I understand that the configuration space in this post isn't "a photon here, a photon there", but rather "a photon with this polarization here, a photon with that polarization there".
More like "photon with polarization up-down" and photon with polarization "left-right".
In this post, we can think of space as discretized to consist of, say, three positions: (1) in between the first filter and the second filter, (2) at the second filter, and (3) beyond the second filter.
Actually, this is more complicated than necessary, just the polarization states are enough.
What kind of mathematical object is "an amplitude"? It had always been a scalar, not a vector. But, in this post, Eliezer seems to be using "amplitude" to mean "a vector from one of the C² components of the Hilbert space".
I suppose I wasn't clear, either. Amplitude is a map from the Hilbert space to C. It is always a complex scalar, but potentially a different one at each point in the Hilbert space. When this (infinitely dimensional) space includes continuous position (call it x), we write the amplitude (wave function) as psi(x), and it is a map R^3->C. When we are talking about polarization of a single photon, the Hilbert space is 2 dimensional, so the map is {up-down, left-right} ->C. Because the polarization space is so small, we can write the whole function explicitly as {psi1, psi2}, instead of writing psi(p), where p ={up-down, left-right}. The amplitude is still a scalar at each of these two points, just like it is a scalar at each spacetime point.
I understand that the configuration space in this post isn't "a photon here, a photon there", but rather "a photon with this polarization here, a photon with that polarization there".
More like "photon with polarization up-down" and photon with polarization "left-right".
Okay, thanks. I think I'm starting to make some progress now. That makes more sense than what I wrote, though I'm not sure why you aren't including any location information. Why isn't it "a photon here with polarization up-down, and a pho...
Today's post, Decoherence as Projection was originally published on 02 May 2008. A summary (taken from the LW wiki):
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