The inclusion of the quantum fluctuations of the metric in the geometric action is a promising avenue for the understanding of the quantum properties of gravity. In this approach the metric is decomposed in the sum of a classical and of a fluctuating part, of quantum origin, which can be generally expressed in terms of an arbitrary second order tensor, constructed from the metric, and from the thermodynamic quantities describing matter fields. In the present paper we investigate the effects of the quantum fluctuations on the spherically symmetric static black hole solutions of the modified field equations, obtained from a variational principle, by assuming that the quantum correction tensor is given by the coupling of a scalar field to the metric tensor. After reformulating the field equations in a dimensionless form, and by introducing a suitable independent radial coordinate, we obtain their solutions numerically. We detect the formation of a black hole from the presence of a singularity in the metric tensor components. Several models, corresponding to different functional forms of the scalar field potential, are considered. The thermodynamic properties of the black hole solutions (horizon temperature, specific heat, entropy and evaporation time due to Hawking luminosity) are also investigated.

在几何作用中包含度量的量子涨落是理解重力量子性质的有前途的途径。在这种方法中，度量被分解为量子起源的经典部分和波动部分的总和，通常可以用任意二阶张量来表示，由该度量和描述物质场的热力学量构成。在本文中，我们通过假设标量场与标量场耦合给出了量子校正张量，从而研究了量子涨落对修正场方程的球对称静态黑洞解的影响。公制张量。在以无量纲形式重构场方程之后，通过引入合适的独立径向坐标，我们得到了数值的解。我们从度量张量分量中存在奇异点来检测黑洞的形成。考虑了几种与标量场势的不同功能形式相对应的模型。还研究了黑洞溶液的热力学性质（温度，比热，熵和霍金光度引起的蒸发时间）。