Mastodawn

John Carlos Baez 4d ago

The Higgs boson gives elementary particles their mass, but 98% of the visible mass in the Universe (not dark matter) comes from a less famous mechanism: chiral symmetry breaking. This is why protons and neutrons are so much heavier than their quarks!

Briefly, protons and neutrons act like bags full of a soup of virtual quark-antiquark pairs, which give them most of their mass. This soup, called a 'quark condensate', breaks a certain symmetry that exists outside the bag: 'chiral symmetry', where you change the phase of the clockwise and counterclockwise rotating quarks separately. In the quark condensate, the clockwise spinning virtual quarks are entangled with counterclockwise spinning virtual antiquarks.

https://en.wikipedia.org/wiki/Chiral_symmetry_breaking

Chiral symmetry breaking - Wikipedia

Andrej Bauer 4d ago

@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?

John Carlos Baez 3d ago

@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.

Anger Issues 3d ago

@johncarlosbaez @andrejbauer I always thought that mathematics is like language, physics like thought. There are things that you can think but hardly express with language. Thought is far more complex and inexplicable. As a feeling or an intuition as is well said. And yes, the great revolutions in physics were ignited by intuition, not by rigorous logic. An apple falling on someone head or Einstein's dream of cows.

John Carlos Baez 3d ago

@Cazzandro @andrejbauer - that's a nice way to think of it. I often think something a bit different: physics is a massive extension of our ability to guess things like "how hard will I have to bend this branch before it snaps" or imagine what will happen when a thrown rock hits a wall. There are a lot of bodily sensations and visualization built into doing physics, refined by all the math and physics we learn. So, for example, if you can't imagine a bag full of virtual quarks, you probably won't be able to guess what protons do under various conditions.

@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?

John Carlos Baez 2d ago

@papalex @andrejbauer @Cazzandro - your mathification is a good preliminary intro to the Standard Model treatment of the proton and neutron. I could easily expand it for page upon page: for example, it says almost nothing about how pertubatively multiparticle states are described as vectors in Fock space, or how condensates are most simply (and too crudely) described as coherent states, or how confinement works in QCD, due to specific features of gluon self-interactions.

Would this expansion help @andrejbauer much? I doubt it.

Btw, I didn't say "soup of quarks", I said "soup of virtual quark-antiquark pairs". That's the condensate that carries most of the mass. It arises from chiral symmetry breaking. That's what I was actually trying to summarize, and I pointed to a Wikipedia article for more detail.

Andrej Bauer 2d ago

@johncarlosbaez @papalex @Cazzandro

> Would this expansion help @andrejbauer much? I doubt it.

It would not, especially not without a general definition. To my mind, perturbations have no place in the basic conceptual setup, they may at best be technique.

I looked at https://en.wikipedia.org/wiki/Fock_space It's quite low-level from a mathematical point of view. Surely there's a universal property that the Fock space is trying to capture, all those sums seem very computation-oriented (not a surprise).

Fock space - Wikipedia

@johncarlosbaez ah, sorry. Of course the soup was of a different taste. That was unintended sloppyness on my side.

John Carlos Baez 2d ago

@papalex - that's okay, soup is often sloppy.

Andrej Bauer 2d ago

@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?)

John Carlos Baez 2d ago

@andrejbauer @papalex @Cazzandro - classically, fields are sections of bundles over spacetime (which in the Standard Model is ℝ⁴).

"Classically" means that things get more complicated when we "quantize", which we must do... but that nonetheless, this is a good place to start the story.

@johncarlosbaez @andrejbauer @Cazzandro

To add some details on the “complicated” side of things:

In standard QM, we work with a complex Hilbert space where every vector has a clear physical meaning. Moving to QFT, we shift to Fock space, which aggregates these states. We replace individual particle states with field operators (ladder operators) that create the vacuum excitations. But we still work with a conventional complex Hilbert space.

However, there is no single "correct" Fock space here. Thanks to Haag’s Theorem, there is an infinite choice of unitarily inequivalent representations, meaning the way we define our particles is fundamentally linked to the state of the vacuum.

Then, things get messy when we move to gauge theory (which apparently we need to describe nature). We start by quantizing the fields, but this introduces massive redundancy because we quantized redundant degrees of freedom. The raw Fock space gets flooded with "unphysical" states that don't satisfy gauge symmetry. In many cases, these raw spaces contain negative-norm states, making a direct probabilistic interpretation impossible. To recover a physical Hilbert space, we have to "mod out" the gauge orbits and ghosts (aka undo the mistake of quantizing too much).

Finally, what I did not yet mention: transitioning to relativistic QM (aka Dirac equation) forces us to switch from complex scalar fields to slices of spinor bundles, which accounts to space time geometry.

Andrej Bauer 2d ago

@papalex @johncarlosbaez @Cazzandro @johncarlosbaez @papalex @Cazzandro Thanks for humoring me, I appreciate it.

ℝ⁴ to model a small portion of the universe, presumably, not too close to a black hole and such?

It is hard to reconcile the two replies. One tells me fields are sections of bundles, and the other tells me we work with complex Hilbert spaces. What's going on?

John Carlos Baez 2d ago

@andrejbauer wrote: "ℝ⁴ to model a small portion of the universe, presumably, not too close to a black hole and such?"

We're talking about the Standard Model, which does not include gravity. Nobody knows the right theory that includes particle physics and gravity. But yeah, we're talking about an approximate model of spacetime in a region around here.

"One tells me fields are sections of bundles, and the other tells me we work with complex Hilbert spaces."

This one is easy. We work with complex Hilbert spaces of sections of vector bundles.