[SEQ RERUN] On Being Decoherent

I think it is a good way to map what people have commonly called "superposition," but the sentence should probably be "The system is in the superposition STATE1 + STATE2, relative to STATE 3, where STATE 3 roughly factors out". STATE 3 in this case is usually an observer. I mean, if I flip a "quantum coin" and I have not told you if it is heads or tails, then the coin (and I) is in a superposition of "HEADS + TAILS" relative to you, but due to decoherence on my end, it is not in a superposition relative to me. For me this was an important concept to learn, as it helped me see that "many worlds" is a local and non-discrete phenomenon.

For all these types of experiments, how do they "aim" the particle so it hits its target from far away? It would seem that the experimenters would know pretty much where the particle is when it shoots out of the gun (or whatever), so would not the velocity be all over the place?

Only if they make the departing aperture small. A wider aperture allows the departing wave to be tight.

Does the polarization vector change as the photon moves along?

It depends which basis you look at it in. It is conventional to consider a photon's 'polarization' to be ploarization subspace that contains all of its time dependence. The phase then indicates the rest of its state. However, you can look at it other ways. A circularly polarized photon moving +z can be considered as a rapid shift between various orientations of +x and +y polarization... but it's simpler to just let it be in a circular polarization state and let the phase vary. A photon's state in this sense IS its 'main' wavefunction as you call it. There is no distinction. People usually shorthand think of a photon to have perfectly-defined momentum, but of course that would mean the photon extends through all of space. Real photons have multiple momentum components, and form a wavepacket or a static state. In particular, and very relevantly, you can construct electromagnetic field states (photons) that are inverse square laws - the static electrical field from a charge - and these have a very broad momentum distribution.

why can we put in the minus sign in the eqation that you say "we will need" later, instead of a + sign?

I can't find any minus signs in this post, but to take a stab in the dark at whatever it is you're referring to, subtraction is the special case of addition after one of a particular set of phase shifts.

Shminux, I trust you do know the actual answer to this, based on your demonstrated knowledge of QM. The essay does a qualitative job of this, here:

There are no plausible Feynman paths that end up with both LEFT and RIGHT sending amplitude to the same joint configuration. There would have to be a Feynman path from LEFT, and a Feynman path from RIGHT, in which all the quadrillions of differentiated particles ended up in the same places. So the amplitude flows from LEFT and RIGHT don't intersect, and don't interfere.

In order for the joint observer-observed system to be coherent, the two cases need to be reconcilable.

How is this a magical password? He pulls out the guts of decoherence and shows them to the reader!

If there are two nearly identical copies of me in the same place, why is there no further interaction between them

Your two copies differ by states of many neurons, that's billions of particles. They are not "nearly identical".

It is tempting to think about "one different thought" or "one different perception" as very small changes. But on particle level those are huge changes. A small change on a particle level is something you can't notice, and therefore you can't notice as those copies of you interact... and when the small change becomes big enough, your copies are already decoherent.

It seems like the relevant section is

By hypothesis, Sensor-LEFT is a different state from Sensor-RIGHT—otherwise it wouldn't be a very sensitive Sensor. So the final state doesn't factorize any further; it's entangled.

But this entanglement is not likely to manifest in difficulties of calculation. Suppose the Sensor has a little LCD screen that's flashing "LEFT" or "RIGHT". This may seem like a relatively small difference to a human, but it involves avogadros of particles—photons, electrons, entire molecules—occupying different positions.

So, since the states Sensor-LEFT and Sensor-RIGHT are widely separated in the configuration space, the volumes (Sensor-LEFT Atom-LEFT) and (Sensor-RIGHT Atom-RIGHT) are even more widely separated.

The question still left in my mind is what is meant by "widely separated", and why states that are widely separated have volumes that are widely separated.

For example, take a chaotic system evolving from an initial state. (Perhaps an energetic particle in a varying potential field.) After evolving, the probability that was concentrated at that initial state flows out to encompass a rather large region of configuration space. Presumably in this case the end states can be widely separated, but the probability volumes are not.

There is a (very brief) account in that post of what decoherence is and why it leads to non-interaction. There is a much more extensive discussion of the point in the previous (linked) post on decoherence.

You do realise that the one-paragraph summary here is only a one-paragraph summary, and that there's a lot more in the original post, yes?