Brick Wall and Quantum Statistical Entropy of Black Hole

Since Bekenstein introduced the thermodynamical analogy in black hole physics and Hawking discovered thermal radiation from a black hole confirming this analogy , it is an intriguing problem as to what degrees of freedom are counted by the entropy of a black hole. Equivalently, what (if any) statistical mechanics is responsible for the Bekenstein-Hawking entropy? According to ’t Hooft the statistical-mechanical entropy SSM arises from a thermal bath of quantum fields propagating outside the horizon. It should be noted that every calculation of statistical entropy encounters the problem of dealing with the very peculiar behavior of the physical quantities near the horizon where they typically diverge. To remove these divergences ’t Hooft introduced “brick wall”: a fixed boundary near the horizon within which the quantum field does not propagate. Essentially, this procedure (as it formulated in) must be implemented in addition to the removing of standard ultra-violet divergences. Contrary to statistical entropy the thermodynamical entropy of black hole is finite (after UV-renormalization) quantity not possessing any kind of the “brick wall” divergence. This fact has inspired to argue in 3 that black holes provide us with a unique exapmle of a specific system for which these entropies do not necessarily coincide. However, dealing with quantum field, that is system having infinite number of degrees of freedom, we have to take into account that not all of these freedoms are, in fact, physical. Indeed, the high energy modes lead to unphysical infinities and must be subtracted. The well-defined UV-renormalization is invariant procedure of such a subtraction. Therefore, formulating the statistical mechanics for a quantim field we need in the similar subtraction of unphysical modes which must be excluded in the statistical ensemble. How this really happens was demonstrated in 4 by using the Pauli-Villars (PV) regularization scheme. It consists in introducing a number of fictitious fields (regulators) of