The Isomorphism of Complex Numbers and Latent Worlds 5; The Rigorization and Higher-Dimensional Generalization of Latent World Algebra: An Algebraic Foundation of Quantum Structure
The indispensability of the complex structure in quantum mechanics has beenexperimentally confirmed, but its algebraic origin—the fact that the complex number field is the only commutative, algebraically closed division algebra over the realnumbers—has not yet been rigorously linked to an interpretive framework of quantum reality. This paper proposes an axiomatic system for a latent world algebra,decomposing the descriptive structure of quantum states into a direct sum of anobservable subalgebra and a latent world subalgebra, and defining the multiplication rules and commutation relations between them. It is proved that this algebraicstructure is a concrete realization of a C*-algebra, where self-adjoint elements correspond to the algebraic representation of observables, and non-self-adjoint elementsencode the algebraic information of modal dimensions. Furthermore, the square-lawclassification of the latent world subalgebra and its correspondence with Cliffordalgebras are explored, interpreting the difference between J2 = −I (complex structure) and J2 = +I (real structure) as two distinct physical modalities. On thisbasis, the algebraic framework is extended to quaternionic and octonionic cases,arguing that the non-associativity of octonions may correspond to the non-localityof quantum entanglement. Finally, topological invariants such as Chern numbersand characteristic classes are represented as numerical values obtained from thepairing of central cohomology classes of the latent world algebra with K-theory,establishing an equivalence relation between topological transitions and algebraicprojection mappings. This study provides a rigorous mathematical foundation forunderstanding the algebraic necessity of quantum structures and opens a formalizedpath for exploring hypercomplex quantum theories.
The indispensability of the complex structure in quantum mechanics has beenexperimentally confirmed, but its algebraic origin—the fact that the complex number field is the only commutative, algebraically closed division algebra over the realnumbers—has not yet been rigorously linked to an interpretive framework of quantum reality. This paper proposes an axiomatic system for a latent world algebra,decomposing the descriptive structure of quantum states into a direct sum of anobservable subalgebra and a latent world subalgebra, and defining the multiplication rules and commutation relations between them. It is proved that this algebraicstructure is a concrete realization of a C*-algebra, where self-adjoint elements correspond to the algebraic representation of observables, and non-self-adjoint elementsencode the algebraic information of modal dimensions. Furthermore, the square-lawclassification of the latent world subalgebra and its correspondence with Cliffordalgebras are explored, interpreting the difference between J2 = −I (complex structure) and J2 = +I (real structure) as two distinct physical modalities. On thisbasis, the algebraic framework is extended to quaternionic and octonionic cases,arguing that the non-associativity of octonions may correspond to the non-localityof quantum entanglement. Finally, topological invariants such as Chern numbersand characteristic classes are represented as numerical values obtained from thepairing of central cohomology classes of the latent world algebra with K-theory,establishing an equivalence relation between topological transitions and algebraicprojection mappings. This study provides a rigorous mathematical foundation forunderstanding the algebraic necessity of quantum structures and opens a formalizedpath for exploring hypercomplex quantum theories.
The indispensability of the complex structure in quantum mechanics has beenexperimentally confirmed, yet its ontological status—whether complex numbers aremerely a mathematical tool or a constitutive element of physical reality—has longremained undecided. Based on recent breakthrough experiments in complex quantum theory, this paper proposes a modal realist interpretation framework for thecomplex structure. The core thesis is that the mathematical relationship betweenthe complex and real number fields provides a rigorous ontological model for understanding the quantum-classical transition: the observable classical world is thereal projection of the complex structure, while the ”imaginary part” of the complexstructure corresponds to the modal reality dimension of the physical system. Thisframework reconceptualizes the measurement problem as a problem of the transition of the existence mode of a mathematical structure, rather than a problem ofdynamical collapse or world splitting. By establishing a rigorous axiomatic foundation, the modulus-squared projection mapping is established as the mathematicalessence of measurement, and the imaginarity measure is introduced as a quantitative indicator of the modal dimension. The inherent compatibility of this frameworkwith quantum resource theory, quantum metrology, and the branched Hilbert subspace interpretation is demonstrated. Based on this, three testable predictions withquantum metrological precision are proposed, concerning the extractability boundof complex phase information, the correlation between imaginarity resources andadvantages in quantum state discrimination, and the partial reconstructability ofmodal information via threshold tomography. The framework presented here possesses an inherent capacity for generating open questions and a layered, modifiableresilient structure, providing a research program for foundational quantum issuesthat combines mathematical rigor with experimental testability.
The indispensability of the complex structure in quantum mechanics has beenexperimentally confirmed, yet its ontological status—whether complex numbers aremerely a mathematical tool or a constitutive element of physical reality—has longremained undecided. Starting from mathematical necessity, this paper argues thatthe complex structure is not only the descriptive language of quantum mechanicsbut also an ontological constraint condition of physical reality. Based on Frobenius’ theorem on the classification of division algebras over the real numbers, wereveal: if the descriptive framework is required to be both commutative and algebraically closed, then the complex structure is the unique logically necessary choice.This mathematical constraint, when mapped onto physical reality, implies that the”imaginary part” corresponds to a modal dimension that is not directly observablebut constrains observable phenomena. The measurement process is thus reconceptualized as a projection from the complex structure onto a real subspace, whereinformation loss is an inherent feature of the mathematical structure rather thanan additional dynamical assumption. We demonstrate that this framework is compatible with research on imaginarity as a resource in quantum resource theory andwith analyses of Fisher information limits in quantum metrology, and we proposethree testable predictions with quantum metrological precision. This interpretationadvances the quantum measurement problem from a matter of interpretational debate to a level of structural analysis, providing a rigorous mathematical foundationfor understanding the quantum-classical transition.
This paper proposes a structuralist interpretational framework for the quantummeasurement problem based on the concept of a latent world algebra. The framework decomposes the algebra corresponding to a physical system into a direct sumof an observable quantum algebra and a modal Lie algebra: the former correspondsto the commutative structure of classical observables, while the latter encodes thequantum modal information represented by complex phases. The measurementprocess is rigorously defined as a projection mapping from the latent world algebraonto the observable quantum algebra; this mapping discards all modal informationand is not an algebraic homomorphism, thus precisely capturing, at the algebraiclevel, the information loss and non-unitary character of measurement. Through representation theory and the Gelfand–Naimark–Segal construction, the framework isshown to be fully compatible with the standard formalism of quantum mechanics and naturally yields the Born rule. From the perspective of quantum resourcetheory, the framework reveals the operational meaning of the imaginary part ofcomplex numbers: the imaginary-part coherence measure is positively correlatedwith the success probability advantage of complex measurement bases in quantumstate discrimination tasks, thereby providing an algebraic explanation for recentexperiments establishing the imaginary part as a quantum resource. On the dynamical side, the paper introduces the concept of information potential differenceand establishes its threshold condition, characterizing the emergence of the classical world as the continuous dynamics of irreversible loss of modal informationduring decoherence in open systems. The testability of the framework is embodiedin three quantitative predictions concerning the precision limits of complex phasemeasurements, the correlation between the imaginary-part resource measure andstate-discrimination advantage, and the tomographic reconstructability of modalinformation; each prediction comes with a clearly specified experimental platformand falsification criteria. This work transforms the quantum measurement problemfrom an interpretational debate into an analyzable algebraic-dynamical structure,offering a new theoretical pathway for understanding the fundamental status ofcomplex numbers in physics.
The indispensable nature of complex numbers in quantum mechanics has beenexperimentally confirmed, yet its deep origin — whether complex numbers are anad hoc introduction or a logical consequence of some more fundamental constraint—remains insufficiently clarified. From the perspective of symmetry analysis, thispaper argues that the complex structure is not an accidental choice of quantumtheory, but an inevitable outcome under the joint constraints of spacetime transformation symmetry and algebraic completeness. Through a rigorous analysis ofthe covariance of the Schrödinger equation under Galilean transformations, thispaper reveals: if the transformation rule of the wave function is required to preserve the probability interpretation and satisfy spacetime symmetry, then the wavefunction must take values in the complex number field. This conclusion can beextended to relativistic cases, forming an intrinsic echo with the role of mass asa central charge in Bargmann’s theorem. On this basis, we establish a quantitative criterion for the strength of symmetry constraints, proving that the transitionfrom a real-valued formulation of quantum theory to a complex-valued formulationcorresponds to a central extension of the symmetry algebra, which is the uniqueway for a physical theory to realize symmetry in the sense of projective representations. This framework is compatible with recent experimental advances regardingimaginarity as a quantum resource and multipath interference tests of hypercomplex theories, and proposes a symplectic geometry-based scheme for measuring thestrength of symmetry constraints. This paper advances the foundational issue ofcomplex numbers in quantum theory from an ”ontological debate” to a new levelof ”symmetry constraint analysis.”
This paper constructs a derivation framework for quantum mechanics starting from a physical encoder in a finite-temperature heat bath, without relying onprior quantum assumptions. By strictly distinguishing the signal space from thestate-label space, and adopting information conservation and computability as operational axioms, we prove that the steady-state power response of the encoder isuniquely locked by thermodynamic stability to a positive-definite quadratic formon the signal space. This quadratic form, via the parallelogram law, a posterioriinduces an inner product, which is then pulled back by the encoding map to a metric on the state space. Linearity arises from the composability of physical inputsand injectivity of the encoding map, rather than from causal inference of energy response. Under the requirement of algebraic closure, we analyse the thermodynamiccosts of octonions and quaternions: the non-associativity of octonions necessitatesthe introduction of exceptional constraint algebras for composite-system construction, whose maintenance generates irreversible entropy production exceeding theLandauer limit; the non-commutativity of quaternions leads to ambiguity in spectral decomposition and potential path-dependent entropy production, for which arigorous statistical mechanical derivation is yet to be completed. The complex fieldemerges as the unique natural choice, being the only scalar algebra that simultaneously satisfies finite-dimensionality, algebraic closure, associativity, commutativity,and normed divisibility. The Born rule is not derived from thermodynamic powerweights, but emerges as the unique self-consistent frequency assignment for thecontext-independent response of the encoder, supported by a Gleason-type uniqueness argument and the law of large numbers. Latent-world information is defined asthe difference between Shannon entropy and von Neumann entropy; its geometryis exactly the restriction of the Bures metric along off-diagonal generators, corresponding to a scaled version of the Killing form of the adjoint Lie algebra, and itsdynamics follows a decoherence-driven exponential decay equation, directly interfacing with the master equations of open quantum systems. Finally, we proposefalsifiable experimental predictions comparing the information-energy efficiency ofcomplex encoders with real-number quantum mechanics (RQM), and establish astatistical hypothesis-testing framework.
The indispensability of the complex structure in quantum mechanics has beenexperimentally confirmed, but the geometric description of the latent world (modaldimension) still lacks a rigorous metric theory. This paper elevates the latent worldalgebraic framework to the level of information geometry, proving that the naturalmetric carried by the latent world subalgebra Alat is precisely the Bures-Wassersteinmetric, and that the measurement projection π : A → Aobs corresponds to a gradient flow from the Wasserstein-2 space to the Fisher-Rao space. The core discoveryis that the critical condition for the information potential difference ∆µ ≥ ln2 isequivalent to the curvature singularity of the Bures distance, and the projection dynamics is exactly the blow-up solution of the Wasserstein gradient flow at the divergence of the Ricci curvature. This paper establishes the exact relation Klat = −14FQ between the latent world curvature Klat and the quantum Fisher information FQ,proposes testable predictions of information-geometric phase transitions, and extends the framework to noncommutative geometry (spectral triples) and interfaceswith quantum gravity. This work provides a unified perspective of differential geometry and optimal transport for understanding the quantum-classical transition.