# A Is the wavefunction subjective? How?

#### fluidistic

Gold Member
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.

Consider the example of the probability for a UK male of 25 years of age to die within the next year. Clearly, this probability is well defined and exists regardless of whether person A and person B agree about it. We can get a rather good estimate thanks to statistics, for example. But it seems that this logic does not apply anymore in QM. It seems like according to QM logic, that probability is subjective and only depends on who you ask. It is not a well defined number that exists regardless of the observer(s).

I am trying to convince myself that such a thing is possible, but I am unfruitful thus far. What would be the point then to write down the Schrödinger's equation for a system and solve for $|psi \rangle$ if I can come up with any other wavefunction and claim that it solves the same problem as described by another observer?

Does the subjectivity of the wavefunction in fact imply that the Schrödinger's equation is subjective? Because once the Schrödinger's equation is properly settled, then its solution follows.

I feel like I'm missing something in order to understand Lubos Motl and I feel like he's right. I have also glanced Wigner's friend Wikipedia's article and the QBism viewpoint. I do not want to deal with interpretations unless it is absolutely required.
From what I have read, extracting information out of a system is a subjective thing (though I do not understand how). Arnold Neumaier claims that this is done via an irreversible interaction though Lubos Motl claims that this isn't necessarily true and that irreversibility is also subjective (because even in QM everything is reversible though the probability to go one-way might be extremely small and the threshold is subjective), but I think this is besides the point.

So I am entirely confused about $|\psi \rangle$. Can someone shed some light?

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#### A. Neumaier

Science Advisor
Consider the example of the probability for a UK male of 25 years of age to die within the next year. Clearly, this probability is well defined and exists regardless of whether person A and person B agree about it.
This probability is ill-defined and changes with time. It was different in 1850 from what it is this year, and will again be different in 2050.
We can get a rather good estimate thanks to statistics, for example.
Different samples collected by A (data from 1950 to 2015) or B (data from 2000 to 2018) will give possibly quite different estimates. Even the same data give different estimates when evaluated with different statistical methodologies (frequentist or Bayesian, time-independent or time-dependent, linear or nonlinear models, etc.).

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#### stevendaryl

Staff Emeritus
Science Advisor
Consider the example of the probability for a UK male of 25 years of age to die within the next year. Clearly, this probability is well defined and exists regardless of whether person A and person B agree about it. We can get a rather good estimate thanks to statistics, for example. But it seems that this logic does not apply anymore in QM. It seems like according to QM logic, that probability is subjective and only depends on who you ask. It is not a well defined number that exists regardless of the observer(s).
I guess I have the opposite opinion about the subjectivity/objectivity of probability. I think that classical probability is always subjective. A particular male will either die this year or not. Perhaps if we had enough information about his situation, what's going on in his cells, we could say definitely whether he will die. But the statistics saying that x% of the population will die this year, averages over those details. For a specific male, we can come up with different probabilities depending on how much information we have about him. So it's subjective.

In contrast, it seems to me that wave functions are not subjective. Certainly, people can have different opinions about what the wave function is, but it seems that it can be objectively right or wrong. For a given wave function, there is a corresponding measurement that is guaranteed to give a particular result, if that's the correct wave function. If it gives anything else, that wave function was objectively wrong.

#### A. Neumaier

Science Advisor
A particular male will either die this year or not.
But the example was not about a particular male but about an arbitrary male from the population considered:
Consider the example of the probability for a UK male of 25 years of age to die within the next year.

#### fluidistic

Gold Member
I see that the male example was a bad one. Consider a 2 faced coin. The probability that is lands on either of its face is well defined, even though we do not know it a priori (it may be biased, etc.). In QM, it seems we cannot assert this. I.e. in QM the coin doesn't have a well defined probability to land on either face, and in fact this probability varies across observers.

#### A. Neumaier

Science Advisor
in QM the coin doesn't have a well defined probability to land on either face, and in fact this probability varies across observers.
Not really if the coin has the usual macroscopic size.

What would be the point then to write down the Schrödinger's equation for a system and solve for
$|\psi\rangle$ if I can come up with any other wavefunction and claim that it solves the same problem as described by another observer?
The subjective interpretations take the wave function at time 0 as subjective but the Hamiltonian and its dynamics as objective (though making here a difference is questionable). This allows them to obtain their subjective probabilities as computed by a scattering calculation with this Hamiltonian and the Schrödinger equation.

#### stevendaryl

Staff Emeritus
Science Advisor
I see that the male example was a bad one. Consider a 2 faced coin. The probability that is lands on either of its face is well defined, even though we do not know it a priori (it may be biased, etc.). In QM, it seems we cannot assert this. I.e. in QM the coin doesn't have a well defined probability to land on either face, and in fact this probability varies across observers.
As I said, I think that's backwards. Classical probabilities are subjective, but quantum wave functions are not.

Classical probabilities are consistent with the assumption that EVERYTHING is pre-determined, and that the appearance of probability simply reflects our ignorance about the state of the world. If you knew everything there is to know about the structure of the coin, and the way that the person flips coins, and the air currents, etc, you could predict with certainty what the result would be. The 50/50 probability comes from not knowing all those details.

And, as I said, quantum wave functions have an associated certainty to them. For each wave function there is a measurement that is guaranteed to give a specific result if that wave function is the true wave function, but may give a different result if that wave function is wrong.

• jordi

#### stevendaryl

Staff Emeritus
Science Advisor
But the example was not about a particular male but about an arbitrary male from the population considered:
But what does that mean? If some males are definitely going to die, and some males are definitely going to live, then what does the probability of an arbitrary male dying mean?

#### A. Neumaier

Science Advisor
But what does that mean? If some males are definitely going to die, and some males are definitely going to live, then what does the probability of an arbitrary male dying mean?
Well, in a finite sample space (here: UK males today 25 years of age) with uniform measure (as usually assumed in democratic statistics), this probability is precisely the relative frequency.
It is well-defined and deterministically known if you know the state of the universe (assumed classical) today and its precise dynamics.

Those who take a subjective view of classical probability mistake (in my opinion) probabilities estimated for a population from incomplete information (about the current state and/or the full dynamics) for the true probabilities. Only the estimated probabilities, not the true ones, are subjective since they depend on the way the estimation is done, which always depends on matters of choice that may differ between different scientists.

#### stevendaryl

Staff Emeritus
Science Advisor
Those who take a subjective view of classical probability mistake (in my opinion) probabilities estimated for a population from incomplete information (about the current state and/or the full dynamics) for the true probabilities. Only the estimated probabilities, not the true ones, are subjective since they depend on the way the estimation is done, which always depends on matters of choice that may differ between different scientists.
There is no reason to believe that there is such a thing as "true probabilities". It's certainly consistent to deny the existence of such a thing. In classical physics, anyway, there is no inherent nondeterminism, so probabilities are always due to lack of information about the details.

I would not say that relative frequency is the same thing as probability. One out of every 7 days is Sunday, but that doesn't mean that there is a 1/7 chance that today is Sunday.

#### A. Neumaier

Science Advisor
There is no reason to believe that there is such a thing as "true probabilities". It's certainly consistent to deny the existence of such a thing.
It is also consistent to deny everything. But this does not warrant the conclusion that there is no reason to believe in something.
In classical physics, anyway, there is no inherent nondeterminism, so probabilities are always due to lack of information about the details.
This does not follow. The probability that an arbitrary throw of perfect dice gives a 4 is still 1/6, objectively, and deterministically.
I would not say that relative frequency is the same thing as probability.
Not in general. But I had qualified my statement.

#### stevendaryl

Staff Emeritus
Science Advisor
It is also consistent to deny everything. But this does not warrant the conclusion that there is no reason to believe in something.
But there is no advantage in assuming (classically, anyway) that probabilities are objective.

#### A. Neumaier

Science Advisor
But there is no advantage in assuming (classically, anyway) that probabilities are objective.
There is. People working in insurance companies live from it.

To survive you need to have good approximations to the objective probabilities. This is not different in the classical and in the quantum case.

#### stevendaryl

Staff Emeritus
Science Advisor
But there is no advantage in assuming (classically, anyway) that probabilities are objective.
Saying that probabilities are relative frequencies doesn't really make sense. For one thing, relative frequencies are always rationals, with the denominator dependent on the number of trials performed. You can say that probabilities are the limit of relative frequencies as the number of trials goes to infinity, but there is no guarantee of that. All that you can say is that the probability that the relative frequency departs significantly from the probability goes to zero as the number of trials goes to infinity. But that's a circular definition of probabilities in terms of probabilities.

#### stevendaryl

Staff Emeritus
Science Advisor
There is. People working in insurance companies live from it.
No, decision-making is not affected by what your attitude is toward probabilities. If a bet has only a 1% chance of winning, then you won't take it unless the payoff is at least 100 to 1. Whether that 1% is subjective or objective doesn't make any difference.

#### A. Neumaier

Science Advisor
Saying that probabilities are relative frequencies doesn't really make sense.
You are arguing against a straw man. I wasn't saying that. I was saying:
in a finite sample space (here: UK males today 25 years of age) with uniform measure (as usually assumed in democratic statistics), this probability is precisely the relative frequency.
If a bet has only a 1% chance of winning, then you won't take it unless the payoff is at least 100 to 1. Whether that 1% is subjective or objective doesn't make any difference.
It does, because to survive you need to make bets that live up to their expectation - at least if the bets are about things that make a real difference. The other betters will soon be out of the game.

#### stevendaryl

Staff Emeritus
Science Advisor
It does, because to survive you need to make bets that live up to their expectation - at least if the bets are about things that make a real difference. The other betters will soon be out of the game.
You're saying that a gambler will be more successful if he believes in objective probabilities, than if he believes in subjective probabilities? That doesn't make any sense to me. They make the same bets.

#### A. Neumaier

Science Advisor
You're saying that a gambler will be more successful if he believes in objective probabilities, than if he believes in subjective probabilities? That doesn't make any sense to me. They make the same bets.
No. I am saying that a gambler will be more successful if his subjective probabilities closely match the objective probabilities than if his subjective probabilities are far off. The latter leads to very different bets.

• julcab12, PeroK and fluidistic

#### DarMM

Science Advisor
Gold Member
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. You can then interpret the convergence of different density matrices under observational data of an ensemble in an Objective or Subjective Bayesian manner. That leads one into the interpretation of probability theory.

Regardless the quantum case doesn't seem that different from the classical case in this regard.

• bhobba and dextercioby

#### Strilanc

Science Advisor
In the blog post, Lubos is pointing out (in different words) that the wavefunction along a lightcone originating from Alice at time t1 will be different from a wavefunction along the lightcone originating from Bob at time t2. They will use different numbers in a different order. That is correct. But there is an information preserving transformation between the two perspectives; they are just different representations of the same underlying objective thing.

In the same way that two observers in different reference frames may have different descriptions of the state of the same world, two observers may be storing two different wavefunctions that describe the same world. Assuming the observers don't discard necessary information (e.g. by forgetting the wavefunction they were storing before a measurement), there will be a transformation between their respective stored wavefunctions that shows they are equivalent (in the same way that Lorentz boosts show that different reference frames are equivalent). If this was not the case, they would be able to compare notes and find contradictions where e.g. one of them predicted A with 100% certainty and the other predicted not A with 100% certainty.

#### fluidistic

Gold Member
Ok, I think I start to understand. Thanks to the inherent randomness in QM, there is no "general psi" that everyone must agree on. Instead psi is subjective in that it can (but need not) represent what the observer knows about a system. As an example, the QSHM as seen by 2 observers that observed the system since different times. In order to get psi, they have to solve Schrodinger's equation, but their initial conditions might differ, and hence its solution too, thus psi. And they are both correct, despite psi differing.

Things can be made much more complicated when the 2 observers are dealing with different Hilbert spaces, etc.

In the end it is indeed quite different from the classical mechanics case where every one has to agree about the state of the system.

That was a shotgun to the mind!

#### DarMM

Science Advisor
Gold Member
In the end it is indeed quite different from the classical mechanics case where every one has to agree about the state of the system
Just to be clear, when I said classical case, I meant classical (Kolmogorov) probability theory rather than classical mechanics. The arguments for and against quantum states being subjective are basically exactly the same as those for regular old probability distributions.

#### atyy

Science Advisor
Quantum mechanics requires a designation of something that is the quantum system and something else which is the measurement apparatus. This is of course subjective. The quantum system is included in the wave function, and the measurement apparatus is not included in the wave function. In this sense, the wave function is subjective.

• DarMM and odietrich

#### A. Neumaier

Science Advisor
Quantum mechanics requires a designation of something that is the quantum system and something else which is the measurement apparatus. This is of course subjective. The quantum system is included in the wave function, and the measurement apparatus is not included in the wave function. In this sense, the wave function is subjective.
This is not the standard usage of the term 'subjective'. In this sense, the state in classical mechanics is also subjective, unless you always work with the state of the whole universe, which is unknown.

#### DarMM

Science Advisor
Gold Member
Quantum mechanics requires a designation of something that is the quantum system and something else which is the measurement apparatus. This is of course subjective. The quantum system is included in the wave function, and the measurement apparatus is not included in the wave function. In this sense, the wave function is subjective.
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?

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