@johncarlosbaez As a mathematician I have always imagined that I would understand the universe if someone described the Standard Model as a really fancy mathematical something-or-other, without saying things like "a neutron is a soup of quarks and anti-quarks" – these just distract from the math.

I only got a snippet of such a story from a physicist once, and I loved it. I asked what a particle really was, mathematically, and he said something like "eigenvector of a linear approximation to some-field-thing". I may be misremembering the "approximation part", but the important thing was that it was a straight mathematical answer. I felt comforted.

Where could I find out more of this? Is the Standard Model a huge vector field on a vector space, or on a manifold with Riemann metric? Can someone write down the type of the Standard Model in in type theory?

@andrejbauer - I could give you a fairly formal description of the Standard Model, but it would be

1) long
2) fundamentally nonrigorous, in the sense that nobody knows that all the numbers we want to compute from the Standard Model are uniquely determined from any particular axiomatic framework
3) one of several choices: there are different frameworks, each with their own advantages and flaws, and any particle physicist knows more than one.

So, for example, I could write a formula for the Standard Model involves an integral over an infinite-dimensional manifold... such that the measure being used in this formula is not known to exist, and probably doesn't really exist. And even if I showed you this, you'd have no idea what this had to do with anything - without further study.

We must instead accept that right now, the Standard Model is a network of formalisms, where the holes in the mathematics - the chunks of math we don't understand yet - are filled in by physics intuition.

Thus, it's largely pointless to imagine that the Standard Model is some mathematical structure we could understand without actually learning physics. Physics is very different from math. It takes years of study, and most of this is not learning math.

There are, however, large chunks of beautiful and formalizable math connected to the Standard Model, which one needs to understand to understand the Standard Model. I wrote a book about some of those - but not nearly all. My book doesn't try to explain the Standard Model, just some prerequisites.

@johncarlosbaez @andrejbauer Let me add another metaphor which is somewhat the same as @Cazzandro 's just in different clothing: Often the physics thinking works on the very basic level while the maths thinking is on a more complex level, which comes with heavier machinery than necessary. But the heavy machinery has its value in something else than bare necessity: It is the precision tool to assure the statements are understood exactly right (=non-ambiguous) in a most economic (=few words) way.

I want to make a maths analogy which I hope does fit (and I am aware that some here are lightyears more experienced than me in this topic, please be gentle):
one can do a surprising lot of things in maths without the heavy machinery of category theory but using fairly simple stuff, just constructively. For example, to define real numbers via Dedekind cuts one does not need the category theory notion of limits and to extend functions from finite stages to the full reals one does not need Kan extensions, comma categories etc to formulate and proof any of these things (of course one implicitly uses them, but "trivially" so).
But: the category theory language can be very useful for precision in the written text (i.e. if its not written as Agda program), and it is a very economic way of transporting the information content human to human (conditioned on the CT understanding). But this is at a price which may be precisely the price of (non_ambiguity)+(economy).

@johncarlosbaez @andrejbauer @Cazzandro And while John is of course right about his diagnosis about the SM, I think in the particular case of this post one could make a mathsy phrasing of what the "soup of quarks" means. Let me have a try at it.

Original: "protons and neutrons act like bags full of a soup [...] In the quark condensate, the clockwise spinning virtual quarks are entangled with counterclockwise spinning virtual antiquarks."

Mathsify: "SM Quarks are a local excitation in a 𝔰𝔲(3)⊕𝔰𝔲(2)⊕𝔲(1)-gauge 𝔰𝔬(3,1)-spinor bundle with respect to the fundamental representation. But they do not appear as isolated excitation because the Hamiltonian / Lagrangian (aka dynamics) prohibits this (strong coupling). Protons and Neutrons are a fairly stable form of a collective excitation. One can expand it in terms of representations using three fundamental representations (three quarks) but the dynamics makes things complicated, due to frustration of constraints (quark confinement versus charge repulsion etc). This has the effect of adding a whole bunch of virtual representations to the mix (i.e. the decomposition into three fundamental representations is not that clean but really there is a bit more than just that, and this "bit more" is not a clean representation but a superposition of things => the condensate). This has the effect of both, breaking the symmetry between the different Weyl sectors of the spinor bundle and of increasing the mass from the Higgs mass of the three quarks."

This is still only somewhat mathsy. I would be curious how to make this even sharper @johncarlosbaez?

@papalex @johncarlosbaez @Cazzandro May we back up a bit (or a lot)?

> 𝔰𝔲(3)⊕𝔰𝔲(2)⊕𝔲(1)-gauge 𝔰𝔬(3,1)-spinor bundle

So we'll work with bundles? Over what? What is their physical meaning? (And it seems like there's a relationship between bundles and fields. Are the fields sections of bundles?)