# Spin Path Integrals and Generations

Originality
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Referee this paper: arXiv:1006.3114 by C. A. Brannen

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requested Apr 7

paper authored Jun 16, 2010 to physics

I respect Carl Brannen as someone who discovered new, and I think authentic, implications of Koide's mass relation. On the other hand, this paper, also part of his oeuvre, never made sense to me, even though it made it into "Foundations of Physics" (a journal which does publish a lot of speculation); and at Physics Stack Exchange, it's currently the top answer to the question "why are there three generations?", as posed by pop science writer Chad Orzel.

I still don't understand the paper, but it seems to express the propagator of a spin-1/2 particle, as a kind of path integral of 3x3 matrices... and when you do this, somehow the number of states is multiplied by 3 and that's the 3 generations.

Sections 1-5 are simple linear algebra in finite dimensions (not related to field theory), and Section 6 contains essentially unrelated remarks about generations in the standard model, particle mass formulas, and black holes. Nothing convincing; already talking about black holes in this context shows the level of speculation involved.

By the way I don't think Foundations of Physics is a very high quality journal - standards have to be very loose because it is about foundations, which are (beyond what is conventionally accepted) vague by their nature.

I see a downvote ... This document needs some hours++ to be read and a little checked. Its background thesis is that the algebra/geometry approach is richest than the probabilities one and, incidentally, gives smooth interpretations of our world. The author asks really the foundations as others do ... and while many others publicly say that they must be asked ( for more than 8 reasons, or else at least 9 ). Finally, what is the exact reproach to this document ? At which step does it fail ?

@MitchellPorter I have added to the Physics.SE discussion pointer to what seems to be one of the better explanations for multiple generations from string model building: https://physics.stackexchange.com/a/399171/5603

## 1 Review

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The potentially interesting stuff happens at the top of p.13, where it is postulated that ''We will associate these three propagators with the three generations of elementary fermions''.

No reason is given, no discussion. It is simply an arbitrary setting, based on the superficial coincidence that two mathematical structures both involve the number 3.

All calculations on pp.4-12 happen in a Hilbert space of dimension 2 and a derived Hilbert space of dimension 3. But three generations of elementary fermions live in a tensor product of the Hilbert spaces of three elementary fermions, which consists of wave functions $\psi_{s_1s_2s_3}(x_1,x_2,x_3)$ with three spinor indices. Even if one could ignore the three position arguments (which is difficult to justify) this would require an 8-dimensional space of three spins.

The orthogonal projectors constructed in Section 5 are those of the 3-dimensional Hilbert space of traceless operators. There is no second 3-dimensional Hilbert space around, although p.13 claims there is.

Then it is said: ''This is just three times the number of states in the usual spin-1/2. We associate the tripling with the three generations.'' But what would have been needed for three generations is the number of states in the usual spin-1/2 to the third power.

Thus nothing at all has been demonstrated, except a misunderstanding of what it means to have 3 generations of elementary leptons.

reviewed Apr 10 by (13,209 points)

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