Bridging Quantum and Classical Physics at the Planck Scale

We present a comprehensive review of the application of Planck quantities across various domains of physics. Multiple derivations of Planck units are compared and categorized as fundamental maxima, minima, or mesoscopic values. We analyze their application to black holes, Hawking radiation, and the Unruh effect. Comparisons are made between the Planck scale and other scales such as the cosmological scale, the Schwinger scale, and quantum flux system scales. Additionally, we explore the role of Planck quantities to electromagnetic and gravitational Casimir effects, which utilize the concept of "virtual particles." Additionally, we introduce new Planck-scale constructs involving Ricci invariants, which are used to derive the uncertainty of the space-time metric and thereby quantify the notion of "quantum space-time foam." Furthermore, fine-structure constants for each of the four fundamental forces are used to show how all the fundamental force interactions converge at the Planck scale. Lastly, we show that in the limit ħ → 0, the discreteness of space-time vanishes and classical behavior emerges. This provides a formal demonstration of the Bohr correspondence principle applied to the Planck scale and describes how classical behavior emerges from quantum gravity