Special Relativity from Heisenberg’s Uncertainty Relation

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 Sp(2) group which is isomorphic to the Lorentz group applicable to two space-like dimensions and one time-like dimension 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 with the ten generators for Dirac’s two oscillator system leading to the \((3 + 2)\) de Sitter or O(3, 2) Lorentz group, which is isomorphic to Sp(4). The O(3, 2) Lorentz group can then be contracted to the Poincaré 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 \(E = mc^{2}.\)