On the quantization of thermodynamic action: a step towards quantum gravity

On the quantization of thermodynamic action Tan Jun Hao October 8, 2025 Abstract To date, there aren’t any real and validated theory nor framework, that explains quantum gravity. Current leading candidates are string theory, M-theory and loop quantum gravity, but they lack of physical observations that agrees with them. In this work, we propose a direct connection between thermodynamics, relativity and quantum mechanics through a simple relation: T St =ℏ This equation suggests a natural bridge between the thermodynamic flow of time and quantum action, providing a potential link between entropy and the quantization of physical processes. 1 Introduction One of the hardest, and the most beautiful goals in theoretical physics, are unifying three major pillars in physics, thermodynamics, quantum mechanics and relativity. Each field captures a fundamental aspect of nature: energy and heat, uncertainty and discreteness, the geometry of spacetime. In the course of research, we discover a compact unification among these fields through a simple equation that naturally emerges from dimensional reasoning and physical symmetry. 2 Planck Relation and Energy-Time Symmetry In quantum mechanics, the Planck relation Et =ℏ defines the fundamental quantum of action, connecting energy and time. For a particle with mass m, E=mc2 and thus mc2t=ℏ. This expresses how relativistic energy and time together produce the invariant quantum of action. 1 3 Extension to Thermodynamics In thermodynamics, energy can also be expressed as E=TS, where Tis temperature and Sis entropy. Substituting this into the quantum action relation Et =ℏgives TSt =ℏ. This equation reveals that entropy flow in time is quantized, and the fundamental constant of nature, ℏ, encapsulates both thermal and quantum behaviors. 4 Physical Interpretation The relation TSt =ℏimplies that any thermodynamic process evolving over time carries a minimum quantum of action. Entropy (S) represents the microscopic information content of a system, while temperature (T) describes the macroscopic energy exchange. Thus, time evolution of entropy at a given temperature directly corresponds to quantum action. This bridges: •Thermodynamics: E=TS •Quantum mechanics: Et =ℏ •Relativity: E=mc2 Combining these gives a unified perspective that links the arrow of time, thermal processes, and quantum discreteness. 5 Application to Schwarzschild Black Hole To test the validity of the relation TSt =ℏ, we apply it to the simplest gravitational system known to exhibit both quantum and thermodynamic behavior, the Schwarzschild black hole. The Hawking temperature, Bekenstein–Hawking entropy, and evaporation time of a non-rotating, uncharged black hole of mass Mare given by: T=ℏc3 8πGMkB , S =4πkBGM2 ℏc, t =5120πG2M3 ℏc4. Multiplying these quantities yields: TSt = 2560πG3M4 ℏc6. For the equality TSt =ℏto hold, the mass must satisfy: M=ℏ2c6 2560πG31/4 . 2 Substituting the fundamental constants ℏ= 1.054 ×10−34 J s, c = 2.998 ×108m/s, G = 6.674 ×10−11 m3/kg/s2, we find M≈7.6×101kg. This mass corresponds to a Schwarzschild radius of approximately 1.1×10−25 m—a scale approaching the Planck regime. Remarkably, this suggests that the equality TSt =ℏbecomes exact for black holes near the quantum–gravitational boundary, where thermodynamics, quantum mechanics, and general relativity converge. At larger masses, TSt > ℏ, while at smaller masses, TSt < ℏ, indicating that the relation may serve as a balance point between the quantum and gravitational domains. In this view, the equation TSt =ℏnaturally identifies the Planck-scale limit as a region of fundamental equilibrium, where the thermal, temporal, and quantum aspects of spacetime become inseparable. 6 Implications for Quantum Gravity At the Planck scale, we can define TpSptp=ℏ, suggesting that the Planck temperature, entropy, and time are bound by the same invariant quantum action, connecting three fundamental constants at once. This could form a thermodynamic basis for quantum gravity, where spacetime itself behaves as an evolving thermodynamic system constrained by quantum limits. 7 Predictions and Physical Implications Any groundbreaking equation or theory must predict a new phenomena, just as when Einstein’s General Relativity predicted the existence of black holes and gravitational waves. Thus, from this equation, several predictions might occur or can be tested: •Thermal Time Dilation: At very small scales, the flow of time may depend on temperature and entropy. A system with higher thermal energy could experience a slightly altered rate of time evolution. This effect may be tested using ultracold atoms or superconducting qubits. •Entropy-Induced Quantum Shift: If ℏbalances TSt, then changing entropy may affect the phase of a quantum system. A small variation in entropy could produce measurable phase shifts, similar to a thermodynamic version of the Aharonov–Bohm effect. •Planck-Scale Balance: At the Planck scale, the product of temperature, entropy, and time remains constant. Any increase in temperature or entropy must be balanced by a shorter time interval. This balance could represent a universal equilibrium between heat, information, and quantum time, possibly underlying quantum gravity. 3 8 Discussion We’ve discovered a simple equation, connecting thermodynamics, quantum mechanics and relativity, suggesting when entropy changes with time the result is a fixed quantum action. In other words, time, heat and quantum behavior are linked together at the same level. It holds at Planck-scaled black holes, where gravitational and quantum effects meet, hinting spacetime might have a thermal nature, arising from the flow of entropy under quantum limits. In this way, this relation points to a deeper idea: the smallest action in the universe may come from the natural balance of time, temperature and entropy, bringing a step closer towards quantum gravity. 9 Conclusion The equation TSt =ℏ encapsulates a minimal and elegant connection between thermodynamics, quantum mechanics, and relativity. It implies that the evolution of entropy in time is quantized, and that every process — gravitational, thermal, or quantum — shares a universal limit set by ℏ. This relation may serve as a foundational link toward a consistent framework of quantum gravity. References [1] A. Einstein, On the Electrodynamics of Moving Bodies, Annalen der Physik, 17, 891–921 (1905). [2] M. 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