User:JRSpriggs/Dirac particle in general relativity

Tensor equivalent of Dirac equation

The Dirac equation is invariant under Lorentz transformations, but not (without modification) under the arbitrary curvilinear coordinate transformations used in general relativity. Instead of trying to get spinors to transform under such coordinate transformations, I will try to translate from spinors to tensors.

Replace the spinor, $\psi \,,$ by a sequence of antisymmetric tensors of every rank from 0 to 4, thusly

$\psi =(\psi ^{(0)}\mathbf {1} +\psi _{\alpha }^{(1)}\gamma ^{\alpha }+{\frac {1}{2}}\psi _{\alpha \beta }^{(2)}\gamma ^{\alpha }\gamma ^{\beta }+{\frac {1}{6}}\psi _{\alpha \beta \gamma }^{(3)}\gamma ^{\alpha }\gamma ^{\beta }\gamma ^{\gamma }+{\frac {1}{24}}\psi _{\alpha \beta \gamma \delta }^{(4)}\gamma ^{\alpha }\gamma ^{\beta }\gamma ^{\gamma }\gamma ^{\delta })\,\xi \,$

where ξ is a constant spinor. The Lagrangian is then

$L=({\frac {1}{2}}mc^{2}(\psi ^{(0)}\psi ^{(0)}+\psi _{\alpha }^{(1)}g^{\alpha \beta }\psi _{\beta }^{(1)}+\ldots +{\frac {1}{24}}\psi _{\alpha \beta \gamma \delta }^{(4)}g^{\alpha \epsilon }g^{\beta \zeta }g^{\gamma \eta }g^{\delta \iota }\psi _{\epsilon \zeta \eta \iota }^{(4)})\,+\,$

$i\hbar c(\psi _{\alpha }^{(1)}g^{\alpha \beta }\partial _{\beta }\psi ^{(0)}+\ldots +{\frac {1}{24}}\psi _{\alpha \beta \gamma \delta }^{(4)}g^{\alpha \epsilon }g^{\beta \zeta }g^{\gamma \eta }g^{\delta \iota }\partial _{\epsilon }\psi _{\zeta \eta \iota }^{(3)}))\,{\sqrt {-g}}\,$

where the partial derivative could be replaced by the gauge covariant derivative to link it to electromagnetism.