Modern theoretical physics is formulated through several highly successful mathematical frameworks, including Green-function theory, quantum mechanics, quantum field theory, gauge theory and statistical mechanics. Although these theories employ different mathematical objects and were developed independently, they exhibit remarkable mathematical compatibility. Green functions naturally appear in quantum field theory, Hilbert-space methods underlie quantum mechanics, gauge theories are formulated geometrically, and statistical mechanics connects microscopic dynamics with macroscopic thermodynamics. The mathematical origin of this compatibility remains an open conceptual question. This paper investigates whether these apparently different frameworks can be understood as representations of a common underlying mathematical structure. We introduce the Architecture Principle, according to which the primitive object is not a Green function, wave function or Hilbert space, but a Green-Function Architecture describing propagation, interaction, closed transport and structural consistency. Green functions, quantum-mechanical states, gauge transport and statistical ensembles are interpreted as different mathematical representations of the same underlying architecture. The proposed framework does not modify the established equations of physics. Instead, it provides a common structural language that relates several major branches of theoretical physics while preserving their existing mathematical formalisms. The paper develops the basic concepts of the Architecture Principle, introduces an architectural invariant, formulates the Representation Principle and discusses representative applications. Finally, the principal mathematical problems required for a complete representation theory are identified.
I have proposed OST (Observable System Theory) as a foundational theory of system theory.3 This theory adopts two principles, namely, the Born principle and the Causal principle, and furthermore explicitly uses the one-shot measurement principle as a rule of use. OST is one of the most fundamental scientific theories, alongside mathematics and physics, and it can provide a unified framework for discussing quantum mechanics (information), statistics (system theory), practical logic, epistemology (philosophy), statistical mechanics, and so on.In this paper, I clarify that Fisherian statistics, Bayesian statistics, and quantum mechanics share a common mathematical foundation, namely OST. In statistics, an OST-based understanding of the Kalman filter is presented. Moreover, the so-called “interpretation problem” of quantum mechanics after measurement is unnecessary, and the “Wigner’s friend paradox” is resolved within OST. In addition, the FR paradox (= Frauchiger–Renner paradox) is understood as a paradox arising from the violation of the one-shot measurement principle, and thus it suggests the validity of OST. Through these discussions, I claim the following two points:(i) OST-based statistics will, in the future, become the standard understanding of statistics.(ii) OST-based quantum mechanics is the final form of von Neumann-style quantum mechanics
Classical statistical average values are generally generalized to average values of quantum mechanics, it is discovered that quantum mechanics is direct generalization of classical statistical mechanics, and we generally deduce both a general new continuous eigenvalue equation and a general discrete eigenvalue equation in quantum mechanics, and discover that a eigenvalue of quantum mechanics is just an extreme value of an operator in possibility distribution, the eigenvalue f is just classical observable quantity. A general classical statistical uncertain relation is further given, the general classical statistical uncertain relation is generally generalized to quantum uncertainty principle, the two lost conditions in classical uncertain relation and quantum uncertainty principle, respectively, are found. We generally expound the relations among uncertainty principle, singularity and condensed matter stability, discover that quantum uncertainty principle prevents from the appearance of singularity of the electromagnetic potential between nucleus and electrons, and give the failure conditions of quantum uncertainty principle. Finally, we discover that the classical limit of quantum mechanics is classical statistical mechanics, the classical statistical mechanics may further be degenerated to classical mechanics, and we discover that only saying that the classical limit of quantum mechanics is classical mechanics is mistake. As application examples, we deduce both Shrodinger equation and state superposition principle, deduce that there exist decoherent factor from a general mathematical representation of state superposition principle, and the consistent difficulty between statistical interpretation of quantum mechanics and determinant property of classical mechanics is overcome.