Pythagoras's theorem on a two-dimensional lattice from a ``natural'' Dirac operator and Connes's distance formula (Q2757874)

A Dirac operator \({\mathcal D}\) on the two-dimensional lattice \(\mathbb{Z}^2\) whose Connes' distance gives usual Euclidean distance, is presented. It is a genealization of a Dirac operator of [\textit{A. Dimakis} and \textit{F. Müler-Hoissen}, Int. J. Theor. Phys. 37, No.~3, 907-913 (1998; Zbl 0934.58005)] whose Connes distance recovers the linear distance of an one-dimensional lattice, and defined as follows: Let \(\gamma^i\) be the \(\gamma\)-matrices, the generators of \(\text{Cl}(E^4)\), and set NEWLINE\[NEWLINE\gamma^1_\pm= \textstyle{{1\over 2}} (\gamma^1\pm i\gamma^2),\quad \gamma^2_\pm= \textstyle{{1\over 2}} (\gamma^3\pm i\gamma^4),\quad T^\pm_i f(x)= f(x\pm\widehat i),NEWLINE\]NEWLINE \(\widehat i,i= 1,2\), the generators of \(\mathbb{Z}^2\), and \(\partial^\pm_i= T^\pm_i- {\mathbf{1}}\). Then NEWLINE\[NEWLINE{\mathcal D}= \sum^2_{i=1} \sum_{s=\pm} \gamma^i_s \partial^s_i.NEWLINE\]NEWLINE It acts on \({\mathcal H}= \ell^2(\mathbb{Z}^2)\otimes \mathbb{C}^2\) (Section 3). To compute Connes' distance of \({\mathcal D}\), necessary and sufficient condition to hold \([{\mathcal D},f]\|\leq 1\) is given in terms of the boundedness condition of the eigenvalues of the \(f\)-Hamiltonian \(H(df)= [{\mathcal D}, f]^\dag[{\mathcal D}, f]\) (Section 4). By using this condition, Pythagoras's theorem on a two-dimensional lattice Connes' metric \(d_{\mathcal D}\) of \({\mathcal D}\) NEWLINE\[NEWLINEd_{\mathcal D}((0, 0),(p, q))= \sqrt{p^2+ q^2},NEWLINE\]NEWLINE is proved in Section 5 (Theorem 1). Other properties of \(d_{\mathcal D}\) such as translation invariance, are proved in Section 3. In Section 6, by using generators \(\Gamma^i\) of \(\text{Cl}(E^{2d})\), the operator \({\mathcal D}_d= \sum^d_{k=1} \sum_{s=\pm} \Gamma^i_s \partial^s_k\) is introduced and conjectured its Connes' distance is natural for a \(d\)-dimensional lattice.