Fundamental physics contains a family of reciprocal relations that appear in distinct theoretical contexts yet share an identical mathematical structure. The wave propagation relation fλ = c [1], the photon energy relation Eλ = hc [3], the matter-wave relation pλ = h [4], and the quantum of action ET = h [2] each preserve the product of two complementary observable quantities through one or more universal constants. These relations arise independently within classical electrodynamics, photon physics, and quantum mechanics, and their common reciprocal structure is generally attributed to mathematical coincidence rather than an underlying common principle. This work investigates whether these relations can be understood as different representations of a common invariant physical content, connected through a principle of representation preservation. The development proceeds in three stages. First, the general mathematical condition imposed by invariant physical content under changes of representation is derived, establishing that complementary representations must transform reciprocally (Theorem 1). Second, a geometric construction based on synchronized concentric circular representations is introduced, from which a representation length emerges through arc differences. Third, the experimentally established constants c [1, 16] and h [2] are introduced as independent normalization principles governing propagation and action, respectively, connecting the abstract geometric structure to measurable physical quantities. Within this framework, the standard wave relation, the photon energy relation, the de Broglie matter-wave relation, and the quantum of action are recovered as complementary representations of the same invariant physical content (Sect. 9). A representation operator is formally defined and shown to form a one-parameter commutative group of content-preserving transformations (Propositions 1–4, Sect. 7). The representation coordinate is computed for the hydrogen ground state, yielding the fine-structure constant α as the dimensionless representation coordinate of the electronic ground state (Sect. 11). The framework does not replace established physical theories nor derive the numerical values of the fundamental constants. It proposes instead a unified geometric interpretation in which wavelength, frequency, momentum, energy, and time appear as reciprocal representations connected through propagation and action normalization.
Fundamental physics contains a family of reciprocal relations that appear in distinct theoretical contexts yet share an identical mathematical structure. The wave propagation relation fλ = c [1], the photon energy relation Eλ = hc [3], the matter-wave relation pλ = h [4], and the quantum of action ET = h [2] each preserve the product of two complementary observable quantities through one or more universal constants. These relations arise independently within classical electrodynamics, photon physics, and quantum mechanics, and their common reciprocal structure is generally attributed to mathematical coincidence rather than an underlying common principle. This work investigates whether these relations can be understood as different representations of a common invariant physical content, connected through a principle of representation preservation. The development proceeds in three stages. First, the general mathematical condition imposed by invariant physical content under changes of representation is derived, establishing that complementary representations must transform reciprocally (Theorem 1). Second, a geometric construction based on synchronized concentric circular representations is introduced, from which a representation length emerges through arc differences. Third, the experimentally established constants c [1, 16] and h [2] are introduced as independent normalization principles governing propagation and action, respectively, connecting the abstract geometric structure to measurable physical quantities. Within this framework, the standard wave relation, the photon energy relation, the de Broglie matter-wave relation, and the quantum of action are recovered as complementary representations of the same invariant physical content (Sect. 9). A representation operator is formally defined and shown to form a one-parameter commutative group of content-preserving transformations (Propositions 1–4, Sect. 7). The representation coordinate is computed for the hydrogen ground state, yielding the fine-structure constant α as the dimensionless representation coordinate of the electronic ground state (Sect. 11). The framework does not replace established physical theories nor derive the numerical values of the fundamental constants. It proposes instead a unified geometric interpretation in which wavelength, frequency, momentum, energy, and time appear as reciprocal representations connected through propagation and action normalization.
Several fundamental physical relations share a mathematical structure in which the product of two conjugate quantities equals an invariant constant. Representative examples include the energy-period relation ET = h (where T = 1/f is the oscillation period and h is Planck’s constant), the momentum-wavelength relation pλ = h (the de Broglie relation), and the frequency-wavelength dispersion relation fλ = c. Although these expressions originate from quantum mechanics, wave mechanics, and electromagnetism respectively, each belongs to the generalized class XY = K, where K is an invariant constant. Their shared algebraic form raises the question of whether a common organizational principle underlies physical representation. This paper proposes the Foundational Principle of Closure Representation as an interpretational framework. Within this framework, action is adopted as the foundational quantity, and closure is introduced as a representation-independent measure of cyclic completion, defined by the condition Δθ = 2π. Planck’s constant h is interpreted as the action associated with one completed closure—consistent with the Bohr–Sommerfeld quantization condition ∮p dq = nh. The Planck relation E = hf is accordingly understood as the product of action per closure and closure rate per unit time. A representation-transfer relation (RX)(Y/R) = XY is introduced, where R is a dimensionless representation coordinate. A measurable realization is given by R = λC/λdB = p/(mc), coinciding with the standard relativistic quantity γβ. This relation is a mathematical identity; its physical significance lies in the proposed interpretation of R as a representation coordinate characterizing the distribution of closure action between conjugate observables. The proposed framework does not introduce new physical laws, modify established equations, or claim to supersede existing theories. It provides a unified interpretational architecture within which several established reciprocal relations—connecting action, closure, wavelength, frequency, momentum, and propagation—may be understood as instances of a common closure-action structure.
We present a group-theoretic reconstruction of quantum mechanics in which the standard formalism emerges from the representation theory of symmetry groups rather than being introduced axiomatically. Starting from physically motivated symmetry principles, we show how the conventional postulates of quantum mechanics including the Hilbert space structure of states, observables as generators of symmetry transformations, the Born probability rule, unitary time evolution, the composition of multipartite systems, and the measurement framework can be recovered within a unified representation-theoretic framework. We then examine the measurement problem by interpreting wave-function collapse as the scattering of quantum states, thereby providing an account of the apparent loss of information. Finally, we investigate the origin of Planck's constant, proposing that it arises as the conserved charge of the Lorentz symmetry group. In this formulation, ℏ is an invariant that relates Lie algebra generators to observable quantities and fixes the scale of quantum action. This framework offers a conceptually unified perspective on the foundations of quantum mechanics, connecting its postulates, measurement theory, and the emergence of the quantum of action to a common symmetry-based origin while identifying conditions under which the conventional quantum formalism is recovered.
This work presents a two-stage first-principles construction for foundational physics. In the first stage, an axiomatic framework is introduced in which physical objecthood is defined exclusively through participation in observation–feedback (OF) processes. At this axiomatic level, no background spacetime, metric, Hilbert space, or symmetry group is assumed a priori. In the second stage, a minimal faithful realization of this framework is adopted, consisting of a set of explicitly stated, controlled algebraic choices that support finite operational distinguishability. Within this minimal realization, a hierarchy of physically significant structures is consistently represented: an internal O(3) symmetry with oriented SO(3) sectors; a Lorentz-type kinematical structure with invariant group SO⁺(3,1) (the proper orthochronous Lorentz group) and an associated invariant propagation scale that occupies the structural role of the conventional speed c; and a minimal closure scale that, under standard action-dimensional normalization, plays a structural role analogous to that of the reduced Planck constant ℏ. This construction thus provides a unified relational setting in which the core kinematical architectures of special relativity and quantum mechanics admit a consistent, operationally grounded representation, realized under explicitly stated minimal assumptions. No derivation of dynamical equations or direct identification with empirical spacetime geometry is claimed.
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.
Abstract We propose physical distinguishability as the fundamental primitive underlying quantum mechanics, general relativity, and thermodynamics. A state is defined operationally: two states are physically distinct if a physical interaction or measurement can reveal a difference between them. From this single principle, the distinguishability divergence D generates a smooth state space endowed with a Riemannian metric , a geometric velocity, an accessibility scale Λ, an accessibility ratio , and a geometric time . The framework provides a unified geometric description of the three major domains of physics. Quantum mechanics emerges through the Fubini–Study geometry of projective Hilbert space, thermodynamics through Fisher–Rao and Ruppeiner information geometry, and general relativity through the geometry of geodesic deviation. The threshold condition A=1acquires a universal physical meaning and appears in all three realizations as the boundary between physically accessible and inaccessible evolution. Variational formulations of both the Einstein and Schrödinger equations are constructed within the same geometric framework. The Lorentzian signature of spacetime emerges from a single sign reversal associated with the geometric-time direction, while the coupling constant of the distinguishability action is connected to the observed vacuum energy density. The framework further predicts a characteristic vacuum distinguishability scale ≈8.8×10^(-5) m, corresponding to an energy scale≈2.25" " meV. The theory is tested experimentally on IBM Quantum superconducting processors. Across one-qubit, two-qubit, 127-qubit, and 156-qubit experiments performed on three independent Heron-generation backends, excitation probabilities collapse onto universal curves when parameterized by endpoint Fubini–Study distinguishability. Protocols with identical laboratory control parameters but different endpoint distinguishabilities produce excitation probabilities differing by up to a factor of 3.8, while protocols with identical distinguishability produce identical outcomes across independent processors. Cross-backend transfer errors remain below 1%, providing direct experimental support for endpoint distinguishability as a universal physical variable. Taken together, the theoretical construction, variational formulations, vacuum-sector predictions, and large-scale quantum-processor experiments support distinguishability geometry as a unified and experimentally testable foundation for physical dynamics.
Modern theoretical physics is formulated through several highly successful mathematical frameworks. Green functions describe propagation, Hilbert spaces provide the mathematical language of quantum mechanics, gauge theories describe transport and symmetry, quantum field theory combines relativistic fields with quantum mechanics, and statistical mechanics relates microscopic dynamics to thermodynamic observables. Although these theories employ different primitive mathematical structures, they exhibit remarkable mathematical compatibility. The structural origin of this compatibility remains largely unexplained. The purpose of this paper is to investigate whether these apparently distinct frameworks admit a common mathematical description. We introduce the Architecture Principle, according to which the primitive mathematical object is a Green-Function Architecture rather than an isolated Green function, Hilbert space, quantum state, or particle. Green functions, Hilbert-space representations, gauge transport, quantum field theoretic propagators, and statistical ensembles are interpreted as different representations of a common underlying architecture. The associated Architectural Multiplicity provides a structural invariant that organizes these representations within a unified mathematical framework. The present work should be viewed as a structural reconstruction program rather than a replacement for established physical theories. Existing mathematical formalisms remain unchanged; the Architecture Principle instead proposes a common structural foundation that explains their mathematical coherence. The paper develops the basic definitions, introduces the principal architectural invariant, formulates the Representation Principle, discusses applications to several branches of theoretical physics, and identifies the principal mathematical problems that remain open.