路径积分测度与宇宙常数

我们考虑在Einstein-Hilbert截断下的(欧几里得)量子引力，using a spherical background, we calculate the one-loop effective action $\Gamma^{1l}_{\rm grav}$ using a spherical background. Usually, this calculation is performed resorting to proper-time regularization within the heat kernel expansion and gives rise to quartically and quadratically UV-sensitive contributions to the vacuum energy $\rho_{\rm vac}=\frac{\Lambda_{\rm cc}}{8\pi G}$, with $\Lambda_{\rm cc}$ and $G$ cosmological and Newton constant, respectively. We show that, if the measure in the path integral that defines $\Gamma^{1l}_{\rm grav}$ is correctly taken into account, and the physical UV cutoff $\Lambda_{\rm cut}$ properly introduced, $\rho_{\rm vac}$ presents only a (mild) logarithmic sensitivity to $\Lambda_{\rm cut}$. We also consider a free scalar field and a free Dirac field on a spherical gravitational background, and find that the same holds true even in the presence of matter. These results are found without resorting to any supersymmetric embedding of the theory, and shed new light on the cosmological constant problem.