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 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 Colourful Poincaré symmetry, gravity and particle actions

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Joaquim Gomis ◽  
Euihun Joung ◽  
Axel Kleinschmidt ◽  
Karapet Mkrtchyan
Keyword(s):  
Field Theory ◽  
Free Particle ◽  
Three Dimensional ◽  
Background Field ◽  
Quantum Field ◽  
Symmetry Factor ◽  
Starting Point ◽  
Poincaré Algebra ◽  
Poincaré Symmetry ◽  
Three Space

Abstract We construct a generalisation of the three-dimensional Poincaré algebra that also includes a colour symmetry factor. This algebra can be used to define coloured Poincaré gravity in three space-time dimensions as well as to study generalisations of massive and massless free particle models. We present various such generalised particle models that differ in which orbits of the coloured Poincaré symmetry are described. Our approach can be seen as a stepping stone towards the description of particles interacting with a non-abelian background field or as a starting point for a worldline formulation of an associated quantum field theory.

scholarly journals Conjugate spinor field equation for massless spin-3/2 field in de Sitter ambient space

2021 ◽  
Vol 67 (3 May-Jun) ◽  
pp. 465
Author(s):  
S. Falahi ◽  
S. Parsamehr
Keyword(s):  
Field Equation ◽  
Lagrange Equation ◽  
Ambient Space ◽  
Quantum Field ◽  
De Sitter ◽  
Comprehensive Description ◽  
Sitter Group ◽  
Gauge Invariant ◽  
Massless Spin ◽  
Spinor Field Equation

The quantum field theory in de Sitter ambient space provide us with a comprehensive description of massless gravitational field. Using the gauge-covariant derivative in the de Sitter ambient space, the gauge invariant Lagrangian density has been found.In this paper, the equation of the conjugate spinor for massless spin-$\frac{3}{2}$ field is obtained by Euler-Lagrange equation. Then the field equation is written in terms of the Casimir operator of the de Sitter group. Finally, the gauge invariant field equation is presented.

scholarly journals Strict deformations of quantum field theory in de Sitter spacetime

2021 ◽  
Vol 62 (6) ◽  
pp. 062302
Author(s):  
M. B. Fröb ◽  
A. Much
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
De Sitter Spacetime ◽  
Quantum Field ◽  
De Sitter ◽  
Sitter Spacetime

scholarly journals Interacting quantum field theory in de Sitter vacua

2003 ◽  
Vol 67 (2) ◽  
Author(s):  
Martin B. Einhorn ◽  
Finn Larsen
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Field ◽  
De Sitter ◽  
De Sitter Vacua

scholarly journals Analyticity Properties and Thermal Effects for General Quantum Field Theory on de Sitter Space-Time

1998 ◽  
Vol 196 (3) ◽  
pp. 535-570 ◽  
Author(s):  
Jacques Bros ◽  
Henri Epstein ORF RID="a3"> ◽  
Ugo Moschella
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Thermal Effects ◽  
De Sitter Space ◽  
Space Time ◽  
Quantum Field ◽  
De Sitter ◽  
General Quantum
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