Where does a quantum experiment *begin*?

[martinbn]

No. In MWI, particles do not have wave functions. In fact, particles do not exists at all in MWI. According to MWI, there is only a wave function and nothing but the wave function. Only one wave function, not many wave functions. However, evolution by Schrodinger equation is such that wave function often splits into branches, such that the overlap between the branches is very small. When the overlap is small, then each branch can approximately be thought of as an object by its own, not depending on the existence of other branches. In this case, each branch can approximately be thought of as a separate "world". That's what the world in MWI is.

What is the relation of this to experiments and observation? Can you phrase the example above in MWI? Also what happened to space and time? When you say that there is only a wave function and nothing else, it seems meaningless.

 

[Demystifier]

What is the relation of this to experiments and observation? Can you phrase the example above in MWI?

Can you be more specific, which example do you have in mind?

Also what happened to space and time? When you say that there is only a wave function and nothing else, it seems meaningless.

Well, like any other function in mathematics, the wave function is a map from a domain D to a codomain C. The space and time are in D. (I'm sure you will appreciate such a Bourbaki-like answer. )

 

[vanhees71]

Well, for a N-body system the wave function's domain is $\mathbb{R}^{6N+1}$ and $C=\mathbb{C}$.

 

[martinbn]

Can you be more specific, which example do you have in mind?

Well, like any other function in mathematics, the wave function is a map from a domain D to a codomain C. The space and time are in D. (I'm sure you will appreciate such a Bourbaki-like answer. )

The example from #103.

 

[Demystifier]

Well, for a N-body system the wave function's domain is $\mathbb{R}^{6N+1}$ and $C=\mathbb{C}$.

No, it's $\mathbb{R}^{3N+1}$.
And even $C=\mathbb{C}$ is not correct for systems with spin.
(I know you know that, I'm just splitting hairs. )

 

[martinbn]

(I know you know that, I'm just splitting hairs. )

But do you split worlds or not?

 

[Demystifier]

But do you split worlds or not?

Me? No. I'm just explaining what do world splitters do.

 

[vanhees71]

True. Sorry, but I still don't get why I should introduce "many worlds". While causally disjoint regions of GR spacetime are unavoidable, I don't need unobservable many worlds in QT to make physics out of the mathematical formalism. You have a plethora of examples in theoretical physics, where you have unobservable mathematical elements, e.g., the vector potential in classical electrodynamics. It's unobservable (and thus "unphysical") but simplifying the calculations, but I think we are splitting hairs indeed, and we are once more far from discussing relevant physics, which brings me again to my idea to split off the philosophical discussions of the qmech subforum.

 

[Demystifier]

While causally disjoint regions of GR spacetime are unavoidable, I don't need unobservable many worlds in QT to make physics out of the mathematical formalism.

But multiple wave-function branches with a negligible overlap are a prediction of Schrodinger evolution for systems with many degrees of freedom. They are unavoidable just as disjoint regions of GR spacetime are unavoidable. The only controversial part is whether all these mathematical branches are ontic or epistemic. In minimal interpretation they are epistemic. In MWI they are ontic. In Bohmian interpretation they are something in between. But as mathematical objects they are an unavoidable part of the theory in all interpretations.

 

[vanhees71]

I don't care whether it's ontic or epistemic. For me mathematical objects in physical theories are epistemic anyway. It's completely irrelevant for physics.

 

[Demystifier]

The example from #103.

That's in fact a standard example to describe the idea of MWI. I will assume that you are familiar with the bra-ket notation.

The measured system in the superposition can be written as
$$|\psi\rangle = |\psi_A\rangle + |\psi_B\rangle$$

But this cannot be a full description, because there is also a macroscopic measuring apparatus. The measuring apparatus can be in 3 different states:
$|\phi_0\rangle$ - the apparatus does not show any result (e.g. because it is turned off).
$|\phi_A\rangle$ - the apparatus shows that the "particle" is in region $A$.
$|\phi_B\rangle$ - the apparatus shows that the "particle" is in region $B$.
These 3 states are macroscopically distiniguishable so their overlaps are negligible, e.g.
$$\langle\phi_A|\phi_B\rangle \approx 0$$

Taking the measuring apparatus into account, the full state at the initial time $t_0$ is
$$|\Psi(t_0)\rangle = (|\psi_A\rangle + |\psi_B\rangle) |\phi_0\rangle $$
At this point there is no yet "world splitting". But at later time $t_1$ the state evolves into
$$|\Psi(t_1)\rangle = |\psi_A\rangle |\phi_A\rangle+ |\psi_B\rangle |\phi_B\rangle $$
The macroscopic branches $|\Psi_A\rangle =|\psi_A\rangle |\phi_A\rangle$ and $|\Psi_B\rangle =|\psi_B\rangle |\phi_B\rangle$ have a negligible overlap
$$\langle\Psi_A|\Psi_B\rangle \approx 0$$
so they can be thought of as separate "worlds". That's the essence of MWI.

 

[Demystifier]

I don't care whether it's ontic or epistemic. For me mathematical objects in physical theories are epistemic anyway. It's completely irrelevant for physics.

But then space behind the horizon in GR is epistemic, other worlds in MWI are epistemic, and non-measurable trajectories in BM are epistemic. By what general criteria is the first more "physical" than the second or the third?

 

[vanhees71]

Physical is what's observable.

 

[Demystifier]

Physical is what's observable.

So neither space behind cosmological horizon of GR nor other worlds of MWI are physical. Yet, somehow you feel that space behind horizon is more "acceptable" than other worlds of MWI. Still, you cannot explain why exactly do you feel so. Am I right?

 

[martinbn]

That's in fact a standard example to describe the idea of MWI. I will assume that you are familiar with the bra-ket notation.

The measured system in the superposition can be written as
$$|\psi\rangle = |\psi_A\rangle + |\psi_B\rangle$$

But this cannot be a full description, because there is also a macroscopic measuring apparatus. The measuring apparatus can be in 3 different states:
$|\phi_0\rangle$ - the apparatus does not show any result (e.g. because it is turned off).
$|\phi_A\rangle$ - the apparatus shows that the "particle" is in region $A$.
$|\phi_B\rangle$ - the apparatus shows that the "particle" is in region $B$.
These 3 states are macroscopically distiniguishable so their overlaps are negligible, e.g.
$$\langle\phi_A|\phi_B\rangle \approx 0$$

Taking the measuring apparatus into account, the full state at the initial time $t_0$ is
$$|\Psi(t_0)\rangle = (|\psi_A\rangle + |\psi_B\rangle) |\phi_0\rangle $$
At this point there is no yet "world splitting". But at later time $t_1$ the state evolves into
$$|\Psi(t_1)\rangle = |\psi_A\rangle |\phi_A\rangle+ |\psi_B\rangle |\phi_B\rangle $$
The macroscopic branches $|\Psi_A\rangle =|\psi_A\rangle |\phi_A\rangle$ and $|\Psi_B\rangle =|\psi_B\rangle |\phi_B\rangle$ have a negligible overlap
$$\langle\Psi_A|\Psi_B\rangle \approx 0$$
so they can be thought of as separate "worlds". That's the essence of MWI.

But this is just quantum mechanics. The interpretation is supposed to say what a world is and what it means for them to be thought as separate.

It seems that these are just words, which are useless and misleading.

 

[vanhees71]

No, I explained it. If I want to use GR I must live with the fact that it predicts unobservable parts of spacetime, because there are horizons. I only don't like purely speculative assumptions, made for some vague philosophical reasons, which don't help to describe the observable world.

 

[Demystifier]

But this is just quantum mechanics. The interpretation is supposed to say what a world is and what it means for them to be thought as separate.

It seems that these are just words, which are useless and misleading.

You are right that these are just "words", in the sense that their meaning is not defined precisely. Indeed, this is why MWI is a branch of philosophy of physics, and not a branch of mathematical physics. But I don't think that it makes it completely useless. Many important aspects of human life cannot be precisely defined (e.g. emotions or ethics), but any insight on them can be very useful.

A similar idea has been brilliantly expressed by (my avatar) Bertrand Russell:
"Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover."

 

[atyy]

We should clarify this point, because I think it's wrong what you claim. The collapse in the usual meaning of textbooks claims that the interaction of the measured system with the measurement device (a local interaction according to local relativistic QFT) acts instantaneously over the entire space, which is contradicting itself, because that would mean the interaction is not local. In the mathematical foundations, however it is built in that the interaction is local. So collapse cannot be part of relativistic local QFT.

Well, we always disagree - but collapse can be a part of relativistic QFT for either definition of collapse. I don't believe this is a matter of opinion.

 

[vanhees71]

But it's a contradiction to the very foundations! So how can it be acceptable at all?

 

[martinbn]

So neither space behind cosmological horizon of GR nor other worlds of MWI are physical. Yet, somehow you feel that space behind horizon is more "acceptable" than other worlds of MWI. Still, you cannot explain why exactly do you feel so. Am I right?

You are right that these are just "words", in the sense that their meaning is not defined precisely. Indeed, this is why MWI is a branch of philosophy of physics, and not a branch of mathematical physics. But I don't think that it makes it completely useless. Many important aspects of human life cannot be precisely defined (e.g. emotions or ethics), but any insight on them can be very useful.

A similar idea has been brilliantly expressed by (my avatar) Bertrand Russell:
"Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover."

When I said useless I meant useless because they don't carry any additional meaning. It seems that a world is a summand in a certain representation of the wave function, perhaps associated to a basis. And separate means that they are orthogonal (or nearly so). That is just substituting words.

I thought that each "world", when separate, comes with their own space and time, hence the choice of the word. Otherwise why not call it terms or something like that.

 

[Demystifier]

If I want to use GR I must live with the fact that it predicts unobservable parts of spacetime, because there are horizons.

If you want to use Schrodinger equation, you must live with the fact that it predicts unobservable parts of wave function, because there is decoherence. There is nothing speculative or vague about that.

 

[vanhees71]

What do you mean by "unobservable parts of wave function"? $|\psi|^2$ is the probability distribution for the position of the particle, and that's measurable by preparing an ensemble and detecting particles as a function of position. I think our discussion becomes weirder and weirder... :-(.

 

[martinbn]

If you want to use Schrodinger equation, you must live with the fact that it predicts unobservable parts of wave function, because there is decoherence. There is nothing speculative or vague about that.

Ay, there's the rub. The system is completely described by only part of the wave function, if the other parts are to be treated equally, then what do they describe? Alternatives of the system in other worlds (but somehow in the same space-time!). In GR you need the whole space-time.

 

[atyy]

But it's a contradiction to the very foundations! So how can it be acceptable at all?

Because the foundations are not about local physical interactions.

In QFT, the foundation is "no faster than light signalling". This is guaranteed by spacelike observables commuting. The "local interactions" of relativistic theories is one way of guaranteeing that spacelike observables continue to commute under time evolution.

In the minimal interpretation, there is nothing "physical" at all. So we are free if we want to add collapse as something physical (or non-physical), and there is no contradiction with relativistic QFT.

There is only a contradiction if one mistakenly think that the local interactions of relativistic QFT are "physical" interactions.

 

[Demystifier]

What do you mean by "unobservable parts of wave function"? $|\psi|^2$ is the probability distribution for the position of the particle, and that's measurable by preparing an ensemble and detecting particles as a function of position. I think our discussion becomes weirder and weirder... :-(.

I mean unobservable after the update of the probability distribution.

 

[vanhees71]

QFT, as it is formulated and used on the Standard Model, realizes the "no faster than light signalling" by the (probably) stronger assumption of local interactions, and I'm discussing this standard flavor of relativistic QFT. I'm not aware of any alternative successful formulation of relativistic QT. Are you denying that particles interact? Why then does the Standard Model work better than many HEP people like?

 

[vanhees71]

I mean unobservable after the update of the probability distribution.

Then I have a new probability distribution according to a new state-preparation procedure. So what?

 

[atyy]

QFT, as it is formulated and used on the Standard Model, realizes the "no faster than light signalling" by the (probably) stronger assumption of local interactions, and I'm discussing this standard flavor of relativistic QFT. I'm not aware of any alternative successful formulation of relativistic QT. Are you denying that particles interact? Why then does the Standard Model work better than many HEP people like?

The mathematics does not mean what you think it does! You are reading far more into the mathematics than allowed by the minimal interpretation and by Bell's theorem.

 

[Demystifier]

Then I have a new probability distribution according to a new state-preparation procedure. So what?

Likewise, I can "cut out" the space behind the horizon and get a new spacetime. So what?

In the spacetime case, the cutting is not described by Einstein equation. In the probability distribution case, the transition from the old to the new probability distribution is not described by the Schrodinger equation. So what?

 

[martinbn]

Likewise, I can "cut out" the space behind the horizon and get a new spacetime. So what?

In the spacetime case, the cutting is not described by Einstein equation. In the probability distribution case, the transition from the old to the new probability distribution is not described by the Schrodinger equation. So what?

But you can see galaxies disappear behind the horizon. If the there is no space-time there, it means that matter must vanish. It is not the same with the wave function. If a particle goes through a Stern-Gerlach apparatus, there is just one particle, it is found either up or down. And if it is found up, nothing went down. So, saying that the "world" in which it is down doesn't exist is very different from saying that nothing beyond the horizon exists.

 

[Demystifier]

When I said useless I meant useless because they don't carry any additional meaning.

The meaning is in the eyes of the beholder. (For instance, I know people who will say that category theory doesn't carry any additional meaning. Or those who will say that transfinite numbers don't carry any additional meaning.)

It seems that a world is a summand in a certain representation of the wave function, perhaps associated to a basis. And separate means that they are orthogonal (or nearly so). That is just substituting words.

Yes.

I thought that each "world", when separate, comes with their own space and time, hence the choice of the word. Otherwise why not call it terms or something like that.

You missed the crucial meta-physical assumption of MWI that there is only wave function and nothing but the wave function. If that assumption is correct, then all other objects we observe (tables, chairs, galaxies, space, time) must be a part of the wave function. The collection of all objects that we can observe is called "the world". But wave function contains separated parts which we cannot observe, so it makes sense to call them "other worlds".

 

[Demystifier]

But you can see galaxies disappear behind the horizon. If the there is no space-time there, it means that matter must vanish. It is not the same with the wave function. If a particle goes through a Stern-Gerlach apparatus, there is just one particle, it is found either up or down. And if it is found up, nothing went down. So, saying that the "world" in which it is down doesn't exist is very different from saying that nothing beyond the horizon exists.

Here you tacitly assume that the particle (just like galaxy) exists all the time, and has some position at any time, despite the fact that the position is not measured at all times. But this is a hidden-variable assumption, and Bell theorem implies that such an assumption implies action at a distance. One of the motivations for MWI is to avoid action at a distance.

 

[martinbn]

The meaning is in the eyes of the beholder. (For instance, I know people who will say that category theory doesn't carry any additional meaning. Or those who will say that transfinite numbers don't carry any additional meaning.)

That is very different. If you use a new word for something that already has a name, what new does this bring? Except, of course, possible confusion. You agreed that terms and orthogonal is fine, then worlds and separate is useless. That's what I meant by useless.

You missed the crucial meta-physical assumption of MWI that there is only wave function and nothing but the wave function. If that assumption is correct, then all other objects we observe (tables, chairs, galaxies, space, time) must be a part of the wave function. The collection of all objects that we can observe is called "the world". But wave function contains separated parts which we cannot observe, so it makes sense to call them "other worlds".

Ok, those other worlds, do they have things that could be observed (tables, chairs, galaxies, space, time)? It seems that the answer must be yes. So, do they come with their own space and time? Or are the space and time shared? Since there is just one wave function it must be that space and time are unique, they are the domain of the wave function. But in my world, where I can observe things, I have access to all of space and I observe things only in some places (say a particle whent up). The other worlds will have observers that observe other things elsewhere (a particle went down). Yet, it there is just one space where all this happens.

 

[martinbn]

Here you tacitly assume that the particle (just like galaxy) exists all the time, and has some position at any time, despite the fact that the position is not measured at all times. But this is a hidden-variable assumption, and Bell theorem implies that such an assumption implies action at a distance. One of the motivations for MWI is to avoid action at a distance.

No, I only assume that at one given moment of time(when detected) the particle can exists at one place only.

 

[rubi]

I think vanhees is right about the incompatibility of the collapse with relativity. However, it's not because of locality, but rather because of the incompatibility with the Poincare group. There is an interplay between time evolution, translations, rotations and boosts, which is encoded in the Poincare group relations and its Lie algebra. In quantum theory, compatibility with relativity is guaranteed by the use of unitary representations of the (centrally extended universal cover of the) Poincare group. If you claim that collapse is compatible with relativity, you must explain in what sense it is supposed to satisfy the Poincare group relations (representation theory won't work, since it is a non-linear operation) and then show that it actually satisfies them. I don't think anyone has done this and I don't see how it is supposed to work. For example, in the Poincare group, two time evolutions always commute, but projectors of non-commuting observables don't commute in general. How do you resolve this issue? Another example: There is a Poincare group element corresponding to a time translation followed by a Lorentz boost. What is the Poincare group element corresponding to a collapse followed by a boost?