Dirac equation in curved spacetime

In mathematical physics, the Dirac equation in curved spacetime generalizes the original Dirac equation to curved space.

Main article: Quantum field theory in curved spacetime

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]

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

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

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

where σ_ab is the commutator of Dirac matrices:

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

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

1. 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". Acta Physica Polonica B. 41 (8): 1827.
• 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.