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 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 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 .

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 p-ADIC AND ADELIC MINISUPERSPACE QUANTUM COSMOLOGY

2002 ◽  
Vol 17 (10) ◽  
pp. 1413-1433 ◽  
Author(s):  
GORAN S. DJORDJEVIĆ ◽  
BRANKO DRAGOVICH ◽  
LJUBIŠA D. NEŠIĆ ◽  
IGOR V. VOLOVICH
Keyword(s):  
Quantum Mechanics ◽  
Cosmological Constant ◽  
Quantum Cosmology ◽  
Quantum Effects ◽  
De Sitter Space ◽  
Cosmological Models ◽  
De Sitter ◽  
Adelic Quantum Mechanics ◽  
Matter Fields

We consider the formulation and some elaboration of p-adic and adelic quantum cosmology. The adelic generalization of the Hartle–Hawking proposal does not work in models with matter fields. p-adic and adelic minisuperspace quantum cosmology is well defined as an ordinary application of p-adic and adelic quantum mechanics. It is illustrated by a few cosmological models in one, two and three minisuperspace dimensions. As a result of p-adic quantum effects and the adelic approach, these models exhibit some discreteness of the minisuperspace and cosmological constant. In particular, discreteness of the de Sitter space and its cosmological constant is emphasized.

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