Content-Type: multipart/mixed; boundary="-------------1508281337778" This is a multi-part message in MIME format. ---------------1508281337778 Content-Type: text/plain; name="15-87.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="15-87.comments" 3 pp. Presented at several mathematical conferences. ---------------1508281337778 Content-Type: text/plain; name="15-87.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="15-87.keywords" Non-commutativity, Dirac, quantum mechanics ---------------1508281337778 Content-Type: application/x-tex; name="Diraclike-NONCOM.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Diraclike-NONCOM.tex" %\documentstyle[12pt,aps]{revtex} \documentclass[12pt]{article} %\documentclass{Rinton-P9x6} %\widetext %\draft %\tighten %\oddsidemargin5mm %\evensidemargin15mm %\usepackage{amsmath} \usepackage{bm} \begin{document} \title{A Note on the Dirac-like Equation in Non-commutative Geometry} \author{Valeriy V. Dvoeglazov\\ Universidad de Zacatecas\\ Apartado Postal 636, Suc. 3\\ Zacatecas 98061, Zac., M\'exico\\ E-mail: valeri@fisica.uaz.edu.mx} %\date{June 17, 2003} \date{\empty} \maketitle \begin{abstract} We postulate the non-commutativity of 4-momenta and we derive the mass splitting in the Dirac equation. The applications are discussed. \end{abstract} The non-commutativity~\cite{snyder,amelino} manifests interesting peculiarities in the Dirac case. Recently, we analized Sakurai-van der Waerden method of derivations of the Dirac (and higher-spins too) equation~\cite{Dvoh}. We can start from \begin{equation} (E I^{(2)}-{\bm \sigma}\cdot {\bf p}) (E I^{(2)}+ {\bm\sigma}\cdot {\bf p} ) \Psi_{(2)} = m^2 \Psi_{(2)} \,, \end{equation} or \begin{equation} (E I^{(4)}+{\bm \alpha}\cdot {\bf p} +m\beta) (E I^{(4)}-{\bm\alpha}\cdot {\bf p} -m\beta ) \Psi_{(4)} =0.\label{f4} \end{equation} As in the original Dirac work, we have \begin{equation} \beta^2 = 1\,,\quad \alpha^i \beta +\beta \alpha^i =0\,,\quad \alpha^i \alpha^j +\alpha^j \alpha^i =2\delta^{ij} \,. \end{equation} For instance, their explicite forms can be chosen \begin{eqnarray} \alpha^i =\pmatrix{\sigma^i& 0\cr 0&-\sigma^i\cr}\,,\quad \beta = \pmatrix{0&1_{2\times 2}\cr 1_{2\times 2} &0\cr}\,, \end{eqnarray} where $\sigma^i$ are the ordinary Pauli $2\times 2$ matrices. We also postulate the non-commutativity relations for the components of 4-momenta: \begin{equation} [E, {\bf p}^i]_- = \Theta^{0i} = \theta^i\,, \end{equation} as usual. Therefore the equation (\ref{f4}) will {\it not} lead to the well-known equation $E^2 -{\bf p}^2 = m^2$. Instead, we have \begin{equation} \left \{ E^2 - E ({\bm \alpha} \cdot {\bf p}) +({\bm \alpha} \cdot {\bf p}) E - {\bf p}^2 - m^2 - i ({\bm\sigma}\otimes I_{(2)}) [{\bf p}\times {\bf p}] \right \} \Psi_{(4)} = 0 \end{equation} For the sake of simplicity, we may assume the last term to be zero. Thus, we come to \begin{equation} \left \{ E^2 - {\bf p}^2 - m^2 - ({\bm \alpha}\cdot {\bm \theta}) \right \} \Psi_{(4)} = 0\,. \end{equation} However, let us apply the unitary transformation. It is known~\cite{Berg,Dvoe} that one can\footnote{Some relations for the components ${\bf a}$ should be assumed. Moreover, in our case ${\bm \theta}$ should not depend on $E$ and ${\bf p}$. Otherwise, we must take the non-commutativity $[E, {\bf p}^i]_-$ into account again.} \begin{equation} U_1 ({\bm \sigma}\cdot {\bf a}) U_1^{-1} = \sigma_3 \vert {\bf a} \vert\,.\label{s3} \end{equation} For ${\bm \alpha}$ matrices we re-write (\ref{s3}) to \begin{eqnarray} {\cal U}_1 ({\bm \alpha}\cdot {\bm \theta}) {\cal U}_1^{-1} = \vert {\bm \theta} \vert \pmatrix{1&0&0&0\cr 0&-1&0&0\cr 0&0&-1&0\cr 0&0&0&1\cr} = \alpha_3 \vert {\bm\theta}\vert\,. \end{eqnarray} The explicit form of the $U_1$ matrix is ($a_{r,l}= a_1\pm ia_2$): \begin{eqnarray} U_1 &=&\frac{1}{\sqrt{2a (a+a_3)}} \pmatrix{a+a_3&a_l\cr -a_r&a+a_3\cr} = \frac{1}{\sqrt{2a (a+a_3)}} [ a+a_3 + i\sigma_2 a_1 - i\sigma_1 a_2]\,,\nonumber\\ {\cal U}_1 &=&\pmatrix{U_1 &0\cr 0& U_1 \cr}\,. \end{eqnarray} Let us apply the second unitary transformation: \begin{eqnarray} {\cal U}_2 \alpha_3 {\cal U}_2^\dagger = \pmatrix{1&0&0&0\cr 0&0&0&1\cr 0&0&1&0\cr 0&1&0&0\cr} \alpha_3 \pmatrix{1&0&0&0\cr 0&0&0&1\cr 0&0&1&0\cr 0&1&0&0\cr} = \pmatrix{1&0&0&0\cr 0&1&0&0\cr 0&0&-1&0\cr 0&0&0&-1\cr}\,. \end{eqnarray} The final equation is \begin{equation} [E^2 -{\bf p}^2 -m^2 -\gamma^5_{chiral} \vert {\bm \theta}\vert ] \Psi^\prime_{(4)} = 0\,. \end{equation} In the physical sense this implies the mass splitting for a Dirac particle over the non-commutative space, $m_{1,2} =\pm \sqrt{m^2 \pm \theta}$. This procedure may be attractive for explanation of the mass creation and the mass splitting for fermions. \begin{thebibliography}{99} \footnotesize{ \bibitem{snyder} H. Snyder, Phys. Rev. {\bf 71}, 38 (1947); ibid. {\bf 72}, 68 (1947). \bibitem{amelino} A. Kempf, G. Mangano and R. B. Mann, Phys. Rev. D{\bf 52} , 1108 (1995); G. Amelino-Camelia, Nature {\bf 408}, 661 (2000); gr-qc/0012051; hep-th/0012238; gr-qc/0106004; J. Kowalski-Glikman, hep-th/0102098; G. Amelino-Camelia and M. Arzano, hep-th/0105120; N. R. Bruno, G. Amelino-Camelia and J. Kowalski-Glikman, hep-th/0107039. \bibitem{Dvoh} V. V. Dvoeglazov, Rev. Mex. Fis. Supl. {\bf 49}, 99 (2003) ({\it Proceedings of the DGFM-SMF School, Huatulco, 2000}) \bibitem{Berg} R. A. Berg, Nuovo Cimento {\bf 42}A, 148 (1966). \bibitem{Dvoe} V. V. Dvoeglazov, Nuovo Cimento A{\bf 108}, 1467 (1995). } \end{thebibliography} \end{document} ---------------1508281337778--