# Instanton contributions in quantum gravity

Suppose a low-energetic System, i.e. a System, where the presence of "classical" gravitational fields can be assumed to be Zero. Classically we would have e.g. the ordinary Minkowski metric or more general some other Solutions of the vacuum Einstein-Cartan equations

$$\epsilon_{abcd}e^b\wedge D \omega^{cd}=0;$$ (energy-momentum Tensor $T^a$ can be neglected)

$$\epsilon_{abcd}e^c\wedge D e^{d}= 0$$ (spin-angular-momentum Tensor $S^{ab}$ can be neglected).

Here, $e^a$ is the tetrad one-form, $\omega^{ab}$ the spin Connection one-from and $D$ the exterior covariant derivative.

A trivial solution would be $e_\mu^a = \delta_\mu^a, \omega_\mu^{ab} = 0$, but there may exist some nontrivial Solutions of These equations which are called instantons.

1st question: Will wormhole metrics be Solutions of the vacuum Einstein-Cartan field equations? I would say yes, since some wormhole metrics are Solutions of Einstein vacuum equations.

Now gauge invariance (diffeomorphism and Lorentz invariance) can create a whole bunch of Solutions; if we fix the gauge, we obtain topological distinct Solutions that can be classified by the topological Action (corresponding to spacetime $M$)

$$S_t = \alpha_1 \int_M \epsilon_{abcd}D \omega^{ab}\wedge D \omega^{cd} + \alpha_2\int_M D \omega_{cd}\wedge D \omega^{cd}$$

where first term is the Euler term and the second term is the Pontryagin term (with coupling constaints $\alpha_1, \alpha_2$).

Now the 2nd question: I want to compute Quantum observables with the path integral, i.e. for an observable $O$ I want to compute

$$<O> := \int \int \mathcal{D}[e^a]\mathcal{D}[\omega^{ab}]Oe^{iS_c+iS_t}\delta(\epsilon_{abcd}e^b\wedge D \omega^{cd}) \delta(\epsilon_{abcd}e^c\wedge D e^{d}) \delta_{Gauge-fix}$$

with the classical general-relativistic action $S_c$ and Delta functions for some gauge fixing $\delta_{gauge-fix}$. For simplification I gauge fix by setting some fields to Zero (no ghost Terms will appear). Now I think I have a discrete set of Solutions of vacuum Einstein-Cartan equations and can take a sum instead of an integral. In $S_t$ topological invariants are assigned and on $O$ and $S_c$ the corresponding solutions are substituted. By computing the Delta functions, I pick up some weight (a Jacobian for the solution).

Is it correct so?

Now a third question: The sum over all possible instanton configurations will give different scattering amplitudes than a Quantum or classical field theory in flat spacetime. Could it be possible that we may observe these low-energy Quantum-gravitational deviations from the classical predictions?

Maybe yes. We use the energy-momentum balance $<D*T^a> = 0$ instead of $<d*T^a> = d*<T^a> = 0$ and thus, we may observe a vacuum-fluctuation induced quasi-Violation of energy conservation that has the contribution $\sigma^a = d*<T^a> = -<\omega^{a}_b *T^b>$.