Non-Unitary Behavior of Three-Dimensional Complex Numbers in Quantum Evolution: Numerical Experiments and Controlled Verification 三维复数在量子演化中的非幺正行为:数值实验与对照验证
Abstract This report systematically investigates the numerical behavior of the three-dimensional complex algebra under the standard quantum mechanical evolution framework. The three-dimensional complex number is a commutative and associative real algebra spanned by , satisfying . By replacing the conventional complex-valued wave function with a -valued wave function while preserving the formal structure of the Schrödinger equation, numerical evolution under the explicit Euler scheme exhibits continuous norm drift and a directional component transfer from to . To isolate algebraic effects from numerical artifacts, we design controlled experiments with standard two-dimensional complex numbers under identical computational conditions. The results show that both 2D complex and 3D complex systems produce nearly identical norm drift magnitudes and growth patterns. Both systems exhibit exponential numerical divergence after approximately 600 iteration steps under the explicit Euler method. We conclude that the observed non-unitary drift primarily originates from the inherent instability of the explicit Euler discretization, rather than from the algebraic structure of . Further high-precision verification using unitary-preserving algorithms such as the Crank–Nicolson scheme is required to confirm whether possesses intrinsic non-unitary quantum evolution behavior. 摘要 本报告系统记录了三维复数 在标准量子力学演化框架下的数值行为。三维复数 是实数域上的三维交换结合代数,基底 满足 。将标准薛定谔方程中的波函数值域从复数替换为三维复数后,在显式欧拉格式下观察到范数的线性漂移及 的方向性分量转移。 为排除数值方法的影响,本文设计了标准二维复数对照组实验。结果表明:在相同显式欧拉格式下,二维复数与三维复数表现出几乎完全一致的范数漂移模式与量级;在本次对照轮次的仿真运行当中,两套系统均在约 600 步后发生指数级数值爆炸。本报告的核心结论是:在显式欧拉格式下观察到的非幺正行为,主要可归因于数值方法本身的缺陷,而非三维复数的代数结构。三维复数作为开放量子系统数学基础的定位,需要在更换保幺正演化算法(如 Crank‑Nicolson 或谱方法)后重新评估。
Abstract This report systematically investigates the numerical behavior of the three-dimensional complex algebra under the standard quantum mechanical evolution framework. The three-dimensional complex number is a commutative and associative real algebra spanned by , satisfying . By replacing the conventional complex-valued wave function with a -valued wave function while preserving the formal structure of the Schrödinger equation, numerical evolution under the explicit Euler scheme exhibits continuous norm drift and a directional component transfer from to . To isolate algebraic effects from numerical artifacts, we design controlled experiments with standard two-dimensional complex numbers under identical computational conditions. The results show that both 2D complex and 3D complex systems produce nearly identical norm drift magnitudes and growth patterns. Both systems exhibit exponential numerical divergence after approximately 600 iteration steps under the explicit Euler method. We conclude that the observed non-unitary drift primarily originates from the inherent instability of the explicit Euler discretization, rather than from the algebraic structure of . Further high-precision verification using unitary-preserving algorithms such as the Crank–Nicolson scheme is required to confirm whether possesses intrinsic non-unitary quantum evolution behavior. 摘要 本报告系统记录了三维复数 在标准量子力学演化框架下的数值行为。三维复数 是实数域上的三维交换结合代数,基底 满足 。将标准薛定谔方程中的波函数值域从复数替换为三维复数后,在显式欧拉格式下观察到范数的线性漂移及 的方向性分量转移。 为排除数值方法的影响,本文设计了标准二维复数对照组实验。结果表明:在相同显式欧拉格式下,二维复数与三维复数表现出几乎完全一致的范数漂移模式与量级;在本次对照轮次的仿真运行当中,两套系统均在约 600 步后发生指数级数值爆炸。本报告的核心结论是:在显式欧拉格式下观察到的非幺正行为,主要可归因于数值方法本身的缺陷,而非三维复数的代数结构。三维复数作为开放量子系统数学基础的定位,需要在更换保幺正演化算法(如 Crank‑Nicolson 或谱方法)后重新评估。
We present Unified Quantum Mechanics (UQM) by unifying real and standard complex quantum theories via real energy and momentum operators derived from the Hilbert Transform. This paper especially presents a fully real-valued formulation of quantum mechanics, showing that while complex numbers are commonly used in the standard theory, they are not fundamentally necessary for describing quantum phenomena. We develop Real Quantum Mechanics (RQM) by defining the real energy and momentum operators using the temporal Hilbert transform which shifts the phases of wave functions while preserving all essential quantum behavior. These real operators unify the proposed RQM and standard Complex Quantum Mechanics (CQM), as these are valid for both real and complex wavefunctions. We introduce a real-valued algebraic structure constructed from the identity operator and the Hilbert transform operator. This framework is mathematically equivalent to the system of complex numbers, preserving all fundamental algebraic properties while operating entirely within the realm of real operators. The isomorphism provides a novel representation of complex quantum mechanics using only real-valued components, offering potential advantages for both theoretical analysis and physical implementations. Importantly, we establish that the Hilbert transform operator is a generalization of the imaginary unit. The key insight is that the Hilbert transform generalizes the role of imaginary numbers in standard quantum mechanics, allowing us to construct real operators for both real and complex wave functions that produce the same physical predictions. We derive real versions of energy and momentum operators, leading to a modified wave equation that replaces the Schrödinger equation. Despite its non-local time evolution, this framework retains local spatial behavior and reproduces the usual energy-momentum relations. We rigorously test this approach on fundamental quantum systems, including the hydrogen atom and relativistic particles, confirming that it yields identical energy levels and wave functions as conventional quantum mechanics. Notably, we establish a consistent probabilistic framework by introducing real analogues of quantum probabilities via the analytic Born rule. We derive a unified uncertainty principle that coincides with the standard Heisenberg uncertainty bound for complex-valued wave functions, while admitting a vanishing lower bound in the case of real-valued wave functions. The implications of this work are significant: (1) it demonstrates that complex numbers are a mathematical convenience rather than a fundamental requirement in quantum theory, (2) it introduces new tools for analyzing quantum dynamics using signal-processing techniques, (3) it provides insights into non-local quantum evolution, (4) it connects quantum mechanics to real-field theories. Ultimately, RQM reproduces all known quantum predictions while offering a fresh perspective on operator structure, time evolution, and the mathematical foundations of the theory, and (5) it provides a real qubit representation on the modified Bloch sphere for quantum computing.
非厄米量子系统哈密顿量因具有不同的对称性，在量子态快速演化、量子态区分、量子精密测量等方面展现出超越标准量子力学系统的优势和新奇特性，近二十多年来受到持续关注，成为前沿领域。量子模拟非厄米系统是主要方向之一，又是研究非厄米系统的重要手段。典型的非厄米系统包括宇称–时间(Party-Time，简称PT)对称系统和赝厄米对称系统等。已有研究集中于对非厄米系统的时间演化开展量子模拟。文章围绕非厄米系统的虚时演化算符进行研究，通过计算非厄米系统的虚时演化算符，为开展典型非厄米系统的虚时演化量子模拟理论研究打下基础。我们的研究从理论上拓展了量子计算机可模拟新奇系统的范畴，并将虚时演化拓展至非厄米系统。基于我们对非厄米系统的虚时演化理论研究，对利用不同的量子系统和量子真机开展实验研究具有指导作用。The Hamiltonian of non-Hermitian quantum systems, due to its different symmetries, has shown advantages and novel properties beyond standard quantum mechanical systems in terms of rapid quantum state evolution, quantum state differentiation, quantum precision measurement, etc., which has attracted continuous attention and become a frontier field in the past two decades. Quantum simulation of non-Hermitian systems is one of the main directions and an important means to study non-Hermitian systems. Typical non-Hermitian systems include parity-time (PT) symmetric systems and pseudo Hermitian symmetric systems. Previous studies have focused on the quantum simulation of the time evolution of non-Hermitian systems. In this paper, the virtual time evolution operators of non-Hermitian systems are studied. By calculating the virtual time evolution operators of non-Hermitian systems, it lays a foundation for the theoretical study of virtual time evolution of typical non-Hermitian systems. Our research theoretically expands the range of novel systems that quantum computers can simulate, and extends virtual time evolution to non-Hermitian systems. Based on our research on the virtual time evolution theory of non-Hermitian systems, it has a guiding role to carry out experimental research with different quantum systems and quantum real machines.
Quantum mechanics is routinely simulated on classical digital computers for applications ranging from molecular design to quantum algorithm prototyping.Yet every such simulation confronts a fundamental mismatch: the mathematical structure of quantum theory is defined over infinite-dimensional Hilbert spacesand continuous operators, while real hardware possesses only finite memory and finite precision. Standard numerical libraries must therefore impose artificialtruncations, basis cutoffs, or discretization schemes that introduce approximation error and numerical instability. The Conservation of Complexity (COC) framework provides a self-consistent algebraic alternative that eliminates the infinity problem at its source. By identifying the maximum length k of ascending ideal chains in a commutative ring with the maximum depth of a finite game tree, and by imposing the Equiopposition Axiom together with normalized complex outcome states, the theory guarantees that every differential operator application remains inside a uniformly k-bounded Noetherian ring. The resulting operator-exponential solution of the associated cascading-feedback PDE is represented exactly by a finite sum of at most k Fourier modes, with higher-order terms identically zero by the ring axioms and normalization preserved unitarily at every step. This paper derives the explicit trigonometric wave-function representation of the COC model and demonstrates its direct utility for classical quantum simulations. The framework does not replace or reinterpret the established laws of quantum mechanics; rather, it supplies a computationally exact, resourcebounded description that is immediately implementable in any modern programming language. By choosing the complexity bound ℓ(R) = k appropriate to agiven physical subsystem, one obtains a closed algebraic expression whose evaluation requires no infinite series, no floating-point approximations to infinity,and no manual truncation—offering a practical bridge between the continuous formalism of quantum theory and the finite reality of digital computation.
Real-time path integrals provide a natural formulation of quantum dynamics also in situations where a Hamiltonian description is unavailable or impractical, but their direct evaluation is obstructed by severe phase cancellation from the oscillatory weight.A guiding theme of this thesis is that, after complexifying the integration variables, real-time path integrals are naturally organized by complex saddle points.In favorable cases the original contour can be replaced by steepest-descent manifolds (Lefschetz thimbles) attached to the relevant saddles, providing a controlled saddle-based description of real-time observables.In Part I, we study quantum decoherence in a closed-system real-time path-integral formulation: the total system (subsystem plus environment) evolves unitarily, and the environment is traced out only at the final time to obtain the reduced density matrix.For the Caldeira-Leggett model with Gaussian initial conditions, the discretized effective action becomes a quadratic form, so that the reduced density matrix is characterized by a complex saddle point and the thimble integral reduces to a finite-dimensional Gaussian integral.The complex saddle is obtained by solving a large but sparse linear system, allowing an exact evaluation of the reduced density matrix without relying on the usual master-equation approximations.We then extend this methodology to a three-dimensional cubic lattice toy model by separating center-of-mass and relative coordinates, treating the center-of-mass mode as the subsystem and the relative modes as an effective environment.This enables controlled numerical studies of how the suppression of off-diagonal matrix elements depends on the coupling, inverse temperature, and system size, and allows us to extract the early-time scaling behavior of decoherence diagnostics from the complex-saddle evaluation.In Part II, we investigate spacetime dynamics in the Lorentzian IKKT matrix model.To define a Lorentzian formulation that is not continuously connected to the Euclidean model by the standard contour deformation, we include a Lorentz invariant mass term with coefficient γ, for which the classical equations admit real, smooth, expanding solutions when γ > 0. Since the Lorentz group is noncompact, the path integral can contain a divergent volume from the boost directions; we remove this volume nonperturbatively by implementing a boost "gauge-fixing" that selects the frame minimizing tr(A 0 ) 2 , leading to the condition tr(A 0 A i ) = 0 with the associated Faddeev-Popov determinant.We then apply the complex Langevin method to the resulting complex-weight matrix integral.Time and spatial observables are defined from matrix configurations, with holomorphic time ordering implemented by auxiliary variables, and spatial structure characterized through block observables such as the extent R 2 (t) and the moment-of-inertia tensor.In the bosonic model we find approximately real time and no indication of spontaneous breaking of the spatial symmetry in the i parameter range studied.Including fermions, we control the singular-drift problem induced by the Pfaffian by introducing a fermionic mass regulator m f and an anisotropic mass deformation, which enhances fermionic effects while maintaining numerical stability.In this setup we observe a real, expanding spatial extent and a dynamical dominance of three spatial directions at late times, consistent with a (3 + 1)-dimensional expanding spacetime.We finally discuss the intended limiting procedure-taking the large-N limit first and then approaching the undeformed theory-and comment on future directions, including a comparison with partially supersymmetric deformations that preserve 16 supercharges.
宣言 Manifesto 标准量子力学的数学基础是结合代数:算子乘法满足 (AB)C = A(BC),态空间是希尔伯特空间(正定内积空间),演化由幺正群 U(N) 描述。 The mathematical foundation of standard quantum mechanics is associative algebra: operator multiplication satisfies (AB)C = A(BC), the state space is a Hilbert space (positive-definite inner product space), and evolution is described by the unitary group U(N). 然而,自然界存在深刻的非结合结构:从八元数在弦论中的角色,到 Jordan 代数在量子逻辑中的起源,非结合性不是病态,而是更一般的对称性破缺。中道积分理论已经证明:非结合代数上的围道积分可以通过算术平均唯一地定义,结合子精确量化每条分支的偏差。 However, nature harbors profound non-associative structures: from octonions in string theory to Jordan algebras in quantum logic, non-associativity is not pathological but a more general symmetry breaking. The Mid-Path Integral theory has proven: contour integrals on non-associative algebras can be uniquely defined via arithmetic averaging, with the associator precisely quantifying each branch’s deviation. 本文提出一个根本性的转向:不再把非结合代数塞进希尔伯特空间的模子里,而是直接从复化三维数(ℂ³)的内生结构出发,重建量子力学的全部公理。标准量子力学将是本文框架在“结合性极限”(δ → 0)和“标量子空间限制”(ψ₁ = ψ₂ = 0)下的子理论。 This paper proposes a fundamental pivot: instead of forcing non-associative algebras into the Hilbert-space mold, we reconstruct all axioms of quantum mechanics directly from the endogenous structure of the complexified ternary numbers (ℂ³). Standard quantum mechanics will be a sub-theory of this framework in the “associative limit” (δ → 0) and the “scalar subspace restriction” (ψ₁ = ψ₂ = 0). 预备知识:ℂ³ 代数结构 Preliminaries: ℂ³ Algebraic Structure 乘法表 Multiplication Table 基为 {1, e₁, e₂},乘法由以下关系定义: Basis {1, e₁, e₂}, multiplication defined by: ✅ e₁² = −1 + δ e₂, e₁ e₂ = e₂ e₁ = α e₁, e₂² = β e₂ ✅ e₁² = −1 + δ e₂, e₁ e₂ = e₂ e₁ = α e₁, e₂² = β e₂ 基本性质 Basic Properties • 交换性:X·Y = Y·X(乘法表对称) Commutativity: X·Y = Y·X (symmetric multiplication table) • 分配律:X·(Y+Z) = X·Y + X·Z(严格成立) Distributivity: X·(Y+Z) = X·Y + X·Z (strictly holds) • 非结合性:一般地 (X·Y)·Z ≠ X·(Y·Z) Non-associativity: generally (X·Y)·Z ≠ X·(Y·Z) • 结合子:Δ(X,Y,Z) = (X·Y)·Z − X·(Y·Z) Associator: Δ(X,Y,Z) = (X·Y)·Z − X·(Y·Z) • 严格定理:Δ(X,X,X) ≡ 0(幂结合性在自乘时成立) Strict theorem: Δ(X,X,X) ≡ 0 (power-associativity holds for self-multiplication) • 严格定理:Δ(X,X,Y) ≠ 0 当 Y ≠ X(混合非结合性) Strict theorem: Δ(X,X,Y) ≠ 0 when Y ≠ X (mixed non-associativity) • 定理 2.5(非结合激活条件):当 x₀=y₀=z₀=0 时 Δ(X,Y,Z) ≡ 0。非结合性只在至少一个元素的标量分量非零时显现。 Theorem 2.5 (Non-associative activation): Δ(X,Y,Z) ≡ 0 when x₀=y₀=z₀=0. Non-associativity only manifests when at least one element has non-zero scalar component.
One of the unitary forms of the quantum mechanical time evolution operator is given by Cayley's approximation. A numerical implementation of the same involves the replacement of second derivatives in Hamiltonian with the three-point formula, which leads to a tridiagonal system of linear equations. In this work, we invoke the highly accurate five-point stencil to discretize the wave function onto an Implicit-Explicit pentadiagonal Crank-Nicolson scheme. It is demonstrated that the resultant solutions are significantly more accurate than the standard ones. We also discuss the resolution of bipartite wavepacket dynamics and derive conditions under which a product state from the laboratory perspective remains a product state from the center-of-mass point of view. This has profound applications for decoupling complicated bipartite dynamics into two independent single-particle problems.