Dirac equation, curved space time

[pleasehelpmeno]

Hi when trying to derive this equation, i am stuck on:
$[\Gamma_{\mu}(x),\gamma^{\nu}(x)]=\frac{\partial \gamma^{\nu}(x)}{\partial x^{\mu}} + \Gamma^{\nu}_{\mu p}\gamma^{p}$.

This $[\Gamma_{\mu}(x)$ term is the spin connection, if this is an ordinary commutator:
a) is it a fermionic so + commutator
b) how can one solve to find the Gamma term whilst cancelling away the
c) can anyone give a qualitative description of what the spin connection is?
thanks

 

[Bill_K]

In curved space the gamma matrices are position-dependent, since they obey γ_μγ_ν + γ_νγ_μ = 2 g_μν. But we can and do take them to be covariantly constant.

A gamma matrix is a hybrid quantity, having one vector index, and two matrix indices referred to a tetrad basis. The covariant derivative contains a correction term for each index - a Christoffel symbol for the vector index and a spin connection (Ricci rotation coefficient) for each tetrad index.

The equation you wrote is the statement that the covariant derivative vanishes. You can take a look at this paper for more details.

 

[pleasehelpmeno]

Hi
I have reread the paper: http://arxiv.org/abs/gr-qc/0501077v1 (eq 8 and 10) and i see what you mean. How else then can one find $\Gamma_\mu$?

 

[Bill_K]

Eqs. (8) and (9) in the paper I referred you to shows how to solve for Γ_μ.

 

[pleasehelpmeno]

thx,

am i right in thinking $\bar{S_{ab}}$ is always equal to zero when b =0 because all my answers involved $\bar{S_{a0}}$ terms?

 

[pleasehelpmeno]

sorry i should explain more,

I assumed that $t^{a}_{b}$ is only non zero if a = b (from 0 to 3). and the same for $t_{ab}$.

If this is correct then all my answers for $w_{abj}$ contain either a =0 and b=1,2,3 or vice versa meaning that when multiplied with $S_{ab}$ then will surely go to zero if $s^{0b} = 0$.
Are any of these assumptions incorrect?

 

[FunkyDwarf]

Sorry for the thread necro, but could someone please provide more details regarding eqn (8) in that reference, specifically why the commutator of the spin connection with the curved space gamma matrix is defined that way.

 

[Bill_K]

Sorry for the thread necro, but could someone please provide more details regarding eqn (8) in that reference, specifically why the commutator of the spin connection with the curved space gamma matrix is defined that way.

A gamma matrix is a hybrid quantity, having one vector index, and two matrix indices referred to a tetrad basis. The covariant derivative contains a correction term for each index - a Christoffel symbol for the vector index and a spin connection (Ricci rotation coefficient) for each tetrad index.

The equation you wrote is the statement that the covariant derivative vanishes.

The covariant derivative of gamma contains a correction term for each index: a Christoffel symbol for the vector index, and a spin connection for each of the two spinor indices.

∂γ^ν/∂x^μ + Γ^ν_μρ γ^ρ + Γ_μ γ^ν - γ^ν Γ_μ

Eq.(8) says that this quantity vanishes (we choose to define our γ's that way) and then the terms are rearranged slightly to make them look like a commutator.

 

[nuclearhead]

It can be regarded as the gauge field generated by local Lorenz transformations. Although we find it is not independent of the metric and must be equal to combinations of vierbeins and their derivatives.

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