Alright, this makes me question whether the mental model I’ve been using can be correct, because I’ve assumed that no entanglement with the apparatus happens, but I guess for the action of the apparatus on the atom there probably has to be an equal and opposite reaction of the atom on the apparatus, of some form...
Here’s my guess of why the entanglement between the atom and the apparatus may not cause decoherence. (Although it turns out something else does.) First consider a 10000-dimensional unit ball. If we shift this ball by two units in one of the dimensions, it would no longer intersect at all with the original volume. But if we were to shift it by 1/1000 units in each of the 10000 dimensions, the shifted ball would still mostly overlap with the original ball even though we’ve shifted it by a total of 10 units (because the distance between the centers of the balls is only sqrt(10000)/1000 = 0.1).
Now consider the amplitude blob of the apparatus and the shift provided to it by an aligned atom as it moves through. The shift is divided among all of the dimensions of the blob (i.e., all the particles of the apparatus), and in each dimension the shift is tiny compared to the spread of the blob, so the shifted blob almost completely overlaps with the original blob. This means the two possible shifted blobs (for aligned and anti-aligned atoms) can interfere with each other just fine.
(This was me trying to answer “how would the math have to work if the entanglement between atom and appartus doesn’t cause decoherence?” Maybe someone with better math skills than me can do the actual math and confirm this?)
Now what if we add an accelerometer? I have no understanding of the physics of accelerometers so I don’t know if one that can detect such a small acceleration is even theoretically possible, but if we assume that it is, then when an atom passes through the new apparatus (with the accelerometer), its amplitude blob will be shifted a lot in some of the dimensions (namely the dimensions representing the particles that make up the accelerometer output), which would prevent the shifted blobs from interfering with each other.
First consider a 10000-dimensional unit ball. If we shift this ball by two units in one of the dimensions, it would no longer intersect at all with the original volume. But if we were to shift it by 1/1000 units in each of the 10000 dimensions, the shifted ball would still mostly overlap with the original ball even though we’ve shifted it by a total of 10 units (because the distance between the centers of the balls is only sqrt(10000)/1000 = 0.1).
Actually no, it doesn’t mostly overlap. If we consider a hypercube of radius 1 (displaced along the diagonal) instead of a ball, for simplicity, then the overlap fraction is 0.9995^10000 = 0.00673. If we hold the manhattan distance (10) constant and let number of dimensions go to infinity, then overlap converges to 0.00674 while euclidean distance goes to 0. If we hold the euclidean distance (0.1) constant instead, then overlap converges to 0 (exponentially fast).
For the ball, I calculate an overlap fraction of 5.6×10^-7, and the same asymptotic behaviors.
(No comment on the physics part of your argument.)
For the ball, I calculate an overlap fraction of 5.6×10^-7, and the same asymptotic behaviors.
Hmm, my intuition was that displacing a n-ball diagonally is equivalent to displacing it axially, and similar to displacing a hypercube axially. I could very well be wrong but I’d be interested to see how you calculated this.
The intuition:
For a high dimensional ball, most of the volume is near the surface, and most of the surface is near the equator (for any given choice of equator). The extremity of “most” and “near” increases with number of dimensions. The intersection of two equal-size balls is a ball minus a slice through the equator, and thus missing most of its volume even if it’s a pretty thin slice.
The calculation:
Let
%20=%202\int%20_{y=0}%5E{r}%20v(n-1,\sqrt{r%5E2-y%5E2})%20dy%20=%20(2%20\pi%5E{n/2}%20r%5En)%20/%20(n%20\Gamma(n/2))) which is the volume of a n-dimensional ball of radius r. Then the fraction of overlap between two balls displaced by x is
%20dy}{v(n,r)}) (The integrand is a cross-section of the intersection (which is a lower-dimensional ball), and y proceeds along the axis of displacement.) Numeric result.
The position of the apparatus has to be uncertain enough for you to be able to measure momentum (i.e. acceleration) precisely enough. It works out just fine to patterns being smeared, an interesting exercise to do mathematically though.
edit: didn’t see context, thought you were speaking of the regular double slit experiment. It still applies though.
With regards to the M1 I don’t quite understand the question as the spin is not an arrow that snaps from arbitrary orientation to parallel or anti-parallel. When it interacts with field, after the speed of light lag, there’s recoil.
Here’s my guess of why the entanglement between the atom and the apparatus may not cause decoherence. (Although it turns out something else does.) First consider a 10000-dimensional unit ball. If we shift this ball by two units in one of the dimensions, it would no longer intersect at all with the original volume. But if we were to shift it by 1/1000 units in each of the 10000 dimensions, the shifted ball would still mostly overlap with the original ball even though we’ve shifted it by a total of 10 units (because the distance between the centers of the balls is only sqrt(10000)/1000 = 0.1).
Now consider the amplitude blob of the apparatus and the shift provided to it by an aligned atom as it moves through. The shift is divided among all of the dimensions of the blob (i.e., all the particles of the apparatus), and in each dimension the shift is tiny compared to the spread of the blob, so the shifted blob almost completely overlaps with the original blob. This means the two possible shifted blobs (for aligned and anti-aligned atoms) can interfere with each other just fine.
(This was me trying to answer “how would the math have to work if the entanglement between atom and appartus doesn’t cause decoherence?” Maybe someone with better math skills than me can do the actual math and confirm this?)
Now what if we add an accelerometer? I have no understanding of the physics of accelerometers so I don’t know if one that can detect such a small acceleration is even theoretically possible, but if we assume that it is, then when an atom passes through the new apparatus (with the accelerometer), its amplitude blob will be shifted a lot in some of the dimensions (namely the dimensions representing the particles that make up the accelerometer output), which would prevent the shifted blobs from interfering with each other.
Actually no, it doesn’t mostly overlap. If we consider a hypercube of radius 1 (displaced along the diagonal) instead of a ball, for simplicity, then the overlap fraction is 0.9995^10000 = 0.00673. If we hold the manhattan distance (10) constant and let number of dimensions go to infinity, then overlap converges to 0.00674 while euclidean distance goes to 0. If we hold the euclidean distance (0.1) constant instead, then overlap converges to 0 (exponentially fast).
For the ball, I calculate an overlap fraction of 5.6×10^-7, and the same asymptotic behaviors.
(No comment on the physics part of your argument.)
Hmm, my intuition was that displacing a n-ball diagonally is equivalent to displacing it axially, and similar to displacing a hypercube axially. I could very well be wrong but I’d be interested to see how you calculated this.
The intuition: For a high dimensional ball, most of the volume is near the surface, and most of the surface is near the equator (for any given choice of equator). The extremity of “most” and “near” increases with number of dimensions. The intersection of two equal-size balls is a ball minus a slice through the equator, and thus missing most of its volume even if it’s a pretty thin slice.
The calculation: Let
Then the fraction of overlap between two balls displaced by x is
Thanks for both the math and the intuitive explanation. Now I’m really curious what the right answer is to the physics question...
The position of the apparatus has to be uncertain enough for you to be able to measure momentum (i.e. acceleration) precisely enough. It works out just fine to patterns being smeared, an interesting exercise to do mathematically though.
edit: didn’t see context, thought you were speaking of the regular double slit experiment. It still applies though.
With regards to the M1 I don’t quite understand the question as the spin is not an arrow that snaps from arbitrary orientation to parallel or anti-parallel. When it interacts with field, after the speed of light lag, there’s recoil.