Question about quantum back action in hypothetical scenario:
We know that, at $t_0$, a certain kind of particle, with spin initially prepared to be “spin right” in the x basis, goes through a Stern-Gerlach magnet that affects spin in the z basis (as “spin up” or “spin down”).
Eventually, the particle’s z-basis spin will be measured by a detector. Hence, there is a 50/50 chance that the particle will end up being measured as spin up or spin down, after exiting the magnet and reaching the detector.
Well above the magnet (in the z-up direction and between the magnet and the detector) is a second, hypothetical, environment which experiences no internal forces causing its wave function, or density matrix, to change with time. It is also made to be completely isolated from all external influences, except for this particle, which may locally interact with it (pass through it). Assume if the particle passes through this second environment (after exiting the Stern-Gerlach magnet), the environment will experience a substantial back-action, but the environment will have no affect on the spin of the particle.
We know that, at $t_1 > t_0$ , the particle is measured to be spin down.
What happened with regard to the back-action of the second environment? Some different possibilities are:
1.)After exiting the magnet, the particles's wave function split into two packets, one going up and one going down. The "up" wave function interacted with the second environment. The second environment’s wave function, or density matrix, evolved according to the backaction caused by the particle, between $t_0$ and $t_1$. At $t_1$, when the particle was measured to be spin down, the evolution of the second environment was “undone”, so that, at $t_1$, the second environment is in the same state that it was in at $t_0$.
2.) The particle's spin wave function collapsed to spin down, upon exiting the magnet, so it never interacted with the second environment. Therefore, the second environment’s wave function, or density matrix, did not change between $t_0$ and $t_1$ (so, no undoing is necessary).
3.) Since there is no such thing as trajectory in QM, even a spin down particle might have gone through the second environment and so we might see back-action in the second environment even if the particle is measured to be spin down. But if the particle is measured to be spin down,then the z-momentum of the particle must have been down, so I do not see this possibility being good.
I suppose which possibility is correct may depend on one’s interpretation of quantum mechanics. What does your favorite interpretation say?