We consider special classes of Palatini $f(\mathcal{R})$ theories, featured by additional Loop Quantum Gravity inspired terms, with the aim of identifying a set of modified Ashtekar canonical variables, which still preserve the $SU(2)$ gauge structure of the standard theory. In particular, we allow for affine connection to be endowed with torsion, which turns out to depend on the additional scalar degree affecting Palatini $f(\mathcal{R})$ gravity, and in this respect we successfully construct a novel Gauss constraint. We analyze the role of the additional scalar field, outlining as it acquires a dynamical character by virtue of a non vanishing Immirzi parameter, and we describe some possible effects on the area operator stemming from such a revised theoretical framework. Finally, we compare our results with earlier studies in literature, discussing differences between metric and Palatini approaches. It is worth noting how the Hamiltonian turns out to be different in the two cases. The results can be reconciled when the analysis is performed in the Einstein frame.

我们考虑帕拉蒂尼的特殊班级 $F（\mathcal{[R}）$ 这些理论以附加的“环量子引力”启发性术语为特征，旨在识别出一组经过修改的Ashtekar典型变量，这些变量仍保留了 $\mathrm{小号}ü（2）$标准理论的量具结构。特别是，我们允许仿射连接具有扭转力，这取决于影响Palatini的附加标量度$F（\mathcal{[R}）$引力，在这方面，我们成功地构造了一种新颖的高斯约束。我们分析了附加标量场的作用，概述了附加标量场通过不消失的Immirzi参数获得动态特性的过程，并描述了来自这种修订后的理论框架的对区域算子的一些可能影响。最后，我们将我们的结果与文献中的早期研究进行了比较，讨论了度量方法和Palatini方法之间的差异。值得注意的是，两种情况下的哈密顿量有何不同。在爱因斯坦框架中执行分析时，结果可以协调。