Reduction of representations of the complementary series of the 2+3 de Sitter group with respect to the Lorentz group

1978 ◽  
Vol 37 (2) ◽  
pp. 1017-1022
Author(s):  
V. F. Molchanov
Keyword(s):  
Lorentz Group ◽  
De Sitter ◽  
Complementary Series ◽  
Sitter Group

scholarly journals Einstein’s E=mc2 Derivable from Heisenberg’s Uncertainty Relations

2019 ◽  
Vol 1 (2) ◽  
pp. 236-251 ◽  
Author(s):  
Sibel Başkal ◽  
Young S. Kim ◽  
and Marilyn E. Noz
Keyword(s):  
Commutation Relation ◽  
Lorentz Group ◽  
Hermitian Matrix ◽  
Uncertainty Relations ◽  
De Sitter ◽  
Oscillator Representation ◽  
Sitter Group ◽  
Hermitian Conjugate ◽  
Step Down ◽  
The One

Heisenberg’s uncertainty relation can be written in terms of the step-up and step-down operators in the harmonic oscillator representation. It is noted that the single-variable Heisenberg commutation relation contains the symmetry of the S p ( 2 ) group which is isomorphic to the Lorentz group applicable to one time-like dimension and two space-like dimensions, known as the O ( 2 , 1 ) group. This group has three independent generators. The one-dimensional step-up and step-down operators can be combined into one two-by-two Hermitian matrix which contains three independent operators. If we use a two-variable Heisenberg commutation relation, the two pairs of independent step-up, step-down operators can be combined into a four-by-four block-diagonal Hermitian matrix with six independent parameters. It is then possible to add one off-diagonal two-by-two matrix and its Hermitian conjugate to complete the four-by-four Hermitian matrix. This off-diagonal matrix has four independent generators. There are thus ten independent generators. It is then shown that these ten generators can be linearly combined to the ten generators for Dirac’s two oscillator system leading to the group isomorphic to the de Sitter group O ( 3 , 2 ) , which can then be contracted to the inhomogeneous Lorentz group with four translation generators corresponding to the four-momentum in the Lorentz-covariant world. This Lorentz-covariant four-momentum is known as Einstein’s E = m c 2 .

scholarly journals Poincaré Symmetry from Heisenberg’s Uncertainty Relations

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 409 ◽  
Author(s):  
Sibel Başkal ◽  
Young Kim ◽  
Marilyn Noz
Keyword(s):  
Lorentz Group ◽  
Uncertainty Relations ◽  
Quantum Field ◽  
De Sitter ◽  
Sitter Group ◽  
Fundamental Symmetry ◽  
Group I ◽  
Poincaré Symmetry ◽  
Three Space ◽  
Inhomogeneous Lorentz Group

It is noted that the single-variable Heisenberg commutation relation contains the symmetry of the S p ( 2 ) group which is isomorphic to the Lorentz group applicable to one time-like dimension and two space-like dimensions, known as the S O ( 2 , 1 ) group. According to Paul A. M. Dirac, from the uncertainty commutation relations for two variables, it possible to construct the de Sitter group S O ( 3 , 2 ) , namely the Lorentz group applicable to three space-like variables and two time-like variables. By contracting one of the time-like variables in S O ( 3 , 2 ) , it is possible to construct the inhomogeneous Lorentz group I S O ( 3 , 1 ) which serves as the fundamental symmetry group for quantum mechanics and quantum field theory in the Lorentz-covariant world. This I S O ( 3 , 1 ) group is commonly known as the Poincaré group.

scholarly journals Integration of Dirac’s Efforts to Construct a Quantum Mechanics Which is Lorentz-Covariant

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1270
Author(s):  
Young S. Kim ◽  
Marilyn E. Noz
Keyword(s):  
Quantum Mechanics ◽  
Lie Algebra ◽  
Lorentz Group ◽  
Gaussian Function ◽  
Uncertainty Relations ◽  
De Sitter ◽  
Sitter Group ◽  
Fundamental Symmetry ◽  
Oscillator Wave Function ◽  
Inhomogeneous Lorentz Group

The lifelong efforts of Paul A. M. Dirac were to construct localized quantum systems in the Lorentz covariant world. In 1927, he noted that the time-energy uncertainty should be included in the Lorentz-covariant picture. In 1945, he attempted to construct a representation of the Lorentz group using a normalizable Gaussian function localized both in the space and time variables. In 1949, he introduced his instant form to exclude time-like oscillations. He also introduced the light-cone coordinate system for Lorentz boosts. Also in 1949, he stated the Lie algebra of the inhomogeneous Lorentz group can serve as the uncertainty relations in the Lorentz-covariant world. It is possible to integrate these three papers to produce the harmonic oscillator wave function which can be Lorentz-transformed. In addition, Dirac, in 1963, considered two coupled oscillators to derive the Lie algebra for the generators of the O(3,2) de Sitter group, which has ten generators. It is proven possible to contract this group to the inhomogeneous Lorentz group with ten generators, which constitute the fundamental symmetry of quantum mechanics in Einstein’s Lorentz-covariant world.

De Sitter symplectic spaces and their quantizations

1974 ◽  
Vol 76 (2) ◽  
pp. 473-480 ◽  
Author(s):  
J. H. Rawnsley
Keyword(s):  
Lorentz Group ◽  
Bound States ◽  
Kepler Problem ◽  
Classical System ◽  
De Sitter ◽  
Sitter Group ◽  
Group Spin ◽  
Theoretical View ◽  
Simply Connected ◽  
Inhomogeneous Lorentz Group

The de Sitter group, Spin (4, 1), is a simply connected, semi-simple, ten-dimensional Lie group which can be contracted to the inhomogeneous Lorentz group. Physical systems with the de Sitter group as asymmetry group should resemble those of the inhomogeneous Lorentz group and may provide an alternative to these systems of special-relativistic physics. Details of physics in de Sitter space from the group theoretical view-point are given in (3). The de Sitter group is also known to be a symmetry group for the bound states of the hydrogen atom, and recent work has shown how this group acts on the corresponding classical system, the Kepler problem. See (8).

scholarly journals Wavefunctions in dS/CFT revisited: principal series and double-trace deformations

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Hiroshi Isono ◽  
Hoiki Madison Liu ◽  
Toshifumi Noumi
Keyword(s):  
Principal Series ◽  
Light Fields ◽  
Quadratic Term ◽  
De Sitter Spacetime ◽  
Cosmological Horizon ◽  
De Sitter ◽  
Complementary Series ◽  
Point Function ◽  
Sitter Spacetime ◽  
Simple Scaling

Abstract We study wavefunctions of heavy scalars on de Sitter spacetime and their implications to dS/CFT correspondence. In contrast to light fields in the complementary series, heavy fields in the principal series oscillate outside the cosmological horizon. As a consequence, the quadratic term in the wavefunction does not follow a simple scaling and so it is hard to identify it with a conformal two-point function. In this paper, we demonstrate that it should be interpreted as a two-point function on a cyclic RG flow which is obtained by double-trace deformations of the dual CFT. This is analogous to the situation in nonrelativistic AdS/CFT with a bulk scalar whose mass squared is below the Breitenlohner-Freedman (BF) bound. We also provide a new dS/CFT dictionary relating de Sitter two-point functions and conformal two-point functions in the would-be dual CFT.

Foldy-Wouthuysen transformations, Lorentz transformations and the De Sitter group

Physica ◽  
1969 ◽  
Vol 43 (1) ◽  
pp. 45-49 ◽  
Author(s):  
E. De Vries
Keyword(s):  
Lorentz Transformations ◽  
De Sitter ◽  
Sitter Group

scholarly journals Spherical functions on the de Sitter group

2006 ◽  
Vol 40 (1) ◽  
pp. 163-201 ◽  
Author(s):  
V V Varlamov
Keyword(s):  
Spherical Functions ◽  
De Sitter ◽  
Sitter Group

scholarly journals A CHERN–SIMONS E8 GAUGE THEORY OF GRAVITY IN D = 15, GRAND UNIFICATION AND GENERALIZED GRAVITY IN CLIFFORD SPACES

2007 ◽  
Vol 04 (08) ◽  
pp. 1239-1257 ◽  
Author(s):  
CARLOS CASTRO
Keyword(s):  
Gauge Theory ◽  
Grand Unification ◽  
De Sitter ◽  
Sitter Group ◽  
Yang Mills ◽  
Theory Of Gravity ◽  
M Theory ◽  
Chern Simons ◽  
Topological Part ◽  
Gauge Theory Of Gravity

A novel Chern–Simons E8 gauge theory of gravity in D = 15 based on an octicE8 invariant expression in D = 16 (recently constructed by Cederwall and Palmkvist) is developed. A grand unification model of gravity with the other forces is very plausible within the framework of a supersymmetric extension (to incorporate spacetime fermions) of this Chern–Simons E8 gauge theory. We review the construction showing why the ordinary 11D Chern–Simons gravity theory (based on the Anti de Sitter group) can be embedded into a Clifford-algebra valued gauge theory and that an E8 Yang–Mills field theory is a small sector of a Clifford (16) algebra gauge theory. An E8 gauge bundle formulation was instrumental in understanding the topological part of the 11-dim M-theory partition function. The nature of this 11-dim E8 gauge theory remains unknown. We hope that the Chern–Simons E8 gauge theory of gravity in D = 15 advanced in this work may shed some light into solving this problem after a dimensional reduction.

scholarly journals Dimensionally Compactified Chern-Simon Theory in 5D as a Gravitation Theory in 4D

2017 ◽  
Vol 45 ◽  
pp. 1760005 ◽  
Author(s):  
Ivan Morales ◽  
Bruno Neves ◽  
Zui Oporto ◽  
Olivier Piguet
Keyword(s):  
Dimensional Space ◽  
Cosmological Solution ◽  
Space Time ◽  
Gravitation Theory ◽  
Simons Theory ◽  
De Sitter ◽  
Field Equations ◽  
Sitter Group ◽  
Dimensional Space Time ◽  
Chern Simons

We propose a gravitation theory in 4 dimensional space-time obtained by compacting to 4 dimensions the five dimensional topological Chern-Simons theory with the gauge group SO(1,5) or SO(2,4) – the de Sitter or anti-de Sitter group of 5-dimensional space-time. In the resulting theory, torsion, which is solution of the field equations as in any gravitation theory in the first order formalism, is not necessarily zero. However, a cosmological solution with zero torsion exists, which reproduces the Lambda-CDM cosmological solution of General Relativity. A realistic solution with spherical symmetry is also obtained.

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