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?
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.
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).
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:
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:
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?