Quantum Mechanics of the Inverse Cube Force Law

1 May, 2026

In the last episode of my column in Notices of the American Mathematical Society, we looked at a particle moving in an attractive central force whose strength is proportional to the inverse cube of the distance from the origin. Among other things, we saw that a particle moving in such a force can spiral in to the origin in a finite time. But that was classical mechanics. What about quantum mechanics?

Here things get more tricky. The uncertainty principle tends to prevent the particle from falling in to the origin. But when the attractive force is strong enough, the particle can still fall in. We can make up a theory where the particle shoots back out, but there are choices involved: we need to say how the particle changes phase when it shoots back out. So there is not just a single theory, but many!

Why does the particle come back out? We can make up theories where it does not. In these theories time evolution is usually nonunitary: that is, the probability of finding the particle somewhere or other does not stay equal to 1, because the particle simply disappears when it hits the origin. Here we focus on theories where time evolution is unitary and the particle comes back out. Many people have written about these, running into ‘paradoxes’ when they weren’t careful enough. Only rather recently have things been straightened out.

Let us dig into the details. In quantum mechanics, the Hilbert space of states of a particle in \mathbb{R}^3 is L^2(\mathbb{R}^3). In a central force whose strength is proportional to 1/r^3, such a particle has a Hamiltonian of this form:

H = -\nabla^2 + c r^{-2}

The first term describes the particle’s kinetic energy, while the second describes its potential energy: remember, taking the gradient of an inverse square potential gives an inverse cube force. I have set some constants to 1 to remove irrelevant clutter, but we need the constant c to say how strong the force is. When c \, < \, 0, the force is attractive.

In this game, analysis is paramount. We should interpret H as a densely defined linear operator on L^2(\mathbb{R}^3). For this, we choose a dense linear subspace D \subset L^2(\mathbb{R}^3) and treat H as a linear map from D to L^2(\mathbb{R}^3). Different choices of D correspond to different physical assumptions: for example, assumptions about what happens when the particle falls into the origin.

To get unitary time evolution in quantum mechanics, we need the Hamiltonian to be self-adjoint. But adjoints of densely defined operators are tricky. Let us briefly recall how they work. Given a Hilbert space \mathcal{H} and a linear operator A from a dense linear subspace D(A) \subseteq \mathcal{H} to $\mathcal{H},$ we define D(A^*) to be the set of all \psi \in \mathcal{H} for which there exist \psi' \in \mathcal{H} such that

\langle \psi' , \phi \rangle = \langle \psi, A \phi \rangle \; \text{ for all } \; \phi \in D(A)

If such a vector \psi' exists, it is unique, and it depends linearly on \psi. Thus, for \psi \in D(A^\ast) we define A^\ast \psi to be the vector \psi' with the above property. The adjoint of A is then the linear operator A^\ast \colon D(A^\ast) \to \mathcal{H}. We say A is self-adjoint if A = A^\ast. We say that A is essentially self-adjoint if it has a unique extension to a self-adjoint operator. If it does, this extension must be A^\ast.

All this raises the question of whether the Hamiltonian H for the inverse cube force law can be made self-adjoint with a suitable choice of domain. It turns out we can always do it, but sometimes in more than one way. There are three regimes:

• c \ge 3/4. In this case we can start with the domain C_0^\infty(\mathbb{R}^3 - \{0\}) consisting of smooth functions that are compactly supported on \mathbb{R}^3 minus the origin. The operator H is unambiguously defined on this domain, and it is essentially self-adjoint.

• -1/4 \le c \, < \, 3/4. In this case H is still well-defined on the domain C_0^\infty(\mathbb{R}^3 - \{0\}), but it is not essentially self-adjoint. In fact, it admits more than one self-adjoint extension! However, H is bounded below: there is a constant E_0 such that

\langle \psi, H \psi \rangle \ge E_0 \langle \psi, \psi \rangle

for all \psi \in C_0^\infty(\mathbb{R}^3 - \{0\}). Physically, this means that the particle’s energy is bounded below by E_0. Mathematically, this implies that H has a canonical choice of self-adjoint extension called the ‘Friedrichs extension’, with the smallest possible domain. But there is another canonical choice, the ‘Krein extension’, with the largest possible domain.

• c \, < \, -1/4. In this case H is well-defined on the domain C_0^\infty(\mathbb{R}^3 - \{0\}), and it has more than one self-adjoint extension, but it is not bounded below.

These strange results demand explanation. For example, what is special about c =-1/4? In classical mechanics, the energy of a particle in the inverse cube force ceases to be bounded below as soon as c \, < \,0. Quantum mechanics is different. To get a lot of negative potential energy, the particle’s wavefunction must be peaked near the origin, but that gives it kinetic energy. The tradeoff is captured by Hardy’s inequality. This says that for any \psi\in C_0^\infty(\mathbb{R}^3) we have

\langle \psi, (-\nabla^2 - \tfrac{1}{4} r^{-2}) \psi \rangle \ge 0

This is why H is bounded below when c \ge -1/4.

On the other hand, the constant 1/4 in Hardy’s inequality cannot be improved, so if c \, < \, 1/4 we can find \psi with \langle \psi, H \psi \rangle \, < \, 0. Then we can use a remarkable property of the r^{-2} potential to show that H is not bounded below. Namely, H has a kind of symmetry under dilations. You can guess this by noting that both the Laplacian and r^{-2} have units of 1/length{}^2. Indeed, if you take any smooth function \psi, dilate it by a factor of \alpha, and then apply H, you get \alpha^{-2} times what you get if you do these operations in the other order. This implies that if

\langle \psi, H \psi \rangle = E \langle \psi, \psi \rangle

we can dilate \psi and get a function obeying the same equation with E replaced by \alpha^{-2} E. Thus, as soon as E can be negative, it can be made arbitrarily large and negative by choosing \alpha to be very small. Thus H is not bounded below.

Next, what is special about c = 3/4? This is more subtle. For any value of c \in \mathbb{R} we can find spherically symmetric solutions of

( -\nabla^2 + c r^{-2})\psi = i \psi

on \mathbb{R}^3 - \{0\} that are nonzero and smooth. When c \, < \, 3/4, and only in this case, some of these solutions \psi lie in L^2(\mathbb{R}^3). This dooms the chance of H being essentially self-adjoint, because it implies H^\ast \psi = i \psi. If H were essentially self-adjoint H^\ast would be self-adjoint, and it is easy to see that a self-adjoint operator cannot have i as an eigenvalue.

When c \, < \, \frac{3}{4} the operator H has more than one self-adjoint extension from C_0^\infty(\mathbb{R}^3 - \{0\}) to some larger domain. To classify these we can use separation of variables, writing \nabla^2 as a sum of a radial part and an angular part, assuming the angular dependence of \psi is given by a spherical harmonic Y_{\ell m}, and doing a change of variables u = \psi/r to reduce H to the ordinary differential operator

\displaystyle{ - \frac{d^2}{d r^2} + \left(c + \ell(\ell+1)\right) \frac{1}{r^2} }

on the half-line (0,\infty). We can completely classify self-adjoint extensions of this differential operators from C_0^\infty(0,\infty) to larger domains; the answer depends on c and \ell. A choice of self-adjoint extension is a choice of boundary conditions at r = 0, and this says how the phase of an incoming wave changes as it reflects off the origin and bounces back. Finally, we can assemble the results for different spherical harmonics to classify self-adjoint extensions of H.

There exist many self-adjoint extensions of H that respect the rotational symmetry of the inverse cube force law, but for c \, < \, -1/4 the extension must break the dilation symmetry discussed above. This is what physicists call an ‘anomaly’: a symmetry of a classical system that fails to be a symmetry of the corresponding quantum system. But intriguingly, for -5/4 \le c \, < \, -1/4 one can choose a Hamiltonian that is symmetrical under a discrete subgroup of dilations! For c < -5/4 one cannot even do this.

To explore this topic thoroughly, I recommend first this:

• S. Gopalakrishnan, Self-Adjointness and the Renormalization of Singular Potentials, B.A. Thesis, Amherst College, 2006.

then this:

• D. M. Gitman, I. V. Tyutin and B. L. Voronov, Self-adjoint extensions and spectral analysis in Calogero problem.

and finally this:

• J. Dereziński and S. Richard, On Schrödinger operators with inverse square potentials on the half-line, Ann. Henri Poincaré 18 (2017), 869–928.

The first is an excellent overview of problems associated to singular potentials, including the inverse cube force. The second delves into self-adjoint extensions of the ordinary differential operators mentioned above, and the third works them out with exquisite thoroughness.

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Posted by John Baez

Vector Meson Dominance

29 March, 2026

I’m only now learning about ‘vector meson dominance’—a big idea put forth by Sakurai and others around 1960.

Here’s a family of 9 mesons called the ‘vector nonet’. Each one is made of an up, down or strange quark and an antiup, antidown or antistrange antiquark. That’s 3 × 3 = 9 choices.



In this chart, S is strangeness (the number of strange quarks minus the number of antistrange antiquarks in the particle) and Q is electric charge. I’ll focus on the neutral rho meson, the ρ⁰, which has no strangeness and no charge.

But why are these called ‘vector’ mesons? It’s because the quark and antiquark have spin 1/2, and in this kind of meson their spins are lined up, so together they have spin 1. A spin-1/2 particle is described by a spinor, which is a bit weird, but spin-1 particle is described by something more familiar: a vector!

The most familiar spin-1 particle is a photon. And in fact, the photons around us are slightly contaminated by neutral rho mesons! That in fact is the point of vector meson dominance. But more on that later.

First, if you’ve read a bit about mesons, you may wonder why your friends the pion and kaon weren’t on that last chart. Don’t worry: they’re on this chart! This is the ‘pseudoscalar nonet’.



In these mesons, the spins of the quark and antiquark point in opposite directions, so the overall spin of these mesons is 0. That means they don’t change when you rotate them, like a ‘scalar’. But these mesons do change sign when you reflect them, because then you’re switching the quark and antiquark, and those are fermions so you get a minus sign whenever you switch two of them. So these mesons are ‘pseudoscalars’.

If you don’t get that, don’t worry. I’m going to tell the tale of rho mesons and especially the neutral one, the ρ⁰.



A photon will sometimes momentarily split into a quark-antiquark pair. Since the neutral rho meson is the lightest meson with the same charge, spin and other quantum numbers as a photon, this quark-antiquark pair will usually be a neutral rho! This is basic idea behind ‘vector meson dominance’.

In short, the light you see around you is subtly spiced by a slight mix of neutral rho mesons!

More precisely, real-world photons are a superposition of the ‘bare’ photons we’d have in a world without quarks, and neutral rho mesons.

But you might ask: how do we know this?

When you shoot a low-energy photon at a proton, its wavelength is long, so it sees the proton almost as a point particle.

But a high-energy photon has a short wavelength, so it notices that the proton is made of quarks. And the photon may interact with these as if it were a rho meson—because sometimes it is! This changes how high-energy photons interact with protons, in a noticeable way.

The same thing happens when you slam charged pions at each other. You’d expect them to interact electromagnetically, by exchanging a photon. But if you collide them at high energies you get deviations from purely electromagnetic behavior, since the photon is slightly contaminated by a bit of neutral rho!

In fact this is how the neutral rho was found in the first place. In 1959, William Franzer and Jose Fulco used results of pion collisions to correctly predict the existence and mass of the neutral rho!



They used a lot of cool math, too—complex analysis:

• William R. Frazer and Jose R. Fulco, Effect of a pion-pion scattering resonance on nucleon structure, Phys. Rev. Lett. 2 (1959), 365–368.

Then in 1960, Sakurai argued that the three rho mesons ρ⁺,ρ⁰,ρ⁻ form an SU(2) gauge field!

The idea is this: since they’re vector mesons each one is described by a vector field, or more precisely a 1-form. But these rho mesons are made only of up and down quarks and antiquarks—not strange ones. And isospin SU(2) is a symmetry group that mixes up and down quarks. So we expect SU(2) to act on the three rho mesons, and it does: it acts on them just like it does on its Lie algebra 𝔰𝔲(2), which is 3-dimensional.

So: we can combine these 3 vector mesons into an 𝔰𝔲(2)-valued 1-form… which describes an 𝔰𝔲(2) connection! If you don’t know what I mean, just take my word for it: this is how gauge theory works.

Now, Sakurai’s paper showed up before quantum chromodynamics appeared (1973), or even quarks (1964). But Yang–Mills theory had been known since 1954, so it was natural for him to cook up a Yang–Mills theory with rho mesons as the gauge bosons.

Just one big problem: they’re not massless, as Yang–Mills theory says they should be.

This didn’t stop Sakurai. He tried to treat the rho mesons as gauge bosons in a Yang–Mills theory of the nuclear force, and give them a mass ‘by hand’.

You can tell he was very excited, because he starts by mocking existing work on particle physics, with the help of a long quote by Feynman:

• J.J. Sakurai, Theory of strong interactions, Ann. Phys. 11 (1960), 1–48.

Sakurai’s theory had successes but also problems.

The Higgs mechanism for giving gauge bosons mass was discovered around 1964. People tried it for the rho mesons, but it was never clear which particle should play the role of the Higgs boson!

Only in 1985, after quantum chromodynamics had solved the fundamental problem of nuclear forces, did people come up with a nice approximate theory in which the rho mesons were gauge bosons for the strong force, with a Higgs serving to give them mass.

• M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, Is the ρ meson a dynamical gauge boson of hidden local symmetry?, Phys. Rev. Lett. 25 (1985), 1215–1218.

Later a subset of these authors developed a theory where all 9 vector mesons serve as gauge bosons for a U(3) gauge theory:

• Masako Bando, Taichiro Kugo and Koichi Yamawaki, Composite gauge bosons and “low energy theorems” of hidden local symmetries,
Prog. Theor. Phys. 73 (1985), 1541–1559.

In my youthful attempts to learn particle physics I skipped over most of the long struggle to understand mesons, and went straight for the Standard Model. And that’s what many textbooks do, too. But this misses a lot of the fun, and a lot of physics that’s important even now. I just learned this stuff about the rho mesons today, and I find it very exciting!

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Posted by John Baez

Geometry and the Exceptional Jordan Algebra

27 March, 2026

I’m giving a talk online tomorrow at the 2026 Spring Southeastern Sectional Meeting of the American Mathematical Society, in the Special Session on Non-Associative Rings and Algebras. The organizers are Layla Sorkatti and Kenneth Price. I doubt the talk will be recorded, but here are my slides:

• Projective geometry and the exceptional Jordan algebra.

Abstract. Dubois-Violette and Todorov noticed that the gauge group of the Standard Model of particle physics is the intersection of two maximal subgroups of \text{F}_4, which is the automorphism group of the exceptional Jordan algebra \mathfrak{h}_3(\mathbb{O}). Here we conjecture that these can be taken to be any subgroups preserving copies of \mathfrak{h}_2(\mathbb{O}) and \mathfrak{h}_3(\mathbb{C}) that intersect in a copy of \mathfrak{h}_2(\mathbb{C}). Given this, we show that the Standard Model gauge group consists of all isometries of the octonionic projective plane that preserve an octonionic projective line and a complex projective plane intersecting in a complex projective line. This is joint work with Paul Schwahn.

This is an introductory talk for mathematicians. Physicists may prefer the two talks here. Those go much further in some ways, but they don’t cover the new ideas that Paul Schwahn and I are in the midst of working on.

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Posted by John Baez

Standard Model 7: Pions

23 March, 2026

This time I’m talking about pions:

Pions were a revolutionary discovery in the 1930s—part of the first wave of the ‘particle zoo’—but I’m explaining them as a way to work toward the math and physics concepts needed for the Standard Model.

As soon as the neutron was discovered in 1932, Heisenberg invented the idea of ‘isospin’, and the idea that the proton and neutron are two different isospin states of a single particle, the ‘nucleon’. This is why I spent 3 videos explaining the math of spin-1/2 particles: in order to talk about isospin.

Three years later, Yukawa came up with the idea that the force holding nuclei together is carried by a new particle. Even better, he predicted the mass of this yet-unseen particle! It’s a fun bit of physics but also a step toward the concept of gauge bosons.

Later, the pion was discovered: in fact, three kinds of pions! I begin to explain how these three pions form a basis of \mathfrak{sl}(2,\mathbb{C}) just as the proton and neutron form a basis of \mathbb{C}^2.

These theories are outdated, but their math gets reused in the Standard Model. We’ll eventually get around to that!

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Posted by John Baez

Standard Model 6: Pauli Matrices

16 March, 2026

Wolfgang Pauli invented his famous matrices to describe the angular momentum of a spin-1/2 particle back in 1927. You’ll see them in most courses on quantum mechanics. We tend to take them for granted. But where do they come from? Here I derive them from scratch!

There are lots of ways to derive them, and the method I use is not ultimately the best, but it’s the easiest—given that we already have a recipe to describe states of a spin-1/2 particle where it spins in any direction we want.

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Standard Model 4: Quantum Physics

6 March, 2026

This is a crash course on the basic principles of quantum physics! In a self-contained way, I explain quantum states and the basic rule for computing probabilities.

It was a fun challenge stripping down everything to the bare minimum. Of course there is much more to say, but I was focused on leaving out everything that was not absolutely essential—to get to the real core of things.

There’s a huge fog of confusion surrounding most popular introductions to quantum mechanics, and I wanted to avoid all that. To do this, we have to use language in a pretty careful way.

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Posted by John Baez

Sylvester and Clifford on Curved Space

10 January, 2026

Einstein realized that gravity is due to the curvature of spacetime, but let’s go back earlier:

On the 18th of August 1869, the eminent mathematician Sylvester gave a speech arguing that geometry is not separate from physics. He later published this speech in the journal Nature, and added a footnote raising the possibility that space is curved:

the laws of motion accepted as fact, suffice to prove in a general way that the space we live in is a flat or level space […], our existence therein being assimilable to the life of the bookworm in a flat page; but what if the page should be undergoing a process of gradual bending into a curved form?

Then, even more dramatically, he announced that the mathematician Clifford had been studying this!

Mr. W. K. Clifford has indulged in more remarkable speculations as the possibility of our being able to infer, from certain unexplained phenomena of light and magnetism, the fact of our level space of three dimensions being in the act of undergoing in space of four dimensions (space as inconceivable to us as our space to the supposititious bookworm) a distortion analogous to the rumpling of the page.

This started a flame war in letters to Nature which the editor eventually shut off, saying “this correspondence must now cease”. Clifford later wrote about his theories in a famous short paper:

• William Clifford, On the space-theory of matter, Proceedings of the Cambridge Philosophical Society 2 (1876), 157–158.

It’s so short I can show you it in its entirety:

Riemann has shewn that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space in which we live belongs. In particular, the axioms of plane geometry are true within the limits of experiment on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges and furrows, upon which (the total curvature not being zero) these axioms are not true. Similarly, he says, although the axioms of solid geometry are true within the limits of experiment for finite portions of our space, yet we have no reason to conclude that they are true for very small portions; and if any help can be got thereby for the explanation of physical phenomena, we may have reason to conclude that they are not true for very small portions of space.

I wish here to indicate a manner in which these speculations may be applied to the investigation of physical phenomena. I hold in fact

(1) That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them.

(2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave.

(3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or etherial.

(4) That in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity.

I am endeavouring in a general way to explain the laws of double refraction on this hypothesis, but have not yet arrived at any results sufficiently decisive to be communicated.

To my surprise, the following paper argues that Clifford did experiments to test his ideas by measuring the polarization of the skylight during a solar eclipse in Sicily on December 22, 1870:

• S. Galindo and Jorge L. Cervantes-Cota, Clifford’s attempt to test his gravitation hypothesis.

Clifford did indeed go on such an expedition, and did indeed try to measure the polarization of skylight as the Moon passed the Sun. I don’t know of any record of him saying why he did it.

I’ll skip everything the above paper says about why the polarization of skylight was interesting and mysterious in the 1800s, and quote just a small bit:

The English Eclipse Expedition set off earlier in December 1870, on the steamship H.M.S. Psyche scheduled for a stopover at Naples before continuing to Syracuse in Sicily. Unfortunately before arriving to her final call, the ship struck rocks and was wrecked off Catania. Fortunately all instruments and members of the party were saved without injury.

Originally it was the intention of the expedition to establish in Syracuse their head-quarters, but in view of the wreckage the group set up their base camp at Catania. There the expedition split up into three groups. The group that included Clifford put up an observatory in Augusta near Catania. The leader of this group was William Grylls Adams, professor of Natural Philosophy at King’s College, London.

In a report written by Prof. Adams, describing the expedition, we learn that the day of the eclipse, just before the time of totality, “… a dense cloud came over the Moon and shut out the whole, so that it was doubtful whether the Moon or the clouds first eclipsed the Sun […] Mr. Clifford observed light polarized on the cloud to the right and left and over the Moon, in a horizontal plane through the Moon’s centre [….] It will be seen from Mr. Clifford’s observations that the plane of polarization by the cloud…was nearly at right angles to the motion of the Sun”.

As was to be expected, Clifford’s eclipse observations on polarization did not produce any result. His prime intention, of detecting angular changes of the polarization plane due to the curving of space by the Moon in its transit across the Sun´s disk, was not fulfilled. At most he confirmed the already known information, i.e. the skylight polarization plane moves at right angles to the Sun anti-Sun direction.

This is a remarkable prefiguring of Eddington’s later voyage to the West African island of Principe to measure the bending of starlight during an eclipse of the Sun in 1919. Just one of many stories in the amazing prehistory of general relativity!

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Posted by John Baez

Standard Model 3: Qubits

10 November, 2025

Physics is really bizarre and wonderful. Here I start explaining why the Standard Model has U(1) × SU(2) × SU(3) as its symmetry group. But I don’t assume you know anything about groups or quantum mechanics! So I have to start at the beginning: how the electromagnetic, weak, and strong force are connected to the numbers 1, 2, and 3. It’s all about quunits, qubits and qutrits.

You’ve heard of bits, which describe a binary alternative, like 0 and 1. You’ve probably heard about qubits, which are the quantum version of bits. The weak force is connected to qubits where the 2 choices are called “isospin up” and “isospin down”. The most familiar example is the choice between a proton and a neutron. A better example is the choice between an up quark and a down quark.

The strong force is connected to qutrits—the quantum version of a choice between 3 alternatives. In physics these are whimsically called “red”, “green” and “blue”. Quarks come in 3 colors like this.

The electromagnetic force is connected to “quunits” – the quantum version of a choice between just one alternative. It may seem like that’s no choice at all! But quantum mechanics is weird: there’s just one choice, but you can still rotate that choice.

Yes, I know this stuff sounds crazy. But this is how the world actually works. I start explaining it here, and I’ll keep on until it’s all laid out quite precisely.

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Posted by John Baez

The Inverse Cube Force Law

5 November, 2025

Newton’s Principia is famous for its investigations of the inverse square force law for gravity. But in this book Newton also did something that remained little-known until fairly recently. He figured out what kind of central force exerted upon a particle can rescale its angular velocity by a constant factor without affecting its radial motion. This turns out to be a force obeying an inverse cube law.

Given a particle in Euclidean space, a central force is a force that points toward or away from the origin and depends only on the particle’s distance from the origin. If the particle’s position at time t is \mathbf{r}(t) \in \mathbb{R}^n and its mass is some number m > 0, we have

m \, \ddot{\mathbf{r}}(t) = F(r(t)) \,\hat{\mathbf{r}}(t)

where \hat{\mathbf{r}}(t) is a unit vector pointing outward from the origin at the point \mathbf{r}(t). A particle obeying this equation always moves in a plane through the origin, so we can use polar coordinates and write the particle’s position as \bigl(r(t), \theta(t)\bigr). With some calculation one can show the particle’s distance from the origin, r(t), obeys

\displaystyle{ m \ddot r(t) = F(r(t)) + \frac{L^2}{mr(t)^3} \qquad \qquad \qquad \qquad (1) }

Here L = mr(t)^2 \dot \theta(t), the particle’s angular momentum, is constant in time. The second term in equation (1) says that the particle’s distance from the origin changes as if there were an additional force pushing it outward. This is a “fictitious force”, an artifact of working in polar coordinates. It is called the centrifugal force. And it obeys an inverse cube force law!

This explains Newton’s observation. Let us see why. Suppose that we have two particles moving in two different central forces F_1 and F_2, each obeying a version of equation (1), with the same mass m and the same radial motion r(t), but different angular momenta L_1 and L_2. Then we must have

\displaystyle{ F_1(r(t)) + \frac{L_1^2}{mr(t)^3} = F_2(r(t)) + \frac{L_2^2}{mr(t)^3} }

If the particle’s angular velocities are proportional then L_2 = kL_1 for some constant k, so

\displaystyle{ F_2(r_1(t)) - F_1(r(t)) = \frac{(k^2 - 1)L_1^2}{mr(t)^3} }

This says that F_2 equals F_1 plus an additional inverse cube force.

A particle’s motion in an inverse cube force has curious features. First compare Newtonian gravity, which is an attractive inverse square force, say F(r) = -c/r^2 with c > 0. In this case we have

\displaystyle{ m \ddot r(t) = -\frac{c}{r(t)^2} + \frac{L^2}{mr(t)^3 } }

Because 1/r^3 grows faster than 1/r^2 as r \downarrow 0, as long as the angular momentum L is nonzero the repulsion of the centrifugal force will beat the attraction of gravity for sufficiently small r, and the particle will not fall in to the origin. The same is true for any attractive force F(r) = -c/r^p with p < 3. But an attractive inverse cube force can overcome the centrifugal force and make a particle fall in to the origin.

In fact there are three qualitatively different possibilities for the motion of a particle in an attractive inverse cube force F(r) = -c/r^3, depending on the value of c. With work we can solve for 1/r as a function of \theta (which is easier than solving for r). There are three cases depending on the value of

\displaystyle{ \omega^2 = 1 - \frac{cm}{L^2} }

vaguely analogous to the elliptical, parabolic and hyperbolic orbits of a particle in an inverse square force law:

\displaystyle{ \frac{1}{r(\theta)} } = \left\{ \begin{array}{lcl} A \cos(\omega \theta) + B \sin(\omega \theta) & \text{if} & \omega^2 > 0 \\ \\ A + B \theta & \text{if} & \omega = 0 \\ \\ A e^{|\omega| \theta} + B e^{-|\omega| \theta} & \text{if} & \omega^2 < 0 \end{array} \right.

The third case occurs when the attractive inverse cube force is strong enough to overcome the centrifugal force: c > L^2/m. Then the particle can spiral in to its doom, hitting the origin in a finite amount of time after infinitely many orbits, like this:

All three curves above are called Cotes spirals, after Roger Cotes’ work on the inverse cube force law, published posthumously in 1722. Cotes seems to have been the first to compute the derivative of the sine function. After Cotes’ death at the age of 33, Newton supposedly said “If he had lived we would have known something.”

The subtlety of the inverse cube force law is greatly heightened when we study it using quantum rather than classical mechanics. Here if c is too large the theory is ill-defined, because there is no reasonable choice of self-adjoint Hamiltonian. If c is smaller the theory is well-behaved. But at a certain borderline point it exhibits a remarkable property: spontaneous breaking of scaling symmetry. I hope to discuss this in my next column.

For more on the inverse cube force law, see:

• N. Grossman, The Sheer Joy of Celestial Mechanics, Birkhäuser, Basel, 1996, p. 34.

For more on Newton’s work involving the inverse cube force law, see:

• Wikipedia, Newton’s theorem of revolving orbits.

• S. Chandrasekhar, Newton’s Principia for the Common Reader, Oxford U. Press, Oxford, 1995, pp. 183–200.

Cotes’ book is

• Roger Cotes, Harmonia Mensuarum, Cambridge, 1722.

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Posted by John Baez

Standard Model 2: The Big Ideas

3 November, 2025

Check out my video on the big ideas that go into the Standard Model of particle physics!

In the late 1800s physics had 3 main pillars: classical mechanics, statistical mechanics and electromagnetism. But they contradict each other! That was actually good – because resolving the contradictions helped lead us to special relativity and quantum mechanics.

I explain how this worked, or more precisely how it could have worked: the actual history is far more messy. For example, Planck and Einstein weren’t really thinking about the ultraviolet catastrophe when they came up with the idea that the energy of light comes in discrete packets:

• Helge Kragh, Max Planck: the reluctant revolutionary, Physics World, 1 December 2000.

Then, I sketch out how deeper thoughts on electromagnetism led us to the concept of ‘gauge theory’, which is the basis for the Standard Model.

This is a very quick intro, just to map out the territory. I’ll go into more detail later.

By the way, if you prefer to avoid YouTube, you can watch my videos at the University of Edinburgh:

• Edinburgh Explorations.

Leave a Comment » | physics | Permalink
Posted by John Baez

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