A compact description of the evolution of a many-body quantum system, e.g., a dilute gas, is provided by the generalization of the usual reaction U matrix to a system U matrix. Using this tool, it is demonstrated that in such systems (i) time-reversal-invariant states are unstable and cannot exist in nature or be ``constructed in the laboratory;'' (ii) the time-reversal invariance of the Hamiltonian leads inexorably to relaxation towards thermal equilibrium of arbitrary nonequilibrium states, i.e., to dissipation, which thus allows the definition of a quantum arrow of time; (iii) the second law of thermodynamics, and hence the complete field of thermodynamics, is a consequence of quantum physics. These results have at their basis the hermiticity of the Hamiltonian and the Heisenberg uncertainty relations.
A compact description of the evolution of a many-body quantum system, e.g., a dilute gas, is provided by the generalization of the usual reaction U matrix to a system U matrix. Using this tool, it is demonstrated that in such systems (i) time-reversal-invariant states are unstable and cannot exist in nature or be ``constructed in the laboratory;'' (ii) the time-reversal invariance of the Hamiltonian leads inexorably to relaxation towards thermal equilibrium of arbitrary nonequilibrium states, i.e., to dissipation, which thus allows the definition of a quantum arrow of time; (iii) the second law of thermodynamics, and hence the complete field of thermodynamics, is a consequence of quantum physics. These results have at their basis the hermiticity of the Hamiltonian and the Heisenberg uncertainty relations.
We consider the temporal evolution of an isolated system of particles from a non-equilibrium state of entropy S = S′ to the equilibrium state of maximum entropy S = S max, S′ ≤ S max. The application of usual density matrix theory to the temporal development of S leads us to the unexpected result P = dS/dt = 0. The entropy does not change in time. This result is valid in case of the equilibrium state. However it is wrong for a non-equilibrium state, which could not irreversibly change from S′ to S max by entropy production P > 0. This contradicts the second law of thermodynamics. The paradoxial result can be traced back to the Liouville theorem. To overcome this difficulty, we introduce the t–P space. The time t and the entropy production P are observables in quantum thermodynamics. They are defined by the eigenvalue equations t|t〉 = t|t〉 and P|P〉 = P|P〉. The operators t and P do not commute. We have a non-vanishing commutator, [t,P] = ik, and consequently a t–P uncertainty relation, ΔtΔP ≥ k/2. The Boltzmann constant k is characteristic of quantum thermodynamics and represents the atomic entropy unit. — The time evolution operator U(τ) = exp[−(i/k)τP] shifts the time t by τ, U(τ)|t〉 = |t + τ〉. The generator of τ is the operator P of entropy production. Then the eigenvector |P〉 is also an eigenket of U(t), and it follows with progressing time τ ≥ 0 that τP ≥ 0 and hence P = dS/dt ≥ 0. This is the second law of thermodynamics, which is correctly reproduced in quantum thermodynamics. — Exchanging −τ for +τ replaces the equations of forward time +τ by the equations of backward time −τ. Independent of the direction of the arrow of time, the entropy of the system always increases or remains constant in the limit of the equilibrium state, dS ≥ 0. This is characteristic of irreversibility, for the system does not return to its initial state on time reversal. — The t–P uncertainty relation allows to discuss the temporal evolution of the expectation value 〈P〉 of entropy production and the lifetime Δt of the transient states, the system passes through on approaching the equilibrium state. In case of a non-equilibrium state 〈P〉, ΔP, and Δt are finite. On approaching the equilibrium state, 〈P〉 and ΔP decrease, consequently Δt increases. Finally, in the case of equilibrium (eq), we have 〈P〉eq = 0, ΔP eq = 0, and Δt eq → ∞. The lifetime of the equilibrium state is unlimited, because the entropy production 〈P〉 and the uncertainty of entropy production ΔP exactly vanish. The temporal evolution of the system can be described by means of a phenomenological equation.
All the laws of physics are time-reversible. Time arrow emerges only when ensembles of classical particles are treated probabilistically, outside of physics laws, and the entropy and the second law of thermodynamics are introduced. In quantum physics, no mechanism for a time arrow has been proposed despite its intrinsic probabilistic nature. In consequence, one cannot explain why an electron in an excited state will "spontaneously" transition into a ground state as a photon is created and emitted, instead of continuing in its reversible unitary evolution. To address such phenomena, we introduce an entropy for quantum physics, which will conduce to the emergence of a time arrow. The entropy is a measure of randomness over the degrees of freedom of a quantum state. It is dimensionless; it is a relativistic scalar, it is invariant under coordinate transformation of position and momentum that maintain conjugate properties and under CPT transformations; and its minimum is positive due to the uncertainty principle. To excogitate why some quantum physical processes cannot take place even though they obey conservation laws, we partition the set of all evolutions of an initial state into four blocks, based on whether the entropy is (i) increasing but not a constant, (ii) decreasing but not a constant, (iii) a constant, (iv) oscillating. We propose a law that in quantum physics entropy (weakly) increases over time. Thus, evolutions in the set (ii) are disallowed, and evolutions in set (iv) are barred from completing an oscillation period by instantaneously transitioning to a new state. This law for quantum physics limits physical scenarios beyond conservation laws, providing causality reasoning by defining a time arrow.
This work proposes a conceptual and ontological reformulation of classical dynamics, thermodynamics, and quantum mechanics based on a single postulate: dissipation is a fundamental property of all physical processes. Traditional dynamics relies on the idealization of isolated, reversible systems, yet no real physical system satisfies such conditions. By introducing an intrinsic dissipative energy term into the equations of motion, we suggest that dissipation entails an irreversible loss of physical information, from which a classical uncertainty principle naturally follows. Stationary states are reinterpreted as zero‑dissipation configurations, lacking internal temporal evolution, thereby offering an ontological basis for quantization, non‑locality, and the Heisenberg uncertainty principle. Quantum mechanics can thus be viewed as the mathematical description of these stationary states and of the dissipative transitions connecting them. The framework is conceptually consistent with fluctuation‑dissipation theory, open‑system quantum dynamics, decoherence theory, and relaxation phenomena in NMR spectroscopy. Several experimental proposals are outlined to discriminate this ontological model from both classical mechanics and standard quantum mechanics.
The irreversible entropy increase described by the second law of thermodynamics is fundamentally tied to thermalization and the emergence of equilibrium. In the first part of our work (Ref: arXiv.2503.04152), we constructed an isolated gas system model and numerically demonstrated irreversible growth of entanglement entropy caused by erasure of spread non-equilibrium state information. Here, we mathematically prove that for a typical macroscopic system in a non-equilibrium state ∣ϕ0⟩, the quantum state ∣ϕ0′⟩=O^(t)∣ϕ0⟩ will inevitably evolve toward equilibrium. Our work demonstrates that the second law of thermodynamics, and consequently the ergodic hypothesis in statistical physics, can be understood and proven from a quantum information perspective. From this perspective, the second law can be stated as: In typical macroscopic physical systems, the spreading and erasure of non-equilibrium information is inevitable.
This work investigates the fundamental connection between quantum mechanics and thermodynamics through the concept of the arrow of time. While the second law of thermodynamics identifies the arrow of time with the monotonic increase of entropy in isolated systems, closed quantum systems evolve unitarily and conserve von Neumann entropy, making the origin of irreversibility less apparent. Here we establish a no-go theorem proving that correlations in unknown states of closed quantum systems cannot be universally removed by any physical operation. This result demonstrates that correlation generation is inherently irreversible, providing a microscopic and universal basis for the arrow of time. In thermodynamic settings, this irreversibility ensures that heat flows spontaneously from hotter to colder subsystems, reproducing the Clausius formulation of the second law. More generally, the irreversible growth of correlations explains entropy increase, the approach to equilibrium, and decoherence as observable manifestations of the same principle. Our findings ground a quantum-mechanical foundation for the second law of thermodynamics and demonstrate the universality of the arrow of time.