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Background: I have written a pop-science book explaining quantum mechanics through imaginary conversations with my dog-- the dog serves as a sort of reader surrogate, popping in occasionally to ask questions that a non-scientist might ask-- and I am now working on a sequel. In the sequel, I find myself having to talk about particle physics a bit, which is not my field, and I've hit a dog-as-reader question that I don't have a good answer to, which is, basically, "What purpose, if any, do higher-generation particles serve?"

To put it in slightly more physics-y terms: The Standard Model contains twelve material particles: six leptons (the electron, muon, and tau, plus associated neutrinos) and six quarks (up-down, strange-charm, top-bottom). The observable universe only uses four, though: every material object we see is made up of electrons and up and down quarks, and electron neutrinos are generated in nuclear reactions that move between different arrangements of electrons and up and down quarks. The other eight turn up only in high-energy physics situation (whether in man-made accelerators, or natural occurances like cosmic ray collisions), and don't stick around for very long before they decay into the four common types. So, to the casual observer, there doesn't seem to be an obvious purpose to the more exotic particles. So why are they there?

I'm wondering if there is some good reason why the universe as we know it has to have twelve particles rather than just four. Something like "Without the second and third generations of quarks and leptons, it's impossible to generate enough CP violation to explain the matter-antimatter asymmetry we observe." Only probably not that exact thing, because as far as I know, there isn't any way to explain the matter-antimatter asymmetry we observe within the Standard Model. But something along those lines-- some fundamental feature of our universe that requires the existence of muons and strange quarks and all the rest, and would prevent a universe with only electrons and up and down quarks.

The question is not "why do we think there are there three generations rather than two or four?" I've seen the answers to that here and elsewhere. Rather, I'm asking "Why are there three generations rather than only one?" Is there some important process in the universe that requires there to be muons, strange quarks, etc. for things to end up like they are? Is there some reason beyond "we know they exist because they're there," something that would prevent us from making a universe like the one we observe at low energy using only electrons, up and down quarks, and electron neutrinos?

Any pointers you can give to an example of some effect that depends on the presence of the higher Standard Model generations would be much appreciated. Having it already in terms that would be comprehensible to a non-scientist would be a bonus.

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Since you've eliminated CP violation I doubt you are going to get a good answer to this. It's just like having a world with cats and bunnies as well as squirrels. It gives some of us humans more to chase. – pho Jan 7 '11 at 19:12
This is still an open question in particle physics. One of the checks on a theory that unifies gravity with the other three forces is, in fact, whether that theory predicts precisely three generations of particles. My understanding is that this constrains the way in which you compactify the extra dimensions in string theories, and it also constrains the way in which you break supersymmetry. But it's also not my field at all, either. – Jerry Schirmer Jan 7 '11 at 19:21
I didn't mean to completely eliminate CP violation as a response, if that is a valid response. It was just a guess at something that might be a possible response (since CP violation was discovered in kaons, which contain strange quarks), with a hedge against people replying "no, no, the Standard Model can't provide enough CP violation..." If that's the best example of something along these lines, I'm happy to go with it (though a pointer to a reasonably accessible explanation would be a bonus). – Chad Orzel Jan 7 '11 at 19:42
CP violation is the best response I know of, in that you just don't have it in the Standard Model with 1 or 2 generations. I agree though that it seems you need other sources of CP violation to explain the cosmological baryon asymmetry. You might have a look at the Nobel web page and/or lectures when Nambu, Kobayashi and Maskawa got the Nobel Prize for an explanation accessible to your dog and/or readers. – pho Jan 7 '11 at 21:55
Nit picked: neutrino oscillation is such that all flavors are present in respectable fractions from all originating reaction flavors, and the muon- and tau- types are not known to "decay" as such. But I think the basic answer is still "Who ordered that?". – dmckee Jan 7 '11 at 23:30

The question: "I'm wondering if there is some good reason why the universe as we know it has to have twelve particles rather than just four."

The short answer: Our current standard description of the spin-1/2 property of the elementary particles is incomplete. A more complete theory would require that these particles arrive in 3 generations.

The medium answer: The spin-1/2 of the elementary fermions is an emergent property. The more fundamental spin property acts like position in that the Heisenberg uncertainty principle applies to consecutive measurements of the fundamental spin the same way the HUP applies to position measurements. This fundamental spin is invisible to us because it is renormalized away. What's left is three generations of the particle, each with the usual spin-1/2.

When a particle moves through positions it does so by way of an interaction between position and momentum. These are complementary variables. The equivalent concept for spin-1/2 is "Mutually unbiased bases" or MUBs. There are only (at most) three MUBs for spin-1/2. Letting a particle's spin move among them means that the number of degrees of freedom of the particle have tripled. So when you find the long time propagators over that Hopf algebra you end up with three times the usual number of particles. Hence there are three generations.

The long answer: The two (more or less classical) things we can theoretically measure for a spin-1/2 particle are its position and its spin. If we measure its spin, the spin is then forced into an eigenstate of spin so that measuring it again gives the same result. That is, a measurement of spin causes the spin to be determined. On the other hand, if we measure its position, then by the Heisenberg uncertainty principle, we will cause an unknown change to its momentum. The change in momentum makes it impossible for us to predict the result of a subsequent position measurement.

As quantum physicists, we long ago grew accustomed to this bizarre behavior. But imagine that nature is parsimonious with her underlying machinery. If so, we'd expect the fundamental (i.e. before renormalization) measurements of a spin-1/2 particle's position and spin to be similar. For such a theory to work, one must show that after renormalization, one obtains the usual spin-1/2.

A possible solution to this conundrum is given in the paper:
Found.Phys.40:1681-1699,(2010), Carl Brannen, Spin Path Integrals and Generations
The paper is a straightforward QFT resummation calculation. It assumes a strange (to us) spin-1/2 where measurements act like the not so strange position measurements. It resums the propagators for the theory and finds that the strange behavior disappears over long times. The long time propagators are equivalent to the usual spin-1/2. Furthermore, they appear in three generations. And it shows that the long time propagators have a form that matches the mysterious lepton mass formulas of Yoshio Koide.

Peer review: The paper was peer-reviewed through an arduous process of three reviewers. As with any journal article it had a managing editor, and a chief editor. Complaints about the physics have already been made by competent physicists who took the trouble of carefully reading the paper. It's unlikely that someone making a quick read of the paper is going to find something that hasn't already been argued through. The paper was selected by the chief editor of Found. Phys. as suitable for publication in that journal and so published last year.

The chief editor of Found. Phys. is now Gerard 't Hooft. His attitude on publishing junk is quite clear, he writes
How to become a bad theoretical physicist

On your way towards becoming a bad theoretician, take your own immature theory, stop checking it for mistakes, don't listen to colleagues who do spot weaknesses, and start admiring your own infallible intelligence. Try to overshout all your critics, and have your work published anyway. If the well-established science media refuse to publish your work, start your own publishing company and edit your own books. If you are really clever you can find yourself a formerly professional physics journal where the chief editor is asleep.

One hopes that 't Hooft wasn't asleep when he allowed this paper to be published.

Extensions: My next paper on the subject extends the above calculation to obtain the weak hypercharge and weak isospin quantum numbers. It uses methods similar to the above, that is, the calculation of long time propagators, but uses a more sophisticated method of manipulating the Feynman diagrams called "Hopf algebra" or "quantum algebra". I'm figuring on sending it in to the same journal. It's close to getting finished, I basically need to reread it over and over and add references:

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Published, peer reviewed article gets two down votes and no comments. – Carl Brannen Jan 19 '11 at 4:15
I did not down vote, but it does seem very convoluted for the request in the question. Maybe a summation in a phrase on the lines :"thus the existence of spin leads mathematically to the existence of three generations". – anna v Feb 16 '11 at 6:37
Okay, so here's a third down-vote and a comment: first four paragraphs don't mention generations at all. This alone would suffice for a down-vote. You also make some pretty weird statements about QM which would need to be clarified (and I would discuss those if the answer was otherwise good; which it isn't). At the end you say that all of the (obviously irrelevant) stuff in first four paragraphs implies three generations. But you don't say anything about how or why. This is one of those answers where I wish to be able to down-vote more than once... – Marek Feb 17 '11 at 22:42
How about instead of discussing why you paper is perfect and can't possibly have any flaw you'd discuss its contents? I suppose that as its author you'd be the most competent to do it. Instead you spend half of your answer talking about trivialities of QM and the other about yourself being perfect. Way to go ;) – Marek Feb 18 '11 at 10:35
By the way, to be more specific: your paper is definitely interesting. Of course I can't tell whether it's correct or not after spending just few minutes with it but if you edited the answer to talk more about the paper itself (e.g. mentioning how precisely does the number 3 arise, mentioning the Koide formula, etc.) rather than talking about the publishing process, I'd consider giving you an up-vote ;) – Marek Feb 18 '11 at 14:17

At present, I'd have to agree with dmckee's "Who ordered that?" quote, in that the Standard Model must take the list of fundamental particles as an input, i.e., it provides no explanation (just as it does not explain color charge). I'd argue that CP violation isn't so much required by the theory (feel free to correct me here), as it is an observation of reality, like the particles themselves.

Some theories in development, such as String Theory, do provide a reason for precisely three families (as Jerry mentions). In the case of String Theory, it comes about because of allowable string oscillations, which themselves are dependent on the number of dimensions (compactified and extended). (The number of dimensions and the way they are compacted is more fundamental than the number of particle families, so I would argue that, while we may have gotten the number of dimensions in part to make the particle families work out, many other things, such as predicting the properties of the still theoretical graviton, also depend on the dimensional parameters. This leads me to make the claim that choosing the number of dimensions is more than a parameter dictated by the number of particle families, i.e. that dimensions predict three families, rather than three families being used to choose the number and form of the dimensions.)

So, from a pop-sci standpoint, I'd have to say that currently accepted theory can't really explain why we have three families of particles, but theorists are hard at work on new theories, some of which can explain it as a consequence of something deeper (with an appropriate side note that the existing theories are fantastically good at explaining our world, but that we know they have very specific shortcomings in very special cases, and we won't be satisfied until we've cleared those up. I add this because I get tired of arguments of a religious nature that take the very small bit we don't understand and use that to claim we don't understand anything.)

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I'm down voting this answer because of several inaccuracies. 1) The massless modes of the string that show up as generations of fermions have nothing to do with the oscillations of the strings, they arise from the zero modes and 2) the answer gives the impression that string theory provides a reason for 3 families, which it does not. The number of families depends on the geometry of the compatification space (the Euler number of the Calabi-Yau manifold for compactifications of the heterotic string) and you can get many answers besides 3. – pho Jan 8 '11 at 15:52
@Jeff Thank you for the clarification. I certainly should have said modes rather than oscillations, but don't the zero modes depend on the number of dimensions? On the second point, the dependence on geometry is a very appropriate addition to a correct answer, but I would still claim that the "choice" of geometry is deeper than simply getting the correct number of observed families, so that it can be claimed as an explanation of that phenomenon. – Mitchell Jan 9 '11 at 1:37
The number of generations depends on the Euler number of the CY manifold which is a different concept than the number of dimensions. I think it is fair to say that compactification of the heterotic string on a CY space explains why there are chiral generations since this a generic phenomenon true for any CY with nonzero Euler number, but not the number of such generations which varies from 0 to 480 for currently known CY spaces. – pho Jan 9 '11 at 14:57

Dear Chad, I thought you were an atheist. Most atheists tend to realize that many things that exist in the Universe have no "purpose". The existence of the Universe has no "purpose" that may be scientifically demonstrated.

Even if life could exist in a Universe with 1 generation of quarks and leptons, which I find plausible (although I couldn't instantly produce any string compactification with 1 generation), one could still ask why the Universe only has 1 of them if it can have several generations.

The idea that 1 generation is inevitably "qualitatively more likely" or "qualitatively more natural" than 3 generations is just flawed. Life could arguably exist somewhere in a Universe with 1 generation. The parameters and molecules relevant for life - and phase diagrams of QCD etc. - would have to be recalculated but no proof is known that would show that life would be impossible in such a Universe.

Otherwise, the fact that there are 3 generations in a particular Universe can be derived from deeper properties of string theory (half of the Euler character of the Calabi-Yau shape, assuming a conventional heterotic compactification for a while), and as I have hinted, even at this very point, it might be possible to show that the number of generations cannot be one, among other forbidden values. While three-generation models are known, it's not fully known at this moment whether 3 generations is a unique solution to some conditions or whether it's a coincidence, as the anthropic reasoning wants us to immediately believe.

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I am looking at this question as a particle physicist and as a reader. I suppose you have explained to your dog about potentials and quantum mechanical solutions which allow electrons to be trapped around nuclei, so the dog is familiar with the quantum nature of the world :). You could illustrate with a harmonic oscillator and show that given different strengths the energy levels change accordingly. Then you are ready to do an analogue . Each energy level is a "particle" in potentia, if the right material is there. If you have a hydrogen atom yo have one proton and one electron, and you have only one atom of hydrogen, even though there are many energy levels. If you get a helium you fill two energy levels and the rest of the potential lines are free. You can talk about adding energy to get to an excited state and still have the same atom.

You can make an analogue of the standard model, see for example the graph in the particle physics book figure 14.4. Energy input raises a nucleon (three quarks) to a higher "quasi stable" excited state, that contains new generations of quarks. This gives the argument that the quarks and leptons that make up our world are the analogue lowest energy levels filled that create the matter we depend on. The extra generations are there in the same way that the extra levels are there in the simple quantum mechanical problem and may be filled and appear given enough energy. They are there because of the form of the "potential" that makes them possible in order and groupings that are necessary given the stable matter solutions we observe, which are still at the frontier of current theoretical studies in physics.

It is true that higher order terms in QCD will include all the generations and it might be that the nucleon solution would not be stable if these higher generations were not there, but maybe somebody else could think of an analogue for that.

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One thing about looking at the generations as excitations is that there's pretty good experimental evidence that there is only three generations. This is completely alien to our experience with radial excitations of an atom. It brings to mind Tamar Friedmann's proposal, "no radial excitations" – Carl Brannen Feb 20 '11 at 0:44

I've decided to elevate my comment on your question to an answer:

As a result of dmckee's comment (+1), I Googled "cosmological effects of neutrino mixing". There are relevant results, though I'm not qualified to sift them for you. Less cosmologically, neutrino mixing might modify the star-forming effects of supernovae in gas clouds nearby. Google for "supernova neutrino mixing" seems to me much more interesting. It may even affect whether there are supernovae at all. If there were none of those, there would be no dogs or bunnies, though, who knows, there might still be squirrels. A more specific question about Supernovae and neutrino mixing might be good.

This is not to say that neutrino mixing causes Supernovae, even if it were the case that Supernovae would not happen if there were no neutrino mixing. It is contingently the case that we observe Supernovae, and most models take Supernovae to be a principal source of heavy metals, particularly iron, and it is contingently the case that we observe neutrino mixing. There's bound to be someone on Physics SE who knows straight off whether neutrino mixing plays a significant role in current astrophysics models for Supernovae.

At the end of the day, however, this is just to say that effects that are very subtle at small scales may have manifest consequences at large scales. In Dog World —which typically doesn't care about butterflies, even if someone speculates that they might cause a hurricane somewhere—, if Emmy doesn't eat anything for 2 hours, she might not notice, but if Emmy doesn't eat for three days, everybody would notice.

I do want to change a detail of my comment — if there might be a metaphysical category of things that behave like “squirrels”, even without iron in the world, because, counterfactually, we modified the Universe so that there is no metaphysical category of things that behave like “neutrino quantum fields that mix”, surely there would also be a metaphysical category of things that behave like “dogs”. I can't pull it off, but I'm envious of your dog trope.

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The number of fermionic helicity states of the supersymetric standard model with massive neutrinos --if they are right neutrinos, as expected from seesaw and GUT-- and massive gauge bosons is 126. Of course the number of bosonic states is the same :-). You can add another two helicity states if the mass mechanism is the MSSM one. And of course, you can add another two helicity states if you put the gravitino in the bag.

So, with 3 generations and now that the neutrinos are massive, you win the possibility of fitting the game in a 128 fermion, which happens to be the dimension of a D=11 fermion.

With any game of neutrinos, if you put them apart the extant fermions, they amount to 84 helicities. With three generations and a massive top, you can consider neutrinos as in the previous paragraphs and simply put apart the top quark and squark; again the extant "light fermions" amount to 84 helicities, and so they superpartners. Nothing useful has emerged of this, but it comes out from having three generations. Note that in D=11 SUGRA, an 84-component bosonic object is forced to exist, that complements the 44-component graviton.

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Would the down-voters care to leave a comment regarding their decisions? This answer appears rock solid to me. It might not qualify the OP's request to be in terms that would be comprehensible to a non-scientist. But then again it is comprehensible to scientists (or at least one scientist, me) and that is who this site is intended to attract. – user346 Jan 16 '11 at 6:52
The first down-voter did it when the answer only had the 3rd paragraph. In any case, the up/down arrows are labeled as "useful/not useful". So I am assuming that it is because the answer, particularly the 3rd p., can be considered "not useful" (of course not useful for the OP bonus goal, as you say, but also not useful for specialists, as this kind of facts do not hint any way to exploit them in model building). Thanks for your +1, space_cadet!. – arivero Jan 16 '11 at 9:59
edited to remove the original 3rd paragraph. – arivero Feb 17 '11 at 21:09
removed also the SU(5) flavour thing, which was the most speculative part. Refer to and references therein ( arXiv:0910.4793 arXiv:0710.1526 arXiv:hep-ph/0512065) if remotely interested on the issue. – arivero Feb 21 '11 at 17:43

This question is related to the problem of family structure. I will restrict this to QCD, where the gauge group is $SU(3)$, and there are the gauge multiples. The irreducible representation of $SU(3)$ are ${\bf 3}\times {\bar{\bf 3}}$ and ${\bf 8}~+~{\bf 1}$. The first irrep describes the quark doublet structure or “flavors,” while the second defines the “color scheme,” which is how quarks may carry two of the gauge coupling charges $r,g,b$ and anti $r,g,b$. The “one” defines “white” which is color neutral. So it turns out that the gauge field has a certain group structure and the carriers of these charges also have the same group structure.

We might think of the gauge potential $A_\mu$ as written according to $A_\mu~=~A^b_\mu\lambda^b$ where the sum is over the $\bf 8$ color scheme. The currents for the theory $J_\mu~=~{\bar\psi}^i\gamma_\mu\psi^i$ are determined by the quarks. Here the index $i$ is with respect to the $\bf 8$ of $SU(3)$. We have from electromagnetism the Maxwell-Faraday equation $$ \nabla\times {\bf H}~=~{\bf J}~+~\frac{\partial {\bf D}}{\partial t} $$ where the time derivative of electric displacement vector is the “displacement current.” In effect the field theory says, “The left hand side sums these up and computes a value for $\bf H$ independently of the particular values of each.” The two play identical roles. With QCD we have a similar gauge covariant form of this expression, and the magnetic intensity analogue in QCD obeys a similar rule. Consequently, the current and gauge field should be interchangeable by changing the irrep --- to put is somewhat loosely I have to admit.

This does not constitute a proof, but it is suggestive of why the source for the fields has a similar structure as the fields themselves. The one exception is electromagnetism, which is a $U(1)$ gauge group and all particles carry a charge. The connection with particle families and electromagnetism is with hypercharge and the Gell-Mann–Nishijima formula that interchanges the centers of higher groups with $U(1)$.

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Except for the hint in the last line, this seems a reply to a different question, the structure of particles as representations of a gauge group. But gauge group happens for each generation, question is: What Do We Get From Having Higher Generations of Particles? – arivero Feb 18 '11 at 14:58
I was specifically addressing the question of generations of particles. It turns out that for an SU(n) gauge group the fermionic source fields have the same symmetry. – Lawrence B. Crowell Feb 19 '11 at 0:50
So your short answer should we that "3 generations allow us to arrange them with a SU(3) horizontal flavour group", is it? – arivero Feb 19 '11 at 1:39
In a nutshell that is about it. The family structure or group system of flavours is isomorphic to the gauge symmetry they act as a source for by carrying a color charge. This is actually one of those unclosed questions. What I indicate is suggestive, but it is not a proof. It might be that the flavours and the colours form mixed eigenstates in some way, though I am not sure how to make this work in an irrep. – Lawrence B. Crowell Feb 19 '11 at 13:35

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