查看“弯曲时空中的狄拉克方程”的源代码

在[[数学物理]]中，'''弯曲时空中的狄拉克方程（英文：the Dirac equation in curved spacetime）'''指的是将原始的[[狄拉克方程]]推广至[[廣義相對論|弯曲时空]]的情形后得到的方程。

该方程可以通过[[标架场]]与引力[[自旋]][[联络]]写出，标架场给出了一个定域的[[参考系|静止系]]，使得恒定的[[狄拉克矩陣|狄拉克矩阵]]可以作用于每一个时空点。这样一来，弯曲时空中的狄拉克方程就可以写作如下形式：<ref>{{Cite book|title=A Unified Grand Tour of Theoretical Physics|last=Lawrie|first=Ian D.}}</ref>
: <math>i\gamma^a e_a^\mu D_\mu \Psi - m \Psi = 0.</math>
其中{{Math|''e<sub>a</sub><sup>μ</sup>''}}为标架场，{{Math|''D<sub>μ</sub>''}}为[[狄拉克場|狄拉克场]]对应的[[共变导数]]，其定义如下
: <math>D_\mu = \partial_\mu - \frac{i}{4} \omega_{\mu}^{ab} \sigma_{ab},</math>
这里{{Math|''σ<sub>ab</sub>''}} 为狄拉克矩阵的[[交換子|交换子]]：
: <math>\sigma_{ab}=\frac{i}{2} \left[\gamma_{a},\gamma_{b}\right], </math>
且{{Math|''ω<sub>μ</sub><sup>ab</sup>''}} 为自旋联络组件。

注意到此处拉丁字母角标表示的是洛伦兹标架，希腊字母角标则对应[[流形]]坐标。

== 另见 ==
* [[物理空间的代数中的狄拉克方程]]
* [[狄拉克旋量]]
* [[弯曲时空中的麦克斯韦方程组]]
* [[两体狄拉克方程]]

== 参考资料 ==
{{Reflist}}
* {{Cite arXiv|year=2013|author=M. Arminjon, F. Reifler|title=Equivalent forms of Dirac equations in curved spacetimes and generalized de Broglie relations|eprint=1103.3201}}
* {{Cite journal|title=on the dirac equation in curved space-time |author=M.D. Pollock |url=http://www.actaphys.uj.edu.pl/_cur/store/vol41/pdf/v41p1827.pdf |journal=Acta Physica Polonica B |issue=8 |doi= |year=2010 |volume=41 |page= |arxiv= |bibcode= |deadurl=yes |archiveurl=https://web.archive.org/web/20131214155201/http://www.actaphys.uj.edu.pl/_cur/store/vol41/pdf/v41p1827.pdf |archivedate=2013-12-14 }}
* {{Cite book|url=http://books.google.co.uk/books?id=T8Fde0u_XyIC&pg=PA117&dq=Dirac+equation+in+curved+spacetime&redir_esc=y#v=onepage&q=Dirac%20equation%20in%20curved%20spacetime&f=false|title=Einstein's Unification|last=J.V. Dongen|publisher=Cambridge University Press|year=2010|isbn=0-521-883-466|page=117}}
* {{Cite book|url=http://books.google.co.uk/books?id=5nNuGMBBTjMC&printsec=frontcover&dq=L.+Parker,+D.+Toms+%282009%29.+Quantum+Field+Theory+in+Curved+Spacetime:+Quantized+Fields+and+Gravity#v=onepage&q&f=false|title=Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity|last=L. Parker, D. Toms|publisher=Cambridge University Press|year=2009|isbn=0-521-877-873|page=227}}
* {{Cite book|url=http://books.google.co.uk/books?id=Zo5_3cmtFEUC&pg=PA223&dq=Dirac+equation+in+curved+spacetime&redir_esc=y#v=onepage&q=Dirac%20equation%20in%20curved%20spacetime&f=false|title=Aspects of Quantum Field Theory in Curved Spacetime|last=S.A. Fulling|publisher=Cambridge University Press|year=1989|isbn=0-521-377-684}}
[[Category:费米子]]
[[Category:偏微分方程]]
[[Category:量子场论]]
[[Category:旋量]]