The classical ergodic hypothesis has served as a foundational assumption in statistical mechanics for more than a century. However, its core claims---that a physical system explores all accessible microstates over long times, and that time averages equal ensemble averages---are incompatible with the behavior of many real systems, including glasses, biological molecules, and complex non-equilibrium systems. Moreover, the classical formulation lacks a mechanism for information loss, observer dependence, and hierarchical coarse-graining. This paper introduces the \emph{New Ergodic Principle} (NEP), a unified projection framework that integrates (i) unitary time evolution from quantum mechanics, (ii) measurement-induced projection and information loss, and (iii) hierarchical projection from the OS-model, which formalizes observer-dependent coarse-graining. NEP replaces the classical ergodic hypothesis with a three-layer projection structure, yielding a generalized ergodic average:\[\langle A \rangle_{\mathrm{NEP}}= \mathrm{Tr}\!\left[\pi_k\!\left(\sum_i P_i U(t)\rho U^\dagger(t) P_i\right) A\right].\]This formulation naturally incorporates non-ergodic systems, clarifies the origin of irreversibility, and provides a mathematically coherent foundation for equilibrium and non-equilibrium statistical mechanics. The NEP framework also suggests a unified mathematical basis for physics grounded in projection theory rather than classical set-theoretic structures.
The classical ergodic hypothesis has served as a foundational assumption in statistical mechanics for more than a century. However, its core claims---that a physical system explores all accessible microstates over long times, and that time averages equal ensemble averages---are incompatible with the behavior of many real systems, including glasses, biological molecules, and complex non-equilibrium systems. Moreover, the classical formulation lacks a mechanism for information loss, observer dependence, and hierarchical coarse-graining. This paper introduces the \emph{New Ergodic Principle} (NEP), a unified projection framework that integrates (i) unitary time evolution from quantum mechanics, (ii) measurement-induced projection and information loss, and (iii) hierarchical projection from the OS-model, which formalizes observer-dependent coarse-graining. NEP replaces the classical ergodic hypothesis with a three-layer projection structure, yielding a generalized ergodic average:[\langle A \rangle_{\mathrm{NEP}}= \mathrm{Tr}!\left[\pi_k!\left(\sum_i P_i U(t)\rho U^\dagger(t) P_i\right) A\right].]This formulation naturally incorporates non-ergodic systems, clarifies the origin of irreversibility, and provides a mathematically coherent foundation for equilibrium and non-equilibrium statistical mechanics. The NEP framework also suggests a unified mathematical basis for physics grounded in projection theory rather than classical set-theoretic structures.
In this paper, we study and answer the following fundamental problems concerning classical equilibrium statistical mechanics: 1): Is the principle of equal a priori probabilities indispensable for equilibrium statistical mechanics? 2): Is the ergodic hypothesis related to equilibrium statistical mechanics? Note that these problems are not yet answered, since there are several opinions for the formulation of equilibrium statistical mechanics. In order to answer the above questions, we first introduce measurement theory (i.e., the theory of quantum mechanical world view), which is characterized as the linguistic turn of quantum mechanics. And we propose the measurement theoretical foundation of equili-brium statistical mechanics, and further, answer the above 1) and 2), that is, 1) is “No”, but, 2) is “Yes”.
We build a framework for the thermodynamics of macroscopic quantum systems. In contrast with approaches requiring access to the full density matrix, our framework relies on a coarse-grained description, based on measurement statistics of a few observables. When these observables commute, the outcomes define classical macrostates whose entropy is quantified by observational entropy, accounting for uncertainty about both the macrostate and the microstate within it. We extend this notion to non-commuting observables forming a subalgebra of the operator space, and use Jaynes' principle to define an algebra-dependent entropy interpolating between von Neumann and observational entropies. Given initial and final measurement sets, connected by internal and/or environment-induced dynamics, we derive a second law for the coarse-grained dynamics. Unlike formulations based on von Neumann entropy, our inequality captures irreversibility from both non-unitary environment-induced dynamics and internal equilibration. It takes the usual form of a positive entropy production when the system is initially at internal equilibrium, while correction terms capture nonequilibrium resources ignored by the coarse-graining. We also derive fluctuation theorems for coarse-grained thermodynamic quantities. Along a quasi-static path of measurement schemes, we identify quantum macroscopic notions of work and heat fulfilling the first and second laws, including an additional work contribution from manipulating the algebra to which the system is confined, through external constraints or quantum measurement backaction. Finally, we apply our framework to examples illustrating the impact of varying the coarse-graining scheme. Our approach unifies macroscopic and stochastic thermodynamics in a genuinely quantum framework, laying the basis for a versatile, experimentally friendly toolbox to analyze complex quantum dynamics.
Quantum mechanics is widely recognised as being incomplete. It is not consistent with the second law of thermodynamics and does not provide a scientifically credible physical account of the measurement process, the means by which coherence is broken and classically observable states are recorded. This has led to many ad hoc assumptions being used to account for various properties of quantum systems, among which is the coherence time of quantum devices that determines their ability to perform computations. Here, we show that all these properties can be accommodated naturally and consistently in the context of mixing quantum systems which exhibit continuous spectra, as arises in the thermodynamic limit in quantum statistical mechanics and quantum gravity. In particular, for isolated systems we show that the time-reversal symmetry associated with unitary time evolution of the quantum state gives rise to time-symmetry breaking and a semi-group evolution which attains thermodynamic equilibrium at long times. Moreover, the emergence of this non-unitary time-asymmetry leads to microcanonical equilibrium states in which all quantum coherence is lost and is accompanied by the transformation of pure states into mixtures, leading in turn to an increase in entropy. Inclusion of a macroscopic measurement apparatus shows how the outcome of a measurement corresponds to the von Neumann projection postulate, arising with probabilities in conformance with the Born rule. The mathematical structure of the theory which applies to quantum systems with continuous spectra is closely analogous to the classical ergodic theory of dynamical systems and the conditions under which they attain equilibrium states.
This paper presents a novel theoretical framework that aims to unify the core principles of quantum mechanics, general relativity, and thermodynamics by introducing an extended spatial geometry incorporating a compactified fourth spatial dimension. The theory proposes that many of the counterintuitive behaviours observed in quantum systems, such as wave function collapse, superposition, entanglement, tunnelling, and decoherence, can be naturally explained as consequences of 4D quantum objects being projected into 3D space. In this framework, observable quantum behaviour emerges not from intrinsic randomness but from the limitations of a lower-dimensional perspective on higher-dimensional structures. At the heart of this unification is the introduction of a new scalar field, termed the entropion field, which governs the rate and nature of quantum decoherence while simultaneously encoding the thermodynamic arrow of time. The entropion field interacts with known quantum fields and gravitational curvature, modifying the Einstein field equations and extending the standard model of quantum field theory. The result is a coherent description of how classical reality emerges from quantum substrates, how entropy and time are fundamentally linked, and how spacetime geometry influences quantum behaviour. The paper develops this hypothesis through a structured hierarchy of conceptual foundations, mathematical formalism, physical interpretation, and predictive consequences. Particular attention is given to testable deviations from standard models, including quantum interference in 4D-projected systems, entropion-driven decoherence rates, and new interpretations of tunnelling and entanglement. Broader implications are explored in quantum information, cosmological evolution, material science, and theoretical computation. This work serves as a foundational step toward a geometrically grounded theory of everything and offers a unified path forward by bridging the three major pillars of modern physics.
The goal of this thesis is to define the concept of coarse-graining in quantum systems through measurements and to develop a unified mathematical framework for macroscopic states, operations, correlations, and entropy.Within this framework, irreversible phenomena arise from the loss of statistical information that is accessible to macroscopic observers.To achieve this goal, we rigorously define macroscopicity.This thesis investigates observational entropy, a concept that unifies Boltzmann entropy, Gibbs entropy, von Neumann's macroscopic entropy, and diagonal entropy.Our approach to explaining how irreversibility arises in isolated quantum systems is based on this unified perspective.When the initial state is "macroscopic," we use Petz's theory of statistical sufficiency to show that observational entropy strictly increases with unitary time evolution.Furthermore, we show that, for any initial state, the probability of the system's observational entropy increasing and rapidly reaching its maximum value under random unitary time evolution is extremely high.Specifically, we demonstrate that when measurements on a quantum system are sufficiently coarse-grained relative to its size, its initial state (whether pure or mixed) becomes indistinguishable from the maximally mixed state through random time evolution.We show that this holds not only for random time evolutions sampled according to the Haar measure, but also for approximate 2-designs, which are considered more physically realistic models.Next, we extend this framework by introducing positive operator-valued measures (POVMs) and quantum prior states.This allows us to construct a unified inferential framework for macroscopic states.A macroscopic state can be characterized using equivalent algebraic and constructive formulations based on quantum statistical sufficiency.This approach is based on the concept of an observational deficit, which quantifies the degree to which a state is "irretrodictable" relative to prior and measurement.This structure is formulated as an "inferential reference frame" that encapsulates the observer's informational perspective.Furthermore, we treat macroscopic states as free states and construct the resource theory of microscopicity.This theory unifies existing theories of coherence, athermality and asymmetry.Finally, we extend this framework to the study of quantum correlations under observational constraints.This enables us to show that quantum correlations, such as entanglement and quantum discord, are context-dependent resources that depend on the observer's reference frame.I would like to express my deepest gratitude to my supervisor,
The quantum measurement problem arises from the apparent incompatibility between unitary evolution, definite outcomes, and the Born rule. Standard formulations treat measurement as a special physical process, introducing either collapse, branching, or hidden variables. In this paper we adopt a different approach. We do not modify quantum dynamics. Instead, we redefine the structure to which measurement applies. We introduce a minimal continuation-based framework in which physical systems are described by admissible succession structures and observable outcomes arise through coarse-graining projections. Measurement is defined as a non-injective projection from the space of admissible continuations to a finite outcome space. We show that non-injectivity generically induces apparent indeterminacy, while conditioning on the projection fiber replaces collapse. Within the weak-correlation regime, the Born rule emerges as the unique aggregation-stable weighting under additive continuation composition. Decoherence is modelled as constraint-induced suppression of continuation multiplicity, recovering the phenomenology of environmental decoherence without collapse. Under these conditions, three components of the measurement problem (definite outcomes, probabilistic weights, and classical stability) admit structural reformulations as consequences of projection and aggregation rather than postulates. A fourth component, the projection selection problem: which coarse-graining is realised in a given physical context: is identified as a residual open question of comparable difficulty to the preferred basis problem; the dissolution claim is conditional on its eventual resolution.