Position operators in terms of converging finite-dimensional matrices: Exploring their interplay with geometry, transport, and gauge theory
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arXiv
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2024
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| author | Song, B. Q. Smith, J. D. H. Wang, J. |
| author_facet | Song, B. Q. Smith, J. D. H. Wang, J. |
| contents | 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 $[s_i,s_j]=ε_{i,j,k}s_k$, the $r$-matrix cannot be a solution for $[\hat{r},p]=i\hbar$, 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 $\hat{r}{\rightarrow}i{\partial_p}$) 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., $r_{m,n}=r_{n,m}^*{\nLeftrightarrow}\hat{r}=\hat{r}^{\dagger}$. We also discuss how such a non-representation CRM can contribute to building a unified transport theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_02519 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Position operators in terms of converging finite-dimensional matrices: Exploring their interplay with geometry, transport, and gauge theory Song, B. Q. Smith, J. D. H. Wang, J. Quantum Physics 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 $[s_i,s_j]=ε_{i,j,k}s_k$, the $r$-matrix cannot be a solution for $[\hat{r},p]=i\hbar$, 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 $\hat{r}{\rightarrow}i{\partial_p}$) 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., $r_{m,n}=r_{n,m}^*{\nLeftrightarrow}\hat{r}=\hat{r}^{\dagger}$. We also discuss how such a non-representation CRM can contribute to building a unified transport theory. |
| title | Position operators in terms of converging finite-dimensional matrices: Exploring their interplay with geometry, transport, and gauge theory |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2403.02519 |