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.

scholarly journals Some Properties of Massless Particles in Arbitrary Dimensions

1998 ◽  
Vol 10 (08) ◽  
pp. 1079-1109 ◽  
Author(s):  
Mourad Laoues
Keyword(s):  
Tensor Product ◽  
Arbitrary Spin ◽  
Space Time ◽  
De Sitter ◽  
Sitter Group ◽  
Massless Particles ◽  
Arbitrary Dimensions ◽  
The One

Various properties of two kinds of massless representations of the n-conformal (or (n+1)-De Sitter) group [Formula: see text] are investigated for n≥2. It is found that, for space-time dimensions n≥3, the situation is quite similar to the one of the n=4 case for Sn-massless representations of the n-De Sitter group [Formula: see text]. These representations are the restrictions of the singletons of [Formula: see text]. The main difference is that they are not contained in the tensor product of two UIRs with the same sign of energy when n>4, whereas it is the case for another kind of massless representations. Finally some examples of Gupta–Bleuler triplets are given for arbitrary spin and n≥3.

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 Hamiltonian charges in the asymptotically de Sitter spacetimes

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Maciej Kolanowski ◽  
Jerzy Lewandowski
Keyword(s):  
Phase Space ◽  
Initial Data ◽  
Exact Solutions ◽  
Gravitational Waves ◽  
Physical Meaning ◽  
De Sitter ◽  
Killing Vectors ◽  
Asymptotically Flat ◽  
Angular Momenta ◽  
The One

Abstract We generalize a notion of ‘conserved’ charges given by Wald and Zoupas to the asymptotically de Sitter spacetimes. Surprisingly, our construction is less ambiguous than the one encountered in the asymptotically flat context. An expansion around exact solutions possessing Killing vectors provides their physical meaning. In particular, we discuss a question of how to define energy and angular momenta of gravitational waves propagating on Kottler and Carter backgrounds. We show that obtained expressions have a correct limit as Λ → 0. We also comment on the relation between this approach and the one based on the canonical phase space of initial data at ℐ+.

scholarly journals Pathways to relativistic curved momentum spaces: de Sitter case study

2016 ◽  
Vol 25 (02) ◽  
pp. 1650027 ◽  
Author(s):  
Giovanni Amelino-Camelia ◽  
Giulia Gubitosi ◽  
Giovanni Palmisano
Keyword(s):  
Momentum Space ◽  
Hopf Algebras ◽  
Affine Connection ◽  
Planck Scale ◽  
Test Case ◽  
De Sitter ◽  
Group Manifold ◽  
The One ◽  
Natural Test

Several arguments suggest that the Planck scale could be the characteristic scale of curvature of momentum space. As other recent studies, we assume that the metric of momentum space determines the condition of on-shellness while the momentum space affine connection governs the form of the law of composition of momenta. We show that the possible choices of laws of composition of momenta are more numerous than the possible choices of affine connection on a momentum space. This motivates us to propose a new prescription for associating an affine connection to momentum composition, which we compare to the one most used in the recent literature. We find that the two prescriptions lead to the same picture of the so-called [Formula: see text]-momentum space, with de Sitter (dS) metric and [Formula: see text]-Poincaré connection. We then show that in the case of “proper dS momentum space”, with the dS metric and its Levi–Civita connection, the two prescriptions are inequivalent. Our novel prescription leads to a picture of proper dS momentum space which is DSR-relativistic and is characterized by a commutative law of composition of momenta, a possibility for which no explicit curved momentum space picture had been previously found. This momentum space can serve as laboratory for the exploration of the properties of DSR-relativistic theories which are not connected to group-manifold momentum spaces and Hopf algebras, and is a natural test case for the study of momentum spaces with commutative, and yet deformed, laws of composition of momenta.

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.

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