Todd Decker

Stokes’ Beautiful Theorem: Differential Forms, Boundaries, Exterior Derivatives, and Manifolds

Stokes’ Theorem in its general form is a remarkable theorem with many applications in calculus, starting with the Fundamental Theorem of Calculus. The pattern in each case is that the integral of a function over a region is equal to the integral of a related function over the boundary of the region. We can use information about the boundary of a region to get information about the entire region, which is both useful and mathematically elegant.

For the visual aids that go with this episode see the video version on YouTube.

I’ve been reviewing some calculus recently but I’ve been approaching it in a different way than I have before, like when I learned it in school to be an engineer. Instead of studying it for the purpose of solving specific problems I’m looking at it more from a high-level, trying to understand it conceptually to see the general structure and patterns. It’s in line with my philosophical penchant to take things up a level or look behind things at another level of abstraction. One of the patterns I’ve seen across calculus has been something that falls under the generalized version of Stokes’ Theorem. I find the generalized Stokes’ Theorem quite beautiful, in that way that mathematics can be beautiful. It can express concisely and compactly a concept that has broad and varied applications in its more particular forms.

I’ll go through the generalized Stokes’ Theorem and some of its special applications. Since a lot of this is better understood visually I’ve made the YouTube video to go along with this so those listening to the podcast might want to check it out as well if this stuff is hard to picture.

The generalized Stokes’ Theorem states that the integral of some differential form of dimension k-1 over the boundary of some orientable manifold of dimension k is equal to the integral of that differential form’s exterior derivative over the whole of that orientable manifold.

The four key concepts here (the ones listed in the subtitle of this episode) are:

1. Differential forms
2. Boundaries
3. Exterior derivatives
4. Manifolds

That’s very abstract, not that there’s anything wrong with that. We need to be abstract to be general. The key concept, the idea I want to drive home with this episode, is that under certain conditions information about a boundary can give you information about the entire region that it bounds. In the generalized form we’re getting information about a whole manifold from the boundary of that manifold. That will be the case in all these examples. But now let’s look at examples to illustrate. Interestingly enough this comes into play from the very beginning of calculus, with the Fundamental Theorem of Calculus.

The Fundamental Theorem of Calculus

As a very quick crash course in calculus for the uninitiated, in calculus the two most important operations are derivatives and integrals. When you take the derivative of a function it produces another function that tells you the instantaneous rate of change of the original function at any point. So for example, if you have an equation for distance from some starting point with respect to time, the derivative of that function will tell you what the velocity is at any point in time. Very useful. You can also do the opposite of that, which is an antiderivative, or integral. Say you were starting with the function of velocity with respect to time. You could take the antiderivative of that function and get a function for the position at any point in time. You’d just need to know what your starting point was and add that to it. One of the most important applications of these operations uses the Fundamental Theorem of Calculus.

The Fundamental Theorem of Calculus is probably the most recognizable thing to calculus students, even if they don’t remember that that’s what it was called. It’s the principle behind finding the area under a curve. For example, if you have a function for the velocity with respect to time plotted on a graph, the area under that curve, between the curve and the horizontal axis, sweeps out an area that will give you the value of the distance traveled between any two points in time you choose. The cool thing is that you only need to know the values of the antiderivative at your starting time and at your ending time. You don’t need to know the values in between them.

So finding the area under a curve, finding it analytically as opposed to numerically, is really quite a remarkable thing if you think about it. Because you’re basically taking some function, doing an operation on it, and then applying the resulting function only to the two points bounding the region you’re interested in. If you’re integrating from point a to point b you’re only paying attention to points a and b, not to any of the points in between them. But you’re still getting information about the whole region. You’re getting the area under all those points in between a and b. I think that’s quite remarkable. And that’s what happens in each particular version of Stokes’ Theorem. You’re able to get information about an entire region from its boundaries.

We might forget this sometimes if we’re using numerical methods more than analytical methods of integration. Using numerical methods like the trapezoidal or the rectangle method we actually do go in and add up all the regions between the boundaries that approximate the total area under the curve. But for analytical integration we don’t need to do that. We only need the antiderivative and the boundary points. And that tells us everything about the region in between those points. It seems almost magical.

Circling back to those four key concepts in the generalized Stokes’ Theorem, with the Fundamental Theorem of Calculus what we have is a differential form of dimension 0 and a manifold of dimension 1. For a function, lower case f(x), the antiderivative, upper case F, is the (0-form) differential form. The closed interval from a to b is a 1-dimensional manifold. (For the sake of simplicity think of a manifold as a surface of some dimension. A 1-dimensional manifold here being a line.) The boundaries a and b are 0-dimensional; they’re points.

And the exterior derivative is lower case f(x)dx.

Green’s Theorem

What other applications are there of this generalized Stokes’ Theorem? There’s also Green’s Theorem. Green’s Theorem has a similar form to the Fundamental Theorem of Calculus but instead of looking at a curve bounded by points we’re looking at a plane region bounded by a curve. With the Fundamental Theorem of Calculus we have the integral of a 0-dimensional differential form over the boundary of a 1-dimensional manifold. With Green’s Theorem we have the integral of a 1-dimensional differential form over the boundary of a 2-dimensional manifold.

As with all these theorems we’re looking at, in Green’s Theorem we are able to determine features of a region by looking at its boundaries. If we have a closed curve C that surrounds a region D we can figure out the area of D from the closed curve C. This is actually how planimeters work. And planimeters are pretty cool. A planimeter is a device that you can use to trace out, with a mechanical arm, a curve of any shape, and when you return to the position you started at it calculates the area that that shape encloses. This is exactly what Green’s theorem does.

So now to state Green’s Theorem: Say we have curve C, and functions M and N defined on a region containing D, the region enclosed by C. Green’s Theorem states that the double integral of partial derivative of M with respect to y, minus the partial derivative of L with respect to x, δM/δy – δL/δx over a region D is equal to the line integral of Ldx + Mdy over the curve C. This is that perimeter to area connection, where we can get an area, normally found using a double integral, from a perimeter, found using a line integral. The differential form is Ldx + Mdy. The region D is the two-dimensional manifold. The curve C is the 1-dimensional boundary of the manifold. And the exterior derivative is δM/δy – δL/δx.

The Divergence Theorem

We can bump all this up another dimension to get the Gauss’s Theorem, also known as the Divergence Theorem. With Green’s Theorem we have the integral of a 1-dimensional differential form over the boundary of a 2-dimensional manifold. With the Divergence Theorem we have the integral of a 2-dimensional differential form over the boundary of a 3-dimensional manifold.

With the Divergence Theorem we’re able to get information about a 3-dimensional region from it’s 2-dimensional boundary. Call the 3-dimensional region V and the 2-dimensional surface S. Then we have a vector field F. The Divergence Theorem states that the triple integral or volume integral of the divergence of vector field F over the 3-dimensional region V is equal to the surface integral of F over the surface S.

In terms of the generalized Stokes’ Theorem, with the Divergence Theorem the differential form is F·dS. The space E is the 3-dimensional manifold. The surface S is the 2-dimensional boundary of the manifold. And the exterior derivative is the divergence div F dV.

One way to understand this is that the net flux out of the region gives the sum of all sources of the field in a region. And this has many physical applications. For example, the Divergence Theorem has application to the first two of Maxwell’s four equations in physics. All four of Maxwell’s Equations have an integral form and a differential form. But the integral and differential forms are really equivalent. For the first two of Maxwell’s Equations the Divergence Theorem shows the equivalence between these two forms.

The first of Maxwell’s Equations, Gauss’s Law, relates an electric field to its source charge. This is a perfect application for the Divergence Theorem because the divergence operator gives information about sources and sinks. And an electric charge is a source. Gauss’s Law states that the net outflow of the electric field through any closed surface is proportional to the charge enclosed by the surface. In the integral form the way this is expressed is that the surface integral of electric field E over an enclosed boundary is equal to the charge divided by the permittivity of free space. In the differential form the way this is expressed is that the divergence of the electric field is equal to the charge density divided by the permittivity of free space.

The Divergence Theorem tells us that the triple integral of the divergence of electric field E over a volume is equal to the surface integral of the electric field E over the surface boundary. Since by the differential form of Gauss’s Law the divergence of the electric field is equal to the charge density divided by the permittivity of free space, if we take the triple integral of both sides we see that the triple integral of the divergence is equal to the charge divided by the permittivity of free space. By the integral form of Gauss’s Law the surface integral of electric field E is also equal to the charge divided by the permittivity of free space. So both the surface integral of the electric field E over the surface boundary and the triple integral of the divergence of the electric field E are equal to the charge divided by the permittivity of free space, and so they are equal to each other, which is exactly what the Divergence Theorem says. So these two forms are actually equivalent.

The second of Maxwell’s Equations, Gauss’s Law for Magnetism has a similar form but demonstrates that there are no magnetic monopoles. The surface integral of a magnetic field B over some surface S is always equal to 0. Magnetic field lines neither begin nor end but make loops or extend to infinity and back. Any magnetic field line that enters a given volume must somewhere exit that volume. In the integral form the way this is expressed is that the surface integral of magnetic field B over an enclosed boundary is equal to 0. In the differential form the way this is expressed is that the divergence of the magnetic field is equal to zero. The Divergence Theorem tells us that the triple integral of the divergence of magnetic field B over a volume is equal to the surface integral of the magnetic field B over the surface boundary. If we take the triple integral of the divergence of the magnetic field this is still equal to zero, as is the surface integral of the magnetic field over the enclosed boundary. So again these two forms are also equivalent.

Kelvin-Stokes’ Theorem

The last of the particular applications of the generalized Stokes’ Theorem is also called Stokes’ Theorem or Kelvin-Stokes’ Theorem. With Kelvin-Stokes’ Theorem as with Green’s Theorem we have the integral of a 1-dimensional differential form over the boundary of a 2-dimensional manifold, but in R3, i.e. 3-dimensional space. Given a vector field F the theorem states that the double integral or surface integral of the curl of the vector field over some surface is equal to the line integral of the vector field around the boundary of that surface. Here again, the boundary gives us information about the region inside it.

In terms of the generalized Stokes’ Theorem, with Kelvin-Stokes’ Theorem the differential form is F·dr. The surface S is the 2-dimensional manifold. The curve C is the 1-dimensional boundary of the manifold. And the exterior derivative is curl F·dS.

Curl is another vector operator, like divergence, and it’s easier to get the gist of it from physical examples, which we can get from the other two of Maxwell’s Equations.

The third of Maxwell’s Equations is also known as Faraday’s Law of Induction. Faraday’s Law describes how a time varying magnetic field creates, or induces, an electric field, which is the reason we’re able to generate electricity from turbines. In the integral form the way this is expressed is that the line integral of electric field E is equal to the negative derivative with respect to time of the surface integral of the magnetic field B. In the differential form the way this is expressed is that the curl of the electric field E is equal to the negative derivative of the magnetic field B with respect to time. Kelvin-Stokes’ Theorem tells us that the surface integral of the curl of the electric field E over some surface is equal to the line integral of the electric field E around the boundary of that surface. If we take the surface integral of the curl of the electric field E this is equal to the surface integral of the negative partial derivative of the magnetic field B with respect to time. And by the integral form of Faraday’s Law this is also equal to the line integral of the electric field around the surface boundary. So these two forms are also equivalent.

The fourth of Maxwell’s Equations is also known as Ampère’s Law. Ampère’s Law describes how a magnetic field can be generated by (a) an electric current and (b) a changing electric field. In the integral form the way this is expressed is that the line integral of magnetic field B is equal to the permeability of free space times the surface integral of current density J, plus the permittivity of free space, times the derivative with respect to time of the surface integral of the electric field E. In the differential form the way this is expressed is that the curl of the magnetic field B is equal to the permeability of free space times the current density J plus the permittivity of free space times the partial derivative of the electric field E with respect to time. Kelvin-Stokes’ Theorem tells us that the surface integral of the curl of the electric field E over some surface is equal to the line integral of the electric field E around the boundary of that surface. If we take the surface integral of the curl of the magnetic field B this is equal to the permeability of free space times the surface integral of current density J, plus the permittivity of free space, times the derivative with respect to time of the surface integral of the electric field E. And by the integral form of Ampere’s Law this is also equal to the line integral of magnetic field B around the surface boundary. So these two forms are also equivalent.

Maxwell’s Equations and Differential Forms

All of Maxwell’s equations actually simplify considerably in the language of differential forms. I’m just going to brush over this quickly without going into detail. We can describe both the electric and magnetic fields jointly by a 2-form, F, in a 4-dimensional spacetime manifold. And we can describe electric current by a 3-form, J. Then we’ll need the exterior derivative operator, d, and the Hodge star operator, *. And Maxwell’s Equations are just:

dF = 0
d*F = J

That’s it. And one benefit of this is that thinking of the equations in terms of differential forms lets them generalize more easily to manifolds and relativistic settings.

To summarize, the general pattern with all these forms of Stokes’ Theorem is that the integral of a function over a region is equal to the integral of a related function over the boundary of the region. We can get information about an entire region from its boundary. And this is something that applies in interesting ways at different dimensions. Mathematically it’s aesthetically satisfying and elegant.

“But Also For the Interests of Others”

“I gotta go home. What do I owe ya?”
“The real question, Eleanor, is what do we owe to each other?”
(The Good Place, Season 2, Episode 12)

What do we owe to each other? This is the question that runs through all of what is now one of my favorite television shows, The Good Place. This question has stuck in my head the last few days. And as a Christian it’s got me thinking about what it means to be a disciple of Christ and live in imitation of Christ.

In his letter to the Philippians Paul said: “Do nothing from selfishness or empty conceit, but with humility of mind regard one another as more important than yourselves; do not merely look out for your own personal interests, but also for the interests of others.” (Philippians 2:3-4) I think there is a radical shift in perspective, call it a new life, being born again, in coming to look out not for your own personal interests, but also for the interests of others. We could call it maturity, while noting that it’s a kind of maturity that we don’t just reach naturally. It’s not maturation of the old man into a more developed version of the old man. It’s a complete rebirth and transformation into a new creature.

For all of this Christ is the model and the means. Paul continues in his exhortation to the Philippians saying, “Have this attitude in yourselves which was also in Christ Jesus, who, although He existed in the form of God, did not regard equality with God a thing to be grasped, but emptied Himself, taking the form of a bond-servant, and being made in the likeness of men. Being found in appearance as a man, He humbled Himself by becoming obedient to the point of death, even death on a cross.” (Philippians 2:5-8) These are some of the most important verses in the New Testament and have been foundational to the development of the doctrine of Christ’s nature and Godhood. But what I’d like to focus on is the exhortation embedded in it; the ethic.

Jesus’s teachings repeatedly feature a theme of reversal, particularly in the ways that we esteem ourselves and others.

“And whoever exalts himself will be humbled, and he who humbles himself will be exalted.” (Matthew 23:12)

“For whoever wishes to save his life will lose it; but whoever loses his life for My sake will find it.” (Matthew 16:25)

There is a repeated message of shifting focus away from self and onto others. Paul said to the Philippians that this was exactly Christ did at a level fundamental to the very nature of his mortal existence. He “emptied Himself”; ἑαυτὸν ἐκένωσεν (heauton ekenōsen). That verb, κενόω (kenóō), “to empty out”, is significant in Christian theology for the “kenosis” of Christ, the ‘self-emptying’ of Jesus’ own will to become entirely receptive to the will of the Father. But this isn’t only teaching about Christ’s nature. Paul is calling for the Church to do this as well, to imitate Christ in his kenosis of self. Rather than being self-centered we are to de-center ourselves from our own circle of concerns, maybe even put ourselves on the outside edge of that circle, looking in, to center the interests of others.

Another thing Jesus said was that, “whoever wishes to become great among you shall be your servant” (Matthew 20:26). This could be understood, perhaps in the first instance, as a warning. If you try to be greater than others you’ll end up being a servant instead. But I think the positive interpretation, that we ought to act as servants to others, that this is true greatness, is also consistent with Jesus’s teachings. And it’s consistent with Paul’s message to the Philippians. Christ took “the form of a bond-servant”, a δοῦλος (doulos). And I think here of the image of Jesus kneeling and washing his own disciples’s feet.

What does it look like to “not merely look out for your own personal interests, but also for the interests of others”? A few years ago in my church’s worldwide general conference one of the church’s leaders, Dallin H. Oaks, quoted Alexandr Solzhenitsyn who said: “It is time, in the West, to defend not so much human rights as human obligations.” I come back to that idea a lot. Was Solzhenitsyn saying that we shouldn’t defend human rights? Certainly not. Having been a victim of the Soviet gulags himself he would be the last person to say that. But I think he was onto something quite astute. Rights don’t carry much force without human obligations. We have to think about others. We have to think about one another.

A community has to have people who look out for the interests of others. This is what Paul wanted to see among the Philippians. He said these things would make his “joy complete” (Philippians 2:2).

The author of comparative religion Karen Armstrong has focused a lot on kenosis in her writing and it is clearly on view in the 2009 Charter for Compassion that she spearheaded. The second sentence in that Charter states that, “Compassion impels us to work tirelessly to alleviate the suffering of our fellow creatures, to dethrone ourselves from the centre of our world and put another there.”

These are ideas that impact me deeply:

“Regard one another as more important than yourselves; do not merely look out for your own personal interests, but also for the interests of others.”

“Dethrone ourselves from the centre of our world and put another there.”

“What do we owe to each other?”

Education

Jack and Todd discuss education: the state of gifted education, how education can be structured in a way for students with various needs to get the kind of education they need, how to assess achievement for a broader range of skills, the ends and purposes of education.

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, c_i, of the basis vectors. So for vector |φ> the components will be c₁|α₁> and c₂|α₂>. In the more generalized form with an unspecified number of dimensions we can say that the vector |φ> is equal to the sum of c_i|α_i> for all i.

|φ> = ∑c_i|α_i>

The complex numbers c_i are amplitudes, or probability amplitudes, though we should note that it’s actually the square of the absolute value of c_i that is a probability. Specifically, the quantity |c_i|² is the probability that the eigenvalue a_i 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 c_i. We could say that the application of this operator A to vector |φ> extracts c_i and multiplies it by the eigenvalue a_i. 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 a_i, amplitude c_i, and eigenvector |α_i>, for all i.

A|φ> = ∑a_ic_i|α_i>

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 {a_ic_i} can be called an A-amplitude and is, using the eigenvalues, expressed in terms of the probability amplitude c_i. And this is where Auyang locates the properties of quantum systems. She interprets the probability amplitude c_i 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 c_i, 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 |c_i|² is the probability that the eigenvalue a_i is observed in a measurement of A on the state |φ>. But it is the probability amplitude c_i 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.