The paper by Mark Srednicki explores the entropy of a massless free field when the degrees of freedom inside an imaginary sphere are traced out. The resulting entropy, \( S \), is found to be proportional to the area of the sphere, \( A = 4\pi R^2 \), rather than the volume. This result is derived by considering the ground state density matrix and tracing over the degrees of freedom inside the sphere, leading to a density matrix that depends only on the degrees of freedom outside the sphere. The entropy is then computed using the trace of the logarithm of this density matrix. The author argues that the entropy should depend on properties shared by both regions, such as the boundary, and suggests that it might be a function of the ultraviolet cutoff \( M \) and the infrared cutoff \( \mu \). Through numerical simulations and theoretical analysis, it is shown that \( S \) scales as \( S = \kappa M^2 A \), where \( \kappa \) is a numerical constant. This result is strikingly similar to the entropy formula for a black hole, \( S_{\text{BH}} = \frac{1}{4}M_{\text{Pl}}^2 A \), where \( M_{\text{Pl}} \) is the Planck mass. The paper discusses the implications of this connection and explores its potential relevance to the number of quantum states accessible to a black hole. The author also notes that the entropy in one-dimensional systems depends on the infrared cutoff, while in higher dimensions, the sum over partial waves does not converge with radial lattice regularization.The paper by Mark Srednicki explores the entropy of a massless free field when the degrees of freedom inside an imaginary sphere are traced out. The resulting entropy, \( S \), is found to be proportional to the area of the sphere, \( A = 4\pi R^2 \), rather than the volume. This result is derived by considering the ground state density matrix and tracing over the degrees of freedom inside the sphere, leading to a density matrix that depends only on the degrees of freedom outside the sphere. The entropy is then computed using the trace of the logarithm of this density matrix. The author argues that the entropy should depend on properties shared by both regions, such as the boundary, and suggests that it might be a function of the ultraviolet cutoff \( M \) and the infrared cutoff \( \mu \). Through numerical simulations and theoretical analysis, it is shown that \( S \) scales as \( S = \kappa M^2 A \), where \( \kappa \) is a numerical constant. This result is strikingly similar to the entropy formula for a black hole, \( S_{\text{BH}} = \frac{1}{4}M_{\text{Pl}}^2 A \), where \( M_{\text{Pl}} \) is the Planck mass. The paper discusses the implications of this connection and explores its potential relevance to the number of quantum states accessible to a black hole. The author also notes that the entropy in one-dimensional systems depends on the infrared cutoff, while in higher dimensions, the sum over partial waves does not converge with radial lattice regularization.