This paper establishes a strict isomorphism between the supersymmetric modulispace and the quantum state space, identifying the moduli space under BPS constraints as a Lagrangian submanifold in the quantum state space. By constructingan embedding map Φ : M → P(H) from the moduli space to the space of density operators, we prove that this map satisfies three core conditions: the imageis a Lagrangian submanifold, the pullback of the Bures metric equals the specialKähler metric, and the tangent space is isomorphic to the image of the commutator of the latent world algebra. Furthermore, we reveal the relationship betweenthe Bures metric and the quantum Fisher information matrix, demonstrating thatits reduced form under BPS states is completely consistent with the special Kähler metric. Based on this isomorphism, the objective reality of complex numbersis reinterpreted as a geometric projection of the modal dimensions in the latentworld algebra, providing a unified mathematical foundation for understanding thequantum measurement problem and the emergence of spacetime.
This paper establishes a strict isomorphism between the supersymmetric modulispace and the quantum state space, identifying the moduli space under BPS constraints as a Lagrangian submanifold in the quantum state space. By constructingan embedding map Φ : M → P(H) from the moduli space to the space of density operators, we prove that this map satisfies three core conditions: the imageis a Lagrangian submanifold, the pullback of the Bures metric equals the specialKähler metric, and the tangent space is isomorphic to the image of the commutator of the latent world algebra. Furthermore, we reveal the relationship betweenthe Bures metric and the quantum Fisher information matrix, demonstrating thatits reduced form under BPS states is completely consistent with the special Kähler metric. Based on this isomorphism, the objective reality of complex numbersis reinterpreted as a geometric projection of the modal dimensions in the latentworld algebra, providing a unified mathematical foundation for understanding thequantum measurement problem and the emergence of spacetime.
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.
This paper constructs an interpretative framework of quantum mechanics thattreats the complex structure as a modal dimension, reconstructing quantum measurement as an information-geometric phase transition rather than an instantaneouswavefunction collapse. By introducing a rigorous decomposition of Jordan algebrasand Lie algebras, the latent-world subspace is proven to form a bi-module structureunder the Jordan product and the projected Lie bracket, thereby being compatible with finite-dimensional algebraic frameworks. In the pure-state limit, the sectional curvature of the Bures metric and the quantum Fisher information satisfy anexact differential-geometric identity; the environmental corrections for the mixedstate case are honestly labeled as scaling conjectures constrained by renormalizationgroup. The information potential difference is rigorously defined via the relative entropy of coherence, and the Landauer principle yields the minimal thermodynamiccost of erasing coherent information in the quasi-static limit, with the ln 2 threshold re‑positioned as a calibratable benchmark in non‑equilibrium, finite‑temperatureenvironments. The physical inadequacy of the real-number framework under thesame Hilbert-space dimension is established by the exclusion principle of Bell-typeexperiments, proving that imaginarity is the physical origin of quantum interferencerather than a notational convenience. By introducing the Frobenius theorem as aselection constraint and Bell-type experiments as empirical constraints, the ternarycircular argument is broken into a unidirectional implication chain, establishingthe epistemological foundation of constraint structural realism. Four experimental predictions with explicit falsification conditions provide an operable interfacefor experiments, completing the proof closure from heuristic program to axiomaticsystem.
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, 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.
This paper establishes the foundational differential-geometric architecture for the coordinate-independent tracking of pure quantum states within closed physical systems, validated via explicit computational modeling of a driven two-level system. While standard quantum mechanics relies on the linear algebraic structure of complex Hilbert spaces, the unique identification of physical states requires factoring out non-zero complex scaling factors, positioning the true arena of quantum dynamics as the projective Hilbert space P(H). We analyze this space as a compact complex manifold possessing an intrinsic Kähler structure that fuses a Riemannian metric tensor, a symplectic form, and a complex structure into a unified geometric framework. Through this manifold lens, we systematically derive the Fubini–Study metric tensor gF S and its corresponding connection geometries. By evaluating the evolution of state vectors under a time-dependent driving Hamiltonian, we show that the natural propagation of a closed quantum system traces out a geodesic path on the projective manifold. This geometric tracking defines a localized vector gauge field—the Berry connection 1-form—which gives rise to an antisymmetric curvature tensor that modifies the effective classical mechanics of the parameter space. Furthermore, we state and prove three core theorems establishing the geometric constraints of pure state transport: the Holonomy Theorem for cyclic evolutions, the Quantum Cramér–Rao Bound, and the Mandelstam–Tamm Quantum Speed Limit. To bridge theory with physical architectures, we implement an analytical, parameter-shift algorithmic pipeline to extract these metrics natively. Numerical grid analysis confirms the exact structural scaling relation gQF IM = 4 · gF S down to machine precision (1.24 × 10−15). Finally, by tracking the metric volume form det(gij ) alongside the inverse metric components, we visually map out and isolate the topological caustic singularities (θ → 0, π) where the embedding map drops rank. This computational diagnostic provides a robust framework to identify and structurally bypass non-immersive control bottlenecks during the initial Hamiltonian design phase.
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.