Abstract
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.
Highlights
As early as in 1927 [1], Paul A
In 1949 [3], Dirac pointed out that the task of constructing relativistic dynamics is to construct a representation of the inhomogeneous Lorentz group
In 1963 [4], Dirac considered two coupled harmonic oscillators and constructed an algebra leading to the Lie algebra for the SO(3, 2) de Sitter group, which is the Lorentz group applicable to three space dimensions and two time-like variables
Summary
As early as in 1927 [1], Paul A. In 1949 [3], Dirac pointed out that the task of constructing relativistic dynamics is to construct a representation of the inhomogeneous Lorentz group He wrote down the ten generators of this group and their closed set of commutation relations. In 1963 [4], Dirac considered two coupled harmonic oscillators and constructed an algebra leading to the Lie algebra for the SO(3, 2) de Sitter group, which is the Lorentz group applicable to three space dimensions and two time-like variables. It is pointed out that this group, with three generators, is isomorphic to the Lorentz group applicable to two space dimensions and one time variable. This closed set is called the Lie algebra of the Sp(2) group, isomorphic to the Lorentz group applicable to two space dimensions and one time dimension. Lorentz transformations applicable only to the three-dimensional space of ( x, z, t), while the y variable remains invariant
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