Paper Review: Unified dynamics for microscopic and macroscopic systems

In the wake of the quantum measurement problem there is a constellation of competing theories of quantum mechanics. Any adequate solution to the measurement problem must make predictions that agree with the observed quantum statistics. In particular, the theory must explain why small particles exhibit counter-intuitive behaviour and why this seems to go away at the macroscopic scale of our everyday experience.

I’ve written on a few different theories of quantum mechanics before. One I haven’t yet written about is GRW, proposed in the mid 80s by Ghirardi, Rimini, and Weber (hence GRW). Interestingly enough, even though Bell was a major fan and proponent of Bohmian mechanics, later in his career he started considering GRW-style theories quite seriously.

***The original paper can be found here.***

The paper itself is quite long and detailed, as one would expect from a sophisticated proposal for a new theory of quantum mechanics. Thus what I will do for this post is explain theory in a more intuitive way, putting aside most of the mathematics and detail of the original paper.

Recall from the previous post that we can think of the quantum measurement problem as arising from the disagreement between the two dynamical laws of the standard theory. Stated fairly informally:

  1. When a measurement occurs, the system being measured collapses probabilistically and instantly into a state such that there is a definite measurement outcome.
  2. When no measurement occurs, the system evolves deterministically and linearly according to the energy properties of the system.

The details of the dynamics don’t really matter here too much. The key idea is that one applies when a measurement occurs, and one occurs otherwise. The two dynamics seem to track the following: we notice that whenever we measure the state of a physical system, we always find that it has a definite property. That is, even though the standard theory predicts that physical systems can at times lack the property we are trying to measure, for example position, whenever we actually measure that property we always find that the particle has one.

Any adequate theory of quantum phenomena must explain why this is the case. The standard theory fails to do this adequately because the term “measurement” appears in the theory as an undefined primitive term. The term is too vague to play a key role in a foundation physical theory, and it invites an odd kind of dualism.

Let’s see how GRW tries to solve the measurement problem. Remember, in broad strokes, we seem to get one type of behaviour when we observe small systems, and another when we observe big systems. Understanding this makes it easier to follow what the GRW proposal is.

Another thing we have to understand about quantum mechanics is the phenomenon of entanglement. It sounds more mystical and spooky than it is, but it definitely still is a little weird. I’ll just try to provide a quick and dirty intuition here; for a better understanding of entanglement it really helps to understand the mathematics. Once we get entanglement down we are ready to get a sense of the GRW theory and how it tried to account for the standard quantum statistics.

Consider two physical systems, maybe two electrons. Electrons can have a property known as spin. For this example I’ll provide a very cursory discussion of spin. In general, along one axis, for example the x-axis, a particle could be spin-up or spin-down. It is just a physical property.

If we were thinking about this property classically, without quantum entanglement, each particle would have its spin. The two particles might be correlated because of a past event—for example I might have prepared one so it is up and one so it is down but you don’t know which one is which. When you measure the spin of one and get up, you know the spin of the other because of this correlation. However, each particle in fact has its own spin even while correlated. This is what can happen classically.

However, in quantum mechanics, the spin of two particles can be correlated in a way such that it is impossible to describe the spin of a single particle. According to the theory, the single particle doesn’t even have a spin state of its own. When this happens we say that the two systems are entangled.

Classically, if you measure the state of the particles with a correlated spin (again, maybe because I correlated the spin in some way), you know the spin of the other because of this correlation. However, you are just learning what the spin already was. In the entanglement case, neither particle has a spin until you measure the spin of one of the particles (according to the standard theory). Once you measure the spin of one, the collapse dynamics kicks in since a measurement is made, and both particles collapse into a state in which they have a definite spin. This happens even if the entangled particles are physically separated by a large distance. This kind of phenomena is what Einstein called “spooky action at a distance”. The collapse of the wave function happens instantaneously, however great the distance between the particles.

Okay. Now that we have built a bit of an intuition for entanglement, let’s see how GRW works. In GRW measurement plays no fundamental role. Instead, we keep the linear dynamical law from the standard theory, but modify it a bit. Recall from above that the linear dynamics was

  • When no measurement occurs, the system evolves deterministically and linearly according to the energy properties of the system.

Since measurement plays no fundamental role in GRW, we can scrap the part about no measurement concurring. So we are left with:

  • A system evolves deterministically and linearly according to the energy properties of the system.

This is good. Measurement is never mentioned, so we see we are making progress at solving the measurement problem. However, right now we have no way to account for our determinate measurement results when we measure, say, the spin of a particle. If we measure spin then we find we always get a determinate result. Thus, in GRW we add to the dynamics:

  • A system evolves deterministically and linearly according to the energy properties of the system. However, each particle that is part of the system has a small probability \lambda per unit of time to collapse to having a definite position. Furthermore, the probability of where the particle ends up is given by the standard quantum probabilities.

This is interesting. We’ve mainly kept the deterministic linear dynamics, but we’ve also added two probabilistic processes. Or rather, one process with two different probabilities. The first probability, \lambda, is the probability that each particle has of collapsing into having a definite position. We can choose lambda such that an individual particle collapses only about once every 10^{8} years. Thus, if we were to consider the evolution of the wave function of a single particle not entangled with any other particle, it would seem as if the particle were entirely governed by the normal linear dynamics, since the probability of collapse is so so so small. However, if the particle is part of a larger object like a book (by “part of” I mean sufficiently entangled with the other particles in the object), then, since whenever one particle collapses into having a definite position so do all the particles with which it is entangled, the macroscopic object will almost certainly look as if it behaves classically. This is because there are so many particles in a macroscopic object that even though a single given particle will almost never collapse, at least one particle in the macroscopic object will collapse in a very small time interval, causing them all to collapse. Thus, we would expect a system with very few particles to exhibit qualitatively different behaviour than systems with a large number of particles. There is no vagueness issue either; the probability \lambda gives us our expectation for how stable we expect systems of different sizes to be.

The second probability that is part of this process tells us how the particles collapse. Or, rather, it gives us the probability that a given particle has of collapsing into a certain position. Since this probability is constructed such that it would agree with the standard quantum statistics, this allows it to reproduce the standard quantum statistics.

This, then, is a toy version of the theory. The actual theory is more complicated in that a particle cannot actually collapse into having a definite position, but must instead collapse such that its wave function is sharply peaked around a certain area. This detail need not concern us here, but it is worth thinking about for those who want to do more research and thinking about GRW.

There are three final things I want to discuss briefly about the theory. These are measurement, position, and ontology.

Measurement: How does the theory ensure that whenever we measure something we always get a determinate result? That is, if we measure the position of a single particle, why do we always see it with a definite position? Whenever we measure the position of the particle, even if the particle initially is not entangled with any other particles, as soon as we use a machine to measure its position, all the particles making up both the machine and us become entangled with that single particle. Thus, the particle will (almost certainly) collapse to having a definite position during the measurement process.

Position: In GRW, it is the position of the particle that collapses. What about other properties like spin, energy, momentum, et cetera? How do we account for measurements of those? The theory is meant to account for measurements of those because, at the end of the day, all measurements are really measurements of position. This is a subtle point. The account must be something like the following. Whenever we measure something, what this really is physically is correlating the position of various particles in our brain with the position of particles in the world. So a measurement about the energy of a system really is a process that correlates my mental state, ultimately described as the position of particles in my brain, with the position of other particles, such as those composing the needle of a device used to measure energy. Whether or not this works can be debated. This is further complicated by the fact that GRW does not predict with certainty that an observer will have a definite mental record, but rather only with high probability. What would it feel like to not have a determinate mental record? I do now know.

Ontology: Although I have described GRW in terms of particles, this was mainly for ease of description. There are different variations of GRW that have a different base ontology. For example, one might take matter, or something called flashes, to be the fundamental ontology. The same kind of dynamical process would describe the evolution of the ontology. Interesting, different ontologies face different problems and have different virtues. For example, most versions of GRW are incompatible with the constraints of special relativity, which is a strike against them. However, it seems that GRW with a flash ontology is compatible with special relativity.

It is interesting to see the variety of ways in which we have tried to solve the quantum measurement problem. Though there is no canonical solution yet, understanding the different proposals can help us see what open problems remain.

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