The cosmological constant problem roughly states that the predicted value of $\Lambda$ becomes mostly infinite even though we only measure negligible values for it.
In classic General Relativity, the zero-point energy of a system is identically zero, so if there are zero particles then there is zero energy. In the case of QFT the Hamiltonian of a vacuum state tends to diverge. This would, in principle, infinitely curve spacetime producing the net effect of an infinite expansion.
Nonetheless, one could break down the total energy-momentum tensor $\mathcal{T}_{\mu\nu}$ into its classical $T_{\mu\nu}$ version and a quantum mechanical one $\Psi_{\mu\nu}$ that includes the divergence corresponding to the zero-point energy of the quantum system. Now, given that the diverging Hamiltonian is independent of spacetime coordinates then we should have
$$\Psi_{\mu\nu} = \xi g_{\mu\nu},$$
So we can “renormalize” the theory’s Lagrangian by including this diverging term into the action
$$S = \int{\sqrt{-g}\left[\frac{1}{2\kappa}R + \frac{\Lambda}{\kappa} + \mathcal{L}_m - \xi\right]},$$
(here using the mostly minus convention). The EFEs become
$$G_{\mu\nu} = \Lambda g_{\mu\nu} + \kappa \mathcal{T}_{\mu\nu} - \xi\kappa g_{\mu\nu} = \Lambda g_{\mu\nu} + \kappa T_{\mu\nu}.$$
Could this renormalization procedure help with the cosmological constant problem? Is it equal to normal-ordering the right hand side?
