AI 中文总结
该研究在范畴量子力学框架内，基于装备希尔伯特空间，提出波粒二象性与德布罗意关系的表述，利用傅里叶-庞特里亚金对偶关联位置与动量的可观测量结构，确定了二象性的结构形式。
AI 中文摘要
我们在装备希尔伯特空间上的范畴量子力学框架内，提出了波粒二象性与德布罗意关系λ=h/p的表述。位置与动量由与局部紧阿贝尔群ℝ及其庞特里亚金对偶相关联的连续 dagger-Frobenius 结构表示，傅里叶变换是庞特里亚金对偶的幺正实现，关联这两种可观测量结构。我们证明，一般特征配对χ_p^(α)(x)=exp(ipx/α)产生一族幺正等价的傅里叶变换，因此庞特里亚金对偶决定了二象性的结构形式，但不决定普朗克常数的数值；将α=ℏ的认定是由魏尔对易关系确定的物理输入，经此输入，特征的空间周期给出λ=h/p。
英文摘要
The position--momentum correspondence combines several mathematically distinct identifications that are often conflated. For the additive group $(\mathbb R,+)$, every continuous character has the form $x\mapsto \mathrm e^{ikx}$. After choosing a coordinate $p=αk$ on the dual group, one obtains $\mathrm e^{ipx/α}$, so Pontryagin duality alone does not identify a particular dual coordinate with physical momentum. For the corresponding unitary Fourier transform, multiplication in position space is transported to convolution in momentum space, while the diagonal distribution is transported to the specific composition with the addition map $(p,q)\mapsto p+q$; it is not transported to pointwise multiplication in momentum space. The family $\hat P_α=-iα\,d/dx$ has commutator $[\hat X,\hat P_α]=iαI$ on $\mathcal S(\mathbb R)$, and the standard quantum-mechanical normalization $[\hat X,\hat P]=i\hbar I$ therefore selects $α=\hbar$ within this family. Equivalently, the standard normalization of the translation generator gives $\hat P=-i\hbar\,d/dx$. The resulting characters $\mathrm e^{ipx/\hbar}$ have spatial period $h/|p|$ for $p\neq0$. All distributional statements are formulated in the Schwartz rigging and no product or pullback of arbitrary tempered distributions is used.
