## January 8, 2009

### the measurement problem

Shortly after Newton proposed his new mechanics, the "shut up and calculate" approach of Newton, Halley and others produced the first astonishing results. However, it did not take long until the foundational debate about the interpretation of the new physics began. In particular, the true meaning of the position coordinates x(t) was heavily discussed. The x(t) were of course projections onto a holonomic basis in the 3-dimensional Euclidean vector space. But how exactly would they be determined in a measurement process?

It came down to measuring distances between point masses (1). But how does one actually measure such a distance? Suppose we use a ruler in the simplest case (2). We have then only replaced one distance measurement with two distance measurements, because instead of measuring the distance between two mass points we need to measure now the distance of each mass point to the markings on the ruler (3).
Now we could use another two rulers to measure those distances etc. - an infinite regress. (Notice the superposition of rulers at 3!)

There were soon two main groups of opinion. The first was known as realists, assuming that the x(t) represented the real position of a mass point, and even if human beings had a problem to comprehend the infinite regress of the measurement process, the omniscient God would necessarily know it.
A small subgroup proposed that the infinite regress is the position, but could not really explain what this means.
The other group insisted that the x(t) were only a subjective description of reality but not part of reality itself. They emphasized the important role of the conscious observer who would terminate the otherwise infinite regress of the measurement process; This introduced the issue of subjective uncertainty into the debate.

Careful analysis showed that x(t) was only known with finite uncertainty dx and in general this uncertainty would increase with time. Astronomers noticed that the dx for some planets was larger than the whole Earth! The realists assumed that there was still one true x(t), even if we do not know it, while Sir Everett 1st proposed the stunning interpretation that *all* positions within dx were equally real, rejecting the idea of random measurement errors. The world was really a multitude of infinitely many worlds and the infinite regress of the measurement problem reflected this multitude!
Subsequently, this type of analysis became known as decoherence program: The position of a mass point can be determined only if the mass point interacts with other mass points. But this means that in order to reduce the uncertainty dx, one necessarily increases the uncertainty of the position of all mass points in the environment.
While it was not clear if decoherence really helped to solve the foundational problems, the complicated calculations were certainly very interesting.

In a devilish thought experiment, a cat was put in a large box and then the lid closed. Obviously the cat would move around inside the box (some would even suggest that the cat moved around randomly, since no law was known that could determine the movement of the cat!), but one could not observe it.
The stunning question was, and still is, if the cat had a position x(t) if one waited long enough.
The realists again insisted that the position of the cat was a real property of the cat, even if it was unknown to everybody. But others insisted that it made no sense to assign a position, since the rays emitted by the eyes of the observer were not able to reach the cat; Furthermore, the animal itself has no conscious soul and thus cannot determine its own position.

While the "shut up and calculate" approach celebrated many more successes, the foundational issues of the new physics were never resolved.

wolfgang said...

Apparently, one reader understood this post as "A parable".

But I think it really is just a (more or less accurate) report about an important historical debate about the foundations of classical mechanics.

The fact that it never happened does not mean it was not important.

(Physicists who believe in the many [classical] worlds interpretation will understand what I mean.)

Dave Bacon said...

This is a classic post, wolfgang, I greatly enjoyed it.

wolfgang said...

Thank you.
And thank you for the link and the nice words...

Cosma said...

Actually, the position that the infinite regress is the measurement reminds me of a rather strange-sounding remark in one of Bertrand Russell's books (The Analysis of Matter, I think), that it is possible to start with extended spatio-temporal events and a relationship "overlaps with", and recover point-instants as a kind of equivalence class of overlapping events. I should revisit this now that I know more math.

wolfgang said...

There is perhaps a similarity with Dedekind cuts.

And perhaps I could have added some Bergson to this thing 8-)

Bryan said...

Nice post! The second part about the cat in the box doesn't seem too pressing. If you believe our best theories (GR and QM under some important constraints), then dynamical evolution is deterministic. So there's no question about the value of x(t). If you don't, then the point is moot.

The first part about the measurement regress is very interesting. However, I don't see why you think "we need to measure the distance of each mass point to the markings on the ruler". I've got a negative and a positive comment about this.

Negative. Using a ruler to measure another ruler in this case is very bad methodology (indeed, the very idea smacks of absurdity). I'm reminded of Hilbert's maxim: we cannot use just any mathematical methods to justify mathematical practice itself, for that would be circular; a practice can only be justified by more stringent methods of justification. Similarly here -- if we want to justify the practice of measuring length with rulers, we can't use a ruler to do it!

Positive. I actually don't think 'measurement with rulers' is the kind of thing that needs much justification. When we learn to make measurements, we learn a reliable, standardized practice. This could involve a ruler, or it could involve the Stern-Gerlach apparatus. But much of the practice itself is conventional -- we define what counts as a measurement, and then proceed to compare 'measurements' according to that definition. No further justification seems possible, and none seems necessary.

wolfgang said...

>> If you believe our best theories (GR and QM under some important constraints), then dynamical evolution is deterministic. So there's no question about the value of x(t).

Do you believe we could use GR+QM to predict the movement of a cat in a closed box?
I doubt that this is possible even in principle. Measuring the initial state of all neurons (or perhaps of all molecules) in the cat's brain would most likely kill it.

I even doubt that it makes sense to assign a wave function to the macroscopic brain of the cat - assuming it exists but we just dont know it.
The wave function of the cat would be entangled with the environment in a most complicated way and one would ultimately have to deal with the wave function of the universe.

Anonymous said...

Just out of curiousity, why can't taking the limit of the sequence of infinite regress givee for a well-defined position?

wolfgang said...

Perhaps I should be more explicit about the main point of this example.

There is of course no infinite regress, nobody uses other rulers to determine the distance from a mass point to the markings of the 1st ruler.

The measurement ends at step 2 when you look at the ruler and 'read the result from the measurement device'.
Your eyes and finally your brain terminate the 'infinite regress' early on - and we honestly dont really know how they do it 8-)

The 'translation' (or perhaps better: correspondence) of classical mechanics to our direct conscious experience is usually not considered to be an important issue - and is usually not even discussed as part of classical physics. (Ernst Mach was the only classical physicist who emphasized that there even is an issue.)

But when physicists discuss the measurement process within quantum theory, all of a sudden there is an issue, because the 'translation' from QM to our conscious experience is no longer so easy to ignore.

Bryan said...

>> Do you believe we could use GR+QM to predict the movement of a cat in a closed box?
>> I doubt that this is possible even in principle. Measuring the initial state of all neurons (or perhaps of all molecules) in the cat's brain would most likely kill it.

We have to be careful not conflate schrodinger/unitary evolution with the act of measuring. The former is deterministic, even though the latter isn't. That is, given an initial state (say, inside the box before we close it), quantum theory says it will undergo a unique time-evolution. There's only one possible future for the state of the cat, as long as state reduction does not occur!

So, I still don't see how our 'in principle' inability to interact with every particle in the cat is a problem.

wolfgang said...

> given an initial state

In order to determine or prepare the initial state you must interact with (each molecule of) the cat.

Anonymous said...

I'd like to put what Bryan said above slightly differently. The values taken by a mathematical symbol representing a physical quantity, e.g. temperature or position, are intended to represent the numbers that experimenters report and call 'temperature' or 'position' (I'm paraphrasing Truesdell here). And this correspondence cannot really be made more precise than that. It is a matter of agreement and discussion, and is subject to historical development and change; it is 'soft' and cannot be precisely formalised. I find it very interesting and exciting precisely because of this. In some cases we analyse those experiments by means of the theory itself in which the name 'position' is used, or by means other theories. But in doing so we use other physical quantities, for which the same 'problem' and soft correspondence remains. This is the 'regress', but is not infinite because at some point it resolves in agreement, convention, or is subject to further discussion - indeed, I don't mean that it should not be discussed and analysed in every instance. Only, it cannot be dispelled once and for all. This remark is similar to Bryan's and Hilbert's.

After all, the same 'problem' appears when we apply geometry. When you calculate the area of a geometrical figure in front of you, how do you know whether to use the formula for the triangle or that for the circle? Justify that. Strange that people insist on discussing about what certain mathematical entities (like 'position') correspond to, but not others (like 'triangle'). It is a social and historical matter of how strong the respective conventions are.

This 'soft' correspondence has always been there, and has been made very explicit in classical mechanics from around the fifties, after the work of mathematicians like Noll, Truesdell, Serrin, Coleman, Owen, and many others (whose existence is ignored by most quantum physicists). And I think that many classical authors were aware of that (I am ingnorant here, since I haven't read their works and don't believe in textbook summaries and other second-hand information of that kind). But some quantum physicists one day awoke and thought they had found a new problem; while, as Wolfgang say if I understand him/her correctly, this 'problem' or non-'problem' had always been there.

This in part explains also why we use different mathematical objects to represent e.g. 'momentum' in classical and quantum physics. In classical physics an experiment is understood as giving always the same numerical result given the same experimental conditions (otherwise we make a more detailed study of the conditions until we find what is the origin of the fluctuations). In quantum mechanics istead it is assumed that an experiment can give different results, given the same experimental conditions, and so we need a mathematical object that encodes not only the numerical results, but also their frequencies - or sometimes the latter alone. This is the role of Hermitean operators - or of positive-operator-valued measures (resolutions of identity). In principle we could play the same game without these mathematical objects though, using the classical symbols together with probability distributions over them.

Interestingly, here we can have again the discussion of whether it is justified to call the Hermitean operators ^x and ^p 'position' and 'momentum'. Some think that these are misnomers because the experiments they correspond to are not in the same 'category' as the classical ones with the same name. (For example, in Bohmian mechanics the operator ^x may not directly correspond to the position of any particle in that theory).

Sorry for the partly polemic and partly lecturing tone, folks!

wolfgang said...

>> But some quantum physicists one day awoke and thought they had found a new problem; while, as Wolfgang say if I understand him/her correctly, this 'problem' or non-'problem' had always been there.

Just to clarify this one point. I think quantum physicists certainly found new problems, but they also stumbled over some 'problems' that have always been there, just not made explicit.

wolfgang said...

I will try to find out more about Clifford Truesdell.

The title "The Tragicomical History of Thermodynamics" sounds interesting...

Anonymous said...

Also 'Six Lectures in Modern Natural Philosophy', 'A First Course in Rational Continuum Mechanics', and, probably above all, 'Rational Thermodynamics' (2nd edition).

Ponder Stibbons said...