Notification e-mails are wonderful, particularly when they bring a table of contents for Studies in History and Philosophy of Modern Physics. I found the highlight this month to be James R.Henderson's "Classes of Copenhagen interpretations: Mechanisms of collapse as typologically determinative", which classifies some of the versions of the Copenhagen Interpretation quite nicely, in terms of a class of four Physicists, Bohr, von Neumann, Heisenberg, and Wheeler. Henderson is nicely careful to say that each of these four has a view of CI that has spawned its own industry of claims about what each of the big names really said and meant.
The citation is: Studies in History and Philosophy of Modern Physics 41 (2010) 1–8.
Henderson's very clear presentation points out for me the way in which the discussion so much starts with QM and tries to construct classical physics from it, because QM is supposed to be better and more fundamental than the old classical mechanics. As indeed it is, but when discussing foundations it's perhaps better not to start with such a strong assumption. Starting, conversely, from a purely classical point of view, discrete events can be taken as thermodynamic transitions, without any causal account for why they happen (that being the nature of thermodynamics, in contrast to statistical mechanics), so that Heisenberg's or Wheeler's records are the given experimental data, from which unobserved causes might be inferred. There's no question of there being a philosophically based measurement problem in this view, because there is so far no such thing as QM, there are just records in an experimenter's notebook or computer memory.
If we start from the recorded data that comes from an experiment, the question is how we come to have QM? The fundamental issue is that we have to split the world into two types of pieces, pieces that we will model with operators Si, which ordinarily we call states, and pieces that we will model with operators Mj, which ordinarily we call observables or measurement operators. When we record a number Vij when we use the piece of the world that we model with the operator Si with the piece of the world that we model with the operator Mj, we write the equation Vij=Trace[Si Mj]. When we've got a few hundred or million such numbers Vij, we solve the few hundred or million simultaneous nonlinear equations for Si and Mj.
This is a little strange, because QM is supposed to be a linear theory, but these are nonlinear equations --- if we know what the measurement operators should be a priori, we would have a set of linear equations for the states, and vice versa if we know what the states should be a priori, but it's not clear that we know either a priori, so in fact and in principle we have a nonlinear system of equations to solve. In practice, we solve these nonlinear equations iteratively, alternately as linear equations for the set of states, guessing what the measurement operators are, so that after a while we know what the states we are using are (a process often known as characterization), and then as linear equations for the measurement operators, but this is just an approximation method for solving a system of nonlinear equations.
Also peculiarly, the dimensions of the state and measurement operators are not determined, except by experience. In quantum optics there are some choices of experimental data for which it is enough to use 2-dimensional matrices to get good, useful models, but sometimes we have to introduce higher dimensional matrices, sometimes even infinite dimensional matrices, which is rather surprising given that we only have a finite number of numbers Vij. Indeed, given any finite set of experimental results, QM is not falsifiable, because we can always introduce a higher dimensional matrix algebra or set of abstract operators, so we can always solve the equations Vij=Trace[Si Mj].
Instead of solving the equations Vij=Trace[Si Mj], we can minimize the distance between Vij and Trace[Si Mj], using whatever norm we think is most useful. With this modification, we introduce an interesting question: what dimensionalities give us good models? We might find that 2-dimensional matrices give us wonderfully effective models even for millions of data points, in which case we might be tempted not to introduce higher dimensional matrices. Higher dimensionality will certainly allow greater accuracy, a smaller distance between the data and the models, but it may not be worth the trouble. If we find that some matrix dimensions work very well indeed for a given class of experiments, however, we are tempted to think that the world is that way, even though a better model that we haven't discovered yet may be possible.
This is far from being all of QM. We can also introduce transformations that can be used to model the effect of intermediate pieces of the world, so that we effectively split the world into more than two pieces, and we can introduce transformations that model moving pieces of the world relative to each other instead of introducing different operators for different configurations.
The point, I suppose, is that this is a very empirical approach to what QM does for us. There is a fundamental critique that I think QM in principle may not be able to answer, which is that when we use a piece of the world S1 with two pieces of the world M1 and M2, one after the other, we cannot be sure that the piece of the world that we model using S1 is not changed, so we appear to have no warrant for using the same S1 in two equations, V11=Trace[S1 M1] and V12=Trace[S1 M2]. We can say that we can use the same operator in the two different experimental contexts as a pragmatic matter, which we have to do to obtain a set of equations that we can solve for the states and measurement operators, but ultimately everything surely interacts with everything else, indeed QM is more insistent about this than is classical Physics, so as a matter of principle we cannot use the same operator, we cannot solve the equations, and we cannot construct ultimately detailed QM models for experimental apparatuses.
Finally, this way to construct QM does not explain much. For that we have to introduce something between the preparation apparatus and the measurement apparatus. The traditional way of talking is in terms of particle trajectories, but that can only be made to work by introducing many prevarications, the detailed content of which can be organized fairly well in terms of path integrals. An alternative, random fields, a mathematically decent way to bring probability measures and classical fields into a single structure, is the ultimate topic of this blog.
7 comments :
Peter, I have some layman questions for you, and to add to the difficulty, I have little theoretical understanding of algebra.
1. When you are constructing QM models, what is it that you notice when you decide a particular model isn't adequate?
2. As I read through your ideas here, and working to infer implications, the implications folded back into the math itself, which unfortunately I'm blind to.
However, without any translation yet, am I understanding you when you want to begin with fields as they present a means of orientation that is more sure footed than "quantum objects?"
Hi, Mike,
To (1), to be able to decide anything, we have to introduce a way to measure how far a given model is from the experimental data. If a model produces a list of 100 numbers and experiment produces a list of 100 numbers that are different, the root mean square of the 100 differences is often the simplest measure to take (but by no means the only measure that people use).
If the root mean square difference is small enough, we might happily accept the model. If the root mean square difference is a little larger, whether we accept the model may well depend on how much work it will take to construct a better model. If it's easy to get better accuracy, we might do it just for the sake of it, but if there's a deadline and the accuracy is already good enough, we won't.
Finally, modeling tools can be more or less adaptable to new situations. In particular, a given modeling technique may be unable to construct a model that is accurate enough. If we construct a model of a real bridge using Lego, it might look almost the same, but it probably won't be accurate in ways that would allow us to discover whether the bridge will fall down in 100mph winds. In that case, we may have to construct a new type of model. There is a very sophisticated discipline associated with constructing scale models that are physically accurate enough despite being many times smaller than the original.
To (2), I'm trying to find ways to talk about the Physics in elementary ways, but really being able to judge whether some aspects of my approach are nonsense is only possible at this stage to someone who has quite a lot of expert knowledge. Certainly for over a hundred years no-one has found a simple way to understand QM, and I don't believe there's going to be one now. What I'm doing can be given a simple gloss, but if there were not deep currents that matter, it would be hopeless. The introduction of random fields is fundamental, but they're sophisticated enough that I doubt the details will ever be accessible to everybody. It remains a very sharp question whether I am wrong that this is an appropriate mathematical subtlety to introduce. I have seen too many crank papers that introduce some elementary mathematics that they think is powerful, and it's quite possible that the theoretical physics elite will equally scorn my supposed mathematical sophistication. The possibility of proving that at least some quantum fields are equivalent to a random field gives some hope.
To (3), I am shifting my position somewhat, which is partly the point of the post you have commented on. I want to talk about experimental data as classical records in experimental notebooks or in computer memory. With this data, making a jump to continuity is can be done in many ways, with different results. Particles that have continuous trajectories don't work as a way to explain the statistics we observe unless we introduce something like de Broglie-Bohm trajectories, which raises questions of whether we like the particular form of nonlocality that this requires. On the other hand, continuous fields are also problematic, because the concept of probability can't be added to continuous fields without considerable mathematical care. One way to be mathematically careful is to use random fields, which are in many ways a half-way house between classical physics and quantum physics. They're a bridge, which in time I think will certainly be used by other people. I think they're too elegant and powerful not to be used. I presume that people will find ways to use them that I haven't thought of, some of which it will be beyond my mathematical abilities to do, and some of which it will be beyond my mathematical abilities to understand.
So are the "building blocks in "play" here, "particulate," waves, and fields? And am I understanding things right that probability replaces the classical notions of location and vector?
As far as my modeling is concerned, there is a classical level, where we have to describe the geometrical configuration of the experimental apparatus and the materials of which its components are made, and we also have to describe the properties of whatever events happen in the active components of the apparatus (essentially, their timings and where they happen, but the shape of the waveform of the current, say, during a single event is a more detailed property that is sometimes significant).
So, at this level there is the classical apparatus and there are the events, and there are the classically described properties thereof.
When we come to talk about putative explanations, or causes, for the events, my preference for now is to say that the events are caused by properties of a (random) field. We construct models using random fields and see what they look like. My preference is that we won't introduce point-like particles unless we discover that we have to in order to get good models.
The proviso to this is I presume that discrete structures that correspond to the discrete structures we know of, such as electric and other charges, must be emergent from the mathematics of random fields.
The random fields cause the events, but not as straightforwardly as we would suppose classical particles would cause events, just by hitting them.
For a simple analogy, suppose we watch a coastline for cliff erosion events. There is no exact pattern to rock fall events, but we notice that far more rock fall events happen when the seas are running high, far fewer when the sea is calm. If we couldn't see the sea, we might explain that rock fall events happen when a ship fires its guns at the cliff. If in each case the only data we have to go on is the timing of rock fall events, we could choose either explanation.
In QM, we have built up enormous quantities of information that rule out any simple-minded ship firing its guns at the coastline, but the alternative of waves against the edge of the detection devices has been too little considered, I claim.
Needless to say, don't take the waves against the cliff analogy too seriously. In particular, we have to expect that a classical apparatus is not as different from empty space as a cliff face is from the sea, because a classical apparatus creates no wake when it moves at different speeds. This analogy is looking pretty dangerous, in fact.
This is so interesting. I had a aha moment reading a Trent Lott quote that might tie in here- in addressing evolutionary theory and morality, he didn't think there could be moral motivation when people saw themselves as only coming from apes and slime... here's the aha connection:
Lott was poised to see the gorilla and slime as the effective referents. Poised differently I see the "creative field" as the effective referent- that slime and apes and anything emerge from the "same field".
I'm not equating fields here, I'm only making an analogy of something...I'm beginning to grasp more of what your doing as you turn your attention to the field rather than the stuff that emerges from a field?
I should add that the relationship of the classical level model of an experimental apparatus is so far unanalyzed. It's not necessary for it to be analyzed for a field model of what happens between the preparation apparatus and the measurement apparatus to be workable.
If we were using classical continuous field models, there would be a presumption that the field equations would be nonlinear and that there would be soliton solutions, however the mathematics of random fields is not sufficiently developed for a simple statement.
Mike, I'm not sure whether I'm comfortable with your analogy or not. Sure, we can say something like "everything is ur-stuff", but in Physics there has to be mathematical model that is quite tightly tied to experimental datasets, and it ultimately has to predict.
I constructed my previous comment before I considered your most recent comment, but the issues are perhaps related. How stuff emerges from a field has to be derived. Or, we don't understand the field unless we understand how the field can be more than just a field. We need, perhaps, to be able to give an obsessive, mathematically detailed account of effective ways to see the forest, the trees, and the beetles for it to be Physics.
Science explicitly denies the inclusion of unrepeatable events, but this is in tension with the possibility that any pair of events has its similarities as well as its differences, even if we haven't yet discovered how to look. Where there is similarity, there is repetition. So, perhaps, we may see slime and apes as similar, but the differences are important too. Indeed, how we describe similarities and differences — what we notice and what we think important — is a significant part of what defines us.
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