贝特曼对偶振子的分裂四元数结构与正则量子化
Split-quaternionic structure and canonical quantization of the Bateman dual oscillator
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中文总结 AI 辅助
研究贝特曼对偶振子，通过分裂四元数公式从经典到量子化保持一致，给出相关动力学、量子化结果等，阐明双重模型是相干零噪声嵌入，非微观储能器理论，还涉及不确定关系及传播子等内容。
中文摘要 AI 辅助
贝特曼对偶振子在一个保守的双重系统中嵌入了一个阻尼谐振子和一个独立的反阻尼伴随坐标。我们发展了一种分裂四元数公式，它从经典方程到普通复正则量子化都保持一致。双重动力学被写成双边超复演化，其守恒的不定二次型是分裂范数，而生成元平方的符号统一了欠阻尼、过阻尼和临界状态。贝特曼拉格朗日量在分裂复配置子代数上作为实标量作用被恢复。量子化后，算符值分裂四元数属于复化代数\(\mathbb C\otimes_{\mathbb R}\mathbb H_s\simeq M_2(\mathbb C)\)。非对易力学分量需要对称化的量子分裂范数，它精确地重现了对称排序的正则哈密顿量。零幂等元解决了双重相空间的交叉正则结构。特别是，阻尼产生\([m\dot{\widehat y},m\dot{\widehat x}]=\mathrm{i}\hbar m\gamma\)和罗伯逊 - 薛定谔不确定关系。\(\gamma\to0\)的极限保留了双重的正/负几何；只有在选择正的正常模式或在量子化之前解除双重之后，才能恢复普通的单振子海森堡关系。标量分裂作用给出了精确的二次传播子和高斯演化。最后，对辅助坐标的形式迹与高温卡尔德雷拉 - 莱格特动力学进行了比较，阐明了双重模型是一个相干零噪声嵌入而不是微观储能器理论。
英文摘要
Bateman's dual oscillator embeds a damped harmonic oscillator and an independent anti-damped adjoint coordinate in a conservative doubled system. We develop a split-quaternionic formulation that remains consistent from the classical equations through ordinary complex canonical quantization. The doubled dynamics is written as a two-sided hypercomplex evolution whose conserved indefinite quadratic form is the split norm, while the sign of the generator square unifies the underdamped, overdamped, and critical regimes. The Bateman Lagrangian is recovered as a real scalar action on the split-complex configuration subalgebra. After quantization, operator-valued split quaternions belong to the complexified algebra $\mathbb C\otimes_{\mathbb R}\mathbb H_s\simeq M_2(\mathbb C)$. Noncommuting mechanical components require a symmetrized quantum split norm, which exactly reproduces the symmetrically ordered canonical Hamiltonian. Null idempotents resolve the crossed canonical structure of the doubled phase space. In particular, damping yields $[m\dot{\widehat y},m\dot{\widehat x}]=\mathrm{i}\hbar mγ$ and a Robertson--Schrödinger uncertainty relation. The limit $γ\to0$ retains the doubled positive/negative geometry; the ordinary single-oscillator Heisenberg relation is recovered only after selecting the positive normal mode, or by undoubling before quantization. The scalar split action gives the exact quadratic propagator and Gaussian evolution. A formal trace over the auxiliary coordinate is finally compared with high-temperature Caldeira--Leggett dynamics, clarifying that the doubled model is a coherent zero-noise embedding rather than a microscopic reservoir theory.
