AUTOMATIC SYMMETRY INVESTIGATION OF SPACE-TIME METRICS

1994 ◽  
Vol 03 (01) ◽  
pp. 323-326 ◽  
Author(s):  
THOMAS WOLF ◽  
GUY GREBOT
Keyword(s):  
Lie Algebra ◽  
Computer Algebra ◽  
Space Time ◽  
Killing Equations

Computer algebra programs, [Formula: see text] and [Formula: see text] for automatically formulating and solving Killing equations and calculating the corresponding Lie algebra are described and examples are given.

SPECTRAL FLOW AND FREE FIELD REALIZATIONS OF BOSONIC STRINGS ON NAPPI–WITTEN GROUP MANIFOLD

2011 ◽  
Vol 26 (01) ◽  
pp. 149-160
Author(s):  
GANG CHEN
Keyword(s):  
Lie Algebra ◽  
Bosonic Strings ◽  
Free Field ◽  
Geometric Interpretation ◽  
Closed String ◽  
Space Time ◽  
Spectral Flow ◽  
Affine Lie Algebra ◽  
Group Manifold ◽  
Free Field Realization

In this paper we study some aspects of closed string theories in the Nappi–Witten space–time. The effects of spectral flow on the geodesics are studied in terms of an explicit parametrization of the group manifold. The worldsheets of the closed strings under the spectral flow of the geodesics can be classified into four classes, each with a geometric interpretation. We also obtain a free field realization of the Nappi–Witten affine Lie algebra in the most general conditions using a different but equivalent parametrization of the group manifold.

scholarly journals GROUND RING OF α-SYMMETRIES AND SEQUENCE OF RNS STRING THEORIES

2009 ◽  
Vol 24 (31) ◽  
pp. 5897-5924 ◽  
Author(s):  
DIMITRI POLYAKOV
Keyword(s):  
Lie Algebra ◽  
Extra Dimensions ◽  
Space Time ◽  
Field Theories ◽  
Superstring Theory ◽  
Local Gauge ◽  
Work In Progress ◽  
Gauge Symmetries ◽  
Solvable Lie Algebra ◽  
Virasoro Type

We construct a sequence of nilpotent BRST charges in RNS superstring theory, based on new gauge symmetries on the worldsheet, found in this paper. These new local gauge symmetries originate from the global nonlinear space–time α-symmetries, shown to form a noncommutative ground ring in this work. The important subalgebra of these symmetries is U (3) × X6, where X6 is solvable Lie algebra consisting of six elements with commutators reminiscent of the Virasoro type. We argue that the new BRST charges found in this work describe the kinetic terms in string field theories around curved backgrounds of the AdS × CP n-type, determined by the geometries of hidden extra dimensions induced by the global α-generators. The identification of these backgrounds is however left for the work in progress.

A FIRST STUDY OF GENERIC CUBIC TERMS IN GRAVITATION THEORIES

1993 ◽  
Vol 02 (03) ◽  
pp. 257-278
Author(s):  
H. CAPRASSE ◽  
J. DEMARET ◽  
P. HOUBA
Keyword(s):  
Exact Solutions ◽  
Computer Algebra ◽  
Space Time ◽  
Cosmological Models ◽  
Gravitation Theory ◽  
Algebraic Equations ◽  
Field Equations ◽  
Time Dimension ◽  
Bianchi I ◽  
The Way

The generic cubic contributions to the Lagrangian of gravitation theory are considered. Field equations are determined and put in their simplest form. In the framework of Bianchi I cosmological models with a metric which is power-like in time, algebraic equations are obtained and their exact solutions are derived exploiting computer algebra techniques. These solutions are fully discussed. The analysis is, presently, essentially restricted to a space-time dimension equal to four but results obtained here open the way to an analysis in any dimension.

scholarly journals Lie Invariants in Two and Three Variables

2014 ◽  
Vol 21 (01) ◽  
pp. 95-116
Author(s):  
Murray R. Bremner ◽  
Jiaxiong Hu
Keyword(s):  
Lie Algebra ◽  
Computer Algebra ◽  
Coefficient Matrix ◽  
Adjoint Representation ◽  
Free Lie Algebra ◽  
Natural Representation ◽  
3 Dimensional

We use computer algebra to determine the Lie invariants of degree ≤ 12 in the free Lie algebra on two generators corresponding to the natural representation of the simple 3-dimensional Lie algebra 𝔰𝔩2(ℂ). We then consider the free Lie algebra on three generators, and compute the Lie invariants of degree ≤ 7 corresponding to the adjoint representation of 𝔰𝔩2(ℂ), and the Lie invariants of degree ≤ 9 corresponding to the natural representation of 𝔰𝔩3(ℂ). We represent the action of 𝔰𝔩2(ℂ) and 𝔰𝔩3(ℂ) on Lie polynomials by computing the coefficient matrix with respect to the basis of Hall words.

scholarly journals Lie algebra in quantum physics by means of computer algebra

2017 ◽  
Author(s):  
Ichio Kikuchi ◽  
Akihito Kikuchi
Keyword(s):  
Quantum Mechanics ◽  
Angular Momentum ◽  
Lie Algebra ◽  
Computer Algebra ◽  
Quantum Physics ◽  
Character Formula ◽  
Weyl Character Formula ◽  
Weight Modules

This article explains how to apply the computer algebra package GAP (www.gap-system.org) in the computation of the problems in quantum physics, in which the application of Lie algebra is necessary. The article contains several exemplary computations which readers would follow in the desktop PC: such as, the brief review of elementary ideas of Lie algebra, the angular momentum in quantum mechanics, the quark eight-fold-way model, and the usage of Weyl character formula (in order to construct weight modules, and to count correctly the degeneracy

THE GENERALIZED SPHERICALLY-SYMMETRIC METRIC

2020 ◽  
Vol 22 (4) ◽  
pp. 223-226
Author(s):  
M.M. Khashaev
Keyword(s):  
Lie Algebra ◽  
Spherical Symmetry ◽  
Vector Fields ◽  
Killing Vector ◽  
Space Time ◽  
Spherically Symmetric ◽  
Killing Vector Fields ◽  
Parameter Group ◽  
Killing Vectors ◽  
Spherically Symmetric Metric

Four parameter group of transformations containing rotations and time translations is consi[1]dered due to spherical symmetry and stationarity of the space-time metric. It is found that there exists such a quartet of Killing vector fields which constitute the Lie algebra of the transforma[1]tion group and in which space-like vectors are not orthogonal to the time-like one. The metric corresponding to the Lie algebra of Killing vectors is composed. It is shown that the metric is non-static.

ADVANCES IN TERNARY AND OCTONIONIC GAUGE FIELD THEORIES

2011 ◽  
Vol 26 (18) ◽  
pp. 2997-3012 ◽  
Author(s):  
CARLOS CASTRO
Keyword(s):  
Field Theory ◽  
Gauge Field ◽  
Lie Algebra ◽  
Space Time ◽  
Time Dependent ◽  
Gauge Field Theory ◽  
Gauge Fields ◽  
Gauge Transformations ◽  
Quadratic Action ◽  
Rigid Transformations

A ternary gauge field theory is explicitly constructed based on a totally antisymmetric ternary-bracket structure associated with a 3-Lie algebra. It is shown that the ternary infinitesimal gauge transformations do obey the key closure relations [δ1, δ2] = δ3. Invariant actions for the 3-Lie algebra-valued gauge fields and scalar fields are displayed. We analyze and point out the difficulties in formulating a nonassociative octonionic ternary gauge field theory based on a ternary-bracket associated with the octonion algebra and defined earlier by Yamazaki. It is shown that a Yang–Mills-like quadratic action is invariant under global (rigid) transformations involving the Yamazaki ternary octonionic bracket, and that there is closure of these global (rigid) transformations based on constant antisymmetric parameters Λab = - Λba. Promoting the latter parameters to space–time dependent ones Λab(xμ) allows one to build an octonionic ternary gauge field theory when one imposes gauge covariant constraints on the latter gauge parameters leading to field-dependent gauge parameters and nonlinear gauge transformations. In this fashion one does not spoil the gauge invariance of the quadratic action under this restricted set of gauge transformations and which are tantamount to space–time dependent scalings (homothecy) of the gauge fields.

A THOUGHTFUL STUDY OF LORENTZ TRANSFORMATIONS BY CLIFFORD ALGEBRAS

1992 ◽  
Vol 07 (08) ◽  
pp. 1793-1817 ◽  
Author(s):  
J. RICARDO ZENI ◽  
WALDYR A. RODRIGUES
Keyword(s):  
Master Equation ◽  
Special Relativity ◽  
Lie Algebra ◽  
Physical Interpretation ◽  
Clifford Algebras ◽  
Space Time ◽  
Lorentz Transformations ◽  
Thomas Precession ◽  
Finite Form ◽  
Exact Product

We give a thoughtful presentation of proper orthocronous Lorentz transformations using the space-time (ℝ1,3) and the Pauli (ℝ3,0) algebras. We present in a closed finite form, a generic [Formula: see text] written as the exponential of elements of the associated Lie Algebra, together with a physical interpretation of the parameters. We call the result obtained the master equation and it is used to study several important topics of Special Relativity, e.g. exact product of two boosts and the Thomas precession. We also derive other results necessary for a dynamic interpretation of Lorentz transformations.

scholarly journals SCHRÖDINGER EQUATIONS WITH TIME-DEPENDENT P2 AND X2 TERMS

2002 ◽  
Vol 17 (11) ◽  
pp. 1559-1575
Author(s):  
MICHAEL MARTIN NIETO ◽  
D. RODNEY TRUAX
Keyword(s):  
Continuous Function ◽  
Lie Algebra ◽  
Coherent States ◽  
Schrodinger Equation ◽  
Space Time ◽  
Time Dependent ◽  
Squeezed States ◽  
Special Cases ◽  
Dependent Mass ◽  
Schrödinger System

We present some general results for the time-dependent mass Hamiltonian problem with H=-½e-2ν(t)∂xx+h(2)(t)e2ν(t)x2, where ν(t) is a continuous function of t. This Hamiltonian corresponds to a time-dependent mass (TM) Schrödinger equation with the restriction that there are only P2 and X2 terms. We give the specific transformations to a different quadratic Schrödinger (TQ) equation and to a different time-dependent oscillator (TO) equation. For each Schrödinger system, we give the Lie algebra of space–time symmetries and (x,t) representations for number states, coherent states and squeezed states. These general results include earlier works as special cases.

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