensional Lorentz group is found to result from a restriction of the six-dimensional treatment to a five-dimensional subspace, and the algebraic significance of Eriksson's work is thereby clarified.

scholarly journals POVM construction: A simple recipe with applications to symmetric states

2017 ◽  
Vol 15 (06) ◽  
pp. 1750042
Author(s):  
Swarnamala Sirsi ◽  
Karthik Bharath ◽  
S. P. Shilpashree ◽  
H. S. Smitha Rao
Keyword(s):  
Orthonormal Basis ◽  
Dimensional Space ◽  
Rotation Group ◽  
Dimensional Subspace ◽  
Irreducible Representations ◽  
Projection Operators ◽  
Simple Method ◽  
Positive Operator Valued Measures ◽  
Higher Dimensional ◽  
Symmetric States

We propose a simple method for constructing positive operator-valued measures (POVMs) using any set of matrices which form an orthonormal basis for the space of complex matrices. Considering the orthonormal set of irreducible spherical tensors, we examine the properties of the construction on the [Formula: see text]-dimensional subspace of the [Formula: see text]-dimensional Hilbert space of [Formula: see text] qubits comprising the permutationally symmetric states. Using the notion of vectorization, the constructed POVMs are interpretable as projection operators in a higher-dimensional space. We then describe a method to physically realize the constructed POVMs for symmetric states using the Clebsch–Gordan decomposition of the tensor product of irreducible representations of the rotation group. We illustrate the proposed construction on a spin-1 system, and show that it is possible to generate entangled states from separable ones.

Rotations in a Non-Orthogonal Frame

2018 ◽  
Author(s):  
Shengnan Lu ◽  
Xilun Ding ◽  
Gregory S. Chirikjian
Keyword(s):  
Lie Group ◽  
Reference Frames ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Rotation Group ◽  
Dimensional Subspace ◽  
Iwasawa Decomposition ◽  
Discrete Family ◽  
Orthogonal Frame ◽  
Orthogonal Coordinates

This paper is concerned with describing the space of matrices that describe rotations in non-orthogonal coordinates. In scenarios such as in crystallography, conformational analysis of polymers, and in the study of deployable mechanisms and rigid origami, non-orthogonal reference frames are natural. For example, non-orthogonal vectors in the direction of atomic bonds in a molecule, the lattice coordinates of a crystal, or the directions of links in a mechanism are intrinsic. In these cases it is awkward to impose an artificial orthonormal reference frame rather than choosing one that is defined by the geometry of the object being studied. With these applications in mind, we fully characterize the space of all possible non-orthogonal rotations. We find that in the 2D case, this space is a three-dimensional subset of the special linear group, SL(2, R), which is itself a three-dimensional Lie group. In the 3D case we find that the space of nonorthogonal rotations is a seven-dimensional subspace of SL(3, R), which is an eight-dimensional Lie group. In the 2D case we use the Iwasawa decomposition to fully characterize the solution. In the 3D case we parameterize this seven-dimensional space by conjugating elements of the rotation group SO(3) by elements of a discrete family of of four-parameter subgroups of GL(3, R), and using this we derive an inversion formula to extract classical orthogonal rotations from those expressed in non-orthogonal coordinates.

scholarly journals Discovering a sparse set of pairwise discriminating features in high-dimensional data

2020 ◽  
Author(s):  
Samuel Melton ◽  
Sharad Ramanathan
Keyword(s):  
Single Cell ◽  
Dimensional Space ◽  
Cell Types ◽  
Dimensional Subspace ◽  
Supplementary Information ◽  
High Dimensional ◽  
Technological Advances ◽  
Data Points ◽  
Low Dimensional ◽  
Sparse Set

Abstract Motivation Recent technological advances produce a wealth of high-dimensional descriptions of biological processes, yet extracting meaningful insight and mechanistic understanding from these data remains challenging. For example, in developmental biology, the dynamics of differentiation can now be mapped quantitatively using single-cell RNA sequencing, yet it is difficult to infer molecular regulators of developmental transitions. Here, we show that discovering informative features in the data is crucial for statistical analysis as well as making experimental predictions. Results We identify features based on their ability to discriminate between clusters of the data points. We define a class of problems in which linear separability of clusters is hidden in a low-dimensional space. We propose an unsupervised method to identify the subset of features that define a low-dimensional subspace in which clustering can be conducted. This is achieved by averaging over discriminators trained on an ensemble of proposed cluster configurations. We then apply our method to single-cell RNA-seq data from mouse gastrulation, and identify 27 key transcription factors (out of 409 total), 18 of which are known to define cell states through their expression levels. In this inferred subspace, we find clear signatures of known cell types that eluded classification prior to discovery of the correct low-dimensional subspace. Availability and implementation https://github.com/smelton/SMD. Supplementary information Supplementary data are available at Bioinformatics online.

scholarly journals Complex Moment-Based Supervised Eigenmap for Dimensionality Reduction

2019 ◽  
Vol 33 ◽  
pp. 3910-3918 ◽  
Author(s):  
Akira Imakura ◽  
Momo Matsuda ◽  
Xiucai Ye ◽  
Tetsuya Sakurai
Keyword(s):  
Dimensionality Reduction ◽  
Parallel Implementation ◽  
Dimensional Space ◽  
Recognition Performance ◽  
Optimization Methods ◽  
Original Data ◽  
Dimensional Subspace ◽  
Reduction Methods ◽  
Low Dimensional ◽  
Matrix Trace

Dimensionality reduction methods that project highdimensional data to a low-dimensional space by matrix trace optimization are widely used for clustering and classification. The matrix trace optimization problem leads to an eigenvalue problem for a low-dimensional subspace construction, preserving certain properties of the original data. However, most of the existing methods use only a few eigenvectors to construct the low-dimensional space, which may lead to a loss of useful information for achieving successful classification. Herein, to overcome the deficiency of the information loss, we propose a novel complex moment-based supervised eigenmap including multiple eigenvectors for dimensionality reduction. Furthermore, the proposed method provides a general formulation for matrix trace optimization methods to incorporate with ridge regression, which models the linear dependency between covariate variables and univariate labels. To reduce the computational complexity, we also propose an efficient and parallel implementation of the proposed method. Numerical experiments indicate that the proposed method is competitive compared with the existing dimensionality reduction methods for the recognition performance. Additionally, the proposed method exhibits high parallel efficiency.

Immersions of space-time

1969 ◽  
Vol 47 (6) ◽  
pp. 607-609 ◽  
Author(s):  
Peter Rastall
Keyword(s):  
Dimensional Space ◽  
Dimensional Subspace ◽  
Space Time ◽  
Elementary Particles ◽  
Conformally Flat ◽  
Group Of Isometries

It is shown that space-time can be regarded as a 4-dimensional subspace of a conformally flat, 6-dimensional space, or of a flat, 8-dimensional space. The group of isometries of the tangent spaces of the 6-dimensional space is O(4, 2), which has recently been used to describe elementary particles.

scholarly journals Dynamical Codimension – 2 Brane in a Warped Six Dimensional Space-time

2018 ◽  
Vol 28 (3) ◽  
pp. 277
Author(s):  
Phan Hong Lien
Keyword(s):  
Einstein Equation ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Space Time ◽  
Cosmological Evolution ◽  
System Of Equations ◽  
Dimensional Space Time ◽  
Six Dimensions ◽  
Bulk Space

In this paper we present the Einstein equation extended in six-dimensions (6D) from the formation of codimension-2 brane, which is created by a 4-brane and 4-anti brane moving in the warped 6D “bulk” space-time. The system of equations of motion for the dynamical codimension - 2 brane has been derived to describe the cosmological evolution on the probe branes. Some cosmological consequences are investigated.

scholarly journals Quaternionic Bézier curves, surfaces and volume

2013 ◽  
Vol 54 ◽  
Author(s):  
Severinas Zube
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Dimensional Subspace ◽  
Bezier Curves ◽  
Bézier Curves ◽  
The Real

We extended the rational Bézier construction for linear, bi-linear and threelinear map, by allowing quaternion weights. These objects are Möbius invariant and have halved degree with respect to the real parametrization. In general, these parametrizations are in four dimensional space. We analyse when a special the three-linear parametrized volume is in usual three dimensional subspace and gives three orthogonal family of Dupine cyclides.

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