Question
--------

I was recently wondering about [semi-classical gravity](https://en.wikipedia.org/wiki/Semiclassical_gravity) :

$$ G\_{\mu \nu} = \frac{8 \pi G}{c^4} \langle \hat T\_{\mu \nu} \rangle\_\psi$$

Does this obey the [equivalence principle](https://en.wikipedia.org/wiki/Einstein%27s_thought_experiments#Falling_painters_and_accelerating_elevators)? 

My intuition
------------

Let's say I am in a lift and I want to measure the standard deviation in the acceleration. Then with Heisenberg's equation of motion I can do so ,however , the left hand side $ G\_{\mu \nu}$ is a classical object and will predict there is none whereas using the Hamiltonian in quantum mechanics $\hat T\_{00}$ will only have zero standard deviation if it is in a acceleration eigenstate. But this view would only work in quantum mechanics and I'm not sure how to argue it for fields. Does this kind of argument still hold in QFT?