Dirac equation in curved spacetime

Main article: Quantum field theory 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 Gamma matrices to act at each spacetime point. In this way, Dirac's equation takes the following form in curved spacetime:

$i\gamma ^{a}e_{a}^{\mu }D_{\mu }\Psi -m\Psi =0.$

Here e_a^μ is the vierbein and D_μ is the covariant derivative for fermionic fields, defined as follows

$D_{\mu }=\partial _{\mu }-{\frac {i}{4}}\omega _{\mu }^{ab}\sigma _{ab},$

where σ_ab is the commutator of Gamma matrices:

$\sigma _{ab}={\frac {i}{2}}\left[\gamma _{a},\gamma _{b}\right],$

and ω_μ^ab are the spin connection components.

The modified Klein–Gordon equation obtained by squaring the operator in the Dirac equation, first found by Erwin Schrödinger as cited by Pollock is given by

$\left({\frac {1}{\sqrt {-\det g}}}\,{\cal {D}}_{\mu }\left({\sqrt {-\det g}}\,g^{\mu \nu }{\cal {D}}_{\nu }\right)-{\frac {1}{4}}R+{\frac {ie}{2}}F_{\mu \nu }s^{\mu \nu }-m^{2}\right)\Psi =0.$

where $R$ is the Ricci scalar, and $F_{\mu \nu }$ is the field strength of $A_{\mu }$.

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

• 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. S2CID 38235437.
• M.D. Pollock (2010). "on the dirac equation in curved space-time". Acta Physica Polonica B. 41 (8): 1827.{{cite journal}}: CS1 maint: url-status (link)
• J.V. Dongen (2010). Einstein's Unification. Cambridge University Press. p. 117. ISBN 978-0-521-883-467.
• L. Parker, D. Toms (2009). Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity. Cambridge University Press. p. 227. ISBN 978-0-521-877-879.
• S.A. Fulling (1989). Aspects of Quantum Field Theory in Curved Spacetime. Cambridge University Press. ISBN 0-521-377-684.