# Comments on Vulgarisation de la physique des particules avec un peu d'algèbre linéaire

## jonas (2018-09-22T23:50:10Z)

The recent relevation in the Order of the Stick wecomic <URL: http://www.giantitp.com/comics/oots1141.html > really reminds me of this spontaneous symmetry breaking. At the start of the Order of the Stick metaverse, there were four quiddities, and (although the comic doesn't explicitly confirm this) I think they were equal to each other. Then the snarl appeared, and made one of the four quiddities impossible. The snarl didn't care which quiddity it chose, it was random. The rest of the quiddities can exist after the snarl exiled one of them.

This probably won't make sense unless you read the whole webcomic. It's long. Sorry.

## jonas (2018-08-30T10:55:05Z)

> But this idea can't be too correct either, because W bosons can't just sit around, they are short-lived,

I don't think that's a problem. The local fluctuations of the Higgs-field that they disguise must be short-lived too. The problem is more like that the way you describe that reaction, the photon can only do a full turn in its isospin direction, turning into a $Z^0$, not a quarter-turn into a $W^+$. But I'll have to re-read your original writeup about the pre-particles slowly and very carefully, because I think I missed some subtlety about the pre-W and pre-B bosons and the pre-forces there.

## Ruxor (2018-08-30T10:22:53Z)

@jonas: In the (again, inadequate) picture I have in mind, the photon doesn't turn into a W. The W is, shall we say, lying around since the beginning of the universe. The W, being charged, can capture the photon. Maybe the W re-emits a Z, which then goes along the same path that the photon was going along. So we get something like γ + W⁻ → W⁻ + Z⁰, and in effect, the photon as turned into a Z⁰ by interacting with the W field. But another way to see the same thing is that the photon has wandered around into a region where the Higgs field has a different orientation, and when measured w.r.t. that different orientation, it is found to be no longer a photon: gauge invariance lets you transform one of these descriptions into the other (absorbing variations of direction of the Higgs field into the W field). So this is the idea I have in mind. But this idea can't be too correct either, because W bosons can't just sit around, they are short-lived, they decay into lots of other stuff (like electron + antineutrino), so in the end I don't know we should describe the overall process.

## jonas (2018-08-29T18:00:15Z)

> from the particle point of view, the photon encounters a (charged) W boson and is absorbed by it and perhaps re-emitted as a Z.
>
> The bosons involved in the direction vibrations of the Higgs field are the W bosons (which also have their own field, but that field somehow combines with part of the Higgs field upon symmetry breaking).

This seems like the right sort of explanation at my level. But the details don't seem to work out. What exactly is the particle balance of the reactions that we observe (after “gauging away the Higgs-field”) with the photon turning into a W boson? By particle balance, I mean the signed sum of particles entering and exiting like $W^+ \\to e^+ + \\nu_e$, but hopefully no fermions figure into this one. I don't see how such a reaction would be possible without observably breaking either the conservation of electric charge or the conservation of spin.

## Ruxor (2018-08-23T15:34:00Z)

@jonas:

There are of course different ways of putting it, but the picture of a photon crossing a large part of the universe and ending in a region where the Higgs field has a different orientation is going to be one where, from the particle point of view, the photon encounters a (charged) W boson and is absorbed by it and perhaps re-emitted as a Z. (Again, we can always "gauge away" the differences in direction of the Higgs field and postulate that it is the same everywhere, but the cost of this "gauge change" is going to manifest itself in the form of W bosons.) The photon is always going to be massless, because, in a sense, the photon is defined to be the part of the gauge field that the Higgs does not feel: the question is not going to be whether the photon acquires mass, but whether it gets absorbed and possibly re-emitted as a Z along the way.

But I realize that this is not a very satisfactory answer to your question, and I'm afraid I don't have a clearer understanding than that. Part of what puzzles me is that W bosons are quite heavy and unstable, so they decay very quickly, and I don't understand what this really means on the W gauge field itself (the electromagnetic field has a non-quantum, macroscopic, effect, the W field apparently does not, and this is probably related to its mass, but I don't really understand why it does not, hence, how a macroscopic difference in the Higgs orientation would appear in the end); and since this is intimately related to the whole problem of what quantum fields are in the first place, I'm just very confused. Maybe the answer will involve the words "quantum superposition of vacuum states" in a way I don't understand.

Just one clarification I can make is that Higgs bosons are the amplitude vibrations in the Higgs field (vibrations in the direction in which it is already nonzero). The bosons involved in the direction vibrations of the Higgs field are the W bosons (which also have their own field, but that field somehow combines with part of the Higgs field upon symmetry breaking).

## jonas (2018-08-22T21:50:55Z)

I was still thinking of this post more than the metrology thing. I have two conclusions.

1. You have previously written posts on why you find symmetries and ordinals fascinating. I didn't need much convincing there, because I already thought that they were interesting topics, but I have definitely learned at least some new things from you about specific topics, especially when thinking about transfinite programming in <URL: http://www.madore.org/~david/weblog/d.2017-08-18.2460.html >.

On the other hand, I wasn't interested in quantum electrodynamics at all. The few popular accounts I read about them made it sound boring. Your writeup gives a perfect answer for why quantum electrodynamics is interesting, and why the Higgs mechanism for breaking its perfect symmetry is interesting. You admit that you don't understand how it really works, but you've seen enough to know why you should be interested, and you also managed to explain enough to me to finally understand why physicists make such a big deal of this Higgs thing.

2. There's something about the uniformity of the direction of the Higgs field that I still find quite confusing, and I can explain why it's surprising even without invoking astronomical distances and relativistic causality limits.

On one hand, The Higgs field really must have a very uniform direction throughout space. If it weren't, then physicists would have already made experiments in which they find that a photon spontaneously oscillates to a W or Z boson and then decomposes to leptons, just because the direction of the Higgs field has had a small local random fluctuation. That would be such a surprising discovery that I would have certainly heard news of it if it had happened. But no, all the descriptions still insist that photons are massless and have exactly $c$ speed.

On the other hand, the important experiment physicists keep talking about is detecting Higgs bosons in their particle accelerators and measuring its properties to make sure what they found really is the Higgs boson. They take this as a very important confirmation that the theories about the Higgs field and Higgs mechanism of breaking electroweak symmetry is the right theory. At the same time, the physicists also insist that their particle accelerators won't cause such unimaginable catastrophes as the decay of the entire universe as we know it, because cosmic rays have particles with much higher energy than anything they create in their particle accelerators, and they sometimes hit the Earth and interact with other particles there. This probably means that a lot Higgs particles are also created just from random cosmic ray collisions to Earth, they're just impossible to detect. And those Higgs particles are supposed to indicate changes in the Higgs field. So then why are photons massless?

## jonas (2018-08-06T13:22:29Z)

> [#8b] […] je prends un électron dans une région de l'Univers, je la transporte dans une autre région de l'Univers, est-ce que c'est toujours un éléctron

I'm not particularly worried about electrons. I'm worried about photons. By now, astronomers have detected photons that come from very far.

## jonas (2018-08-06T13:02:19Z)

I didn't ask why the Higgs particle was a boson. I was asking about the hypothetical particle that would be associated with the earlier spontaneous symmetry breaking that you refer to in the paragraph starting with “Il vaut sans doute mieux que j'arrête là”, just like how the Higgs boson is associated with the later spontaneous symmetry breaking that you talk about in the bulk of the “Exemple nº4” section. You say that that earlier symmetry breaking is what made the sinistro-leptons behave differently from the dextro-leptons, so naively I would assume that at least some such particles have to be fermions, but in retrospect that doesn't make much sense either, now I imagine it's more likely that they are bosons with chiralité, but I'm not certain anyway.

It's also not clear to me if there are even only two levels of spontaneous symmetry breaking speculated about, or three, and what exactly the earlier ones are, which one the hypothetical GUT is associated with, which one the hypothetical supersymmetry is assocated with, when did each happen, and what order of magnitude of mass do we expect the detectable particles associated with each have. You do make it clear that the latest spontaneous symmetry breaking so far was $10^{-12} s$ after the Big Bang, the detectable particle associated with it is the Higgs boson ($H$), and it's easy to look up that it is believed to have a 125 GeV mass. Thus, my question is about the one or more instances of spontaneous symmetry breakings before that. That, however, is clearly not in scope of this blog post, so I can't expect you to answer it.

## Ruxor (2018-08-06T11:59:18Z)

@jonas: I have made a few small changes to address some of your remarks.

• I added a parenthetical to clarify that I'm only assuming some knowledge of linear algebra in finite dimension. (Of course, it's not entirely clear, even now that the entry is written, exactly what I'm assuming the reader is familiar with, but finite-dimensional linear algebra is definitely the main part of it.)

• John Baez noticed this entry (<URL: https://twitter.com/johncarlosbaez/status/1025423987839397889 >), but sadly he doesn't read French, and even though the Google translation is not as bad as I imagined, I don't think he wants to read it in full and I don't feel like asking him to. (But I think there are some readers of this blog who know a good deal more physics than I do, so maybe I'll get some clarifications from them on various points I raise.)

• I'm pretty sure there's no standard term for what I call the vibratory space (if there were, I would have run across it at some point).

• I agree the presence of the graviton (this is what the G stands for) in my table of particles is confusing, but I don't like having several different versions of the same document. Instead, I added a phrase in the text making it clearer that the "G" should be ignored.

• It's not correct to say that the sun gets its energy from turning up quarks into down quarks, which would be from the weak interactions. It gets its energy from packing nucleons closer together, i.e., from the strong interactions (which manifest themselves, at scales intermediate between that of a nucleon and that of a nucleus, as bindings between nucleons mediated by pions and other similar particles). In this process, it has to turn up quarks into down quarks (and destroy electrons and emit neutrinos) through the weak interactions, but this is more like a price to pay for the rest. The first generation of quarks is different from the others in that the −1/3 charged quark is heavier than the +2/3 one, so one would more readily imagine a down quark emitting a W⁻ and turning into an up quark than an up quark emitting a W⁺ and turning into a down quark, but both are possible if energy allows (and I didn't want to get into energy considerations), so I just went with a blanket statement.

• Regarding tauon (/toɔ̃/), it also sounds strange to the French ear, but if you just say « un tau » (which is also possible), it can be confused with « un taux ». I like your sentences illustrating sequences of pure vowels, I imagine they can be hard to process.

• Your objection about the Higgs value throughout the Universe is very serious. I added a note (#8b) to address it, but I only have a partial answer, as I explain there. My thoughts aren't entirely clear on the matter.

• As to how we can be sure that the Higgs had to be a boson, the reason is that fermion fields cannot form a [Bose-Einstein] condensate (= piling up at the lowest energy state). Instead, the Pauli exclusion principle forces them to occupy different quantum states. But there again, I have to admit that I never really wrapped my mind about what condensates are (or formed an intuition about why the way helium behaves at very low temperature and the way the Higgs boson has a vacuum expectation value are part of the same phenomenon), so I'm afraid my explanation will be found wanting, but that's the best I can offer.

## jonas (2018-08-06T06:47:50Z)

> Mais si au moins j'ai réussi à faire un peu passer l'idée que, quand l'univers était très chaud, non seulement les électrons, neutrinos, quarks et bosons Z/W n'avaient pas de masse, mais en plus ils étaient organisés différemment, que l'électron-chiral-gauche et le neutrino-chiral-gauche étaient la même particule (sénestrolepton) et de même le quark-haut-chiral-gauche et le quark-bas-chiral-gauche, alors que l'électron-chiral-droit, l'éventuel neutrino-chiral-droit et les différents quarks chiraux-droits étaient des particules n'ayant pas grand-chose à voir (ni entre eux ni avec les précédents), et que pourtant <i>tout ça était plus symétrique</i>, eh bien ce sera déjà ça.

Wait, so the kicker is supposed to be that the $G$ I saw earlier was never supposed to stand for a particle associated with <i>gravity</i>. It's the very high energy particle that reveals us the <i>Grand</i> unification theory, the one that is actually tiny vibrations in the pre-Grand condensate, where the pre-Grand condensate is the one that spontaneously broke the symmetry between dexo- and sinistro- particles $10^{-36}$ seconds after the Big Bang, the one that perhaps explains some of those strange arbitrary coupling constants you mentioned above. And if Lederman claims that the Higgs is the God particle because the condensate is what told the universe “let there be light” the first day and what I vulgarize as “let there be mass” on the second day, then the pre-Grand condensate plays a level above God and said “when God is ready to say, ‘let there be light’, he shall then the next day say, ‘let there be mass’ ”. And if this were an insult to God's name, then I claim that it was Lenderman who made that insult when he gave that name, and he had it coming because already when he wrote that book he had heard of possible Grand unification theories and mentions them in the book. But what I don't understand is, if this particle is the vibration of a field that tells sinistro- and dextro- particles to behave differently, then why are we sure that it is a boson?

## jonas (2018-08-05T22:32:19Z)

> Or l'algèbre linéaire est quand même quelque chose de moins ésotérique, et sa compréhension est plus répandue, que les arcanes de la physique des particules

Yes, the idea of such an introduction does sound promising, and already when I saw the title (in the context that it's an entry on your blog) it caught my attention. But at this point or earlier you could clarify if we should rejoice because you're talking of linear algebra in finite dimensions, or be afraid of the linear algebra in infinite dimensions coming. It would only take two words.

> Si je pouvais persuader un vrai physicien de prendre les choses vraiment au sérieux, évidemment, ce serait parfait

Hmm. Try with John Baez? His interest in popular writing of mathematics overlaps yours a lot, but he knows more physics than you do. I don't know how you can convince or bribe or threaten him to write about what you want. But if you already have this idea written down, he might be willing to read it and give valuable feedback if you just ask nicely. The only problem is, I don't think he reads French well enough.

> [#] Je ne trouve pas de terme générique pour désigner le ou les espaces vectoriels dans lesquels les différents champs de la théorie (classique ou quantique) des champs prennent leurs valeurs.

That's a good concrete question that you should ask the friendly physicist you talk to.

> En mettant de côté la gravitation ([…] qui ne fait pas partie du modèle standard […]), les forces fondamentales sont

Then remove that $G$ from the chart. Here, <URL: https://i.stack.imgur.com/SmUf1.png >. Not that I doubt your basic image editing abilities, but because of the principle of constructive feedback.

> un quark haut, resp. charme, resp. vrai, peut émettre un boson W⁺ et devenir un quark bas, resp. étrange, resp. beau ([…] il est énergétiquement plus sensé de le dire comme je l'ai dit)

Let's cross-check that. The Sun releases a lot of energy, and in the low level it's powered by up quarks turning to down quarks. And our nuclear fission bombs release a much smaller amount of energy, powered by down quarks turning to up quarks, but we can only build as many of them as there are nucleuses heavier than iron available, and the energy for making those nucleuses was all paid by supernovas, which gained vastly more power than the Sun from gravity even though very little of this was spent on making those heavy nucleuses, if I understand correctly. So yes, that energetic part sounds plausible.

> la chiralité est invariante par changement de référentiel ; […] à vitesse nulle, les deux versions chirales ont la même amplitude dans cette oscillation, mais plus l'électron va vite, plus une composante chirale est importante dans cette superposition et plus l'hélicité se rapproche de la chiralité.

And these two don't contradict, because special relativity is wierd. It's hard to wrap my head around this.

> tauon

And despite that the French language is constructed such that it tries to avoid adjacent vowels, when a phrase does have adjacent vowels in the unlikeliest combinations like that, the French will pronounce and understand them perfectly without batting an eyelid. For us Hungarian speakers, that doesn't come naturally, so teachers illustrate it with sentences like “En haut ou en bas?” and “Où est A1?”. Even though the system of French vowels is intrinsically nice, it's just difficult for me as a Hungarian speaker. My problems start with “Bon matin!”, so to speak. (As a consolation, at least there are two clusters that are easy for Hungarian speakers and appear in French verbs: “-éer” and “-ier”.)

> En attendant, que s'est-il passé (environ 10−12 secondes après le Big Bang) ? […] le champ de Higgs ne peut pas rester autour de zéro : il se condense en une valeur d'énergie minimale, c'est-à-dire non nulle, la même dans tout l'espace, une valeur dans le vide.

How was that possible? Wasn't the universe already too big by that time that the light speed barrier would not let the préHiggs field equalize to the same direction all throughout?

## Fred le marin (2018-08-05T16:39:10Z)

J'ai en fait trouvé une explication partielle : apparemment, émettre une particule alpha constitue (de loin) la façon la plus "économique" qu'à le noyau pour aller vers sa stabilité future.
Cette réponse qualitative (basée sur le principe de moindre action ?) me satisfait car je ne lirais pas des calculs rébarbatifs (avec des lagrangiens dont l'intégrale [=l'action] est à minimiser).
That's all folks !

## Fred le marin (2018-08-05T09:45:52Z)

Simple tout en étant compliqué !

Au niveau des éléments chimiques, la table de Mendeleïev en comporte une bonne centaine (118 en 2018 ?).
Mais c'est sans compter avec leurs isotopes (stables ou non) : plus de mille (à confirmer).
Le rêve de clarté se transforme là aussi en un cauchemar bordélique…
Sinon, je me demande si la MQ sait dire pourquoi c'est une particule alpha qui sort de certains noyaux (et pas autre chose : un proton couplé à un neutron par ex.).
J'ai vu qu'elle explique cela par l'effet tunnel, mais sans plus.
Y a-t-il une raison (explication via les quarks, les mésons…?) mis à part le fait que c'est ce que l'on observe toujours ?
Quitte à invoquer l'Algèbre Linéaire (ou la théorie des Groupes)…

## Geo (2018-08-04T13:56:52Z)

Ton idée de vulgarisation me rappelle un cours de Werner Krauth à l'ENS qui m'avait interpellé sur Coursera.

https://www.coursera.org/learn/statistical-mechanics/home/info

"In this course you will learn a whole lot of modern physics (classical and quantum) from basic computer programs that you will download, generalize, or write from scratch, discuss, and then hand in. Join in if you are curious (but not necessarily knowledgeable) about algorithms, and about the deep insights into science that you can obtain by the algorithmic approach."

L'idée semble similaire, se servir d'un domaine que l'on connait déjà pour en aborder un autre ?

## Le Paradis de Higgs (2018-08-04T08:20:50Z)

Finalement le vrai Zombieland c'est le pays des particules quantiques !

Une question à te poser, selon toi quand faudra-t-il fermer le grand collisionneur du CERN à Genève maintenant que tous ces chasseurs du boson de Higgs ont abattu leur grand mistigri ?

## un peu de géométrie non commutative (2018-08-04T07:13:06Z)

Avant de lire le post, à propos de "faire un portrait correct du modèle standard", pour une vision plus aérienne, le point de vue spectral de Connes est séduisant:
à partir de la non commutativité:
Noncommutative Geometry as a
Framework for Unification of all Fundamental
Interactions including Gravity. Part I.
<URL: https://arxiv.org/pdf/1004.0464.pdf >
le correctif pour le raté sur la prévision de masse du boson de Higgs:
Resilience of the Spectral Standard Model
<URL: https://arxiv.org/pdf/1208.1030.pdf >
état de l'art en 2017 de l'interaction entre la physique et la géométrie non commutative:
Geometry and the Quantum
<URL: https://arxiv.org/pdf/1703.02470.pdf >

## Ruxor (2018-08-03T19:23:55Z)

@ooten: Oui, mais ce n'est pas ce dont je parle. 😛 En gros, je parle de l'espace dans lequel un (ou un ensemble de) champ prend ses valeurs en un point (champ classique ou, avec des pincettes, champ quantique), alors que le cours dont tu parles correspond au vecteur état du système. Tout est confus, parce que lors de la première quantification, les particules deviennent des fonctions d'ondes (« champs classiques », prenant leurs valeurs dans ce que j'appelle un espace vibratoire), ces fonctions d'ondes peuvent être vues comme des états dans un espace de Hilbert, mais la seconde quantification transforme les fonctions d'ondes (qui sont maintenant des « champs quantiques ») en des opérateurs sur un espace de Hilbert d'états beaucoup plus abstrait que lors de la première quantification. Tout ça est bordélique.

## ooten (2018-08-03T18:48:04Z)

@Ruxor : "Je ne trouve pas de terme générique pour désigner le ou les espaces vectoriels dans lesquels les différents champs de la théorie (classique ou quantique) des champs prennent leurs valeurs. Donc je sors de mon chapeau ce terme complètement pourri d'"espace vibratoire". En fait je suis tombé sur ce cours : < URL : http://www.cmls.polytechnique.fr/perso/paul/ceri.pdf > et l'auteur dit que ce sont les espaces d' Hilbert qui sont utilisés en mécanique quantique. Et toute la modélisation ou axiomatisation de la mécanique quantique est principalement due à John von Neumann, voir ce petit historique très bien fait < URL : http://kafemath.fr/2013-2014/1402-20fev/didier-robert-kafemath-fev2014.pdf > à partir du slide 31.

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