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

Sibel Başkal,Young S Kim,Marilyn E Noz

doi:10.3390/quantum1020021

Abstract

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 .

Highlights

IntroductionThe three rotation and three translation generators are contained in, or are derivable from, Heisenberg’s commutation relations, and the time translation operator is seen in the Schrödinger equation
Let us start with Heisenberg’s commutation relations xi, Pj = i δij, (1) with Pi = −i ∂ ∂xi (2)where i = 1, 2, 3, corresponds to the x, y, z coordinates respectively.With these xi and Pi, we can construct the following three operators, Ji = eijk x j Pk . (3)These three operators satisfy the closed set of commutation relations: Ji, Jj = ieijk Jk .Quantum Rep. 2019, 1, 236–251; doi:10.3390/quantum1020021
The purpose of this paper is to show that the Lie algebra of the Poincaré symmetry is derivable from the Heisenberg commutation relations

Summary

Introduction

The three rotation and three translation generators are contained in, or are derivable from, Heisenberg’s commutation relations, and the time translation operator is seen in the Schrödinger equation They are all Hermitian operators corresponding to dynamical variables. It is possible to contract one of those two time variables of this O(3, 2) group into the inhomogeneous Lorentz group, consisting of the Lorentz group applicable to the three space-like dimensions and one time-like direction, plus four translation generators corresponding to the energy-momentum four-vector. This leads to Einstein’s energy–momentum relation of. We shall give a brief review of Dirac’s efforts during the period in Appendix A

Symmetries of the Single-Mode States

Symmetries from Two Oscillators

Concluding Remarks