Question regarding the Many-Worlds Formulation

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As Sean Carroll states here

If the particle can be in a superposition of two states, then so can the apparatus. So nothing stops us from writing down a state of the form

(spin is up ; apparatus says “up”)
+ (spin is down ; apparatus says “down”). (2)

The plus sign here is crucial. This is not a state representing one alternative or the other, as in the textbook view; it’s a superposition of both possibilities. In this kind of state, the spin of the particle is entangled with the readout of the apparatus.
...
Once our quantum superposition involves macroscopic systems with many degrees of freedom that become entangled with an even-larger environment, the different terms in that superposition proceed to evolve completely independently of each other. It is as if they have become distinct worlds — because they have.
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The next question would be, do multiple-world superpositions of the form written in (2) ever actually come into being? And the answer again is: yes, automatically, without any additional assumptions. It’s just the ordinary evolution of a quantum system according to [URL='http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation']Schrödinger’s equation
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superpositions "come into being" due to "the magic of decoherence".

He decides to measure "a spinning particle that can have spin-up or spin-down." He' measures spin-up and he'' measures spin-down. The day before his sister has sent him a letter which will arrive on the next day (after the measurement was performed).

Two possibilities:

The next day he' and he'' receive the letter. This seems to imply that sister' and sister'' have written the letter which however doesn't make sense as the splitting into different branches is future-oriented The letter was written before he decided to make the measurement.

The next day he' or he'' receives the letter. But this seems to make no sense either. The sister exists in his' and in his'' past. So, what's wrong here?[/url]

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andrewkirk
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I would take the first of the two, with the and. One way to think of this is that each object, including each person, is a branching process. When I refer at time t in world W1 to something object O did in my past, I am referring to the sub-process that consists of the part of O that starts with the part of O that is in W1 at time t, and traces process O from there backwards in time to when O commenced (the 'root' of O). Call that subprocess O(t,W1). If my counterpart in world W2 makes the analogous reference, they are referring to O(t,W2), which is the sub-process that traces from t,W2 back to the root of O.

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So, what's wrong here?
The wave function of the universe starts out in the state "the letter is in the mail and the particle is in a superposition of spin-up and spin-down". The measurement evolves this state into a superposition of "He measured spin-up yesterday and received a letter from his sister today" and "He measured spin-down yesterday and received a letter from his sister today".

He decides to measure "a spinning particle that can have spin-up or spin-down." He' measures spin-up and he'' measures spin-down. The day before his sister has sent him a letter which will arrive on the next day (after the measurement was performed).
Let us take the viewpoint of the letter. The recipient has a tendency to split. When the letter arrives to the recipient, then letter' learns that the recipient is recipient', and letter'' learns that the recipient is recipient''.

There is no reason for sister to become sister' and sister'', unless recipient' or recipient'' decide to send a reply. (I assume letters are the only way that information is transferred)

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Thanks for your answers. So you agree that both, he' and he'', get the letter of her sister.

There is no reason for sister to become sister' and sister'', unless recipient' or recipient'' decide to send a reply. (I assume letters are the only way that information is transferred)
Does this mean that the decision to reply creates a superposition. In this case the superposition (sister' get's his' letter') and (sister'' get's his'' letter''), correct?

According to the MWI macroscopic objects are in a superposition (looking classically though). Is the consequence then that not necessarily measuring a quantum object splits into two worlds?
If I am sitting in a restaurant and haven't chosen yet between fish and meat, am I in the superposition in the moment I took the decision: (I eat fish) and (I eat meat), which evolves into two non-interacting worlds?

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Does this mean that the decision to reply creates a superposition. In this case the superposition (sister' get's his' letter') and (sister'' get's his'' letter''), correct?
No, because there is some distance between the people that send letters to each other, and there is no spooky action at a distance in the MWI.

According to the MWI macroscopic objects are in a superposition (looking classically though). Is the consequence then that not necessarily measuring a quantum object splits into two worlds?
Reading a letter is a very large number of very fuzzy measurements of the states of the molecules of the letter. Positions of ink molecules get measured. And energies of said molecules get measured - if you can see any writing, then you know that not every molecule of ink has gained enough energy to evaporate.

If I am sitting in a restaurant and haven't chosen yet between fish and meet, am I in the superposition in the moment I took the decision: (I eat fish) and (I eat meet), which evolves into two non-interacting worlds?
In MWI "to make a decision to eat meat" refers to things that you do in order to increase the number of future yous that eat meat, and decrease the number of future yous that eat fish.

Actually the previous sentence is wrong. Writing "remember to eat meat" on the wall is not decision making, it's propping up a decision. I don't know how you, or anyone, makes decisions, so I can't say anything about it.

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No, because there is some distance between the people that send letters to each other, and there is no spooky action at a distance in the MWI.
Agreed, but then your conclusion "There is no reason for sister to become sister' and sister'', unless recipient' or recipient'' decide to send a reply." is unclear to me. Following that, the decision to send a reply is the the cause "for sister to become sister' and sister''. Could you explain what that means physically?

In MWI "to make a decision to eat meat" refers to things that you do in order to increase the number of future yous that eat meat, and decrease the number of future yous that eat fish.

Actually the previous sentence is wrong. Writing "remember to eat meat" on the wall is not decision making, it's propping up a decision. I don't know how you, or anyone, makes decisions, so I can't say anything about it.
As far as I can tell Everett and MWI refer to quantum objects. So may be speculating about 'taking decisions' in this context is rather speculative.

PeterDonis
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As far as I can tell Everett and MWI refer to quantum objects.
In the MWI, everything is a quantum object.

So may be speculating about 'taking decisions' in this context is rather speculative.
No more so than the MWI in general. According to the MWI, things that can "take decisions" are quantum objects, so everything they do, including taking decisions, can be described in quantum terms.

PeterDonis
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Does this mean that the decision to reply creates a superposition.
No. The superposition was already there; it was created when the spin measurement was made that entangled the states of the particle and "he" (the observer who measured the spin).

It really, really helps to write down the math instead of trying to reason about these things using ordinary language. Ordinary language is built with implicit assumptions that are violated if the MWI is true.

Here is a quick summary of the math; each line below represents a quantum state of the full "universe" (which here consists of the particle whose spin is measured, the observer who measures the spin, and the letter). I have not bothered with normalization factors, since they don't change any of the substance of what's being discussed. Subscripts on each of the factors indicate the subsystem ("P"article, "O"bserver, "L"etter, "R"eply); the symbols inside the kets indicate states of interest.

$$\left( \vert \text{up} \rangle + \vert \text{down} \rangle \right)_P \vert \text{ready} \rangle_O \vert \text{sent} \rangle_L \vert \text{not yet written} \rangle_R$$

$$\left( \vert \text{up} \rangle_P \vert \text{observed "up"} \rangle_O + \vert \text{down} \rangle_P \vert \text{observed "down"} \rangle_O \right) \vert \text{sent} \rangle_L \vert \text{not yet written} \rangle_R$$

$$\left( \vert \text{up} \rangle_P \vert \text{observed "up"} \rangle_O \vert \text{received by observer who observed "up"} \rangle_L + \\ \vert \text{down} \rangle_P \vert \text{observed "down"} \rangle_O \vert \text{received by observer who observed "down"} \rangle_L \right) \\ \vert \text{not yet written} \rangle_R$$

$$\left( \vert \text{up} \rangle_P \vert \text{observed "up"} \rangle_O \vert \text{received by observer who observed "up"} \rangle_L \\ \vert \text{written by observer who observed "up"} \rangle_R + \\ \vert \text{down} \rangle_P \vert \text{observed "down"} \rangle_O \vert \text{received by observer who observed "down"} \rangle_L \\ \vert \text{written by observer who observed "down"} \rangle_R \right)$$

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In the MWI, everything is a quantum object.
No more so than the MWI in general. According to the MWI, things that can "take decisions" are quantum objects, so everything they do, including taking decisions, can be described in quantum terms.
Thanks for clarifying.

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Thanks.
$$\left( \vert \text{up} \rangle + \vert \text{down} \rangle \right)_P \vert \text{ready} \rangle_O \vert \text{sent} \rangle_L \vert \text{not yet written} \rangle_R$$
Is the quantum state of the "universe" described correctly as

$$\left( \vert \text{up} \rangle + \vert \text{down} \rangle \right)_P \vert \text{ready} \rangle_O \vert \text{sent} \rangle_L \vert \text{not yet written} \rangle_R \vert \text{not yet received reply} \rangle_S$$

PeterDonis
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Is the quantum state of the "universe" described correctly as

$$\left( \vert \text{up} \rangle + \vert \text{down} \rangle \right)_P \vert \text{ready} \rangle_O \vert \text{sent} \rangle_L \vert \text{not yet written} \rangle_R \vert \text{not yet received reply} \rangle_S$$

Yes. And then the "S" subsystem would become entangled with the rest when the reply was received.

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Yes. And then the "S" subsystem would become entangled with the rest when the reply was received.

Agreed, but then your conclusion "There is no reason for sister to become sister' and sister'', unless recipient' or recipient'' decide to send a reply." is unclear to me.
Let me rephrase, without the unimportant word "decide": If the Brother' sends a letter to the Sister, then at some time the Sister becomes Sister'. Said time is between the time that the Sister starts opening the letter and the time when she is done reading the letter. (The envelope is such that no information can get out of a closed envelope)

Following that, the decision to send a reply is the the cause "for sister to become sister' and sister''. Could you explain what that means physically?
Let's consider the time when the letter is on it's way. And let's say that only the Brother that measured an up spin sends a letter.

We have a system consisting of a Sister and a Letter.

Said system can be represented by a point in a Hilbert space. The coordinates of the point are a description of the system. So, if I tell how those coordinates change as time passes, would I be explaining what physically happens?

Addition: Oh I have forgotten to say what information the Letter contains. Well it says: "The spin was up". And the guy that wrote the letter never lies. And it's not possible to forge the letter.

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PeterDonis
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I have forgotten to say what information the Letter contains. Well it says: "The spin was up".
That is not a complete description of the Hilbert space vector describing the state. It's only one term. For the complete description, see my post #9. If you stipulate that the brother only sends a letter if spin up is measured, there will still be another term in the superposition, where the "letter" subsystem remains in the "not sent" state instead of the "sent, contains information that spin was measured up" state.

That is not a complete description of the Hilbert space vector describing the state.
The state of what?

A letter is a message to Sister, and an absence of letter is a message to Sister too. So this thing we call "message to Sister" is in superposition of two states: A letter, and an absence of letter.

But I would say that letter is not in superposition of two states, not the letter that i am talking about when I say "letter".

Besides the not sent letter and the sent letter exist in different MWI universes. So I focused on one MWI universe ignoring the other one, isn't that a good approximation?

(I am aware of the fact that when the letter is on its way, the Sister has not yet been split between the two universes.)

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PeterDonis
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The state of what?
Of the system. Read my previous posts in response to @timdeeg.

A letter is a message to Sister, and an absence of letter is a message to Sister too. So this thing we call "message to Sister" is in superposition of two states: A letter, and an absence of letter.
Yes; but this "letter" subsystem is also entangled with other subsystems. It has no definite state on its own.

I would say that letter is not in superposition of two states, not the letter that i am talking about when I say "letter".
I have no idea what you are talking about here.

the not sent letter and the sent letter exist in different MWI universes.
Only if you take the different terms in the superpositions like those I wrote down in post #9 to be "different MWI universes". But that's not part of the MWI; it's just part of how you choose to describe the MWI in ordinary language. The MWI says there is one system with one quantum state, which happens to be an entangled state when you write it in terms of subsystems.

I focused on one MWI universe ignoring the other one, isn't that a good approximation?
Not for this discussion, since it is explicitly about the entire state of the system, not just one MWI "branch" or "universe" or whatever you want to call it.

Of the system. Read my previous posts in response to @timdeeg.
I don't understand.... So I'm wondering if the state of the thing moving towards the Sister could be a superposition of a letter from the brother that measured spin up, and a letter from the brother that measured spin down, where the amplitude of the latter is very close to zero, because the brother was very careful not to send any letter if he measures spin down?

Oh yes, when I say "the thing moving towards the Sister" I leave out some stuff not moving towards the Sister ... and that is not allowed. Well never mind then.

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PeterDonis
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the state of the thing moving towards the Sister
It's only moving towards Sister if the letter was sent. The subsystem described by the "S" subscript does not have a definite location in space or a definite speed. That subsystem, heuristically, is whatever degrees of freedom will end up being used to write and send the letter, if the letter is written and sent. If the letter is not written and sent, those degrees of freedom are still there; they just have something else happen to them.