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It's most likely me being confused.
My picture of it right now is that all the dimensions you need in total, are all the dimensions in state-space + 2 dimensions for the complex amplitude. If this assumption is wrong, then we have found the error in my thinking already!
Note that the two complex amplitude dimensions are of course not like the other dimensions. For every position in the state-space, there is a single point in the amplitude dimensions. Or in my suggestion, a line from origo out to the calculated complex value.
Don't try to think this through with matrices, there's a very real chance that what I'm after cannot be captured by matrices at all. I think you have to do a complete geometric picture of it.
As any other amateur who reads Eliezer’s quantum physics sequence, I got caught up in the “why do we have the Born rule?” mystery. I actually found something that I thought was a bit suspicious (even though lots of people must have thought of it, or experimentally rejected this already.) Note that I'm deep in amateur swamp, and I'll gleefully accept any "wow, you are confused" rejections.
Here is my suggestion:
What if the universes, that we live in, are not located specifically in configuration space, but in the volume stretched out between configuration space and the complex amplitude? So instead of talking about “the probability of winding up here in configuration space is high, because the corresponding amplitude is high”, we would say “the probability of winding up here is high, because there are a lot of universes here”. And here, would mean somewhere on the line between a point in configuration space and the complex amplitude for that point. (All these universes would be exactly equal.) And then we completely remove the Born rule. Of course someone thought of this, but responds: “But if we double the amplitude in theory, the line becomes twice as long, and there would be twice as many universes. But this is not what we observe in our experiments, when we double the amplitude, the probability of finding ourselves there multiplies by four!” This is true, if you study a line between the complex amplitude peak and a point in configuration space. But you are never supposed to study a point in configuration space, you are supposed to integrate over a volume in configuration space.
Calculating the volume between the complex amplitude “surface” and the configuration space, is not like taking all the squared amplitudes of all points of the configuration space and summing them up. The reason is that, when we traverse the space in one direction and the complex amplitude changes, the resulting volume “curves”, causing there to be more volume out near the edges (close to the amplitude peak) and less near the configuration space axis.
Take a look at the following image (meant to illustrate an "amplitude volume" for a single physical property): [http://www.wolframalpha.com] , type in: ParametricPlot3D {u Sin[t], u Cos[t], t / 5}, {t, 0, 15}, {u, 0, 1}
Imagine that we’d peer down from above, looking along the property axis. If we completely ignore what happens in the view direction, the volume (the blue areas) would have the shape of circles. If we’d double the amplitude, the volume from this perspective would be quadrupled.
But as it is, what happens along the property axis matters. The stretching out causes the volume to be less than the amplitude squared. It seems that, the higher the frequency is, the closer the volume is to have a square relationship with the amplitude, while as the frequency lowers, the volume approaches a linear relationship with the frequency. Studying the two extreme cases; with frequency 0 the geometric object would be just a straight plane, with an obvious linear relationship between amplitude and volume, while with an “infinite” frequency, the geometric object would become a cylinder, with a squared relationship between volume and amplitude. This means that the overall current amplitude-configuration-space ratio is important, but as far as I know, it is unknown to us.
In a laboratory environment, where all frequencies involved are relatively low, we would see systems evolving linearly. But when we observe the outcome of the systems, and entangle them with everything else, what suddenly matters is the volume of our combined wave which has a very very high frequency.
Or does it? At this point I'm beginning to lose track and the questions starts piling up.
What happens when multiple dimensions are mixed in? I’m guessing that high-frequency/high-amplitude still approaches a squared relationship from amplitude to volume, but I’m not at all certain.
What happens over time as the universe branches, does the amplitude constantly decrease while the length and frequencies remain the same? (Causing the relationship to dilute from squared to linear?)
Note that this suggestion also implies that there really exists one single configuration space / wave function that forms our reality.
So, what do you think?
Hi! This is my first post. I have a physics/MWI question, and I don't know if LessWrong / Discussion is the right place to post it, so is it OK if I ask it here? Here it goes:
As any other amateur who reads Eliezer’s quantum physics sequence, I got caught up in the “why do we have the Born rule?” mystery. I actually found something that I thought was a bit suspicious (even though lots of people must have thought of it, or experimentally rejected this already.) Note that I'm deep in amateur swamp, and I'll gleefully accept any "wow, you are confused" rejections.
Here is my suggestion:
What if the universes, that we live in, are not located specifically in configuration space, but in the volume stretched out between configuration space and the complex amplitude? So instead of talking about “the probability of winding up here in configuration space is high, because the corresponding amplitude is high”, we would say “the probability of winding up here is high, because there are a lot of universes here”. And here, would mean somewhere on the line between a point in configuration space and the complex amplitude for that point. (All these universes would be exactly equal.) And then we completely remove the Born rule. Of course someone thought of this, but responds: “But if we double the amplitude in theory, the line becomes twice as long, and there would be twice as many universes. But this is not what we observe in our experiments, when we double the amplitude, the probability of finding ourselves there multiplies by four!” This is true, if you study a line between the complex amplitude peak and a point in configuration space. But you are never supposed to study a point in configuration space, you are supposed to integrate over a volume in configuration space.
Calculating the volume between the complex amplitude “surface” and the configuration space, is not like taking all the squared amplitudes of all points of the configuration space and summing them up. The reason is that, when we traverse the space in one direction and the complex amplitude changes, the resulting volume “curves”, causing there to be more volume out near the edges (close to the amplitude peak) and less near the configuration space axis.
Take a look at the following image (meant to illustrate an "amplitude volume" for a single physical property): [http://www.wolframalpha.com] , type in: ParametricPlot3D {u Sin[t], u Cos[t], t / 5}, {t, 0, 15}, {u, 0, 1}
Imagine that we’d peer down from above, looking along the property axis. If we completely ignore what happens in the view direction, the volume (the blue areas) would have the shape of circles. If we’d double the amplitude, the volume from this perspective would be quadrupled.
But as it is, what happens along the property axis matters. The stretching out causes the volume to be less than the amplitude squared. It seems that, the higher the frequency is, the closer the volume is to have a square relationship with the amplitude, while as the frequency lowers, the volume approaches a linear relationship with the frequency. Studying the two extreme cases; with frequency 0 the geometric object would be just a straight plane, with an obvious linear relationship between amplitude and volume, while with an “infinite” frequency, the geometric object would become a cylinder, with a squared relationship between volume and amplitude. This means that the overall current amplitude-configuration-space ratio is important, but as far as I know, it is unknown to us.
In a laboratory environment, where all frequencies involved are relatively low, we would see systems evolving linearly. But when we observe the outcome of the systems, and entangle them with everything else, what suddenly matters is the volume of our combined wave which has a very very high frequency.
Or does it? At this point I'm beginning to lose track and the questions starts piling up.
What happens when multiple dimensions are mixed in? I’m guessing that high-frequency/high-amplitude still approaches a squared relationship from amplitude to volume, but I’m not at all certain.
What happens over time as the universe branches, does the amplitude constantly decrease while the length and frequencies remain the same? (Causing the relationship to dilute from squared to linear?)
Note that this suggestion also implies that there really exists one single configuration space / wave function that forms our reality.
So, what do you think?