Dirac equation in curved spacetime

Main article: Quantum field theory in curved spacetime

Quantum field theory
Feynman diagram
History
Background Field theory Gauge theory Poincaré symmetry Quantum mechanics Spontaneous symmetry breaking
Symmetries Charge conjugation Crossing Parity Time reversal Symmetry in quantum mechanics
Tools Anomaly Effective field theory Expectation value Faddeev–Popov ghosts Feynman diagram Lattice gauge theory LSZ reduction formula Partition function Propagator Quantization Regularization Renormalization Vacuum state Wick's theorem Wightman axioms
Equations Dirac equation Klein–Gordon equation Proca equations Wheeler–DeWitt equation Bargmann–Wigner equations
Standard Model Electroweak interaction Higgs mechanism Quantum chromodynamics Quantum electrodynamics Yang–Mills theory
Incomplete theories Quantum gravity String theory Supersymmetry Technicolor Theory of everything
Scientists C. D. Anderson P. W. Anderson Bethe Bjorken Bogoliubov Brout Callan Coleman DeWitt Dirac Dyson Englert Fermi Feynman Fierz Fock Fröhlich Glashow Gell-Mann Gross Guralnik Heisenberg Higgs Haag Hagen 't Hooft Jordan Kendall Kibble Lamb Landau Lee Majorana Mills Nambu Nishijima Parisi Polyakov Salam Schwinger Skyrme Sudarshan Tomonaga Veltman Ward Weinberg Weyl Wilczek Wilson Yang Yukawa Weisskopf

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).

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.

Dirac equation in curved spacetime

See also

References