One way to think about the quantum measurement is to think about the conflict between subjective and objective probability. We looked at some of Everett’s earlier thoughts along these lines last week. As I mentioned in that post, later on in one version of his thesis he provided a much cleaner account of the measurement problem in terms of an inconsistency between different state attributions.
Another way to wrap one’s mind around the measurement problem is to think about the ambiguity of the term “measurement”. This is what Bell focuses on in his paper Subject and Object, and leads him to propose that a satisfactory quantum mechanics would be in terms of “beables” instead of “observables”. Let’s take a look.
***The original paper can be found here.***
Bell writes that
The subject-object distinction is indeed at the very root of the unease that many people still feel in connection with quantum mechanics. Some such distinction is dictated by the postulates of the theory, but exactly where or when to make it is not prescribed.p. 40
We can see the subject-object distinction creeping into the standard theory of quantum mechanics in how key parts of the theory are stated. In particular, Bell draws our attention to the way Dirac writes a few of the laws (I leave out some of the math) to focus on the language:
…any result of a measurement of a real dynamical variable is one of its eigenvalues…,p. 40
…if the measurement of the observable..the average of all the results obtained will be…,
…a measurement always causes the system to jump into an eigenstate of the dynamical variable that is being measured… .
We see two things coming up often: measurement, and observable. Thus Bell writes
So the theory is fundamentally about the result of ‘measurements’, and therefore presupposes in addition to the ‘system’ (or object) a ‘measurer’ (or subject).p. 40
Remember, quantum mechanics is a physical theory. We might think quite naturally that a physical theory should tell us about how the world is—what it is made of, how it changes over time, etc. However, according to Bell, this is not what the standard theory of quantum mechanics is about. Bell is right, and many of the earlier quantum physicists, most notably Bohr, agreed with this. On this view—often called the Copenhagen view, named after where many of these physicists lived and worked—quantum mechanics does not describe how the world is. Instead, it gives us the tools to organize our experience and calculate our expectations for the future. On this view, to think that we could describe the world of the small is misguided. Instead, we merely use the mathematics of quantum mechanics to figure out what kind of behaviour we should expect in the realm of everyday experience—large, classical objects.
Even this view I sketched is a little too charitable. In reality, there was never such a thing as a single coherent Copenhagen view. Indeed, when reading the writings of the early quantum physicists that we associate with the Copenhagen view, it is clear that they all have different ideas of what the view is supposed to be. The version I sketched above is one very close to viewing quantum mechanics as merely instrumental, for organizing and describing our experience. This follows the view that the Austrian philosopher-physicist Ernst Mach had—Mach was very influential in both philosophy and physics just before quantum mechanics really started taking off. However, others in the Copenhagen group of physicists had different views.
Furthermore, as Bell points out, even if we want to take the preceding view of physical theory as merely calculation tool, there is a deep ambiguity here in terms of the laws:
Now must this subject include a person? Or was there already some such subject-object distinction before the appearance of life in the universe? Were some of the natural processes then occurring, or occurring now in distant places, to be identified as ‘measurements’ and subjected to jumps rather than to the Schrödinger equation? Is ‘measurement’ something that occurs all at once? Are the jumps instantaneous? And so on.p. 40
The “jumps” Bell talks about are what, according to the standard theory, happens to the state of a physical system when a measurement occurs, and the “Schrödinger equation” is what is governs the evolution of the state of a physical system when a measurement doesn’t occur.
So we can get an understanding of the measurement problem like this. We have two laws that tells us how the state of a physical system should change over time. The first law says that if the state of a system is measured then a “jump” occurs, and the second says that if no measurement occurs than the state evolves in a different way. However, as Bell points out, the term “measurement” is very ambiguous. What constitutes a measurement? We seem to get the “jump” results when we measure something, and not when small particles interact. But where exactly does the change from object to subject occur? Would a measurement device when no one is looking cause a jump? How about a dog? Or a flea? Or a very sophisticated computer? And, as Bell points out, what about before there were no people? What was happening then?
Now, despite this ambiguity, Bell thinks that while aware of these questions, the “pioneers of quantum mechanics…quite rightly did not wait for agreed answers before developing the theory. They were entirely justified by the results” (p. 40). The truth is that quantum mechanics is by far one of the most successful physical theories we’ve ever written down, ambiguity and all. By “successful” here I mean successful at predicting the results of measurements.
However, unlike others who were satisfied with this level of success, Bell wanted more, and I think rightfully so. Despite the success of quantum mechanics, as a physical theory, it falls short of being adequate due to the measurement problem. This is where Bell writes one of my favourite quotes:
The snake cannot completely swallow itself by the tail. The awkward fact remains: the theory is only approximately unamabiguous, only approximately self-consistent.p. 41
I love this quote both because it is a striking image, and because it extracts the core of the measurement problem. The standard theory of quantum mechanics is only approximately unambiguous because the theory itself gives no criteria to distinguish an interaction that counts as a measurement and one that doesn’t. Thus, it is unclear which of the two dynamical laws—the “jump” or the other one—should apply. Since this is ambiguous, and since these two laws can lead to different physical states, it is only approximately self-consistent. If there is an interaction that is ambiguous, then, on one reading, the theory would predict that the same physical system would evolve into two different, physically incompatible states. Thus the theory would be inconsistent with itself.
Here is a helpful to think of this. On a weak reading, the theory is incomplete: it does not fully specify what constitutes a measurement. Thus, in order to complete the theory, we would need to clearly specify what counts as a measurement. However, on a strong reading, then we would end up with cases where both dynamical laws would apply, and the theory would be inconsistent with itself, since it would say that the same physical system would evolve to two different physically incompatible states.
There are two broad strategies one might take to solving this problem. One might be to specify precisely what counts as a measurement. Some have tried this, for example, Wigner. However, others have tried to take a different approach—removing a fundamental notion of measurement from the theory entirely. Bell favours this second approach:
[I]t is interesting to speculate on the possibility that a future theory will not be intrinsically ambiguous and approximate. Such a theory could not be fundamentally about ‘measurements’, for that would again imply incompleteness of the system and unanalyzed interventions from outside. Rather it should again become possible to say of a system not that such and such may be observed to be so but that such and such be so. The theory would not be about ‘obervables’ but about ‘beables’.p. 41, emphasis original
Instead of further defining what constitutes a measurement and having our theory described in terms of observables, Bell advocated for a theory that is about things that are, not just things that we might observe. The cute name he gives these in to contrast the spectre of measurement in “observables” is “beables”.
How might we do this? One approach is to “promote some of the ‘observables’ of the present quantum theory to the status of beables” (p. 41). The rest of the short paper sketches how one might do this. This is the approach for which Bell advocated most of his career, and was the main reason he championed the de Broglie/Bohm pilot wave theory of quantum mechanics.
It is in light of the measurement problem, the failure of the snake to swallow itself entirely, that we have the constellation of competing theories of quantum mechanics today. Bell was a key part of moving the physics community past the Copenhagen interpretation, or at least part of it, and making quantum foundations a field. His clarity and rigour of thought is exemplified in this short paper.