About wavefunction collapse and explaining single outcomes in different interpretations

[olgerm]

Suppose one photon goes through an interferometer and then hits a screen. A device detects the hit position, digitizes it, and later a person Bob reads the output.

1. Before detection​

Initially the total wavefunction $\Psi$ describes:

• the photon moving toward the interferometer,
• the screen in its ready state,
• the detector electronics in their ready state,
• Bob in his ready state.

After passing through the interferometer, the photon part of the wavefunction is still coherent. It is a superposition of different path-components, and these path-components can interfere with each other.
So before detection, $\Psi$ describes one coherent quantum process.

2. After the photon lands on the screen​

When the photon interacts with the screen, the total wavefunction becomes approximately $\Psi \approx \sum_i(c_i*\Psi_i)$
where each component $\Psi_i$ corresponds to the photon having triggered the screen near position x_i.

At this stage, the different components of the wavefunction become entangled with many microscopic degrees of freedom of the screen and the environment. Because of this, the components $\Psi_i$ rapidly decohere.
So after the hit, the total wavefunction is a superposition of many decoherent branches, each branch corresponding to amplification beginning at a different screen location. (remember that in real world decorence is just approximate and these really do interact, but the interaction is too small to be mesured)

3. After the device has digitized the coordinate​

The detector electronics amplify the local event and convert it into a digital output. The total wavefunction now has the approximate form $\Psi \approx \sum_i (c_i \Psi_i^{(\text{device outputs }x_i)})$
where each branch corresponds to the device displaying a different coordinate $x_i$.
Because the branches are decohered, they no longer interfere with each other (remember that in real world decorence is just approximate and these really do interact, but the interaction is too small to be mesured). just like light from 2 different lights-bulbs can not form inteference pattern with eachother.

4. After Bob reads the device​

When Bob looks at the device, he becomes entangled with the detector state. The total wavefunction becomes approximately

$\Psi \approx \sum_i (c_i \Psi_i^{(\text{Bob sees }x_i)})$
Each branch now contains:

• a screen hit near $x_i$,
• a detector output equal to $x_i$,
• a Bob who sees and reports the coordinate $x_i$.

So the final total wavefunction contains many decoherent branches, each with a different definite outcome seen by Bob. (remember that in real world decorence is just approximate and these really do interact, but the interaction is too small to be mesured)
A Bob in one branch cannot access or communicate with the Bobs in the other branches, because the branches are decohered and evolve independently. (remember that in real world decorence is just approximate and these really do interact, but the interaction is too small to be mesured). For exmple Bob in branch where device output coordinate=0.3248198481 can not know about Bob in Branch where device outputs coordinate=0.6184932771 nor know how this Bob in this branch reacted. If Bob in some branch of $\Psi$ told his output to some other person then this person would also be in the same Branch as the Bob who told him this. there would be as many branches as there is possibles outputs of the digitalizing device. experiment could be made so that weight of any branch is smaller than sum of weight of other branches. This digitalizing device is similar to Schrödinger cat (makes entageled states to correspond to decoherent macroscopic states), but it enables many different outputs(depending on number of digits in digitalizing device) instead of boolean output(cat dead or alive).

$\Psi$ is one continuous function that takes system-configuration at given time as argument. system-configuration is list of elementary particle coordinates at give time or field-state at given time.And after decoherence it can be written approximately as a sum of almost non-overlapping components(branches). After interaction with the screen and environment, the total wavefunction can be approximately decomposed into a sum of decohered components(branches). Branch component is a continuous wave packet concentrated on a macroscopically distinct region of configuration space (or field-configuration space), corresponding to a different detector outcome.

Interesting question is wether Bob(or any other person in similar experiment setup that can be made in real life) would be councious or there is only some special branch only which there is counciosnes? Or would the Bob's subjective experiences were somehow more vivid in braches that have bigger amplitude? in everyday life the $\Psi$ all the time branches and there are more and more braches that do not remerge.

 

[PeterDonis]

the final total wavefunction contains many decoherent branches, each with a different definite outcome seen by a different Bob-branch.

According to an interpretation like the MWI, yes. But there are other interpretations that would not agree.

In everyday life the all the time branches and there are more and more braches that do not remerge.

Same comment here.

 

[olgerm]

According to an interpretation like the MWI, yes. But there are other interpretations that would not agree.

In standard quantum mechanics (without objective wavefunction collapses) were time-evolution of wavefunction is predicted only by Schrödinger-equation:
in MWI:
In each branch after that branch went to decoherent to branches with different coordinate output-values Bob in that branch says "after my branch went to decoherent to branches with different coordinate output-value I can not interact nor get information from other branches"

In Copenhagen interpretation:
In each branch after that branch went to decoherent to branches with different coordinate output-values Bob in that branch says "Wave function has collapsed! Now only my branch exist. After measurement, one definite outcome occurred, and the wavefunction collapsed to the state corresponding to my branch"

If decoherence is just approximate and some very-very small interactions between branches still exist then interpreting the situation as wavefunction collapse is approximation and Compenhagen interpretation's description of the situation is just an approximation.


There are also some modified QM theories with added objective collapse, where in addition of evolution of wavefunction predicted by Schrödinger-equation there are objective wavefunction collapses that sometimes remove some of branches and affect wavefuntion. where these things may be different.

 

[PeterDonis]

If decoherence is just approximate and some very-very small interactions between branches still exist then interpreting the situation as wavefunction collapse is approximation and Compenhagen interpretation's description of the situation is just an approximation.

No, this is not correct. An interpretation that claims single outcomes (which at least some versions of "Copenhagen" do--that term is pretty broad) claims single outcomes regardless of how decoherence is viewed. Decoherence by itself does not produce single outcomes, so the fact that it might only be approximate is not relevant to whether an interpretation claims single outcomes or not.

 

[olgerm]

No, this is not correct. An interpretation that claims single outcomes (which at least some versions of "Copenhagen" do--that term is pretty broad) claims single outcomes regardless of how decoherence is viewed. Decoherence by itself does not produce single outcomes, so the fact that it might only be approximate is not relevant to whether an interpretation claims single outcomes or not.

I modified my post now and tried to correct mistakes. you can tell me if it is correct now.

 

[PeterDonis]

In standard quantum mechanics (without objective wavefunction collapses) were time-evolution of wavefunction is predicted only by Schrödinger-equation

There is no such thing. In "standard quantum mechanics", time evolution is only predicted by the Schrodinger equation in between measurements. When a measurement is done, the change in the wave function induced by the measurement and its observed result is not described by the Schrodinger equation.

I modified my post now

Please don't do that once a post has been replied to. Make a new post instead with whatever corrections you think need to be made.

you can tell me if it is correct now.

It's not. See above.

 

[olgerm]

There is no such thing. In "standard quantum mechanics", time evolution is only predicted by the Schrodinger equation in between measurements. When a measurement is done, the change in the wave function induced by the measurement and its observed result is not described by the Schrodinger equation.

Do predictions of MWI and copenhagen-interpretation always agree? If yes then effects that are caused by interactions between approxiamately decohered branches that are in QM with MWI should also be predicted by QM with copenhagen-interpretation. So prediction of copenhagen-interpretation that ignores these interactions between branches is approximation. If MWI and copenhagen-interpretation do give different predictions then these are not different interpretations, but different theories. If these do not agree then there must be some objective collapse mechanism that changes the wavefunction $\Psi$. What does ths mechanism do percicely? Does it just randomly select 1 branch and overwrites oher branches with 0 when branches have gone sufficently decoerent?

 

[PeterDonis]

Do predictions of MWI and copenhagen-interpretation always agree?

Yes. But MWI and Copenhagen offer very, very different, and mutually inconsistent, accounts of why the predictions are what they are.

If yes then effects that are caused by interactions between approxiamately decohered branches that are in QM with MWI

There are no such interactions predicted by the MWI. The MWI says that once branches are decohered, they don't interact. That's how the MWI accounts for the fact that we observe single outcomes to measurements--because in each branch there is a single outcome, and the branches can't interfere with each other.

Note that your notion of "approximately decohered branches" is not, AFAIK, a part of decoherence theory as it is actually used by physicists. That's because in any practical sense, decoherence is irreversible. For example, decoherence includes low energy soft photons being sent off into space where they can never be retrieved.

 

[olgerm]

Note that your notion of "approximately decohered branches" is not, AFAIK, a part of decoherence theory as it is actually used by physicists. That's because in any practical sense, decoherence is irreversible.

I agree that in usual macroscopic measurements decoherence is effectively irreversible for all practical purposes. My point was formal(excact not about practical approximations) not practical. Since the global wavefunction $\Psi$ time-evolution is completely described by Schrödinger equation(no objective-collapses affect it) (it is clear in MWI and must be same in all iterpretations, that make same predictions), then $\Psi$ evolves continuously into a sum of components that become entangled with different states of the environment, and the interference between those components becomes extremely small. So branch autonomy is continoulsy increasing in time and approximate, not exact. Collapses may affect $\Psi$ suddenly in addition to normal time-evolution predicted by Scrödinger equation in approximate models where some of degrees of freedom(thing like mesurment devices) are not part of $\Psi$ , not treated as non-quantum environment.
Since the wavefunction evolves continuously, the transition from a coherent superposition to strongly decohered branches must also be continuous, not a sharp jump. So there is an intermediate regime where the components are only approximately decohered. In realistic experiments the residual coherence is negligible, but not literally zero in the exact formalism.

 

[PeterDonis]

Since the global wavefunction Ψ time-evolution is completely described by Schrödinger equation(no objective-collapses affect it)

This is only true in certain interpretations.

(it is clear in MWI and must be same in all iterpretations, that make same predictions)

This is not correct. As noted above, your statement is not true in all interpretations, but all interpretations still make the same predictions.

 

[PeterDonis]

there is an intermediate regime where the components are only approximately decohered

Do you have a reference in the decoherence literature for this?