I think that I may not have made my point clear.
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".
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. (For simplicity, I'm just considering the case where we have only two filters: the 0° one followed by another one with some inclination.)
A state vector will then be an element of a direct sum C² ⊕ C² ⊕ C², where C² is a 2-dimensional complex vector space. That is, we have one C² component for each of the three positions. This is our Hilbert space. If we fix a basis for each of the C² components, then we get an amplitude distribution over six coordinates.
My point has been the following: 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".
You also seemed to be using "amplitude" in two different senses in your earlier comment. On the one hand, you wrote:
Photon state at a spacetime point is actually not a single amplitude, but a pair of them, owing to the classical notion of polarization.
(Emphasis added.) Here, you seem to be calling each of the two coordinates in the polarization (x ; y) an amplitude. Each of these coordinates is a (complex) scalar, so you seem here to be using "amplitude" to mean a scalar.
On the other hand, that comment also contains the following:
So, here, an "amplitude" is still a single complex number.
No, it is already a pair of complex numbers, one for each polarization.
Here, I take it that the "it" in "it is already a pair of complex numbers" is an amplitude. So, here you're calling a vector an amplitude.
I'm used to thinking of scalars and vectors as very different kinds of mathematical objects (notwithstanding the fact that you can put a 1-dimensional vector-space structure on a field). For that reason, it struck me as confusing to use the same word, "amplitude", to refer to these two very different kinds of things, (1) scalars and (2) vectors in C² components.
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 neces...
Today's post, Decoherence as Projection was originally published on 02 May 2008. A summary (taken from the LW wiki):
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