Quantum Properties

Should we understand quantum systems to have definite properties? In quantum interpretations values are usually taken to be the eigenvalues directly revealed in experiments and quantum systems generally have no definite eigenvalues. However, Sunny Auyang argues that this does not mean that they don’t have definite properties. The conclusion that they don’t arises from a restricted sense of what counts as a property. The conceptual structure of quantum mechanics is much richer and an expanded notion of properties facilitates an understanding of quantum properties that are more descriptive and structurally sophisticated.

One of the philosophical problems prompted by quantum mechanics is the nature of quantum properties and whether quantum systems can even be said to have properties. This is an issue addressed by Sunny Auyang in her book How is Quantum Field Theory Possible? And I will be following her treatment of the subject here.

One of the major contributors to the development of quantum mechanics, physicist Neils Bohr, whose grave I happened to visit when I was in Copenhagen, said: “Atomic systems should not even be thought of as possessing definite properties in the absence of a specific experimental setup designed to measure these properties.” Why is that? A lot of this hinges on what counts as a property, which is a matter of convention. For the kinds of things Bohr had in mind he was certainly right. But Auyang argues that it’s useful retain the notion and instead locate quantum properties in different kinds of things, in a way Bohr very easily could have agreed with.

Why are the kinds of things Bohr had in mind not good candidates as definite quantum properties? The upshot, before getting into the more technical description, is that in quantum systems properties like position don’t seem to have definite values prior to observation. As an example, in chemistry the electrons bound in atoms and molecules are understood to occupy orbitals, which are regions of space with probability densities. Rather than saying that a bound electron is at some position we say it has some probability to be at some position. If we think of a definite property as being something like position you can see why Bohr would say an atomic system doesn’t have definite properties in the absence of some experiment to measure it. Atomic and molecular orbitals don’t give us a definite property like position.

Auyang takes these kinds of failed candidates for definite properties to be what in quantum mechanics are called eigenvalues. And this will require some background. But to give an idea of where we’re going, Auyang wants to say that if we insist that properties are what are represented by eigenvalues then it is true that quantum systems do not have properties. However, she is going to argue that quantum systems do have properties, they are just not their eigenvalues; we have to look elsewhere to for such properties.

In quantum mechanics the characteristics of a quantum system are summarized by a quantum state. This is represented by a state vector or wave function, usually with the letter φ. A vector is a quantity that has both magnitude and direction. Vectors can be represented by arrows on a graph. So in a two dimensional graph the arrow would go from the center origin out into what is called the vector space. In two dimensions you could express the vector in terms of the horizontal and vertical axes; and the vector space would just be the plane these sweep out or span. It’s common to represent this in two, maybe three dimensions, but it’s actually not limited to that number; a vector space can have any number of dimensions. Whatever number of dimensions it has it will have a corresponding number of axes, which are more technically referred to as basis vectors. Quantum mechanics makes use of a special kind of vector space called a Hilbert space. This is also the state space of a quantum system. So recall that the description of the quantum system is its state, and this is represented by a vector. The state space then covers all permissible states that this quantum system can have.

Let’s limit this to two dimensions for the sake of visualization. And we can refer here to the featured image for this episode, which is a figure from Auyang’s book. We have a vector |φ> in a Hilbert Space with the basis, vectors {|α1>, |α2>}. So for this Hilbert Space |α1> and |α2> are basis vectors that serve as a coordinate system for this vector space. This is the system but it’s not what we interact with. For us to get at this system in some way we need to run experiments. And this also has a mathematical representation. What we get out of the system are observables like energy, position, and momentum, to name a few. Mathematically observables are associated with operators. An operator is a kind of linear transformation. Basically an operator transforms the state vector in some way. As a transformation, an operator usually maps one state into another state. But for certain states an operator will only result in the same state multiplied by some scaling factor. So let’s take some operator, upper case A, and have it operate on state |φ>. The result is a factor, lower case α multiplied by the original state |φ>. We can write this as:

A|φ> = α|φ>

In this kind of equation the vector |φ> is called an eigenvector and the factor α is called an eigenvalue. The prefix eigen- is adopted from the German word eigen for “proper”, “characteristic”, “own”, in reference to the fact that the original state or eigenvector is the same on both sides of the equation. In quantum mechanics this eigenvector is also called an eigenstate.

Now, getting back to quantum properties, I mentioned before that Auyang takes the kind of definite properties that quantum systems are understood not to have prior to observation to be eigenvalues. Eigenstates are certainly observed and corresponding eigenvalues measured in experiments. But the issue is of properties of the quantum system itself. Any given eigenvalue has only a certain probability of being measured, among the probabilities of other eigenvalues. So any single eigenvalue can’t be said to be characteristic of the whole quantum system.

Let’s go back to the two-dimensional Hilbert space with state vector |φ> and basis vectors |α1> and |α2>. The key feature of basis vectors is that every vector in the vector space can be written as a linear combination of those vectors. That’s how they act as a coordinate system. So if we take our vector |φ> we can break it down into two orthogonal (right angle) components, in this case the horizontal and vertical components, and then the values for the coefficients for those components will be some factor, ci, of the basis vectors. So for vector |φ> the components will be c11> and c22>. In the more generalized form with an unspecified number of dimensions we can say that the vector |φ> is equal to the sum of cii> for all i.

|φ> = ∑cii>

The complex numbers ci are amplitudes, or probability amplitudes, though we should note that it’s actually the square of the absolute value of ci that is a probability. Specifically, the quantity |ci|2 is the probability that the eigenvalue ai is observed in a measurement of the operator A on the state vector |φ>. This is known as the Born rule. Another way of describing this summation equation is to say that the state of the system is a linear combination, or superposition, of all the eigenstates that compose it and that these eigenstates are “weighted” by their respective probability amplitudes. Eigenstates with higher probability amplitudes are more likely to be observed. And this touches again on the idea that observations of certain eigenstates are probabilistic and that’s the reason that the eigenvalues for these eigenstates are not considered definite properties. Because, they’re not definite; they’re probabilistic.

If we apply operator A to state |φ> we have a new vector A|φ>. In our Hilbert space this new vector’s components are expressible in terms of the coordinates, or basis vectors. If the basis vectors are eigenvectors of A then these components are expressible in terms of the probability amplitude ci. We could say that the application of this operator A to vector |φ> extracts ci and multiplies it by the eigenvalue ai. And this is good because remember eigenvalues are what we actually observe in experiments. So now we can express the state of the systems in terms of things we can observe.

This transformed vector A|φ> is equal to the sum of products of eigenvalue ai, amplitude ci, and eigenvector |αi>, for all i.

A|φ> = ∑aicii>

Now we’re ready to get into what Auyang considers what we can properly consider properties of quantum systems. For some observable A and its operator, the sequence of complex numbers {aici} can be called an A-amplitude and is, using the eigenvalues, expressed in terms of the probability amplitude ci. And this is where Auyang locates the properties of quantum systems. She interprets the probability amplitude ci or the A-amplitude as the definite property or the value of a certain quantum system in a certain state for the property type A. And she makes the point that we shouldn’t try to imagine what the amplitudes and A-amplitudes describe because they are nothing like classical feature; “they are literally unimaginable”. But they are calculable. And that’s their crucial, property-type feature.

We might ask why we should locate definite properties in something that we can’t imagine. Classical properties like classical energy, position, and momentum are more easily envisioned, so these prospective, unimaginable quantum properties might seem unsatisfying. But this touches on Auyang’s general Kantian perspective on the sciences, which is that our understanding of scientific concepts relies on a complex underlying conceptual structure. And in this case that underlying conceptual structure includes things like vectors, Hilbert spaces, bases, eigenvectors, eigenvalues, and amplitudes. If that structure is required to comprehend the system it’s not unreasonable that the system’s definite properties would be expressed in terms of that structure.

With that mathematical overview let’s look at the concept of properties more closely and at our expectations of them. And here I’d like to just quote an extended passage directly from Auyang’s book because this is actually my favorite passage:

“In quantum interpretations, the ‘values’ are usually taken to be eigenvalues or spectral values, which can be directly revealed in experiments, although the revelation may involve some distortion so that the veracity postulate does not hold. It is beyond a reasonable doubt that quantum systems generally have no definite eigenvalues. However, this does not imply that they have no definite properties. The conclusion that they have none arises from the fallacious restriction of properties to classical properties, of which eigenvalues are instances. Sure, quantum systems have no classical properties. But why can’t they have quantum properties? Is it more reasonable to think that quantum mechanics is necessary because the world has properties that are not classical?”

“The no-property fallacy also stems from overlooking the fact that the conceptual structure of quantum mechanics is much richer than that of classical mechanics. In classical mechanics, the properties of a system are represented by the numerical values of functions, which assign real numbers to various states of the system. In quantum mechanics, functions are replaced by operators, which are structurally richer. A function is like a fish with only one swaying tail, its numerical value; an operator is like an octopus with many legs. Quantum mechanics employs the octopus with good reason, and we miss something important if we look only at the one leg that reminds us of the fishy tail. Quantum systems generally do not have definite eigenvalues, but they have other definite values. The stipulation that the values must be directly revealable in measurements confuses the empirical and physical meanings of properties.”

“I argue that we cannot give up the notion of objective properties. If we did, the quantum world would become a phantom and the application of quantum mechanics to practical situations sorcery. Are there predicates such that we can definitely say of a quantum system, it is such and so? Yes, the wavefunction is one. The wavefunction of a system is a definite predicate for it in the position representation. It is not the unique predicate; a predicate in the momentum representation does equally well. Quantum properties are none other than what the wavefunctions and predicates in other representations describe.”

And recall here that a wave function is another way of referring to the state of a quantum system. I think of this was moving things up a level. Or down a level depending on how you want to think of it. Regardless, at one level we have the eigenvalues that pop out with the application of an operator on a state vector. These are not definite properties of the system as a whole. In other words, the definite properties of the quantum system do not reside at this level. Rather they reside at the level prior to this, on which these outcomes depend. In the case of an atomically bound electron we could say that it is the orbital, the probability distribution of the electron’s location, that is a property of the quantum system, rather than any particular position. These sorts of properties have a lot more too them. As Auyang says, they are “structurally richer”. They’re not just values. They are amplitudes, from which we derive probabilities for various values. And what Auyang is saying is that there’s no reason not consider that the definite property of the quantum system.

Still, it is different from out classical notion of properties. So what is it that is common to both classical and quantum properties? Auyang borrows a term from Alfred Landé, proposing that a characteristic has empirical ramification if it is observable or “kickable”:

“Something is kickable if it can be kicked and kicks back, or it can be somehow physically manipulated and the manipulation produces observable effects. Presumably the property is remote and obscure if we must resort to the indirect kickability criterion alone. Thus kickability can only work in a well-developed theory in which the property is clearly defined, for we must be able to say specifically what we are kicking and how it is supposed to kick back.”

In the case of quantum properties we are indeed in a situation where the property is “remote and obscure”. But we also have recourse to “a well-developed theory in which the property is clearly defined”. So that puts us in a good position. Because of this it doesn’t matter if properties are easily visualizable. “Quantum properties are not visualizable, but this will no longer prevent them from being physical”. The physical surpasses what we are able to visualize.

So there is a well-developed conceptual structure that connects observables to the definite properties of the quantum system prior to these observables. To review a little how this structure and cascade of connections works:

We start with the most immediate aspect: what we actually observe, which enter into the conceptual structure as eigenvalues. Eigenvalues of an observable can be regarded as labels of the eigenstates. Eigenstates serve as axes of a coordinate system in the state space. This is an important point, so I’ll repeat it again in another way. As Auyang puts it: “An observable coordinates the quantum world in a particular way with its eigenstates, and formally correlates the quantum coordinate axes to classical indicators, the eigenvalues. An observable introduces a representation of the quantum state space by coordinatizing it.” So we have observations to eigenvalues, to eigenstates, to axes in a state space.

The coordinate system in the state space enables us to determine definite amplitudes. The state space is a vector space and any particular state or quantum system in this state space is a vector in this space. We can break this vector down into components which are expressed in terms of the coordinate system or basis, i.e. the eigenstates. This is the coefficient ci, which is a probability amplitude. This is why we’re able to determine definite amplitudes using the coordinate system. A quantum system has no definite eigenvalues but it does have definite amplitudes. When it’s broken down into its basis components a quantum state is series of eigenstate expansion, multiple terms that are added up to define the vector. Each of these terms has an amplitude associated with an eigenstate that is also associated with some observable. Practically, an indicator in the form of an eigenvalue is somehow triggered in measurements and experiments. And the probability of observing any particular eigenvalue will be defined by its amplitude. Specifically, the quantity |ci|2 is the probability that the eigenvalue ai is observed in a measurement of A on the state |φ>. But it is the probability amplitude ci that is the definite property of the quantum system rather than any particular eigenvalue that happens to be observed. What’s more, this is an objective property of the quantum system even in the absence of any experiment. As Auyang puts it: “Unperformed experiments have no results, but this does not imply that the quantum system on which the experiment might be performed has no properties.” Now to show the more complete cascade of kickability: we have physical observations, to eigenvalues, to eigenstates, to axes in a state space, to a state vector, to vector components, to component coefficients, to probability amplitudes. And it’s the probability amplitudes that are the definite properties of the quantum system.

The question of whether or not quantum systems have definite properties is a philosophical question rather than a question of physics, to the extent that those can be separated. It’s not necessary to engage in the philosophy in order to engage in the physics. One can measure eigenvalues and calculate probability amplitudes without worrying about whether any of them count as properties. But it’s arguably part of the scientific experience to step back on occasion to reflect on the big picture. To ask things like, “What is the nature of the work that we’re doing here?”, “What does all this tell us about the nature of reality?”, “Or about the way we conceptualize scientific theories?” For me one of the most fascinating insights prompted by quantum mechanics is of the necessity of the elaborate conceptual structures that support our understanding of the theory. To put it in Kantian terms, these conceptual structures are “transcendental” in the sense that they constitute the conditions that are presupposed and necessary for us to be able to understand the theory in the first place. And to me that seems quite significant.

Spacetime, Individuation, and Fiber Bundles

How can entities be picked out as individual and distinct entities? Sunny Auyang presents a Kantian model of spacetime as an absolute and indispensable structural scheme we project onto the world to organize it and to pick out individual elements, or events in it. Using fiber bundles she packs together a complex structure of individuating qualitative features that she links to individual points in spacetime.

I’d like to talk again about some stuff I’ve been reading in this book by Sunny Auyang, How is Quantum Field Theory Possible? Specifically in this latest chapter I read on the nature of space or spacetime and the possibility of individuation, individuation being the identification and distinction of entities as separate entities.

Both of these issues have a long history in the history of philosophy but Auyang focuses mostly on the work of the modern period of the last few centuries, most especially on Leibniz, Newton, and Kant. There’s a famous dichotomy or division between the models of space put forward by Leibniz and Newton. And the question there is whether space is an independently existing thing or just a way of conceptualizing the relations between actual entities, like their distances and orientations from each other. So Newton’s view was that space has an independent existence. Even if you took out all other entities in the universe space itself would still be there as its own thing. Also time. So both space and time are “absolute”. But for Leibniz these are relative or relational concepts. Lengths, areas, and volumes are relations between entities but if you take away the entities, the actual things there’s nothing left behind, no empty space. Now I’ve read that those are actually drastic simplifications of their views, which doesn’t surprise me. But regardless of that we can at least have those views in mind to start, with the understanding that they’re traditionally associated with Newton and Leibniz. Auyang actually divides both these views further, so that we have four; two Newton-type views and two Leibniz-type views. And I’ll just introduce those so we can use the descriptive names rather than these two proper names.

On the one side we have the substantival view and the absolute view. Spacetime is substantival if it exists independent of material entities. Spacetime is absolute if its concept is presupposed by the concept of individual entities and things. These are similar but slightly different ideas. Substantivalism is ontological, meaning it actually has to do with being, what is. Absoluteness is conceptual; it pertains to the way concepts fit together and what is necessary for certain concepts to work and be intelligible. These can coincide but they don’t have to. And Auyang is going to argue for a model of spacetime that is absolute but not substantival. So in her view spacetime is not a thing that exists independent of material entities but it is a concept that is required to conceptualize material entities.

On the other side Auyang also distinguishes between the relational view and the structural view. I think this is an even more subtle distinction. The difference between these two is a matter of logical priority, looking at what comes first. So recall that with the relational view the concept of space arises from the relations between entities. Dimensions like length, area, and volume are these relations that we perceive between the entities around us. They’re already there and we perceive them. The structural view is the Kantian view, from Immanuel Kant, that space, and we can say also spacetime, are concepts that we project onto the world to organize it bring structure to it. So we as subjects come first. I’m describing that a little differently than she does in the book but that’s the way it makes most sense for me to think about it. And I think it’s consistent with her view. And between these options Auyang is going to argue for a model of spacetime that is structural rather than relational. So it’s more the Kantian model. So bringing these two together her view of spacetime is absolute and structural. In other words, spacetime is a concept that is required for us to conceptualize material entities, and it is a structure that we project onto the world to organize it and make sense of it.

With that in place let’s get to individuation of entities. How do we say that a thing is the same thing across time, something that we can index or label? And how do we say of a thing that it is this thing and not some other thing? “An entity is an individual that can be singly picked out, referred to, and made into the subject of propositions.” Aristotle said that it incorporates two elements. It’s both a this and a what-it-is. These are the notions of individuality and kind. A specific entity is not only a thing but it is this thing. It’s indexed and labeled. It’s also a certain kind of thing. That doesn’t individuate the single entity from other members of that same kind but it distinguishes that class of entities as a kind. Then within that set of that kind of entity they must be further differentiated and identified as individuals. That gets very complex. Other philosophers instead have also argued for the importance of a cluster-of-qualities notion. An entity is no more than the sum of its qualities. If you get specific enough about your qualities maybe that’s all you need. Every entity has a unique spatio-temporal history at least, even if indistinguishable in all other qualities. At least we may so argue. So some important concepts here are individuality, kind, and qualities. These are ways of individuating.

So we’re going to look at a model of these entities. And the first thing to address is that we’re going to look at this through the lens of quantum field theory rather than classical mechanics. So the primary form of matter, the material entities I’ve been talking about before, shift from discrete mass points in space to continuous fields comprising discrete events. Auyang doesn’t mention this but it reminds me a little bit of Alfred North Whitehead’s process philosophy in which he substituted a substance ontology of things to a process ontology of events. Auyang’s quantum field theory is rather different from that, nevertheless, it was something that came to mind. So anyway, the basic entities we’re going to consider now are events.

A field is a continuous system. “The world of fields is full, in contrast to the mechanistic world, in which particles are separated by empty space.” Every point in a field is assigned a value. So say we have a field, that we’ll call the greek letter ψ, for every point x in that field there will be a value ψ(x). And that field variable ψ(x) doesn’t have to be scalar, i.e. just a number. It can be a vector, tensor, or spinor as well. Actually I’m most accustomed to thinking of field variables as vectors like with a gravitational field or an electric field. So with a gravitational field for instance every point in the field around mass M has a vector oriented toward mass M. And then the magnitude of those vectors varies with the distance from mass M. And that’s just an example, the field variable could be any number of things. And that’s important for individuation because we’re going to want to account for the qualities of an individual event with which we can distinguish it. But also one key idea to keep in mind is that the field variable ψ is indexed to some point x in the field. That’s another method of individuation.

So let’s look at how both qualities and numerical identity get taken up in Auyang’s model. To give a bit of a road map before diving into the details her model will include. She’s going to use 6 major pieces: D, G, π, M, x, and ψ.

D is what’s called the total space.
G is a local symmetry group.
π is a map.
M is a base space.
x is a position in the base space M.
And ψ(x) is an event.

All of this will be put together in a fiber bundle structure. And we’ll get into what all that means in a minute.

First let’s talk about symmetry groups, which will be this G in her model. The concept of the this-something, the individuality of events, is incorporated in field theories through two symmetry groups. Symmetry is a key idea in physics. A related term is invariance, also a very important concept. And it’s basically what it sounds like. It’s some property that doesn’t change. More specifically, we’re interested in the very particular circumstances under which it doesn’t change, called transformations. So you have some object, you transform it in some way – say you rotate it for example – the features that don’t change in that transformation are invariants. And this can tell us important things. The big conservation laws in physics come from invariants as we know from what is called Noether’s Theorem. For example, conservation of energy comes from time invariance. Conservation of momentum comes from translational invariance. Conservation of angular momentum comes from rotational invariance. Very significant. Okay, so backing up again to symmetry groups – that was the whole reason for getting into this. A symmetry group is the group of all transformations under which the object is invariant. Some objects have lots of symmetry – they’ll be invariant under many transformation – others have very little. But the key is that the group of all those transformations where it is invariant – that’s a symmetry group.

The two symmetry groups pertinent to the field theories here are the local symmetry group and the spatio-temporal symmetry group. And these embody different aspects of the individuation of entities. “The idea of kinds is embodied in the local symmetry group, which pertains not to spatio-temporal but to qualitative features. The symmetry group circumscribes a set of possible states and defines a natural kind.” So recall one of the important ideas for identification or individuation was quality. Well the state of an entity covers its qualities. But for localization and identification, its numerical identity, we need a global whole, rather than a local whole, and that is represented by a spatio-temporal symmetry group. “The identities of the events are the invariants in the spatio-temporal symmetry structure.” These two symmetries give us the quality and numerical identity of the entities.

To fit this all together Auyang presents a model for the structure of local symmetries. And she does this using fiber bundles. Fiber bundles are great mathematical tools. The most straightforward way I like to think about fiber bundles is that they are a way to relate single points in some base space to more complex structures in another space. And when I say “space” here these can be abstract spaces, though at least one of these in what follows, the base space, will in fact be a spatio-temporal space. The great thing about this is that it lets us sneak a lot of structure into a spatio-temporal position. And that’s good because we need a lot of structure for these individuating elements. A spatio-temporal position is just one of these individuating elements. We want to bring qualities in there too.

So let’s look at Auyang’s model. This is the featured image for this episode by the way if want to look at it. The objects D, G, and M are differential manifolds, which is basically just a kind of space or surface. These manifolds can be actually spatial or spatio-temporal, which will be the case with our base space M. But they can also be, and often are abstract, which will be the case for our total space D and our local symmetry group G in this model. The first manifold, our total space D, is a set of abstract qualities. So this is where we’re going to get the qualities for our entities from. Then she also has a local symmetry group, G, which is also a manifold. We can label the abstract qualities in D as θ, θ’, and so forth. “At this starting point, both D and G are abstract and meaningless. Our aim is to find the minimal conceptual structure in which we can recognize events as individuals”.

The symmetry group G acts on the total space D and collects subsets of elements in D that are equivalent to each other. Each of these subsets we’ll call a G orbit. The elements in a single G orbit are equivalent to each other. We can start with quality θ and θ’ – those will go into one G orbit. Then we can pick out ξ and ξ’. This divides D up into these G orbit subsets until all elements in D are accounted for. None of resultant G-orbits share common elements. D still has all the same elements as before but they are divided into these subsets. This is quite useful for our purposes of individuation. We have some organization here of all this information.

Next we can take a G orbit and introduce a map π that sends all elements in a G orbit, θ,  θ’, for example, sends all those elements onto a single point x. This point x is on another manifold M, a base space. There’s also an inverse map, π-1, that canonically assigns a unique element x in M to each G orbit in D. M is what’s called a quotient of D by the equivalence relation G. It’s not given in advance but falls out from D. Every spacetime point, x, in the spatio-temporal structure, M, is associated with an event, ψ(x), in the total space D. Speaking of this in terms of set theory, D becomes a set with an indexing set M.

So now we have all the pieces put together: D, G, π, M, x, and ψ. And to review, D is the total space, G is a local symmetry group, π is a map, M is a base space, x is a position in the base space M, and ψ(x) is an event. And what’s the significance of all this in the “real world”, so to speak? M is usually called spacetime and x is a point in spacetime, the spatio-temporal position of an event ψ(x). But the identity of an ψ(x) includes more than just it’s spatio-temporal position, even though it’s indexed to that position. All that extra information is in the total space D. It’s divided up by the local symmetry group G. And then it’s mapped onto the spacetime base space M by the map π. The cool thing about the fiber bundle is that it allows us to cram a lot of information into a single point in spacetime, or at least link it to a lot of extra information.

The main goal that Auyang is working toward with this model is individuation. And to do that she needs enough complexity to carry the kind and quality features of individual entities, as well as spatio-temporal position. What happens in this model is that a spacetime position, x, signifies the identity of an event ψ(x). x uniquely designates ψ(x) and marks it out from others. The symmetry group, G, whose features are typical of all ψ(x), signifies a kind; since it collects those features as group. Then the spatio-temporal structure, M, is a system for identifying individuals in that group. So this “sortal concept that individuates entities in a world involves two operations” that will mark out (1) kinds and (2) numerical identity. First the local symmetry group, G, forms identical equivalence classes of qualities for this notion of kinds. Second the projection map, π, introduces numerical identity for each of these equivalence classes. These together secure the individuality of an event, ψ(x).

One thing we can certainly say about this kind of model is that it is analyzable. Events and spacetime positions are not just given in this view. There’s complex interplay between spacetime positions of events and all the qualities of those events. This is what we get with field theories. Even if we look at the world in the most primitive level, as Auyung says, with field theories, “to articulate even this primitive world requires minimal conceptual structure more complicated than that in many philosophies, which regard sets of entities as given.” So is this necessary, are we just making things more complicated than they need to be? Quoting Auyang again: “Field theories have not added complications, they have made explicit the implicit assumptions taken for granted.” I’m not prepared to defend that point but I’m fine with going along with it for the time being.

To wrap things up let’s look at some ways for thinking about this spatio-temporal structure, M. The complexity of the full conceptual structure of this model (D, M, G, π) is what makes it analyzable and it enables us to examine M’s possible meaning. Auyang characteristically promotes a Kantian take on all this. This is to see M as a “scheme of individuation and identification that we project into the world via the inverse map π-1 and by which we present the world to ourselves as comprising distinct entities.” Recall that in Kant’s thought the world is intelligible to us only because we apply categories of understanding to the raw sense data we bring in, and we use these categories to organize it all and make sense of it. Auyung is saying that this is what M does; this is what the spatio-temporal structure, or our concept of spacetime does.

And this idea of space being what individuates things has a long history. For example, speaking of Kant, in Kant’s philosophy space is what makes identity and difference possible. Hermann Weyl called space the “principium individuationis”, which is really fun to say with the classical Latin pronunciation of the ‘v’. But that’s just this idea we’ve been talking about, individuation, the manner in which a thing is identified as distinguished from other things. Weyl also said space “makes the existence of numerically different things possible which are equal in every respect”. So it’s not just the qualities (non-spatial) that are important. You need space to distinguish entities that are otherwise identical. This doesn’t mean that space is substantival, some independently existing substance. But it is conceptually indispensable. So, say it is something that we bring to the scene, something we impose as an organizing tool. It’s still indispensable for the possibility of individuation. So it’s absolute in that sense.

So to review, I’ll put these in Kantian terms. We start off with what is “out there”, just this pre-conceptualized mass of stuff, our total space D. How is that intelligible? We come at it via a conceptual structure, the mental categories of space and time, or spacetime, M. Then we project these spatial and temporal conceptual categories onto the world using the inverse map π-1. This inverse map is able to pick out individual entities in the total space D that are distinguishable by an organizing operation of the local symmetry group G. The local symmetry group G has divided up the total space D into G-orbits with common elements. Our spatial and temporal categories pick these subsets out as events ψ(x) that are mapped onto spacetime M. And that brings the whole structure together in a way that we can see everything together and pick out individual events as individual elements.

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.