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Gravitational waves of object in superposition?

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I recently read a blog post on measuring gravity waves of objects in superposition:http://backreaction.blogspot.in/2016/03/researchers-propose-experiment-to.html

And did some calculations regarding the same.


We go into the heisenberg picture to define velocity:
$$ \hat v = \frac{dU^\dagger x U}{dt} = U^\dagger\frac{[H,x]}{- i \hbar}U$$

where $\hat v$ is the velocity operator $U^\dagger$ is the unitary operator and $H$ is the Hamiltonian.

Now we can again differentiate to get acceleration $\hat a$:

$$ \hat a =  \hat U^\dagger\frac{[[\hat H, \hat x], \hat x]}{-i \hbar} \hat U =  \hat U^\dagger\frac{(\hat H^2 \hat x + \hat x \hat H^2 - 2\hat H \hat x \hat H)}{\hbar^2} \hat U $$ 

We can simplify the calculation by splitting the Hamiltonian into potential  $ \hat V $ and kinetic energy $ \hat T $: $\hat H = \hat T + \hat V$

By noticing (one can also calculate this) that the acceleration of an object in a constant potential is $0$:

$$ \hat 0 = \hat T^2 x + x  \hat T^2 - 2 \hat T  \hat x  \hat T $$

We also know $ [\hat V, \hat x] = 0 $ as potential is a function of position. Thus, we can simplify acceleration as:

$$ \hat a = \hat V \hat T \hat x + \hat x \hat T \hat V - \hat V \hat x \hat T - \hat T \hat x \hat V   $$

Note this acceleration operator also commutes with position:

$$ [\hat a , \hat x ]=0$$


Now by the equivalence principle: "acceleration of the object is indistinguishable from gravity." Hence, I argue if the quantum version of the Ricci curvature tensor is measured then the position operator will also collapse and the superposition will exist no more. 


Is this line of reasoning correct and also existent in the literature? What do other theories of quantum gravity predict for the gravitational waves for objects in superposition?  

Note: I used a similar line of reasoning here (for something else).  However, this is a different question.

asked Jul 14 in Phenomenology by Asaint (30 points) [ no revision ]

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