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My (gifted 8-year-old) son is fascinated by the standard model. We've exhausted my knowledge: he can describe the particles, some of their properties, and understands that quarks make up baryons and mesons. I'd love to provide some better answer to questions like "why are there three lepton flavours?" "why are there three types of colour charge?" and "why do quarks have fractional charge" other than "because group theory".

He has a grasp of what a group is, and it might be interesting to look at $SU(3)$, $SU(2)$ and $U(1)$. But I can't explain the connection between the groups and the particles. All of the resources I've found are either just a list of particles with more or less fancy graphics; or start with tensors and Lie algebras and go from there. Is there any non-mathematical treatment out there which gives an intuition - however metaphorical - for how you get from understandable symmetries to a list of particles?

Edit: thank you for the answers so far. They have helped me understand what I am looking for (rather than a ToE explainable to an 8-year-old!) I think what puzzles us both is this. Imagine we had observed 5 out of the 6 quarks. What is it about the maths that suggests "the way these quarks interact is related to $SU(3)$" and "therefore we might expect there to be a 6th one"?

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    $\begingroup$ The answers might also be useful to people just a bit older than eight $\endgroup$ Dec 7, 2021 at 20:50
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    $\begingroup$ You know, we actually don't have answers to those questions... $\endgroup$
    – rfl
    Dec 7, 2021 at 21:49
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    $\begingroup$ This is not an answer, but you might be interested in this paper aimed at high school teachers written by some people at CERN. It does not answer the whys (I pretty much agree with anna v's answer on this), but it could be a way of going a bit deeper (if you didn't get there yet hahaha) $\endgroup$ Dec 8, 2021 at 5:30
  • $\begingroup$ The CERN paper is great (in fact it probably is an answer). thank you $\endgroup$
    – aucuparia
    Dec 8, 2021 at 11:05

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I will copy from my answer here :

Physics is a science that has a large body of observations, and a limited number of mathematical models/theories that aim to organize and explain those observations and , very important, get validated by predicting the behavior of new observations.

Mathematical theories start with axioms and some tools that develop theorems from those axioms and then various setups can be examined.

and here:

Physics using the appropriate models show "how" one goes from data to predictions for new experimental data. Looking for "why" in the models, one goes up or down the mathematics and arrives at the answer "because that is what has been measured"

The important point is that your young child understands that the standard model of particle physics developed in order to fit data and to mathematically predict new observations. Thus there will always be questions that will hit on the axioms ( postulates, principles, assumed constant values) imposed so that the mathematics fits the data . ( why does the electron have that mass?)

Found this with google, which is again the mathematics of the model. I do not think that the resource you are looking for exists.

The answers to your

"why are there three lepton flavours?" "why are there three types of colour charge?" and "why do quarks have fractional charge" other than "because group theory".

is that that is what fitting the data gives, and makes successful the predictions for new data using the standard model.

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  • $\begingroup$ Thank you. The question of why we expect nature to conform to "beautiful" mathematical patterns, and why it seems to, is an interesting but philosophical one. I agree that it's important to emphasise the empirical basis, that's a helpful reminder. I think I'm most puzzled by the relationship between the mathematical models and the easily understandable description of "there are x fundamental particles of these types". $\endgroup$
    – aucuparia
    Dec 8, 2021 at 10:57
  • $\begingroup$ @aucuparia There exists an infinity of mathematical models that could work with x different fundamental particles. The world they project is not our world. The specific x particle model describes observations in our world $\endgroup$
    – anna v
    Dec 8, 2021 at 11:45
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As Anna said, the "why" of the standard model is because it fits data. Boring answer for an 8yo perhaps, but that's really all there is to it.

In general, I'd suggest trying to connect the standard model to reality and how it could be measured.

As your son already knows the particles, I think a good next step could be to talk about how they interact. Start drawing Feynman diagrams which naturally leads to colliders and other experimental particle physics. They'll form a great common ground between experiments, intuition and mathematics, where you can go as mathematical as you want by attributing factors to each node and line in the diagrams to calculate amplitudes.

Good luck!

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  • $\begingroup$ Good suggestion to look at Feynmann diagrams. thanks. $\endgroup$
    – aucuparia
    Dec 8, 2021 at 10:47
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Loosely, elements of $\mathrm{SU}(3)$ are $3\times 3$ matrices which mix up the components of $3$-component vectors. It also happens to be an $8$-dimensional Lie group, which means e.g. that it possesses eight linearly independent infinitesimal generators. When implemented as the symmetry group of a gauge theory, that means that there are three "types" of charge (corresponding to the $3$-component vectors on which $\mathrm{SU}(3)$ acts) and eight vector bosons (associated with the eight infinitesimal generators) which mediate interactions. In QCD, that means three colors and eight gluons; similar descriptions apply to $\mathrm{SU}(2)$ and $\mathrm{U}(1)$ theories, though as a caveat things like the spontaneous symmetry breaking in the electroweak sector of the full standard model somewhat complicate matters by mixing up which charges/bosons can be attributed to which symmetries.

More importantly, however, attributing the existence of three color charges to the $\mathrm{SU}(3)$ gauge symmetry is putting the cart before the horse. The honest answer is the other way around - we incorporate the $\mathrm{SU}(3)$ symmetry into the standard model because we observe three color charges. If the universe didn't behave that way, then we'd have to come up with some other description of her.

That's not to say that there aren't good mathematical answers to some such questions. For example, the dimension of $\mathrm{SU}(n)$ turns out to be equal to the number of linearly independent, skew-Hermitian $n\times n$ matrices with trace zero; it's not hard to check that this number is $n^2-1$. As a result, a $\mathrm{SU}(n)$ gauge theory will have $n$ "types" of charge and will yield $n^2-1$ vector bosons. However again, whether such a description truly applies to nature is a matter of empirical observation, and the only answer to why that happens to be true is simply "that seems to be how nature behaves."

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  • $\begingroup$ Thank you. This is really the kind of thing I was interested in. $\endgroup$
    – aucuparia
    Dec 8, 2021 at 10:48

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