Turning the measurement problem on its head
TL;DR: Showing that quantum time evolution is equivalent to continuous measurements is much simpler than showing that a measurement is an instance of time evolution
One of the open problems in the foundations of quantum mechanics is the so-called measurement problem. Since, if you look at the details, it doesn’t seem that people completely agree on what the problem is, I’ll characterize it in this way: quantum mechanics gives us two laws of evolution, one for physical processes (i.e. unitary time evolution/Schroedinger equation) and one for measurements (i.e. the Born rule/the collapse of the wave-function). In the end, measurements are physical processes, and so one would expect to be able to model a measurement using time evolution. This can’t be done. Unitary time evolution is deterministic and reversible, in the sense that given one initial state, we get one and only one final state. Measurements are non-deterministic, in the sense that given an initial state, we get multiple possible outcomes, each predicted with a given probability. Many interpretations are there to explain how this is possible.
In the Assumptions of Physics project, particularly in Reverse Physics, we like trying the exact opposite of what people normally do… so, what if, instead of saying that a measurement is a physical process, we say a physical process is a series of measurements, one for each instant in time? That is, the measurement process is the truly fundamental process, while unitary evolution applies only in a particular subcase. If we do that, we find a resolution that is minimal, necessary and interpretation independent. Let’s see how.
To warm up, suppose we prepare an electron with spin up, we make it go through a horizontal polarizer and we measure spin in the horizontal direction. We will get 50% chance of measuring spin left or spin right. Now imagine that we put another polarizer in the middle, rotated 45 degrees. Because the Born rule depends on the angle, \(p = \cos^2(\theta / 2)\), after the first step there will be a greater chance to end up in the upward-right direction. After two 45 degree steps we will see that we have more than 50% chance to end up measuring spin right. We can do that again: put two other polarizers in the middle, with a 22.5 degree angle between each pair. The chance to end up measuring spin right will be again greater. We can imagine we keep putting polarizers, to the point that we have an infinite sequence. In this case, we will be certain to measure spin right: we can get certainty through a series of infinitesimally small projections.
This is typically more counterintuitive for those that have studied quantum mechanics than for those that didn’t, who will just accept the final fact. The puzzle is: how can this work mathematically? Shouldn’t it be that every small variation leaks a small amount of probability? This is not the case because the probability is the square of the inner product. For those that studied linear algebra, unitary evolution is mathematically a rotation. An infinitesimal rotation is, in a sense, like multiplying the original vector by \(1 - \imath \epsilon\), where \(\imath\) is the imaginary unit and \(\epsilon\) is a small number. To calculate the probability, we multiply that number by its complex conjugate, so we have \((1 - \imath \epsilon)(1 + \imath \epsilon)\) which is equal to \(1 + \epsilon^2\). The variation in the probability, then, is second order and can be disregarded. So collapsing the wave-function to an infinitesimally close state is a deterministic and reversible process.
In other words, continuity, in quantum mechanics, is not just small variations, but deterministic and reversible variations.
If we imagine the evolution of an electron in a magnetic field, the spin will rotate much like if there were polarizers in its path. Therefore, we can say that the magnetic field keeps ``measuring’’ the direction of spin. What happens is that the direction of measurement depends on both the magnetic field and the direction of spin of the electron itself. That is, it is not something imposed externally, as is the case when an external agent performs a measurement.
The overall idea, then, is that the projection gives us an infinitesimal black-box time evolution. That is the basic building block. And then, if we assume a quasi-static, deterministic and reversible process, the combination of all those infinitesimal time evolutions become standard time evolution, the Schroedinger equation. From this point of view, there is only one type of process, so it resolves the measurement problem, though not how people typically want to resolve it. It is a minimal resolution because we are adding nothing. It is a necessary resolution because the above thought experiment of infinitely many polarizers is a result of quantum mechanics, so one must concede that. It is interpretation independent because it does not say anything about what happens during the measurement. One can still say that during a measurement the universe divides, consciousness interferes and hidden variables come into play… he just has to grant that that happens, to an infinitesimally small degree, during unitary time evolution as well.