  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 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 the Dirac's two oscillator
system leadingto the group isomorphic to the de Sitter group O(3,2), which can
the 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 = mc^2.
