Sunday, April 07, 2019 ... //

Physics knows a lot about the electron beyond the simple "Standard Model picture"

Ethan Siegel wrote a text about the electron, Ask Ethan: What Is An Electron?, which includes some fair yet simplified standard conceptual facts about the electron's being a particle and a wave, about its properties being statistically predictable, and about the sharp values of its "quantum numbers", some discrete "charges" that are either exactly or approximately conserved in various interactions.

While his statements look overwhelmingly right, there is a general theme that I expected to bother me and that bothers me: Siegel presents a frozen caricature of the particle physicists' knowledge that could be considered "a popularization of the snapshot from 1972 or so". There doesn't seem to be any added value of his text relatively to e.g. the Wikipedia article on the Standard Model. After all, the images such as the list of particles above were just taken from that article.

The final two sentences of Siegel's article suggest that he realizes the "work in progress" character of science – and particle physics is what matters here – and he supports the research:

...Why it works the way it does is still an open question that we have no satisfactory answer for. All we can do is continue to investigate, and work towards a more fundamental answer.
The only problem is that he talks the talk but doesn't walk the walk. This text he wrote – and basically everything he wrote about particle physics – indicates that he has displayed no interest whatsoever to learn something new about the electron that would go e.g. beyond the elementary undergraduate freshman factoids from his article.

The same comment can be said about virtually all popular writers about particle physics these days. They either try to sell pseudoscientific theories that were created from scratch and that have nothing to do with the accumulated knowledge of physics – they're really not capable of reproducing the Standard Model predictions and they are usually very far from passing this test.

Or they avoid such theories but then they avoid all research altogether and spread the misconception that the frozen high school caricature of particle physics from 1972 is what will be enough as "summary of physics" forever.

But the scientific truth and the scientific approach is something completely different – in some sense, it is in between the extreme approaches by the popular writers. Scientific research does take all the established knowledge into account – as compressed e.g. into the Standard Model, an effective theory – but it just didn't freeze when the last terms of the Standard Model Lagrangian were written down.

I think that he was actually asked – pretty much explicitly – about some things that transcend the simplistic picture he has summarized, but his answer included nothing to address that question whatsoever. OK, first, Siegel hasn't really explained any conceptual ideas relevant for the answer, not even at the level of the Standard Model. Note that the original question read:
Please will you describe the electron... explaining what it is, and why it moves the way it does when it interacts with a positron. If you'd also like to explain why it moves the way that it does in an electric field, a magnetic field, and a gravitational field, that would be nice. An explanation of charge would be nice too, and an explanation of why the electron has mass.
OK, Siegel responded with the tables of elementary particles and lists of quantum numbers. But I think that those really don't explain any of these matters – why the electron moves the way it moves in the electromagnetic field (and analogously gravitational field), why it may annihilate with the positron, what gives the electron the mass (the Higgs mechanism and some Yukawa couplings, and those may be approximations of something more fundamental) and more.

Even at the level of field theory, there is a lot of conceptual stuff to say. OK, all the electromagnetic interactions of the electron boil down to the term in the action$S_{\rm int} = \int d^4 x\, j^\mu A_\mu = e\int d^4 x\, \bar \Psi A_\mu\gamma^\mu \Psi.$ Add your standard coefficients or signs or $2\pi$ factors if you hate my schematic picture. OK, there's apparently some electromagnetic 4-potential $A_\mu$ that interacts with the current – the density of the electric charge and the vector-valued density of the flux of that charge. And this term increases or decreases the particles' energy in the electromagnetic potential, bends the paths in magnetic fields, and more.

The existence of the $A_\mu$ electromagnetic gauge field may be derived from the $U(1)$ gauge symmetry, by requiring that the apparent freedom to change the phase of the field $\Psi$ is extended to become the independent freedom at each spacetime point. Once you accept that the field $A_\mu$ is needed for that and you have fields like $\Psi$ for the electron and $A_\mu$ for the electromagnetic field, the structure of the terms in the Lagrangian is mostly dictated by "nonzero physical content", "consistency", "Lorentz covariance", and "gauge symmetry".

Once $\Psi,A_\mu$ are understood to be quantum fields i.e. operators, all the probabilities of all conceivable outcomes may be calculated perturbatively (at least if some good enough approximation is good for you) from expressions such as the annihilation Feynman diagram above. There is quite an interesting question for the beginning: "Why can the point-like electron and the point-like positron exactly hit each other at all?"

If you imagine that they're like point-like planets, the probability that their initial velocities exactly lead them to a head-on collision is infinitesimal i.e. zero. And with a high enough energy, it should also be impossible that they lose enough energy to spiral and fall on one another. In classical physics, the annihilation of point-like particles would be impossible.

But quantum mechanics – its subtype called quantum field theory in this case – saves the day. Quantum mechanics calculates probabilities. They're calculated as squared absolute values of probability amplitudes. And probability amplitudes of quantum field theories involve terms (something that is added with other terms by the plus signs) that are integrals over histories. Alternatively, they may be rearranged as Feynman diagrams which are integrals over the spacetime points where the $\bar\Psi \cdot A\cdot \Psi$ interaction took place.

Because quantum mechanics allows all intermediate histories, it also allows the intermediate history in which the electron and positron precisely collide – their position is the same at some point (which is reflected by continuous electron-positron line in the diagram). This single "infinitely unlikely point" of the history space (which is located at the cubic vertex or vertices of the Feynman diagram) still contributes a finite amount because some terms are weighted by a delta-function (whose integral over the space is one). So just because the precise encounter of the electron and the positron is a possibility, quantum mechanics guarantees that it changes the final probabilities of the elastic collisions, inelastic collisions, or annihilation by a finite amount!

In classical physics, just the existence of an infinitely unlikely possible intermediate history couldn't change the predictions. Classical physicists (or critics of quantum mechanics) could passionately argue that it's impossible for the predictions to change due to infinitely fine-tuned possibilities. But quantum mechanics disagrees and says that the predictions of the probabilities of virtually any outcome are unavoidably affected if there exist some extra potential intermediate histories.

You know, I am sure that this is the kind of the conceptually important wisdom that should be explained by popular articles about similar questions – exactly because the uninformed beginner is likely to make wrong assumptions and incorrect guesses. And because these snippets are great examples showing how some general principles of quantum mechanics (e.g. summation over histories) qualitatively affect particular processes such as the annihilation (they make it possible). But it's never being done – perhaps because none of the popular writers actually understands any of these things.

Obviously, the answer to the question posed to Ethan may be a more or less detailed summary of a quantum field theory course. It makes no sense to try to compress a whole quantum field theory course into this blog post or any other single blog post. But I think that even the point "to understand these things properly, you need to learn quantum field theory well" is simply not being communicated. These laymen clearly maintain some kind of a belief that they may circumvent all the difficult stuff of QFT (and maybe even circumvent all of quantum mechanics) while properly understanding the answers to all questions about Quantum Electrodynamics. But it's simply not the case.

But the laymen – and maybe even young prospective physicists – are much more misdirected when it comes to any "thinking beyond the old-fashioned Standard Model machinery and pictures". They're deliberately indoctrinated by the completely wrong meme that nothing has changed about the physicists' knowledge or thinking about these matters since 1972, a year I semi-randomly picked. Near the beginning, Siegel wrote:
They [electrons] were the first fundamental particles discovered, and over 100 years later, we still know of no way to split electrons apart.
From an appropriate side, this is a perfectly valid statement. And I have almost certainly made the almost identical statement many times, too. The electron still looks like an elementary particle, despite its (discovery's) age over 100 years. In this sense, the electron differs from molecules, atoms, nuclei, protons, and neutrons that have been shown to have some particular internal architecture.

However, in between the lines and in the broader context, you may see that Siegel and others are conveying something stronger that actually isn't right. They want the reader to think that nothing has changed about our view that "the electron could be or should be a structureless point-like particle" from 1919 or from 1972. That is simply untrue.

After 1972, physics has understood lots of deep things that have changed the physicists' understanding of all these questions. It hasn't looked helpful to teach these things to the laymen or high school students, so the laymen and the high school students just remain largely ignorant about them and keep the naive Wikipedia-style or high-school-level pictures as frozen in 1972.

But that doesn't mean that the scientists' opinions and expectations haven't changed since 1972. They have and the popular writers who try to deny these transformations are simply creating an abyss between their readers and the actual scientific research.

If we focus on the "internal architecture of the electron", we may say that the main changes since 1972 took place in two realms:
• renormalization group – and more generally, a deeper, non-perturbative etc. understanding of quantum field theories, what they are, what are their limits, where the parameters come from and which of them are natural etc.
• string theory – and more generally our understanding of particular mechanisms beyond quantum field theory that clarify in what sense the Standard Model is generated or may be generated as an approximate description of a more fundamental theory
OK, Siegel – and most other popular writers – love to deny and hide both. Last week, I discussed some emotional ideas of Eric Weinstein. One of his proposals was that the task for theoretical physicists should be to popularize the renormalization group, a gem they are sitting at, and to spread it to other fields and turning it into a general knowledge.

I have explained that my side is even losing the battle to preserve $x,y$ at the Czech elementary schools, let alone seeds of the renormalization group – and I have argued that top physicists simply shouldn't be understood as teachers or communicators because they're ultimately doing a much more selective and special kind of work. But articles such as Siegel's Forbes column are the perfect examples of the venues in which the renormalization group thinking should be promoted. And the question about the internal architecture of the electron was a perfect opportunity to make a small step for a man but a larger step for the mankind in this direction. Either because Siegel doesn't know or doesn't like the renormalization group, he hasn't used the opportunity at all. It's not just him – it's most of the mass writers. The underlying reason might be that the writers simply do not have a deeper conceptual understanding of the issues than the readers – so the writers just don't have anything substantial to teach to the readers, aside from some boring tables.

So the readers are expected to keep their belief that nothing has changed since 1972 and the electron is just point-like. The end of the story.

In real physics, while we don't have any promising quantum field theories in which the electron is composite, we do understand that the electron could very well be composite – it's a possibility that we must simply always consider viable because we know that seemingly simple theories such as the Quantum Electrodynamics with the simple point-like electrons generally arise as long-distance limits of different (sometimes simpler, sometimes more complex) theories with different degrees of freedom.

Even if "the electron" stayed point-like up to the Planck scale, the field that produces the electron is "renormalized" in between the high-energy regime which are close to the fundamental laws of Nature; and the low-energy regime that is enough for a good enough description of low-energy phenomena involving the electron.

In fact, the electroweak theory that has already been established forces you to isolate the electron's spinor fields from some doublets (in which the neutrino fields start as indistinguishable fields at high energies, before the Higgs has a vev) – and also diagonalize some mass matrix that involves three generations of charged (and neutral) leptons. We also know that the fine-structure constant $\alpha\approx 1/137.036$ isn't really fundamental. First of all, it's a function obtained from two electroweak couplings that mix and these electroweak fine-structure constants are more fundamental than the electromagnetic one; and all these constants "run" with the energy scale, while the high-energy values of the couplings are different than the notorious $1/137$ yet more fundamental.

My point is that Siegel deliberately tries to enforce the readers' belief that there is nothing conceptual about the electron that goes beyond Quantum Electrodynamics. Even the fermion mass matrices of the Standard Model and the doublets – some aspects of the electron beyond Quantum Electrodynamics – are being deliberately obfuscated while the beyond the Standard Model thinking is being hidden altogether.

This brings me to the second class of "hidden secrets". Siegel doesn't want the readers to embrace any insights or arguments from the renormalization group era – which also began in the mid 1970s or so. But he also wants to hide all the actual known types of "more fundamental theories" that produce the Standard Model as an approximation.

They include grand unified theories, supersymmetric theories, and – most ambitiously and rigorously – compactifications of string theory. While no example has been picked as the single, provably right theory beyond the Standard Model, they have provided us with many working proofs of the concept that theories compatible with all the known observations exist where the Standard Model terms are not the "end of the explanatory story".

In particular, the electric charge (of the electron – the lightest charged particle) – may emerge as the quantized Kaluza-Klein momentum $p_5=Q/R$, $Q\in\ZZ$ of a particle moving in a higher-dimensional Universe with a circular dimension $x^5$. The electric charge may also arise as the winding number of a string, or a wrapping number of a membrane, counting how many times the string or brane is wound around a non-contractible circle or a non-contractible higher-dimensional submanifold of the manifold of extra dimensions.

Also, the electric charge may be reduced to a topological invariant in some field configurations – e.g. those involving tachyons within a higher-dimensional annihilating D-branes, along the lines initiated by Ashoke Sen. We could find a few more interesting effects in which "the electric charge has some deeper explanation or geometrization" within string theory. We could divide the stringy compactifications – heterotic, type I, type IIA brane worlds, M-theory with boundaries, M-theory on $G_2$ manifolds, and F-theory of several subfamilies – into the groups. In each group, some particular "geometrization" of the electric charge would be more important than others.

And we could see that the electron itself is most likely to "be" either a vibrating closed string, or a vibrating open string of some kind, although it's really a shrunk wrapped membrane in some M-theory models etc. As a vibrating string, the electron is composite, after all, although the "string bits" – building blocks in a regularized picture – are just a string length away from each other, an inaccessibly tiny distance.

Again, none of them has been established as the final answer yet which is why the research of them is ongoing. But the fact that none of them has been established doesn't mean that the research is meaningless. If one had adopted this utterly stupid anti-research "logic" in the past, then all the scientific progress would have been impossible in the past. Why? Simply because every insight, however rock-solid at the end, must first be investigated by someone who isn't immediately sure that the idea is correct.

If someone spits upon research just because the conclusions aren't rock-solid yet, then he or she is spitting on all research and on science as a whole. I find it amazing that so many people seem unaware of this elementary point.

So Siegel's reader was asking a question of the type "I want to understand electrons at a deeper level" and Siegel responded with "don't ask, nothing interesting to be seen relatively to the high school summaries from 1972". If Siegel and/or his readers were this disinterested in the actual answers and possible answers to these questions, as they are being refined by actual researchers, why do they ask the questions at all? And why do they pretend to answer them?

It makes no sense and this whole question-and-answer ritual seems to be a deception. You are either interested in the deeper origin of the electric charge and the electron – which means that you want to look at some of the best papers in which the best scientists are addressing this question – or you are uninterested. You shouldn't pretend that both answers are possible at the same moment. If you're not following any ideas since 1972 about the deeper explanations of the electron or the electric charge, then you are just a superficial layman uninterested in the research of particle physics. Period. It's just fraudulent for you to pretend something else.

Real theoretical high-energy physicists are very interested and they have made a huge progress, even after 1972.