Carroll and Lodman (2021) consider a very simple setup.
We prepare some isolated system in a pure state which is a superposition of two energy eigenstates:
|ψ_1 > = α |E_1 > + β |E_2 >,
where α^2 + β^2 = 1.
If we measure the energy of the system, the expectation value before the measurement is
|α^2| E_1 + |β^2| E_2.
In the measurement, the system collapses to either the state E_1 or E_2.
After the measurement, the expectation value of energy is either E_1 or E_2. The expectation value changed. We assumed that the system is isolated. Did we witness energy non-conservation?
The authors fail to mention how we prepare the system to a superposition state. If our system is a bowling ball A, we may let A roll close to another bowling ball B whose speed is 300 m/s. Depending on whether our ball hits the super-fast ball, our ball may have energy E_1 or E_2 afterwards. Then we isolate our ball and do the reasoning above.
Now it is obvious what is the error of the authors. Energy is conserved in the combined system A & B. We cannot consider the "isolated" system A alone, because it is entangled with B.
The paper by Carroll and Lodman is essentially a restatement of the Einstein-Podolsky-Rosen paradox.
Lubos Motl (2021) criticizes the authors. He writes that the expectation value of energy is not the energy itself. We agree with Motl on this. The energy of the system A before the measurement is not defined - it is not the weighted sum of E_1 and E_2.
What would it mean to say that the "aliveness" of Schrödinger's cat is 0.5 before we open the box? And the aliveness changes when we have opened the box?
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