There seems to be some confusing terminology in this post. Previously, amplitudes were complex numbers. But, in this post, Eliezer sometimes uses the word "amplitude" to refer to pairs of complex numbers.
He introduces the notation (x ; y) to denote a pair of complex numbers here:
We can represent the polarization of light as a complex amplitude for up-down plus a complex amplitude for left-right. So polarizations might be written as (1 ; 0) or (0 ; -i) or (√.5 ; √.5), with the units (up-down ; left-right).
So, here, an "amplitude" is still a single complex number.
But, later one, Eliezer writes things such as:
This will decohere the vector of (0 ; 1) into a transmission vector of (.5 ; .5) and an absorption amplitude of (-.5 ; .5).
(Emphasis added.) Here, the "amplitude" appears to be the pair (.5 ; .5) of complex numbers.
Does this explanation make sense? Basically, expressing a function as a linear combination of a set of orthogonal functions. A Fourier transform is one of the better known examples.
I not seeing how that resolves the problem. I understood an "amplitude" to be a coordinate of the distribution, understood as a vector, with respect to some orthonormal basis of distributions (functions). Since these distributions are vectors in a complex vector space, their coordinates should be complex numbers, not pairs of complex numbers.
Hmm, I think I understand your question now, sorry for the confusion. Photon state at a spacetime point is actually not a single amplitude, but a pair of them, owing to the classical notion of polarization.
We can represent the polarization of light as a complex amplitude for up-down plus a complex amplitude for left-right. So polarizations might be written as (1 ; 0) or (0 ; -i) or (√.5 ; √.5), with the units (up-down ; left-right).
So, here, an "amplitude" is still a single complex number.
No, it is already a pair of complex numbers, one for each polarization. You can write it as (z_left, z_right). What is commonly done is factoring out a pure phase (a complex number of unit magnitude) and ignoring it (it does not affect the probability). For example, the state (i/sqrt(2), i/sqrt(2)) is "the same" as (1/sqrt(2),1/sqrt(2)) and the state (0,i) is "the same" as (0,1). I put "the same" in quotes because once you have more than one photon their relative phase actually matters.
Consider the polarization (0 ; -i). Eliezer describes this polarization as "a complex amplitude for up-down plus a complex amplitude for left-right". I read him as saying that, in the polarization (0 ; -i), the "complex amplitude for up-down" is the single complex number 0, while the "complex amplitude for left-right" is the single complex number -i. Am I misreading him?
ETA: Perhaps this is his intended reading: You can write (0 ; -i) = 0*(1 ; 0) + (-i)*(0 ; 1). Here, the "complex amplitude for up-down" is the 2-vector 0*(1 ; 0), while the "complex amplitude for left-right" is the 2-vector (-i)*(0 ; 1).
On this reading, he would at least be using "amplitude" to mean a 2-vector throughout this post. But it would still conflict with the use of "amplitude" in all the previous posts, where it always meant a single complex number.
Perhaps this is his intended reading:
Yes, that's the one, and yes, it's different from the previous posts, but not in the way you think. In the previous posts there was a single complex number for every point in space, and its square modulus is probability density. In this case the infinitely dimensional space is replaced with a two-dimensional space (an analogy would be to replace the whole continuous space with just two points, "here" and "there"). Correspondingly, the probability density is replaced with probability at each point.
So, the photon polarization description is actually mathematically much simpler than the wave function in continuous space.
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 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 photon there with polarization left-right"?.
At any rate, I see now that points in the configuration space should correspond to basis vectors in the Hilbert space, so that what I was calling the configuration space wasn't consistent with what I was calling the Hilbert space. Though, my Hilbert space still seems right to me. More on that below.
Actually, this is more complicated than necessary, just the polarization states are enough.
Again, I don't understand how you can do without any location information. Doesn't there need to be different Hilbert basis vectors for having a photon with given polarization between the filters versus past the second filter?
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.
This looks analogous to the distinction between temperature and a temperature. Temperature (in some fixed system of units) is a real scalar field over space, assigning a real number to each point in space. A temperature, on the other hand, is one of the real numbers that is assigned to a point in space by the scalar field. I'm happy to think of "amplitude" as being a dual vector over the Hilbert space, while "an amplitude" is one of the complex numbers yielded by the dual vector when it is evaluated on a given state vector.
If amplitude is a dual vector, then that might resolve the terminological inconsistency I'd claimed. I'll have to think about whether I can make complete sense of Eliezer's post with that reading. Though, Eliezer said that an "amplitude" (x ; y) is usually written as a column vector, which makes me think that he was thinking of it as a vector, not a dual vector (which would normally be represented by a row vector).
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.
So, if we have a 1-dimensional space, and if we discretize it to three positions, shouldn't the Hilbert space contain one of those 2-dimensional components for each of those positions. I.e., shouldn't it be C² ⊕ C² ⊕ C², like I wrote before?
Yes, you can include the position information, it just wasn't necessary for understanding EY's point, the polarization substate of the full photon state is enough. He was talking about 3 disjoint polarization states, one after each polarizer. You can append the position state, but that is messy (seeing how photon is a massless particle, with 2 out of 4 spacetime dimensions suppressed) and does not add much to the point.
This looks analogous to the distinction between temperature and a temperature.
Yes, it's exactly like that, only with complex numbers.
You can treat the polarization state as a column vector. Its Hermitian conjugate would then be a row vector of complex conjugates of each component, such that their scalar product will be the square modulus of the vector (=1, since this is the probability of finding our photon in some polarization state).
You can append the position state, but that is messy (seeing how photon is a massless particle, with 2 out of 4 spacetime dimensions suppressed) and does not add much to the point.
Isn't it necessary to include position information if you actually want to see the vaunted "blobs of amplitude"?
Here is how I was thinking of the state vector's evolution. Let me know if this is getting the physics wrong. For simplicity, suppose that there exist only one photon and three positions: (1) in between the first filter and the second filter, (2) at the second filter, and (3) beyond the second filter. The first filter is at 0°, and the second filter is at θ = 30°.
My state vectors are vectors in the Hilbert space C² ⊕ C² ⊕ C². Within each C² component, I write coordinates with respect to the up-down left-right basis.
Thus, I can represent each state vector as a 6-dimensional column vector, where the first two coordinates correspond to the position between the filters, the second two coordinates correspond to the position at the second filter, and the last two coordinates correspond to the position beyond the second filter. I'll separate each pair of coordinates by a line-break for clarity.
In the initial state, at time t = 1, the photon has passed the first filter and has left-right polarization, but it hasn't yet reached the second filter. The state vector at t = 1 is thus:
0
1 <---- Initially only one blob of amplitude
0
0
0
0
The evolution rule giving my state vector at the next time step t = 2 is the following: zero out the first two coordinates (since, in classical language, the photon has moved on, either to collide with the second filter or to pass beyond it), fill the next two coordinates with the projection in C² of the vector (0 ; 1) onto (−cos θ ; sin θ), and fill the last two coordinates with the projection of (0 ; 1) onto (sin θ ; cos θ). In our case, θ = 30°, so we get the following state vector at time t = 2:
0
0 Initial blob has now decohered into two blobs:
−(√3)/4 \____ One blob of amplitude
1/4 /
(√3)/4 \____ Another blob of amplitude
3/4 /
The first blob is the "world" that sees the photon arrested at the second filter. The second blob is the "world" that sees the photon having moved beyond the second filter. The column vector above is a depiction of the amplitude blobs in configuration space that constitute the "worlds" of the Many Worlds Interpretation.
You can apply the Born rule to the 5th coordinate above to get the probability of observing the photon passed the second filter with up-down polarization: |(√3)/4|² = 3/16. Similarly, looking at the 6th coordinate, you expect to observe the photon passed the second filter and polarized left-right with probability |3/4|² = 9/16. Thus, the total probability of seeing the photon passed the second filter is 3/16 + 9/16 = 3/4.
So, that is how I read Eliezer's post. Is this model a reasonable simplification of the physical reality?
Today's post, Decoherence as Projection was originally published on 02 May 2008. A summary (taken from the LW wiki):
Since quantum evolution is linear and unitary, decoherence can be seen as projecting a wavefunction onto orthogonal subspaces. This can be neatly illustrated using polarized photons and the angle of the polarized sheet that will absorb or transmit them.
Discuss the post here (rather than in the comments to the original post).
This post is part of the Rerunning the Sequences series, where we'll be going through Eliezer Yudkowsky's old posts in order so that people who are interested can (re-)read and discuss them. The previous post was The Born Probabilities, and you can use the sequence_reruns tag or rss feed to follow the rest of the series.
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