# A Is the wavefunction subjective? How?

#### Fra

I have read Lubos Motl blogposts (https://motls.blogspot.com/2012/11/why-subjective-quantum-mechanics-allows.html and https://motls.blogspot.com/2019/03/occams-razor-and-unreality-of-wave.html) stating that the wavefunction is subjective. This means that it is perfectly valid that two different observers use two different wavefunctions to describe the same system. I do not understand how it makes any sense.
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I feel like I'm missing something in order to understand Lubos Motl and I feel like he's right
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So I am entirely confused about $|\psi \rangle$. Can someone shed some light?
I didnt read all the links but as I understand Lubos take on the nature of symmetries, I associate this to basically mean that the choice of observer (in as I envision Lubos thinking here) is thought of a "gauge choice"; and to have a specific information state you need to fix the gauge (observer). The objectivity rather lies in the equivalence class of observers. And psi is not an equivalence class, its gauge dependent.

One can make other comments on this view, ie. objection to reducing the observer to a gauge choice, but this has been discussed elsewhere in the BTSM section so i will not pull that up here. But if you ignore these objections the logic above is i think clear enough and makes perfect sense. Cases where it does not make sense are i think also edges of things where we are forced to BTSM discussions.

/Fredrik

#### atyy

If we're speaking very roughly (ignoring position eigenstate problems), let's say $x$ is a position on a detector screen and $\psi(x)$ is just the wavefunction giving the probability to be detected at a point on the screen.

Would it be a valid way to phrase Bohr's idea of the cut to say that by necessity $x$ has to be classical in order to have the notion of an outcome?

Obviously in classical mechanics we might measure a system with a form of "cut" in that we don't explicitly model our thermometers, meter sticks, etc and simply take them to produce values by interacting with the system under study. So we might be studying a meteorite which is modelled with variables $q_i$ moving under some equations of motion. Our telescopes then record values $A_i$ that we then use to construct $q_i$ etc

However the point of the cut in Quantum Mechanics is that by doing this you're actually treating the system and your devices very differently unlike in classical mechanics. For in QM the modelled systems don't have values like $q_i$, quantum states are very different things. However you have to still consider your devices as producing an $A_i$ in order to still have the notion of an experiment with outcomes.

Would this be accurate do you think?
I'm not sure about Bohr in the strict historical sense (only Bohr in the broad sense that the orthodox interpretation is Copenhagen in the broad sense), but yes, the cut means the measurement apparatus and the quantum system are modelled quite differently. It doesn't seem like we can the extend descriptor of the quantum system to include the measurement apparatus, unless we have yet another measurement apparatus to measure the measurement apparatus.

In classical physics, we can imagine the system as being in a pure state, and that it is only our ignorance that makes things uncertain. In quantum physics, even if the system is assigned a pure state, we can't imagine that the system is "really" in a pure state, unless we attempt something like many worlds or hidden variables.

#### Jehannum

If the wave function is subjective so that two different observers use two different wave functions to describe a system, then any predictions they make must be different - otherwise they're just equivalent wave functions. Two different predictions about the same event cannot both be correct (unless the observers fly off into two alternative realities where one is right and one is wrong - in which case prediction is futile anyway).

#### stevendaryl

Staff Emeritus
If the wave function is subjective so that two different observers use two different wave functions to describe a system, then any predictions they make must be different - otherwise they're just equivalent wave functions. Two different predictions about the same event cannot both be correct (unless the observers fly off into two alternative realities where one is right and one is wrong - in which case prediction is futile anyway).
I made that point earlier. If I believe that the wave function is $\psi$, then that implies an objective fact. I can come with an observable $\Pi_\psi$ that is guaranteed to give the result +1 if the measurement is performed on a system in state $\psi$. If the result is anything other than +1, that objectively proves that I was wrong to say that the wave function was $\psi$. So to me, that shows that there is something objective about the wave function, if you can be proved wrong about it.

#### Lord Jestocost

Gold Member
2018 Award
As remarked by Nick Herbert in “Quantum Reality: Beyond the New Physics”:

The separate images that we form of the quantum world (wave, particle, for example) from different experimental viewpoints do not combine into one comprehensive whole. There is no single image that corresponds to an electron. The quantum world is not made up of objects. As Heisenberg puts it, ‘Atoms are not things.’

This does not mean that the quantum world is subjective. The quantum world is as objective as our own: different people taking the same viewpoint see the same thing, but the quantum world is not made of objects (different viewpoints do not add up). The quantum world is objective but objectless.
” [Emphasis added by LJ]

#### DarMM

Gold Member
I made that point earlier. If I believe that the wave function is $\psi$, then that implies an objective fact. I can come with an observable $\Pi_\psi$ that is guaranteed to give the result +1 if the measurement is performed on a system in state $\psi$. If the result is anything other than +1, that objectively proves that I was wrong to say that the wave function was $\psi$. So to me, that shows that there is something objective about the wave function, if you can be proved wrong about it.
Just to be clear, what's the difference between this and a Bayesian prior $\rho$ with support on a set $A \subset \Omega$ with $\Omega$ the sample space? I could test the random variable $\chi_{A}$, the characteristic function of $A$, and $\rho$ is guaranteed to give $1$ as the response.

i.e. is this anything but Subjective vs Objective Bayesianism without any additional quantum nuances?

#### bhobba

Mentor
The quantum state has a DeFinetti's theorem and other associated subjective Bayesian results, so it's perfectly fine to think of it as subjective.
Exactly - either view - objective or subjective is valid. Have a look at Gleason's Theorem:
http://kiko.fysik.su.se/en/thesis/helena-master.pdf

It shows it exits (providing non-contextuality is assumed) but says nothing about if its just subjective or real.

Thanks
Bill

#### DarMM

Gold Member
Exactly - either view - objective or subjective is valid. Have a look at Gleason's Theorem:
http://kiko.fysik.su.se/en/thesis/helena-master.pdf

It shows it exits (providing non-contextuality is assumed) but says nothing about if its just subjective or real.

Thanks
Bill
I think you might like Cabello's work. It's a sort of weakening of the assumptions of Gleason's theorem. Although be warned heavy duty graph theory is involved.

#### DarMM

Gold Member
I should say, if you want the "Bird's eye view" of the theorem, a quick run down is as follows.

Gleason assumes two things. That observable quantities are related to each other in a specific form, i.e. the algebra of observables is a C*-algebra and also that the world is such that your probability assignments need not take note of the context within which you make measurements. As you known $P\left(\Pi\right)$, where $\Pi$ is a projector, is the same regardless what observable $A$ you measure to examine $\Pi$.

Cabello however only assumes that the algebra of observables has the weaker property that joint measurability of $A$ and $B$ implies there exists an experiment to measure $A$ that doesn't disturb $B$. This isn't quite as strong as assuming the whole C*-algebra structure, but can be shown to imply it if you want a noncontextual probability assignment.

#### stevendaryl

Staff Emeritus
Just to be clear, what's the difference between this and a Bayesian prior $\rho$ with support on a set $A \subset \Omega$ with $\Omega$ the sample space? I could test the random variable $\chi_{A}$, the characteristic function of $A$, and $\rho$ is guaranteed to give $1$ as the response.

i.e. is this anything but Subjective vs Objective Bayesianism without any additional quantum nuances?
Yes, in Bayesian probability, you can be proved objectively wrong if you give an assignment of 0 or 1 to some possibility. So in that sense, Bayesian probability has an objective element to it, which is what is possible and what is not. The exact numbers are subjective.

#### DarMM

Gold Member
Yes, in Bayesian probability, you can be proved objectively wrong if you give an assignment of 0 or 1 to some possibility. So in that sense, Bayesian probability has an objective element to it, which is what is possible and what is not. The exact numbers are subjective.
I think the analogy is fairly direct as the pure states of classical probability theory, the point masses, have exactly the property you mentioned, i.e. always having some observable that can determine if they are wrong. Even a large class of mixed states, i.e. mixed states whose support is not the entire sample space, have this property

#### stevendaryl

Staff Emeritus
I think the analogy is fairly direct as the pure states of classical probability theory, the point masses, have exactly the property you mentioned, i.e. always having some observable that can determine if they are wrong. Even a large class of mixed states, i.e. mixed states whose support is not the entire sample space, have this property
That's the reason I would say that quantum amplitudes are actually objective. They can always be proved wrong by a single measurement. (In contrast, other than 0 or 1 probabilities, no single observation can prove a Bayesian probability assignment wrong.)

#### DarMM

Gold Member
It shows it exits (providing non-contextuality is assumed) but says nothing about if its just subjective or real.
Do you think non-contextuality can be justified via no-signalling? If we made probability assignments that revealed the context and verified them we could know the settings of distant experiments.

#### DarMM

Gold Member
That's the reason I would say that quantum amplitudes are actually objective. They can always be proved wrong by a single measurement. (In contrast, other than 0 or 1 probabilities, no single observation can prove a Bayesian probability assignment wrong.)
In Bayesian probability all pure states and a large class of mixed states (those with support on a strict subset) can be proven wrong with a single observation.

#### stevendaryl

Staff Emeritus
In Bayesian probability all pure states and a large class of mixed states (those with support on a strict subset) can be proven wrong with a single observation.
Isn't that what I said? (Except I said it in the language of probability 0 and probability 1)

#### DarMM

Gold Member
Isn't that what I said? (Except I said it in the language of probability 0 and probability 1)
It's this part:
In contrast, other than 0 or 1 probabilities, no single observation can prove a Bayesian probability assignment wrong
I don't see how that is different from the quantum case or why it is a contrast.

#### stevendaryl

Staff Emeritus
I don't see how that is different from the quantum case or why it is a contrast.
Every quantum case corresponds to the perfect knowledge case of Bayesian probability, and the perfect knowledge case of Bayesian probability is objective.

#### DarMM

Gold Member
I get that, are you saying there is a difference in the quantum case or not?

All I mean is that there doesn't seem to be anything additional that quantum mechanics adds to the subjective/objective probability debate.

#### stevendaryl

Staff Emeritus
I get that, are you saying there is a difference in the quantum case or not?

All I mean is that there doesn't seem to be anything additional that quantum mechanics adds to the subjective/objective probability debate.
I really don't understand what's the difficulty. Bayesian probability becomes objective in the case where all probabilities are either 0 or 1. Quantum mechanics corresponds to this case. So it's objective.

#### DarMM

Gold Member
I really don't understand what's the difficulty. Bayesian probability becomes objective in the case where all probabilities are either 0 or 1. Quantum mechanics corresponds to this case. So it's objective.
There's no difficulty. I'm saying that I don't think QM adds anything to the subjective/objective probability debate, i.e. it doesn't have anything new to say about that issue compared to classical probability theory.

#### stevendaryl

Staff Emeritus
There's no difficulty. I'm saying that I don't think QM adds anything to the subjective/objective probability debate, i.e. it doesn't have anything new to say about that issue compared to classical probability theory.
And I'm saying that you're wrong. If in classical probability, you only allowed 0 or 1 values for the probability, then probability wouldn't be considered subjective, since disagreements could be objectively resolved. Quantum mechanics is in this situation: Disagreements about the value of the wave function can be objectively resolved.

#### DarMM

Gold Member
And I'm saying that you're wrong.
QM does add something to the debate then. What is it?

#### DarMM

Gold Member
And I'm saying that you're wrong. If in classical probability, you only allowed 0 or 1 values for the probability, then probability wouldn't be considered subjective
That's not classical probability theory though, that's Boolean logic.

Quantum mechanics is in this situation: Disagreements about the value of the wave function can be objectively resolved.
I don't get it, what is the feature QM has, mathematically, that classical probability lacks that adds something to the subjective/objective probability debate?

#### stevendaryl

Staff Emeritus
I don't get it, what is the feature QM has, mathematically, that classical probability lacks that adds something to the subjective/objective probability debate?
I really don't get what it is that you don't get. I've answered the question many times, and I guess to no avail. Quantum wavefunctions are objective, not subjective.

#### DarMM

Gold Member
Um. I answered that question several times. I believe that the quantum wave function is objective. If I'm right, then doesn't that resolve the subjective/objective question?
Well obviously if the wave function is objective then it resolves the question.

I'm asking what features are you using that indicate quantum states are objective and classical probability states are not.

Let's look at equivalent states:
1. Pure states. In both cases there are questions that definitively show you are correct or not

2. Somewhat Mixed states. These are states with strict subset support in the classical case and states like $\omega = \frac{1}{2}\left(\omega_1 + \omega_2\right)$ (with $d \geq 3$ of course) in the quantum case. It can be possible to establish one's mixed state is wrong in a single observation.

3. Highly mixed states. Probability measures with support on the whole sample space in the classical case, states like $\mathbb{I}$ in the quantum case. It is not possible to establish you are wrong in one measurement.
To me they seem the same.

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