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

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.