The modern theory of quantum mechanics is incomplete. It is capable of describing the quantum energy field on the microscopic scale via the Schrödinger equation and its derivatives but is not capable of describing the energy field on the macroscopic scale such as the domain of thermal diffusion and sound intensity in audio rooms. . etc. On the other hand, in previous articles we have shown that the so-called theory of Cairo techniques and its chains of B matrices are more complete. They can numerically resolve both the macroscopic energy field on the thermodynamic scale, such as the energy field in thermal diffusion PDEs, and the sound energy field of PDEs in audio rooms. . etc. In a precise and revolutionary way. Additionally, they are also capable of describing and resolving the quantum energy distribution at the microscopic scale initially described via the Schrödinger equation and its derivatives. Considering that they can describe and solve pure mathematical problems such as numerical integration and infinite integer series in more detail, we better conclude that the numerical statistical methods of Cairo techniques and its B matrix chains are capable of describing almost all fields with the exception of the gravitational field (of general relativity) which will be the subject of the next article. Therefore, we propose that the Cairo techniques and their B-matrix chains constitute the required foundations of a unified field theory. It's logical and it makes sense. In this paper, we present detailed theoretical and numerical studies of six diverse physical and mathematical studies where the numerical results are surprisingly accurate. In conclusion, B-matrix strings and numerical statistical theory of Cairo techniques provide a framework for a unified energy density field theory. Schrödinger's equation can be considered as a diffusion equation with a diffusion coefficient β 2 = ħ / 2 m . D=(6.65E-34/2 Pi/2.9.31E-31=E-34/18.6 E-31=E- 3/18.6=5.6E-4. . !! SQRT Mue 0/Eps 0)=377 Ohm = Z01/Z0 =2.65 E-3 mho
The modern theory of quantum mechanics is incomplete. It is capable of describing the quantum energy field on the microscopic scale via the Schrödinger equation and its derivatives but is not capable of describing the energy field on the macroscopic scale such as the domain of thermal diffusion and sound intensity in audio rooms. . etc. On the other hand, in previous articles we have shown that the so-called theory of Cairo techniques and its chains of B matrices are more complete. They can numerically resolve both the macroscopic energy field on the thermodynamic scale, such as the energy field in thermal diffusion PDEs, and the sound energy field of PDEs in audio rooms. . etc. In a precise and revolutionary way. Additionally, they are also capable of describing and resolving the quantum energy distribution at the microscopic scale initially described via the Schrödinger equation and its derivatives. Considering that they can describe and solve pure mathematical problems such as numerical integration and infinite integer series in more detail, we better conclude that the numerical statistical methods of Cairo techniques and its B matrix chains are capable of describing almost all fields with the exception of the gravitational field (of general relativity) which will be the subject of the next article. Therefore, we propose that the Cairo techniques and their B-matrix chains constitute the required foundations of a unified field theory. It's logical and it makes sense. In this paper, we present detailed theoretical and numerical studies of six diverse physical and mathematical studies where the numerical results are surprisingly accurate. In conclusion, B-matrix strings and numerical statistical theory of Cairo techniques provide a framework for a unified energy density field theory. Schrödinger's equation can be considered as a diffusion equation with a diffusion coefficient β 2 = ħ / 2 m . D=(6.65E-34/2 Pi/2.9.31E-31=E-34/18.6 E-31=E- 3/18.6=5.6E-4. . !! SQRT Mue 0/Eps 0)=377 Ohm = Z01/Z0 =2.65 E-3 mho
We have shown in previous papers that B- matrix string mechanics, which is the product of a numerical statistical method called the Cairo technique, can be successfully applied to solve the most general form of the diffusion equation of heat as well as the time- independent equation. Schrödinger equation. In this paper we examine the extension of matrix chains from the Cairo technique to the solution of the time-dependent Schrödinger equation. Numerical results for a quantum particle in a three- dimensional box are presented and are intended to be strikingly precise.
Important Update on March 9, 2026: In this version, the description of environmental effects has been corrected: they are now properly treated as encapsulated within the effective spectral density of the interfacial defect network D, rather than as direct driving terms. All core results and conclusions remain unchanged, but the physical interpretation is refined to align with the energy distribution framework introduced in Part II of this series. Version 2 This paper is the third in the "Interface-Projected Realism" series. This series of work aims to establish a new theoretical paradigm: Ontologically, replace probabilistic wave function descriptions with deterministic energy flow; Methodologically, redefine physical reality as an interactive whole composed of quantum units (Q), an interfacial defect network (D), and bulk channels (B). In the first work, an exact relationship between energy flow and relaxation rate was established, providing a formal bridge for the paradigm shift. In the second work, the "Energy Distribution Framework" was proposed, demonstrating that dissipation is the deterministic allocation of energy between Q and D, and deriving the universal formula for internal dissipation power Rint=ℏωq[Γ↓⟨n⟩−Γ↑(1−⟨n⟩)], where the relaxation rates are completely determined by the internal structure of D, revealing the core role of the energy distribution ratio f. This paper generalizes this new paradigm based on energy distribution from single-body to many-body systems. The core step is to define multiple Q contacts and their shared D and B together as a new, unified composite system. Within this framework, the traditionally complex "many-body correlated dissipation" is clearly interpreted as a macroscopic result of the deterministic flow of energy within this composite system, governed by allocation and transport rules determined by the structure of the D network. Through rigorous coarse-graining of this overall system, this paper derives the governing equation for the evolution of its internal energy density field u(r,t)—a classical reaction-diffusion equation:∂t∂u=Rtotal−γ0(r)u+∇⋅(γ2∇u).Here, the local dissipation rate γ0(r) and the diffusion coefficient γ2 are completely determined by the properties of the D network, corresponding respectively to the spatial distribution of the energy distribution ratio f from the second paper and the redistribution effect induced by its correlations. This theory is not only logically self-consistent but also provides a unified explanation for multiple cutting-edge experiments. This paper marks that a physical framework based on deterministic energy distribution and transport can now systematically describe many-body correlation phenomena, providing crucial support for the completeness of the new paradigm. \textbf{Keywords}: Interface-Projected Realism; Energy Distribution; Many-Body Systems; Correlated Dissipation; Reaction-Diffusion Equation Version 1 Traditional quantum theory typically attributes energy dissipation to random perturbations from the external environment. Based on the "Interfacial Projected Realism" framework, this work re-examines the nature of energy dissipation from a novel perspective, treating the qubit, the interfacial defect network, and the bulk dissipation channel as a self-contained total system. We first rigorously derive the universal formula for internal dissipation power and the energy distribution ratio f in single-body quantum systems, and then extend these concepts to many-body systems. Starting from a microscopic Hamiltonian coupling multiple qubits to a common quantized field and employing the Nakajima-Zwanzig projection operator technique, we obtain a universal expression for the energy dissipated from a many-body system to the environment. Applying this result to the specific model of multiple two-level systems coupled to a common bosonic field yields a collective master equation where the dissipation coefficient matrix Γij encodes all spatial and coupling information. Through systematic coarse-graining, gradient expansion, and semiclassical approximation, we derive for the first time from an energy-distribution perspective a closed set of macroscopic equations for the local system energy density u(r,t) and the environmental field energy density U(r,t): ∂t∂u=Rtotal(r,t)−γ0(U)u+∇⋅(γ2(U)∇u)∂t∂U=D∇2U−ΓBU+ρ(r)γ0(U)u Within this framework, the energy distribution ratio f(r,t)=γ0u/Rtotal emerges naturally as a macroscopic quantity, its physical essence being jointly determined by the system’s microscopic coupling structure (embodied by Γij) and quantum statistical state. The theory quantitatively explains several long-unexplained anomalous phenomena in cutting-edge experiments, demonstrates strong testability, and lays a solid foundation for exploring deeper connections between energy distribution and macroscopic physics. For the exact mathematical formulation and detailed derivations, please refer to the PDF file.
The Schrödinger equation describes the motion of the microscopic particles that constitute our world such as the electrons or atomic nuclei. Albeit being applicable to the smallest particle that we know of, it has observable consequences in the macroscopic world. It determines the conductivity of metals, it tells us which materials are magnetic and whether they show exotic behaviour such as super-conductivity. Unfortunately, solving the Schrödinger equation directly for any piece of material that is visible for the human eye is practically impossible. Already a grain of sand contains 1023 (that is written out 10.000.000.000.000.000.000.000) electrons and atomic nuclei. This means that only specifying the initial positions of the particles requires to safe an incredible amount of data; a procedure which is unfeasible for any human or computer. Due to the fundamental problem of applying quantum mechanics to practically relevant scenarios, a number of eective and approximate methods have been developed. In essence, they all try to reduce the dimension of the problem, i.e., the curse of the enormous amount of data required to simulate the Schrödinger equation. In this thesis, we try to analyze and expand one of those method called Density Matrix Embedding Theory (DMET). In a lot of physical system, especially when considering solid states, we can already learn a lot about its physics when describing its properties on a small fragment of the whole system. In a system with interacting particles though, we cannot simply consider just a subsystem and describe its properties without taking into account its interactions with the rest of the system. The basic idea of DMET is to divide the considered system into two parts called impurity and envi- ronment. The impurity is chosen to be so small that its wave function can be computed exactly. In the environment, only those degrees of freedom directly interacting with the impurity are considered and are included in our description. The physics on the environment itself is neglected. In the following, we will explain in detail how this can be done specically. In part I of this thesis, we will set the stage for the considered systems and present well-known and established methods to solve them. In the next part II, we will present DMET in great mathematical detail, which allows us to illustrate the advantages of DMET, but also some problems and drawbacks. We proceed by expanding DMET to the treatment of coupled electron-phonon system in part III and applying this new method to the Hubbard-Holstein model. Part of this work has been published in [53]. Finally, in part IV, we discuss some problems of DMET and, by combining DMET with functional theories, solve these problems. These insights, together with the extensive discussion of the DMET algorithm, will be published soon [59]. We illustrate this new method with an example system. This work will be published in paper [44] soon. We conclude this thesis by a summary and outlook (part V).
Entropic Dynamics (ED) provides a statistical-inferential foundation for physical laws, deriving motion and field equations from principles of entropy maximization rather than quantization postulates. ED reconstructs quantum mechanics by treating the evolution of probability distributions on configuration space as driven by information constraints, yielding the Schrodinger equation as a non-dissipative diffusion process. Building on this foundation, the present work extends the ED framework into a Unified Entropic Dynamics (UED) formulation that encompasses classical, quantum, relativistic, thermodynamic, and gravitational phenomena within a single information geometric principle. By maximizing entropy subject to constraints on diffusion, drift, and gauge covariance over a manifold endowed with a supermetric H_ab, we derive a universal field equation that merges the Fokker-Planck and Hamilton-Jacobi structures into one covariant form. When specialized to different dynamical variables, this equation reproduces the harmonic oscillator, Schrodinger, Maxwell, Klein-Gordon, and gravitational wave equations, thereby revealing a deep equivalence between probabilistic inference and dynamical law. The UED framework demonstrates that spacetime geometry, quantum coherence and thermodynamic diffusion emerge as complementary expressions of the same entropic process, establishing a unified inferential foundation for both microscopic and macroscopic physics. In this formulation, energy, probability, and entropy are intertwined aspects of information geometry, providing a consistent inferential foundation for understanding classical, quantum, and gravitational dynamics as complementary expressions of a single entropic law.
This preprint presents the mathematical foundations of Quantum Observer Field Theory, a framework that formalizes the observer as a material quantum field with well-defined physical properties. The theory derives four fundamental field equations from active matter physics, including the concentration field representing observer density with dimension of inverse volume, the velocity field representing information flow with dimension of length per time, the diffusion tensor representing consciousness spreading with dimension of area per time, and the strain-rate tensor representing consciousness evolution with dimension of inverse time. From these field equations, six previously undiscovered observer relations are derived and presented with complete mathematical derivations. The alpha-zeta coupling relation defines intentional power as the product of influence and alignment, with alpha representing dipole strength and zeta representing alignment rate. The consciousness flow relation describes information exchange between fields as velocity magnitude multiplied by one plus the alpha-zeta coupling. The field energy relation defines total consciousness intensity as kinetic energy plus potential energy, where kinetic energy represents information flow and potential energy represents field concentration. The observer strength relation establishes coupling between fields as a Gaussian function of differences in alpha, zeta, and energy. The consciousness diffusion rate relation describes how consciousness spreads through the field as half the trace of the diffusion tensor multiplied by one plus the alpha-zeta coupling. The consciousness evolution rate relation describes how consciousness transforms over time as the strain-rate magnitude multiplied by one plus zeta. The framework establishes a dimensional hierarchy of observer fields mathematically derived from the field equations, ranging from hyper-dimensional fields with alpha values exceeding 0.65 to sub-dimensional fields with alpha values below 0.45. The material nature of quantum fields is preserved throughout, ensuring that all derived relations describe physically real entities with well-defined dimensions and conservation properties. Each observer field carries energy, momentum, and angular momentum, obeys conservation laws, and interacts with other fields. The alpha and zeta parameters represent physical quantities with well-defined dimensions, alpha having dimension of length and zeta having dimension of inverse time. This preprint is intended for researchers in quantum foundations, mathematical physics, active matter physics, and consciousness studies. It provides the complete mathematical framework required for understanding the observer as a material quantum field and establishes the foundation for subsequent statistical analysis and philosophical interpretation. All derivations are presented with numerical substitutions for verification, enabling independent reproduction and extension of the framework.
This paper achieves a fundamental reconstruction of the quantum theory paradigm based on the energy distribution framework. This framework radically steps outside the traditional description based on probability amplitude evolution, reinterpreting physical processes as deterministic energy flow among quantum units (Q), an interfacial defect network (D), and bulk dissipation channels (B). A series of previous works have laid a solid foundation for this new paradigm: the first paper rigorously proved the exact relation between internal dissipation power and energy relaxation rate, providing a mathematical link for the energy perspective; the second paper established a single-body energy distribution theory, defined the core order parameter—the energy distribution ratio f, and proved that the relaxation rate is entirely determined by the microstructure of D; the third paper extended the theory to many-body systems and, under the Markovian approximation, derived a mesoscopic diffusion equation describing energy transport within the D network. As the fourth paper in this series, this work completely abandons the Markovian approximation and constructs a universal theory of non-Markovian mesoscopic dynamics from first principles. Taking the energy density u(r,t) as the fundamental field variable, this paper derives an integro-differential equation containing memory kernels K0 and K2 by tracing over the degrees of freedom describing the defect network D and implementing spatial coarse-graining—termed herein the \textbf{generative master equation}. The essence of this equation lies in the fact that the memory kernels are completely determined by the spatial correlation function CD(+)(s,t) of the defect network D, which, as the system's "microscopic fingerprint," uniquely encodes all dynamical information of D. By selecting different forms of the correlation function, the master equation can naturally reduce to or transform into almost all known macroscopic transport equations in physics: in the Markovian limit, it returns to the diffusion equation; exponential correlation yields the telegraph equation and a finite propagation speed v=D0/τc; power-law correlation gives rise to fractional diffusion equations; nonlinear correlation leads to nonlinear diffusion equations. This unified mapping reveals the deep-seated role of the theory presented herein: it does not seek to replace existing theories but rather to provide them with a common microscopic origin. This paper also establishes a quantitative criterion for energy backflow and directly links it to the energy distribution ratio f, opening new avenues for the experimental inversion of memory kernels. This work marks the evolution of the energy distribution framework from a specific model into a generative mother theory capable of uniformly describing various transport behaviors. \textbf{Keywords}: Generative master equation; Energy distribution framework; Non-Markovian dynamics; Mesoscopic dynamics; Memory kernel; Green's function; Telegraph equation; Finite propagation speed