# nLab On Vortex Atoms

Contents

### Context

#### Knot theory

knot theory

Examples/classes:

Types

knot invariants

Related concepts:

category: knot theory

## Surveys, textbooks and lecture notes

The text

• On Vortex Atoms

Proceedings of the Royal Society of Edinburgh, Vol. VI, pp. 94-105.

1867

(web)

presented the hypothesis, developed with Peter Tait, that atoms (elementary particles) are fundamentally knotted vortices in some spacetime-filling substance (the “aether”), specifically that the chemical elements were classified by knot-types.

# Contents

## Impact

As a literal theory of physics the vortex atom hypothesis lasted no more than 30 years, with Thomson himself giving up on it by 1890 (Kragh 02, p. 34). But the vortex atom hypothesis prompted Peter Tait to begin the classification of knots, initiating what became the active modern mathematical field of knot theory.

This history has at times been invoked as a cautionary tale, to warn against contemporary speculations in physics, such as string theory, which may also seem to be mathematically compelling but with uncertain vindication by experiment.

However, on closer inspection there are some ironic twist and turns to this reading:

### Similarity to the Skyrme model for baryons

First, recall that in contemporary understanding (but see below), the chemical elements are (not knots in the aether, as Kelvin hypothesized, but ) types of atomic nuclei, which are bound states of protons and neutrons that, in turn, are thought to be bound states of “constituent quarks” (under the “strong nuclear force”), supposedly described by quantum chromodynamics (QCD).

The irony here is that – due to the unresolved “confinement”- or “mass gap problem” – no good theoretical derivation of these – or any other hadronic bound states – from constituent quarks in QCD is available! The constituent quark-hypothesis works to some extent in computer experiment (see e.g. Durr et al. 2009), but basic properties even of the most common baryons such as the proton remain unexplained (see e.g. the proton spin crisis and the proton radius problem); and experiments such as the LHC keep detecting a long list of novel hadrons which have neither been predicted nor understood by QCD – see e.g. Koppenburg 2021, who accurately reflects the prevalent attitude to this state of affairs by asserting that:

“[Hadron] properties follow from yet unsolved mysteries of the strong interaction” [sic]

In view of this rather disappointing reality, caution may rather be due with respect to dismissing alternative fundamental approaches out of hand.

Indeed, a fairly successful alternative theory for baryon bound states exists – the Skyrme model – which arguably is conceptually closer to the intuition behind Kelvin’s “vortex atoms” than to the quark-model – the main difference being “just” the dimensionality and variance of the “knotted structure”:

Where a knot (in Kelvin’s proposal) is the isotopy class of a map from the circle (hence the “1-sphere”) into 3d Euclidean space (1-point compactified to the 3-sphere), so a Skyrmion has underlying it the homotopy class of a map from 3d space/3-sphere to the group manifold of SU(2), which is itself a 3-sphere. Concretely, the homotopy class of this Skyrmion-map models the global higher-dimensional “knotting” in the pion field, effectively witnessing each baryon state as a soliton in a spacetime-filling pion-field, with its baryon number – hence its position in the periodic table of chemical elements – being the winding number of the map!

(Notice that it is the codomain 3-sphere which is fixed by the physics in this picture, not the domain: On a general spacetime-manifold other than Minkowski spacetime, the Skyrmion-number will be in its 3-Cohomotopy-set, vanishing at infinity.)

Therefore, it is not hard to imagine that Kelvin would have embraced this Skyrmion picture of atoms as confirming the basic intuition behind his proposal. To the extent that the Skyrmion-model is indeed a correct description of baryons, the conclusion could be that Kelvin’s proposal suffered not from having too much guidance by abstract mathematical argument, but too little (missing the full power of homotopy theory beyond plain knot theory). Indeed, according to Ranada-Trueba 01, p. 200:

Skyrme had studied with attention Kelvin’s ideas on vortex atoms.

To add to the irony, it looks like the Skyrme model comes to itself (only) in a context of Randall-Sundum-like non-compact Kaluza-Klein theory and/or string theory, where it finds both refinement compatible with experiment as well as conceptual explanation, potentially connecting it, logically, to all the rest of physics:

Namely, the original model proposed by Skryme 1962 provided a decent qualitative match to low-lying baryons, but the predicted binding energies quickly deviated from observation as the baryon number increased. This was eventually fixed (via an observation due to Atiyah & Manton 1989) in a rather remarkable way: by postulating that Skyrmions are in fact the “non-compact” KK-compactification of instantons in a higher-dimensional gauge theory (D=5 Yang-Mills theory):

In this picture, the lowest KK-mode of the instantons of the higher dimensional “hiddengauge field reproduces Skyrme’s original model, while the higher KK-modes provide corrections (corresponding to adjoining vector meson-contributions to the pion-field) that incrementally move the predicted binding energies ever closer to the observed values (Naya & Sutcliffe 2918a, 2018b, for more see the references at hadrons as KK-modes of 5d Yang-Mills theory).

Moreover, this higher-dimensional refinement of the Skyrme model naturally appears within intersecting brane models within string theory, particularly so in the Witten-Sakai-Sugimoto model for realizing quantum chromodynamics on branes in string theory. In this context the Skyrme model has become part of a broader approach of providing theoretical first-principles foundations for non-perturbative confined quantum chromodynamics (including not just baryons but also mesons, glueballs, etc.), now often known as holographic QCD, or similar.

(In final irony, these results remain almost as underappreciated among string theorists as among the particle/high energy physics-community at large.)

Further reading along these lines may be found in Rho & Zahed 2016 and Manton 2022.

### Similarity with Descartes’ thoughts

According to the Routledge encyclopedia of Philosophy here:

Descartes also rejected atoms and the void, the two central doctrines of the atomists, an ancient school of philosophy whose revival by Gassendi and others constituted a major rival among contemporary mechanists. Because there can be no extension without an extended substance, namely body, there can be no space without body, Descartes argued. His world was a plenum, and all motion must ultimately be circular, since one body must give way to another in order for there to be a place for it to enter ( Principles II: §§2–19, 33). Against atoms, he argued that extension is by its nature indefinitely divisible: no piece of matter in its nature indivisible ( Principles II: §20). Yet he agreed that, since bodies are simply matter, the apparent diversity between them must be explicable in terms of the size, shape and motion of the small parts that make them ( Principles II: §§23, 64) (see G. W. Leibniz §4 ).

However, according to (Kragh 02, p. 33):

In spite of the similarities, there are marked differences between the Victorian theory and Descartes’s conception of matter. Thus, although Descartes’s plenum was indefinitely divisible, his ethereal vortices nonetheless consisted of tiny particles in whirling motion. It was non-atomistic, yet particulate. Moreover, the French philosopher assumed three different species of matter, corresponding to emission, transmission, and reflection of light (luminous, “subtle”, and material particles). The vortex theory, on the other hand, was strictly a unitary continuum theory.

## Similarity to concepts of modern particle physics

### Baryogenesis

It is however striking that the modern concept of baryogenesis via the chiral anomaly and its sensitivity to instantons is not too far away from Kelvin’s intuition.

To play this out in the most pronounced scenario, consider, for the sake of it, a Hartle-Hawking no-boundary spacetime carrying $N$ Yang-Mills instantons. Notice that an instanton is in a precise sense the modern higher dimensional and gauge theoretic version of what Kelvin knew as a fluid vortex.

Then the non-conservation law of the baryon conserved current due to the chiral anomaly says precisely the following: the net baryon number in the early universe is a steadily increasing number – this is the modern mechanism of baryogenesis – such that as one approaches the asymptotically late time after the “Big Bang” this number converges onto the integer $N$, the number of instantons.

Hence while in the modern picture of baryogenesis via gauge anomaly an elementary particle is not literally identified with an instanton, nevertheless each instanton induces precisely one net baryon.

(If one doesn’t want to consider a Hartle-Hawking-type Euclidean no-boundary spacetime but instead a globally hyperbolic spacetime then the conclusion still holds, just not relative to vanishing baryon number at the “south pole” of the cosmic 4-sphere, but relative to the net baryon number at any chosen spatial reference slice. )

For more on this see at baryogenesis – Exposition.

## References

Review and discussion:

Discussion in comparison to knotted states in superconductors:

• Filipp N. Rybakov, Julien Garaud, Egor Babaev, Kelvin knots in superconducting state, Phys. Rev. B 100, 094515 (2019) (arXiv:1807.02509)

Discussion in relation to skyrmions:

• Antonio Ranada, J. Trueba, Force Lines, Vortex Atoms, Topology, and Physics, Section I.A In: Topological Electromagnetism with hidden nonlinearlity in: Myron Evans (ed.) Modern Nonlinear Optics – Part 3, Wiley 2001 (doi:10.1002/0471231495.ch2)
category: reference

Last revised on September 20, 2022 at 12:32:07. See the history of this page for a list of all contributions to it.