The measurement problem

Does the wave function describe reality or only our knowledge of it? Three fundamental problems—the result, the statistics, and the effect—reveal that the issue of measurement is not just a philosophical question.

Countless articles have been written about the measurement problem. Covering them all in one post is an impossible task. I won't pretend to detail everything here, but I thought I'd write about it in order to explain why some of us believe it is indeed a real problem and why some of us think it is just an illusion. The confusion was illustrated by Nobel Prize winner in Physics, Richard Feynman: "We have always had (it's a secret, a secret, close the door!), we have always had the difficulty of explaining the world through quantum mechanics. At least I do. I couldn't come to terms with the fact that this issue is clear to me. Well, I still feel pressured by it..., you know how it is, in any new field, it takes a generation or two to realize that there is really no problem. It's still not clear to me that there is really no problem. I can't define the problem, so I suspect that there is really no problem, but I can't know that there is really no problem."

The quote illuminates the general feeling among the scientific community. On the one hand, philosophers clearly argue that there is a problem with how measurement is defined in quantum mechanics, on the other hand, when you delve deeper into it, it is not clear what the root of the problem is.

In this article I will focus Tim Mauldin's approach (philosopher of science) who summarized the measurement problem in three chapters, but before I detail the problems, I will begin with the Copenhagen interpretation of quantum mechanics that is taught in basic courses at universities around the world. This interpretation assumes that every physical system — from a single particle to the sun, from a single person to the entire universe — is associated with a state vector, or as it is called by many, a wave function. This function carries all the information necessary to describe the physical state of the system. Unlike classical physics, in which a state is determined by deterministic quantities such as well-defined position and momentum, a particle can be in a mixed state – “a little here and a little there.” The new quantum state mixes several classical states together. At the same time, in everyday life we ​​do not see a particle “a little everywhere.” This statement is quite vague, after all, the position of a particle cannot be two at the same time. So what is the meaning of the wave function? The mixing of classical states refers to the probability of an observer measuring, that is, seeing in one way or another, the particle at one of the locations. After the measurement, for a moment, the position of the particle is well defined. After the measurement, there is no longer really a probability that a particle will be in several places, since it is clear that after the measurement, the position of the particle is unique. Therefore, all probabilities collapse to zero except for the one belonging to the particle's true position, which becomes 100%.

Here lies the measurement problem – Schrödinger showed that the wave function varies in time according to a relatively simple, essentially linear equation, and no linear equation could cause such a dramatic collapse. So what are the real dynamics of quantum mechanics? If we are all objects made of particles, and the only dynamics are linear, collapse cannot occur.

First problem – the problem of the result

In more concise terms, the following three arguments cannot exist simultaneously:

1. The wave function describes the reality of the physical system in its entirety. There is no additional information hidden from our view.
2. The wave function varies in time with respect to a linear equation, always. (e.g. Schrödinger equation).
3. Measuring a certain physical quantity will always lead to a deterministic result. For example, if we measure the spin of an electron, it will only be in one of two possibilities – up or down.

We can demonstrate the contradiction between the arguments using the mathematics of quantum mechanics. We will focus on an electron with a spin that we want to measure on the z-axis. Without going into the nature of spin, let's assume that spin has two classical states, an "up" state - which we will denote with an arrow pointing up, and a "down" state - which we will denote with an arrow pointing down. The entire physical system contains not only the spin, but also the measuring device. Therefore, we can describe the system and the spin together before and after the measurement as follows:

Here we have marked the state of the electron with the letter e at the bottom, and the state of the measurement system (device) with the letter d. The word ready marks the system before the measurement, after which the measurement device changes its state according to the spin of the electron. We have chosen to place the spin on the z-axis because our measurement device is able to measure the spin on this axis. If the spin is in the up state, we have marked it with up and if it is down, we have marked it with down.

But as we mentioned, a quantum state can mix several classical states. For example, if the spin is oriented along the x-axis, its quantum state will be described as a superposition of z-states:

And the entire system will start in the following quantum state:

If the second argument is correct, the situation should evolve according to a linear equation, so a measurement would lead to the wave function changing like this-

This raises the question, what situation does it represent?

If the first argument is true, then the wave function should represent all the physical possibilities of the system. But then the third argument cannot hold because here the result is not uniquely deterministic. The Copenhagen interpretation states that in this case the chance of measuring the particle with spin up is 50% and similarly for spin down. In other words the third argument cannot hold in the case where the evolution is linear and when the entire physical state is defined by the wave function.

These contradictions have led researchers to propose several solutions.

Giving up the first argument, namely that the wave function describes reality in its entirety, leads to the existence of "hidden variables." These are physical quantities that we cannot measure directly, but can affect the results of the measurement.

Giving up the second argument leads to the collapse theory. This theory assumes that a measurement collapses the wave function into one of the classical possibilities. For this to happen, the time evolution must be nonlinear, at least in part.

Theories that preserve the first and second arguments but not the third are less well-known. The most popular of these is Everett's many-worlds theory. The theory states that when measuring, there is no collapse, simply that all possibilities exist simultaneously, each in a different universe.

In any case, to resolve the conflict we must add new physics, the question is which one? Sometimes the measurement problem is seen as a purely philosophical problem, but I argue that philosophizing is not enough. It is a deeper problem that requires a new physical principle. Every idea I mentioned is backed by extensive mathematics, not just a literal interpretation. Every revision of the Torah raises new questions and sometimes interesting predictions, that is how science works. At the same time, the interpretations are not immune to criticism, and sometimes they are not consistent with all observations. For example, Bohm's interpretation that adds hidden variables does not fit well with relativistic physics.

Second problem – the problem of statistics

It would seem that we can reconcile the arguments presented above if we decide that the mixed states show a single result on the device's display. For example, we could argue that such a mixed state

The measuring device will show the result "up" as long as beta is greater than alpha, and "down" in the opposite case. The reason we can do this is because we have not necessarily directly linked the wave function to the measurement result. It is true that the quantum state of the measuring device is defined as "up" or "down", but we can treat "state" as a more fluid quantity that does not necessarily dictate the measurement result. The problem with our new formulation is that according to Born's rule, which has been confirmed experimentally, there is still the possibility that we will get both options because the probability that the display will show up is proportional to the beta coefficient and for a spin down the probability is proportional to the alpha coefficient.

Therefore, we conclude that the following three arguments cannot exist together:

1. The wave function perfectly describes reality.
2. The wave function always varies in time according to a linear deterministic equation (the Schrödinger equation).
3. When many copies of the same wave function are measured, a good approximation of the Born principle is obtained.

If we have two identical wave functions, according to Rule 2, they must evolve in time in the same way. So after some seconds the wave functions must remain the same. If they completely describe reality (Rule 1) they cannot describe different measurements, but this is contrary to Rule 3.

This contradiction can be resolved, for example, with hidden variables, but these are not really hidden, otherwise they would have no effect at all. In some way, they have to be generated in the measurement, otherwise we would not be able to get two different results from the same wave function. Those who believe in nonlinear evolution claim that it is itself non-deterministic. In other words, two wave functions that are initialized identically will not look the same after a while. On the other hand, those who deny the rule of 3, that is, those who believe in many worlds, may be perceived as denying empirical results. At the same time, they will say that in fact no measurement occurs at all and all possibilities exist simultaneously.

Third problem – the effect problem

The third problem arises from subsequent measurements. If the electron started in a mixed state, Born's interpretation claims that after the measurement it will collapse to one of the deterministic states - up or down. The collapse actually changes the wave function and in subsequent measurements the spin will remain in the same direction because all other probabilities have collapsed to zero. This is the "effect problem", a phenomenon that preserves the information of the first measurement even in future measurements. Those who favor nonlinear dynamics do not encounter the effect problem. The dynamics preserves the state after it has collapsed. On the other hand, in theories in which there is no collapse like the one formulated by Bohm (hidden variables) the problem can arise. If there is no collapse, how does the system preserve the result of the previous measurement? The solution that Bohm presented to the effect problem is hidden in those hidden variables that preserve the information in their classical behavior. Without going into the range of interpretations that exist in the literature, I will point out that not all of them meet this criterion, and without it, in my opinion, the interpretation has no justification.

So, what is the real interpretation?

recently A survey was published. In a Nature magazine poll of physicists from around the world, the Copenhagen interpretation won 36% of the vote, with only 15% voting for many-worlds and 7% for Bohm. Incidentally, 17% voted for the epistemic approach that the wave function depends on the observer and is just a mathematical description of what might happen after the measurement.

When asked what the wave function was, 47% claimed that it had no actual representation in nature, 19% claimed that it was only a partial description of reality, 17% believed that it was a complete description of reality, 8% claimed that it described a subjective effect of the experimenter, and 8% had no idea.

When asked whether an observer is required for measurement, 9% answered yes, and he must be conscious, 56% answered yes, but he does not have to be conscious, interaction with the environment can also be considered a measurement, 28% answered no, and 8% answered that they did not know.

This survey is remarkable in itself. Only a third of physicists agree with the Copenhagen interpretation. It's a slap in the face to the establishment that has taught generations of physicists on the basis of this assumption. The fact that there is no general agreement proves to us all that quantum mechanics is still not clear to us, even 100 years later. At the same time, and perhaps most surprisingly of all, this has not stopped scientists from discovering new phenomena, investigating the most fundamental structures, and deepening our knowledge of the world. But I ask, will we ever reach saturation, and will new doors only open if we settle this deep problem?

You are welcome to read more articles on my blog. "Why the quanta?"Do you have a question? Want me to write about a topic that interests you? Contact me at the email address: noamphysics@gmail.com
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More of the topic in Hayadan:

• So what do you do there at the university? Chapter 4: Does not stop at yellow - about non-linear optics
• "Students and graduates of computer science must convert to quantum programming, because this is the future"
• Quantum theory affects the process of light emission
• Does gravity solve the measurement problem?
• The negated connection between quanta and consciousness