Position operators in terms of converging finite-dimensional matrices: Exploring their interplay with geometry, transport, and gauge theory

Abstract

Position operator $\hat{r}$ appears as $i{\partial }_p$ in wave mechanics, while its matrix form is well known diverging in diagonals, causing serious difficulties in basis transformation, observable yielding, etc. We aim to find a convergent $r$-matrix (CRM) to improve the existing divergent $r$-matrix (DRM), and investigate its influence at both the conceptual and the application levels. Unlike the spin matrix, which affords a Lie algebra representation as the solution of $[s_i,s_j]={\epsilon }_{i,j,k}{s}_k$, the $r$-matrix cannot be a solution for $[\hat{r},p]=i\hslash$, namely Weyl algebra. Indeed: matrix representations of Weyl algebras prove not existing; thus, neither CRM nor DRM would afford a representation. Instead, the CRM should be viewed as a procedure of encoding $\hat{r}$ using matrices of arbitrary finite dimensions. Deriving CRM recognizes that the limited understanding about Weyl algebra has led to the divergence. A key modification is increasing the 1-st Weyl algebra (the familiar substitution $\hat{r}\to i{\partial }_p$) to the $N$-th Weyl algebra. Resolving the divergence makes $r$-matrix rigorously defined, and we are able to show $r$-matrix is distinct from a spin matrix in terms of its defining principles, transformation behavior, and the observable it yields. At the conceptual level, the CRM fills the logical gap between the $r$-matrix and the Berry connection; and helps to show that Bloch space ${\mathcal{H}}_B$ is incomplete for $\hat{r}$. At the application level, we focus on transport, and discover that the Hermitian matrix is not identical with the associative Hermitian operator, i.e., $r_{m,n}=r_{n,m}^{\ast }\nLeftrightarrow \hat{r}={\hat{r}}^{†}$. We also discuss how such a non-representation CRM can contribute to building a unified transport theory.

arXiv comment37 pages, 2 figures

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