Microscopic origin of Einstein's field equations and the raison d'\^{e}tre for a positive cosmological constant

In the paradigm of effective field theory, one hierarchically obtains the effective action ${\mathcal{A}}_{\mathrm{e}\mathrm{f}\mathrm{f}}[q,\cdots ]$ for some low(er) energy degrees of freedom $q$, by integrating out the high(er) energy degrees of freedom $\xi$, in a path integral, based on an action $$\mathcal{A}$[q,\xi, \cdots]$\mathrm{.}Weshowhowonecanintegrateoutavectorfield$v^a$inanaction$$\mathcal{A}$[\Gamma,v,\cdots ]$andobtainaneffectiveaction$$\mathcal{A}$_{\rm eff}[\Gamma, \cdots]$ which, on variation with respect to the connection $\mathrm{\Gamma }$, leads to the Einstein’s field equations and a metric compatible with the connection. The derivation $\mathit{\text{predicts}}$ a non-zero, positive, \cc, which arises as an integration constant. The Euclidean action $\mathcal{A}[\mathrm{\Gamma },v,\cdots ]$, has an interpretation as the heat density of null surfaces, when translated into the Lorentzian spacetime. The vector field $v^a$ can be interpreted as the Euclidean analogue of the microscopic degrees of freedom hosted by any null surface. Several implications of this approach are discussed.