  The fundamental constants of electromagnetism, gravity and quantum mechanics
can be related empirically by the numerical approximation $\ln(V_e/V_P)\approx
\alpha^{-1}$, where $\alpha$ is the low energy value of the electromagnetic
fine structure constant and $V_e$ and $V_P$ are volumes corresponding to the
classical electron radius, $r_e$, and the Planck length respectively. This
logarithmic relation is used in an ideal gas model to determine the work, $W$,
done when a hypothetical vacuum fluctuation expands relativistically from $V_P$
to $V_e$ in a time limited by the uncertainty principle. It is proposed that
the expansion is a phenomenological representation of a quantum transition from
a Planck-scale initial state into a final virtual photonic state of energy
$W\simeq \hbar c/2r_e$ and lifetime $\simeq r_e/c$, occupying a volume $\simeq
V_e$. The magnitude of the negative gravitational self-energy density,
$\rho_G$, of this virtual state is found to be within $\sim 10\%$ of the
measured value of the positive "dark energy" density, $\rho_\Lambda$. It is
proposed that this is not merely an "accidental" numerical coincidence but has
physical significance, namely that the sum of the two energy densities is zero,
i.e. $\rho_\Lambda+\rho_G=0$. This relation gives a value of the cosmological
constant, $\Lambda$, in agreement with astronomical measurements. The
implications of these inter-relations between $\Lambda$, the gravitational
constant, $G$, and $\alpha$ are outlined.
