Note on $SU(2)$ isolated horizon

We point out that the symplectic structure, written in terms of the Sen-Ashtekar-Immirzi-Barbero variables, of a spacetime admitting an isolated horizon as the inner boundary, involves a positive constant parameter, say $\sigma$, if $\gamma\neq\pm i$, where $\gamma$ is the Barbero-Immirzi parameter. The parameter $\sigma$ represents the rescaling freedom that characterizes the equivalence class of null generators of the isolated horizon. We reiterate the fact that the laws of mechanics associated with the isolated horizon does not depend on the choice of $\sigma$ and, in particular, while one uses the value of standard surface gravity as input, that does not fix $\sigma$ to a particular value. This fact contradicts the claims made in certain parts of the concerned literature that we duly refer to. We do the calculations by taking Schwarzschild metric as an example so that the contradiction with the referred literature, where similar approaches were adopted, becomes apparent. The contribution to the symplectic structure that comes from the isolated horizon, diverges for $\sigma^2=(1+\gamma^2)^{-1}$, implying that the rescaling symmetry of the isolated horizon is violated for any real $\gamma$. Since the quantum theory of $SU(2)$ isolated horizon in the LQG framework exists only for real values of $\gamma$, it is founded on this flawed classical setup. Nevertheless, if the flaw is ignored, then two different viewpoints persist in the literature for entropy calculation. We highlight the main features of those approaches and point out why one is logically viable and the other is not.