Do quantum fluctuations mean metric fluctuations?

[nomadreid]

I suspect the following reasoning is faulty, but I am not sure why. Hence I would appreciate someone pointing out the errors. That is, which, if any, of the following statements are incorrect, and why?
1) Theoretically, albeit not practically due to the large numbers involved, the laws of thermodynamics should follow from the laws governing the interaction of the individual particles.
2) In an analogy to (1), the laws of general relativity should follow from the laws of quantum field theory, which is one of the goals of a theory of quantum gravity.
3) From (2), the metric for the vacuum at a neighborhood in space-time would thus be derivable from the quantum states in that neighborhood.
4) Since the quantum states consist of a superposition of determined states, then the corresponding metric mentioned in (3) would actually be a superposition of metrics.
5) Since the quantum states are fluctuating, so too the corresponding metrics, hence the metrics are also fluctuating, so one cannot really talk of "the" metric of a neighborhood in space-time except as an average.
Thanks.

 

[mfb]

1) Theoretically, albeit not practically due to the large numbers involved, the laws of thermodynamics should follow from the laws governing the interaction of the individual particles.

These derivations are done in university courses.

2) In an analogy to (1), the laws of general relativity should follow from the laws of quantum field theory, which is one of the goals of a theory of quantum gravity.

Not necessarily, and it is not a good analogy. Many physicists expect that gravity can be expressed using QFT, but that is not guaranteed, and it might need extensions of QFT or even something completely new.

3) From (2), the metric for the vacuum at a neighborhood in space-time would thus be derivable from the quantum states in that neighborhood.

As with every good local theory.

4) Since the quantum states consist of a superposition of determined states, then the corresponding metric mentioned in (3) would actually be a superposition of metrics.

Probably.

5) Since the quantum states are fluctuating

They are not. Some measurements related to states can fluctuate, but that is a different statement.