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The fundamental constants of electromagnetism, gravity and quantum mechanicscan be related empirically by the numerical approximation $\ln(V_e/V_P)\approx\alpha^{-1}$, where $\alpha$ is the low energy value of the electromagneticfine structure constant and $V_e$ and $V_P$ are volumes corresponding to theclassical electron radius, $r_e$, and the Planck length respectively. Thislogarithmic 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 thatthe expansion is a phenomenological representation of a quantum transition froma 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 $\simeqV_e$. The magnitude of the negative gravitational self-energy density,$\rho_G$, of this virtual state is found to be within $\sim 10\%$ of themeasured value of the positive "dark energy" density, $\rho_\Lambda$. It isproposed that this is not merely an "accidental" numerical coincidence but hasphysical 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 cosmologicalconstant, $\Lambda$, in agreement with astronomical measurements. Theimplications of these inter-relations between $\Lambda$, the gravitationalconstant, $G$, and $\alpha$ are outlined.

A model to inter-relate the values of the quantum electrodynamic, gravitational and cosmological constants