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An attempt to directly use the synchronous gauge (\$g_{0 lambda} = –
delta_{0 lambda}\$) in perturbative gravity leads to a singularity at \$p_0 =
0\$ in the graviton propagator. This is similar to the singularity in the
propagator for Yang-Mills fields \$A^a_lambda\$ in the temporal gauge (\$A^a_0 =
0\$). There the singularity was softened, obtaining this gauge as the limit at
\$varepsilon to 0\$ of the gauge \$n^lambda A^a_lambda = 0\$, \$n^lambda = (1,
– varepsilon (partial^j partial_j )^{- 1} partial^k ) \$. Then the
singularities at \$p_0 = 0\$ are replaced by negative powers of \$p_0 pm i
varepsilon\$, and thus we bypass these poles in a certain way.

Now consider a similar condition on \$n^lambda g_{lambda mu}\$ in
perturbative gravity, which becomes the synchronous gauge at \$varepsilon to
0\$. Unlike the Yang-Mills case, the contribution of the Faddeev-Popov ghosts to
the effective action is nonzero, and we calculate it. In this calculation, an
intermediate regularization is needed, and we assume the discrete structure of
the theory at short distances for that. The effect of this contribution is, in
particular, to add non-pole terms to the propagator. In overall, this
contribution vanishes at \$varepsilon to 0\$. Thus, we effectively have the
synchronous gauge with the resolved singularities at \$p_0 = 0\$, where only the
physical components \$g_{j k}\$ are active and there is no need to calculate the
ghost contribution.

Conclusion:

The conclusions of the text are that using the synchronous gauge in perturbative gravity leads to a singularity at p_0=0 in the graviton propagator. However, by applying a similar condition as in the Yang-Mills case and by calculating the contribution of the Faddeev-Popov ghosts, the singularities can be bypassed and resolved. This results in the effective synchronous gauge where only the physical components g_jk are active.

In order to move forward with the research in perturbative gravity and the use of the synchronous gauge, the following roadmap can be outlined:

1. Further investigation of the similarities with Yang-Mills fields:

It is important to continue exploring the similarities between perturbative gravity and Yang-Mills fields in terms of their respective gauges and singularities. This can help to gain a deeper understanding of the nature of the singularities in graviton propagators and how they can be resolved.

2. Development of regularization techniques:

Since an intermediate regularization is needed in the calculation of the Faddeev-Popov ghost contribution, it is crucial to develop precise and reliable regularization techniques. This will ensure accurate calculations and a better understanding of the effects of the ghost contributions on the propagator.

3. Study of the discrete structure at short distances:

Assuming a discrete structure of the theory at short distances is an essential step in the calculation of the Faddeev-Popov ghost contribution. It is necessary to investigate this discrete structure further to validate its accuracy and applicability to perturbative gravity.

4. Exploration of non-pole terms in the propagator:

The addition of non-pole terms to the propagator due to the Faddeev-Popov ghost contribution opens up new avenues for research. It is important to explore the implications of these non-pole terms and how they affect the behavior of the propagator in different scenarios.

5. Verification of the vanishing of contribution at ε→0:

The claim that the contribution of the Faddeev-Popov ghosts vanishes at ε→0 needs to be further verified and validated through rigorous calculations and experiments. This will provide more confidence in the effectiveness of the resolved synchronous gauge in perturbative gravity.

Challenges and Opportunities:

While there are potential challenges in terms of developing accurate regularization techniques and validating assumptions about the discrete structure at short distances, there are also exciting opportunities for further exploration and understanding of perturbative gravity. Resolving the singularities in the graviton propagator opens up possibilities for more precise calculations and predictions in gravitational theories.

Overall, with continued research and investigation, the resolved synchronous gauge in perturbative gravity can lead to significant advancements in our understanding of gravity and its behavior at both small and large scales.