Dirac equation in curved spacetime

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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]

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 Dirac matrices:

\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.

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". arXiv:1103.3201 .
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.

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Dirac equation in curved spacetime

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References