Dirac equation in curved spacetime
In mathematical physics, the Dirac equation in curved spacetime generalizes the original Dirac equation to curved space.
It can be written by using vierbein fields and the gravitational spin connection. The vierbein defines a local rest frame, allowing the constant Dirac matrices to act at each spacetime point. In this way, Dirac's equation takes the following form in curved spacetime:^{[1]}
Here e_{a}^{μ} is the vierbein and D_{μ} is the covariant derivative for fermionic fields, defined as follows
where σ_{ab} is the commutator of Dirac matrices:
and ω_{μ}^{ab} are the spin connection components.
Note that here Latin indices denote the "Lorentzian" vierbein labels while Greek indices denote manifold coordinate indices.
See also
- Dirac equation in the algebra of physical space
- Dirac spinor
- Maxwell's equations in curved spacetime
- Two-body Dirac equations
References
- ^ Lawrie, Ian D. A Unified Grand Tour of Theoretical Physics.
- M. Arminjon, F. Reifler (2013). "Equivalent forms of Dirac equations in curved spacetimes and generalized de Broglie relations". Brazilian Journal of Physics. 43 (1-2): 64–77. arXiv:1103.3201 . Bibcode:2013BrJPh..43...64A. doi:10.1007/s13538-012-0111-0.
- M.D. Pollock (2010). "on the dirac equation in curved space-time" (PDF). Acta Physica Polonica B. 41 (8). Archived from the original (PDF) on 2013-12-14.
- J.V. Dongen (2010). Einstein's Unification. Cambridge University Press. p. 117. ISBN 0-521-883-466.
- L. Parker, D. Toms (2009). Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity. Cambridge University Press. p. 227. ISBN 0-521-877-873.
- S.A. Fulling (1989). Aspects of Quantum Field Theory in Curved Spacetime. Cambridge University Press. ISBN 0-521-377-684.
This page is based on the copyrighted Wikipedia article "Dirac equation in curved spacetime"; it is used under the Creative Commons
Attribution-ShareAlike 3.0 Unported License (CC-BY-SA). You may
redistribute it, verbatim or modified, providing that you comply with
the terms of the CC-BY-SA