State Spaces, Representations, and Transformations

Sunny Y. Auyang gives a useful model for thinking about the way states and state spaces of objective reality are represented in ways accessible to us and how transformations between a plurality of representations imply not relativism but common states and state spaces that they represent.

I’ve been reading a fascinating book that’s been giving me lots of ideas that I’ve been wanting to talk about. I was thinking to wait until I had finished it but I changed my mind because there are some ideas I want to capture now. It’s one of the books I call my “eye-reading” books because I’m usually listening to a number of audiobooks simultaneously. And I don’t have much time to sit down and actually read a book in the traditional way. But I sometimes save space for one if it looks really interesting and it’s not available in audio. And that applies to this one. The book is How is Quantum Field Theory Possible?, written by Sunny Y. Auyang. I heard about it while listening to another podcast, The Partially Examined Life, which is a philosophy podcast. One of the guys on there, Dylan Casey, mentioned it in their episode on Schopenhauer. It peaked my interest and I knew I had to get it.

The part of the book I want to talk about today is a model she puts together to think about the different ways an objective state can be represented in our scientific theories. To the extent that our scientific models and measurements are conventional what should we think if they represent things differently? Are we condemned to relativism and the arbitrariness of convention? She argues that we are not gives a model that takes things up a level to see different representations from the outside, how they relate to each other through transformations and how they relate to the objective states that they represent. This is necessarily a philosophical project, particularly a question in the philosophy of science. It is to get behind the work of science itself to think about what it is we’re doing when we do science and what it means when we say that things are a certain way and work in a certain way, as described by some theory.

I’d like to give a brief overview of some of those concepts and the vocabulary Auyang uses. And this will just be to get the concepts in our head. John von Neumann had a very funny quip that “in mathematics you don’t understand things. You just get used to them.” Now, I think that’s an overstatement. But in a way I think it’s kind of helpful whenever we’re getting into a discipline that has lots of unfamiliar terms and concepts that can seem really overwhelming. I think it’s helpful to just relax and not worry about fully understanding everything right away. But to take time to just get used to stuff, which takes time. Eventually things will start to come together and make more sense.

So the first idea I want to talk about is a phase space or state space. A phase space is the set of all possible states of a system. That’s very abstract so I’ll start with a concrete example. Say we have a single particle. At any given time this particle has a position in three-dimensional space that we can specify with three numbers along three spatial axes. For example, you could have a north-south axis, an east-west axis, and an elevation axis. You can also add momentum to this. So a particle’s momentum would be its mass multiplied by its velocity. Mass is scalar quantity – it doesn’t have direction – but velocity is a vector, so it does have direction. And in three-dimensions the velocity has three components along the same spatial axes as position. So you can specify the particle’s position and momentum with six numbers: three numbers to give its position and three numbers to give its momentum.

The really cool move from here is that you can then make use of what’s called a phase space. So for a single particle with these six axes we’ve selected this is a six-dimensional space. This is also called a manifold. Don’t worry about trying to visualize a six dimensional space. It’s not necessary. Just go along with the idea that we’re using such a thing. This is an abstract space. It’s not supposed to represent the kind of space we actually live it with length, width, and height. Any point in this six-dimensional space represents a possible state of the particle. You can represent any combination of position and momentum as a point in this phase space. So for example, the 6-tuple in parentheses with the six numbers (0,0,0,0,0,0) represents a state where a particle is at rest and it is sitting at the origin of whatever spatial reference frame we’ve set up. And you can put in any set of numbers to get any possible state of that particle. If we’re looking at multiple states of this particle through time we can think of it tracing out a trajectory in this state space.

Now, here’s where things get crazy. You can add more than one particle to this system. Say we add a second particle. How many dimensions does our phase space have now? It has twelve dimensions because we have axes for the positions and momentum components for both particles in three-dimensional space. And then we’ll have a 12-tuple, twelve numbers in parentheses, to call out the state of the system. And you can add as many particles as you like. For whatever N number of particles we have in our system the phase space will have 6N dimensions. So you can imagine that dimensions will start to pile up very quickly. Let’s say we take a liter of air. That has something on the order of 1022 molecules in it; over a billion billion. The number of dimensions in our phase space for that system will be six times that. Now, in practice we’d never actually specify a state in this kind of system. With gases for instance we don’t worry about what’s going on with every single particle in the system. We use properties like temperature and pressure to generalize the average behavior of all the particles and that’s much, much more practical. But as a conceptual device we can think of this phase space underlying all of that.

In quantum mechanics the state space of a system is called a Hilbert space. So this is the space of all possible states of a quantum system. Then any particular state of the quantum system is represented by a state vector, usually written with the Greek letter phi: |φ⟩. When we run an experiment to get information about the quantum system we look at a particular property that is called an observable. And you can think of an observable as pretty much what it sounds like, i.e. something that can be observed. And this is associated mathematically with an operator. An operator, as the name implies, operates on a function. And there are all kinds of operators. There are operators for position, momentum, total energy, kinetic energy, potential energy, angular momentum, and spin angular momentum. One way to think of this is that with an operator you’re conducting an experiment to measure the value of some property type. Then the result of that experiment is some number. The name for the resulting value is an eigenvalue. So for all those different operators I just listed off they will spit out corresponding eigenvalues. But an eigenvalue is an actual value. So with a kinetic energy operator, for example, your eigenvalue will actually be a number for the value of kinetic energy in some unit for energy, like Joules or whatever your choice of units.

Recall that in our phase space for particles each dimension, and there were many, many dimensions, had an axis in that phase space. In quantum mechanics the state space, the Hilbert space, has a collection of axes that are called a basis. And the basis of a Hilbert space is composed of eigenstates. And we can think of this as the coordinate system, the axes, of the state place of the system. The eigenvalue is what we get when we run an experiment but one of the interesting things about quantum systems is that we don’t always get the same value when we run an experiment, even if we’re applying the same operator to the same system. That’s because a quantum system is a combination (more specifically a linear combination or superposition) of many eigenstates. And each eigenstate has a certain amplitude. As we repeat several measurements of an observable we’ll observe eigenstates with higher amplitudes more often than eigenstates with lower amplitudes. We can actually quantify this. For any given eigenstate the probability that it will be observed with a measurement of an operator is its amplitude squared. So amplitude is a very important property in a system.

So there are many similarities there between the phase space of the system of classical particles and the Hilbert space of a quantum mechanical system. I just wanted to give an overview of those to introduce and talk about the vocabulary in the interest of starting to “get used to it” as von Neumann said, even if that’s a long way from having a comprehensive understanding of it.

Having laid that groundwork down I want to summarize this section of the book where Auyang introduces a model to analyze the relationship between the objective state space of system and its representations in different scientific theories. The objective state space is what is “out there” independent of our observations or awareness of it. The representations are what we interact with. We could definitely invoke Immanuel Kant here with his concepts of the “thing in itself”, that he calls the “noumena”, and the “phenomena” that we experience of it. And Auyang definitely draws on Kant repeatedly in her book.

There’s a figure she refers to over several pages and I’ve posted this on the website. But for those listening on the podcast I’ll try to describe it in a way that hopefully isn’t too difficult to follow. In her diagram she has three boxes. The top box is the state space, “M”. So that’s the set of all possible states of a system. Then in this state space there’s one state, “x”. x is what is objectively out there, independent of our observations and theories of it. But we don’t observe or interact with x directly. What we observe are the representations of x. And those are the lower two boxes.

These lower two boxes are fα(M) and fβ(M). These are the representations of certain properties of state space M. fα and fβ are property types that we could be looking for and then fα(M) and fβ(M) are the possible representations we can find when we run experiments to measure for those properties. Inside each of these lower boxes is a smaller box for the representation of the single objective state x. So these would be fα(x) and fβ(x). These are the definite predicates or values we get from our experiments. These are the things we come into contact with.

To tie this back to quantum mechanics real quick, in the quantum mechanical case the way this general picture would play out is that M would be a Hilbert space, x would be a state vector (x being one state in that state space), fα would be an observable, fα(M) would be a representation, and fα(x) would be an amplitude of x in the basis of the observable fα.

What’s important to understand here is that the values measured by the two representations are not equivalent. Someone trying to get at the objective state x in the objective state space M from fα will see something different than someone trying to get at it from fβ. One will get fα(x) and one will get fβ(x). Which one is right? Well they’re both right. But what does that mean? It depends on how much of this picture we see.

So we’ll look at parts of this model in pieces before we get back to the whole, comprehensive picture. But first I want to make another comparison because this is all quite abstract. I think it’s helpful to compare this to sentences in different languages. Say we have a sentence in English and Spanish. In English we say “The dog eats his food” and in Spanish we say “El perro come su comida”. These are different utterances. They sound very different. But we want to say that they mean roughly the same thing. We can translate between the two languages. And people can respond to the utterances in similar ways that indicate that there is something in common to both. But whatever it is that is common to both is not expressible. We only express the sentences in particular languages. But because they are translatable into each other it makes sense to think that there is some third thing that is the meaning the both share.

OK, so keep that example in mind as we get back to physical theories and the objective states they represent. Looking at our model again say we look only at one of the lower boxes, fα(M). In this picture as far as we’re concerned this is all there is. So one thing to say about this is that the meaning of fα(x) is what Auyang calls “unanalyzable”. And why is that? Well, it’s because fα(x) is “absolute, self-evident, and theory-free”. It’s just given. There is no objective state space M that fα(M) is representing. Rather fα(M) is the immediate bottom level. So there’s nothing here to analyze. We don’t have to think about the process of representation.

OK, well let’s add the second lower box, fβ(M). So now we have just the two lower boxes but still no objective state space M. What do we have now? Well we have plurality. There are multiple representations of the same thing and we don’t have a way of knowing which one is true. And neither can we say that they point to one common thing. So this gets to be a very confusing situation because we have both plurality and unanalyzability. Plurality in that we have two different values representing a state, fα(x) and fβ(x). Unanalyzability because, as with the previous view with only the one box, there’s not objective state space that either of these correspond to. No process of representation to analyze here. What we have are conventions. This is the kind of picture Thomas Kuhn gives in his book The Structure of Scientific Revolutions. And this is a picture of relativism. The conventions are incommensurate and the choice among them is arbitrary. I think it’s fair to say that there’s much that’s unsatisfying with this picture.

Well, now let’s add the top box back in so we have everything. This brings what I’d consider an explanatorily robust conceptual device. As Auyang says, “the introduction of the physical object whose state is x not only adds an element in our conceptual structure; it enriches the elements discussed earlier,” fα(M) and fβ(M). In other words. fα(M) and fβ(M) look a lot different with M than without it. And I’d say they also make a lot more sense.

For one thing, the picture is no longer unanalyzable but analyzable. We understand that there is a process of representation occurring when we collect numerical data from experiments. When we look at property types fα and fβ we understand that these both represent M in different ways. As Auyang says, “Various representations can be drastically different, but they represent the same object.” She gives a concrete example: “The same electromagnetic configuration that is a mess in the Cartesian coordinates can become simplicity itself when represented in the spherical coordinates. However, the two representations are equivalent.” What’s key to understand here, and what makes this third, fuller picture, more powerful and coherent is that there is one objective state space, one object that the various representations point back to. So we circumvent relativism. The picture only looks relativistic when we only have the partial view. But when we see state space M and that fα(M) and fβ(M) map onto it we can appreciate that even though fα(x) and fβ(x) are different, they both correspond to one objective state x.

Another important thing to consider is that there is a transformation between fα(M) and fβ(M) that Auyang calls fβ•fα-1. The transformation is the rule for transforming from representation fα(M) to fβ(M). That there is such a transformation and that it is possible to transform between representations arguably evinces the existence of the objective state space that they represent. As Auyang states: “Since fα to fβ are imbedded in the meaning of [fα(x) to fβ(x)], the transformation fβ•fα-1 connects the two representations in a necessary way dictated by the object x. fβ•fα-1 is a composite map. It not only pairs the two predicates [fα(x) to fβ(x)], it identifies them as representations of the same object x, to which it refers via the individual maps fα-1 and fβ. Since fβ•fα-1 always points to an object x, the representations they connect not only enjoy intersubjective agreement; they are also objectively valid. To use Kant’s words, the representations are no longer connected merely by habit; they are united in the object.” This is related to the example I gave earlier about two sentences in different languages as representations of a common referent.

And I just think that’s a lovely picture. One of my favorite thinking tools is to take things up a level to try and see things that weren’t observable before. Like how you can see more from an airplane than when you’re on the ground. It’s not that the way things are has changed. But we see more of it. And with a more complete pictures the parts that seemed random or even contradictory make more sense and work together as a rational system.

So that’s one bit of the book I wanted to talk about. There are a few other things I’ve read that I want to talk about later too. And I’m only about halfway through. And if it continues to be as jam-packed with insights as it has been up to now I’m sure there will be more I’ll want to talk about.

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