发表机构
Dartmouth College（达特茅斯学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出无测度的Koopman-von Neumann框架，用RKHS替代L^p空间，构造含自由导子的弱谱三元组，实现可观测量Koopman演化的一致恢复，将非交换几何引入该动力学研究。
AI 中文摘要
经典统计动力学的Koopman-von Neumann公式化，将Liouville方程下L¹中概率密度的等距演化，映射为由Liouville算子反对称部分生成的L²中量子力学波函数的幺正演化。该方法可利用希尔伯特空间技术建模可观测量的统计演化，但作为一种L²方法，它不适于表示动力轨迹上的逐点演化。本文提出一种Koopman-von Neumann框架，将状态空间流形X上与体积测度关联的L^p空间，替换为满足动力学生成元（向量场）V及其RKHS伴随算子联合解析性条件的X上连续函数的再生核希尔伯特空间（RKHS）ℋ。该方案将V的反对称部分扩张为张量积希尔伯特空间=ℓ²(ℕ)⊗ℋ上的本质斜自伴算子L，再将L扩张为作用于生成的加权对称Fock空间𝔉上的本质斜自伴自由导子𝒟。研究表明，由𝒟生成的幺正演化，可在与零均匀有界的时间区间上，一致恢复由联合解析向量生成的ℋ的稠密子代数中可观测量的（通常非幺正的）Koopman演化。随后构建了一族弱谱三元组(𝒜ₙ,𝔉,-i𝒟)，其中n∈ℕ，𝒜ₙ是由产生和湮灭算子生成的B(𝔉)的非阿贝尔*子代数，-i𝒟扮演狄拉克算子的角色，诱导𝒜ₙ状态空间上的扩展伪度量。
英文摘要
The Koopman-von Neumann formulation of classical statistical dynamics maps the isometric evolution of probability densities in $L^1$ under the Liouville equation to a unitary evolution of quantum mechanical wavefunctions in $L^2$ generated by the antisymmetric part of the Liouville operator. This approach enables the use of Hilbert space techniques to model the statistical evolution of observables. However, being an $L^2$ method, it is not suitable for representing pointwise evolution along dynamical trajectories. We propose a Koopman-von Neumann framework that replaces the $L^p$ spaces associated with a volume measure on the state space manifold, $X$, by a reproducing kernel Hilbert space (RKHS), $\mathcal H$, of continuous functions on $X$ that satisfy joint analyticity conditions with respect to the generator (vector field), $V$, of the dynamics and its RKHS adjoint. Our scheme employs a dilation of the antisymmetric part of $V$ to an essentially skew-adjoint operator, $L$, on the tensor product Hilbert space $\mathfrak H = \ell^2(\mathbb N) \otimes \mathcal H$, followed by a dilation of $L$ to an essentially skew-adjoint, free derivation, $\mathcal D$, acting on a weighted symmetric Fock space $\mathfrak F$ generated by $\mathfrak H$. We show that the unitary evolution generated by $\mathcal D$ consistently recovers the (generally, non-unitary) Koopman evolution of observables in a dense subalgebra of $\mathcal H$ generated by jointly analytic vectors, over a time interval that is uniformly bounded away from zero. We then build a family of weak spectral triples $(\mathcal A_n, \mathfrak F, -i \mathcal D)$, $n \in \mathbb N$, wherein $\mathcal A_n$ are non-abelian $*$-subalgebras of $B(\mathfrak F)$ generated by creation and annihilation operators and $-i \mathcal D$ plays the role of a Dirac operator inducing extended pseudometrics on the state spaces of $\mathcal A_n$.
Comments41 pages
