Let's say we are doing the double slit experiment with electrons. We get an interference pattern, and if we put detectors at slits, then we get two piles pattern because we measure electrons' positions when going through slits. But an electron interacts with other particles in a lot of different ways, e.g. electric field, gravity. Seems like the whole universe is receiving information about the electron's position. Why is it not the case and the electron goes through slits "unmeasured"?
Bonus question: in real experiments do we face the problem of not "shielding" particles from "measurement" good enough and thus getting a mix of both patterns on the screen?
Answer
Seems like the whole universe is receiving information about the electron's position.
Yes, the influence that an electron exerts on the rest of the universe does depend on the location of the electron, but that's not enough to constitute a measurement of the electron's location. We need to consider the degree to which the electron's influence on the rest of the universe depends on its location.
Consider something analogous to but simpler than a double slit experiment: consider an electron in deep space, in a superposition of two different locations $A$ and $B$. Even in deep space, the electron is not alone, because space is filled with cosmic microwave background (CMB) radiation. CMB radiation has a typical wavelength of about $1$ millimeter. When CMB radiation is scattered by an electron, the resulting state of the radiation depends on the electron's location, but the key question is how much it depends on the electron's location. If the locations $A$ and $B$ differ from each other by $\gg 1$ millimeter, then the CMB radiation will measure the electron's location very effectively, because an electron in location $A$ will have a very different effect on the CMB radiation than an electron in location $B$ would have. But if locations $A$ and $B$ differ from each other by $\ll 1$ millimeter, then an electron in location $A$ will not have a very different effect on the CMB radiation than an electron in location $B$ would have. Sure, the electron has a significant effect on the CMB radiation regardless of its location, but that key is whether the effect differs significantly when the location is $A$ versus $B$. The CMB radiation measures the electron's location, but it does so with limited resolution. Widely-spaced locations will be measured very effectively, but closely-separated locations will not.
For this to really make sense, words are not enough. We need to consider the math. So here's a version that includes a smidgen of math.
Let $|a\rangle$ denote the state of the universe (including the electron) that would result if the electron's location were $A$, and let $|b\rangle$ denote the state of the universe that would result if the electron's location were $B$. If the electron started in some superposition of locations $A$ and $B$, then the resulting state of the universe will be something like $|a\rangle+|b\rangle$. Whether or not the electon's location is effectively measured, these two terms will be essentially orthogonal to each other, $\langle a|b\rangle\approx 0$, simply because they differ significantly in the location of the electron itself. So the fact that the final state is $|a\rangle+|b\rangle$ with $\langle a|b\rangle\approx 0$ doesn't tell us anything about whether or not the electron's location was actually measured. For that, we need a principle like this:
- The electron's location has been effectively measured if and only if the states $|a\rangle$ and $|b\rangle$ are such that $\langle a|\hat O|b\rangle\approx 0$ for all feasibly-measurable future observables $\hat O$. (Quantifying "$\approx 0$" requires some care, but I won't go into those details here.)
For an operator $\hat O$ to be "feasibly measurable", it must be sufficiently simple, which loosely means that it does not require determining too many details over too large a region of space. This is a fuzzy definition, of course, as is the definition of measurement itself, but this fuzziness doesn't cause any problems in practice. (The fact that it doesn't cause any problems in practice is frustrating, because this makes the measurement process itself very difficult to study experimentally!)
In the example described above, the suggested condition is satisfied if locations $A$ and $B$ differ by $\gg 1$ millimeter, because after enough CMB radiation has been scattered by the electron, the states $|a\rangle$ and $|b\rangle$ differ significantly from each other everywhere, and no operator $\hat O$ that is simple enough to represent a feasibly-measurable observable can possibly un-do the orthogonality of the states $|a\rangle$ and $|b\rangle$. Loosely speaking, the state $|a\rangle$ and $|b\rangle$ aren't just orthogonal; they're prolifically orthogonal, in a way that can't be un-done by any simple operator. In contrast, if locations $A$ and $B$ differ by $\ll 1$ millimeter, then we can choose an operator $\hat O$ that acts just on the electron (and is therefore relatively simple) to obtain $\hat O|a\rangle\approx |b\rangle$, thus violating the condition $\langle a|\hat O|b\rangle\approx 0$. So in this case, the electron's location has not been effectively measured at all. The states $|a\rangle$ and $|b\rangle$ are orthogonal simply because they differ in the location of the electron itself, but they are not prolifically orthogonal because the effect on the rest of the universe doesn't depend significantly on whether the electron's location was $A$ versus $B$.
What I'm doing here is describing "decoherence" in a different way than it is usually described. The way I'm describing it here doesn't rely on any factorization of the Hilbert space into the "system of interest" and "everything else." The way I'm describing it here (after quantifying some of my loose statements more carefully) can be applied more generally. It doesn't solve the infamous measurement problem (which has to do with the impossibility of deriving Born's rule within quantum theory), but it does allow us to determine how effectively a given observable has been measured.
Some quantitative calculations — including quantitative results for the specific example I used here — are described in Tegmark's paper "Apparent wave function collapse caused by scattering" (https://arxiv.org/abs/gr-qc/9310032), which is briefly reviewed in https://physics.stackexchange.com/a/442464. Those calculations use the more traditional description of decoherence, but the results are equally applicable to the way I described things here.
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