I have been thinking recently what will happen if one uses the energy momentum tensor of the Dirac field as a source in the Einstein Field equations. It is well known that in this case

$$ T_{\mu\nu}=i\overline{\psi}\gamma^{\mu}\partial^{\nu}\psi $$

What fallows however causes my confusion. The dimension of the Lagrangian is $\Lambda^d$ in units of mass. Therefore, dimension of the Dirac spinor has to be $\Lambda^{\frac{d-1}{2}}$. According to the Einstein equation we have $$ R_{\mu\nu}=\kappa(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}), \tag{$\star$} $$

where I used contraction with the metric tensor, and since it is dimensionless, the original dimensions hold. So far so good, but let's look at the RHS in 4D. For the dimensions we have, $$[T_{\mu\nu}]=\left[i\overline{\psi}\gamma^{\mu}\partial^{\nu}\psi\right] = \Lambda^4\quad \text{we should have}\quad [\kappa]=\Lambda^{-2}.$$ Due to the equality ($\star$) I can write for the Einstein-Hilbert action

$$ \mathcal{L}=\frac{\kappa}{2s}\int d^4x g^{\mu\nu}(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu})\sqrt{-g} $$

Here, the constants $s$ from the Einstein Hilbert action and $\kappa$ from the Field equations are the same. (I made an error see the edit) Anyway, it does not matter since whatever the constants in the action written in this way, their ratio should be dimensionless. It seems that I have removed the divergent powers of $\Lambda^2$.

Since, I have no constants with dimension mass squared, which will spoil convergence if expansion against flat background is used. I presume this is all fools gold, but can't figure out how it relates to renormalizibility of Gravity.

More precisely by making this substitution I remove some solutions (see one of the answers below), but what would be the General effect on renormalizability and why the approach is wrong?


The constants $s$ and $\kappa$ are the same. So we get no powers of $\Lambda^2$ by cancellation i.e.

$$ \mathcal{L}=\frac{\kappa}{2\kappa}\int d^4x g^{\mu\nu}(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu})\sqrt{-g} $$

where the Einstein-Hilbert action reads

$$ S=\frac{1}{2\kappa} \int d^4x R\sqrt{-g} $$


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