What is new with Koide sum rules?

[mad mathematician]

Aaaand arXiv rejected it after looking at it for 16 days. The basic problem is that it gives a new formulation of QM and if you've been in the business for a while and already know how it's supposed to be done, it's as tough reading as if you'd gone back to school and had to take introductory QM all over again. In other words a lot of work.

So I suppose what happened is that the grad students they have look at things approved it and then an older physicist took a quick look and rejected it.

Another new formulation of QM?
Perhaps I am bit radical but for me a predictive QG should neither be a quantum theory of gravity nor gravitizing QM. Obviously in the right limits it should return to GR.
Is there such a thing as a discrete-manifold?
I guess I need to read more graph theory books...

 

[CarlB]

Schwinger created the formulation which ended up being called "Schwinger's Measurement Algebra". He taught introductory QM with it for years and his students wrote a text book and used it to teach their students. It's kind of not used much these days. I took the formulation and added temperature and statistical mechanics and redid how it used symmetry. To someone educated in the 80s and later it's essentially a new formulation.

 

[mad mathematician]

Schwinger created the formulation which ended up being called "Schwinger's Measurement Algebra". He taught introductory QM with it for years and his students wrote a text book and used it to teach their students. It's kind of not used much these days. I took the formulation and added temperature and statistical mechanics and redid how it used symmetry. To someone educated in the 80s and later it's essentially a new formulation.

Have you read his three volumes on his source theory?
I just read his first volume, and didn't finish reading it...:-(

 

[CarlB]

Before I read Schwinger's papers (and text book) on the measurement algebra, my recollection is that I was already interested in density matrices as they avoid the arbitrary complex phase that state vectors are subject to. Schwinger's foundational description of the measurement algebra are like density matrices in that they avoid arbitrary complex phases, but they are like kets in that they do have complex phases that can cause interference, so called "geometric phases". So I was only interested in Schwinger's work up to where he diverted to follow the vast herd of physicists who work on bra ket or Hilbert space vector work and that doesn't include source theory. Right now I'm just starting to work on gauge bosons and am having a blast. So I've probably read less of his source theory than you have.

 

[mitchell porter]

What can you or Schwinger do with measurement algebra that you can't do with ordinary quantum mechanics? And can you do everything that ordinary nonrelativistic quantum mechanics can do? Like the hydrogen spectrum, or quantum chemistry in general?

 

[CarlB]

Good question. Supposedly all formulations of QM are equivalent but sometimes things are extremely easy in one formulation and extraordinarily difficult in another. For example, the hydrogen spectrum is trivial in Schroedinger's wave function but it took decades and about 100 papers to derive it in QFT. (On the other hand, QFT was easy for deriving corrections to the wave function representation of hydrogen, the problem is in deriving the electric potential.) The reason is that the wave function uses the electric potential to describe an immense number of photons that QFT exhibits with creation and annihilation operators.

The Standard Model is defined using symmetry breaking of a vacuum. The vacuum is a state with no particles where one introduces new particles with creation operators and gets rid of them with annihilation operators. That drove the development of the Standard Model. But in the measurement algebra the vacuum is in Schwinger's word "fictitious". And he should know, he was one of the about 3 developers of QFT.

So for decades my task has been to rewrite the Standard Model in terms of the measurement algebra or equivalently Hilbert operators or equivalently density matrices. In those formulations of QM there are no creation and annihilation operators and symmetry breaking of a vacuum is just mathematical tricks. On the other hand, as Steven Weinberg pointed out, when you switch from state vectors to operators, you get a LOT more interesting things in quantum symmetry. See his paper promoting density matrices over state vectors:

Quantum Mechanics without State Vectors
Steven Weinberg, Phys. Rev. A 90, 042102 – Published 2 October, 2014

"What difference does it make?
There is one big difference, that is our chief concern in this paper. Giving up the definition of the density matrix in terms of state vectors opens up a much larger variety of ways that the density matrix might respond to various symmetry transformations."
https://arxiv.org/abs/1405.3483

This opens the door a little, or it should. To understand you can read my paper. It might help to know that I twice wrote a 990 on the physics GRE and once on the math GRE and reading that paper will be roughly as difficult as relearning quantum mechanics from the bottom up. It took me years to figure out. I'm no dummy and it won't be easy to follow in my footsteps.

Sociologically, what happened is that the symmetry methods used in physics were developed by mathematicians roughly up to the 1930s. From there, the mathematicians continued developing the subject under the name "harmonic analysis" (think generalizations of spherical harmonics) but it became essentially unreadable to physicists when mathematicians generalized from the complex numbers to rings and (mathematical) fields. Similarly, a grad student in math today would laugh at the linear algebra used by physicists. The modern math subject is called "module theory" and is far more sophisticated. Learning even a little about these subjects will help understand my papers but it's well over the head of most physicists. Here, see for yourself:

"Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency."
https://en.wikipedia.org/wiki/Harmonic_analysis

"n mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a module also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers.[1]"
https://en.wikipedia.org/wiki/Module_(mathematics)

If a physicist did understand the Standard Model and Harmonic Analysis, they'd recognize the quarks and leptons as an obvious example of a generalized Fourier transform of a finite group. But without that mathematical understanding they have to grope in the dark and work with symmetry breaking of vacuums. The problem with symmetry breaking of hypothetical vacuums is that you can define an immense number of descriptions of reality and so it's not at all surprising that you can find one that works fairly well. It's a good enough way of representing experimental results but it just isn't predictive and it doesn't tell you where, for example, the symmetry breaking comes from. And to bring it back to the subject of this thread, vacuum and symmetry breaking doesn't give you the Koide formula because that is a formula at low energies and there's no symmetry breaking available to tweak a coincidence.

 

[mitchell porter]

I confess I always avoided reading your "Spin Path Integrals and Generations" because you seemed to be saying that three flavors could emerge from what seemed to be the propagator for a single Weyl fermion, and that made no sense to me. However, with Weinberg proposing that density matrices could be fundamental rather than state vectors (and also saying that this allows for new theories, not just rewrites of state-vector theories), it's making more sense, since if you have a state vector with N components, the corresponding density matrix has N² components. So in a sense, there is an increase in the number of "generalized states" if density matrices are fundamental.

The final stumbling block is that we're dealing with spin-1/2 particles, so I would have thought N=2, but instead N=3. This has something to do with projection onto x,y,z axes, as if we were dealing with an array of 9 objects like |x><x|, |x><y|, |x><z|... I don't quite get it, but I'm avoiding digging into the papers to see if some further realization dawns (or if you step in with a helpful comment). Nonetheless, the idea that a density-matrix formalism can have more "states" than a state-vector formalism, removes my old perplexity.

Incidentally, I mentioned your work (and Lubos's "tripled Pauli statistics") at a famous quantum computing blog the other day. I talked about qutrits, but I understand now that (in terms of quantum information), this is about distinguishing three states of a qubit, not of a qutrits.

 

[CarlB]

I was confused by Motl's tripled Pauli statistics for quite a while. Eventually I realized that it was about a spin-1/2 Pauli state that had 3 possible but exclusive excited states. There's nothing special about this mathematically. But his calculation is about the least possible excitation of a non rotating spherical black hole and if it's not reality, then our understanding of black holes is seriously off. So I assume it's correct. But I don't understand how to write a quantum theory about it. One thing that comes to mind is thinking density matrices but where the "1" part of (1+sigma u)/2 has the Pauli exclusion principle so that there are three choices for u, say x, y and z.

I should add that I thought that a problem with the Spin Path Integral paper is that it implied movement in only one direction, that is (1,1,1)/sqrt(3), but eventually realized that it could be trivially generalized to allow movement (and spin) in any direction. Instead of taking steps in +x, +y and +z, one takes steps in +-x, +-y and +-z with probabilities given by the transition probabilities. Or at least that's how I recall it, I could have gotten something a bit wrong as it's been quite a few years.

The stumbling block on calling the adjoint SU(2) group a "2" or a "3" hit me too. The secret is that the numbers are talking about the basis for the Hilbert vector space. But density matrices have squared number of complex numbers so a spin-1/2 fundamental representation in density matrices is really a 2x2 state which has 4 degrees of freedom but still amounts to the equivalent of a Hilbert vector space with a basis of 2 states. Counting this way, the 2x2 density matrix is made from 2x2 = 3+1 in SU(2) state vector counting and sure enough the 3 is a vector space, but it still amounts to a spin-1/2 state. In other words, when we work in Hilbert vectors we get a simplified arena where the dimensionality of the space is the same as the number of particles. But we already lost our viginity on that when we used SO(3) to represent light where there are only two light basis states (left and right or horizontal and vertical) but SO(3) has three quantum states. Or did I get that confused a bit?

On the interpretation side of this adjoint problem, see the "relational quantum mechanics" papers by Rovelli. This is a lot of papers beginning with https://arxiv.org/pdf/quant-ph/9609002

I wrote a paper on this subject titled "A Relational Analysis of Quantum Symmetry" https://vixra.org/abs/2105.0146 which got rejected at Foundations of Physics after they sat on it for 18 months and is why I no longer bother sending papers to established journals. I'm too old for that BS. Note that the citation to Rovelli in my paper is wrong. The correct Rovelli paper is "Why Gauge?" which I highly recommend: https://arxiv.org/abs/1308.5599

Anyway, (my interpretation or version of) Rovelli's relational idea is that the only thing we can do to detect or measure quantum systems is to look at their interactions. Therefore there is no such thing as SU(2) spin-1/2 as such a state is not an interaction and so is just an assumption (that reality exists without our measuring it). Instead, all our indications of an electron are from the electron interacting with another electron which is by the adjoint state. This implies that a density matrix is the "real" way of representing an electron.

This is the same idea as in my recent paper Complex Time where I have a section titled something like "a matrix introduction to quantum field theory". Standard QFT is about creation and annihilation operators but these operators represent physical processes that are never seen in nature. Instead, where a fermion is created there is another destroyed. Splitting the activity of a fermion into the annihilation of one fermion and the creation of its replacement is equivalent to splitting a density matrix into a bra and a ket. It's a mathematical trick that only works for pure states and introduces an arbitrary complex phase which makes all sorts of calculations more difficult.

And the symmetry part of the Complex Time paper is about "okay, if quantum states have to be relational or density matrix form, then what happens when we apply symmetry to such states?" But I should add that's there's another way of taking this, one that might be more natural to folks who just have to have quantum states be Hilbert space vectors, and that is to note that density matrices are Hilbert space operators so the part of my paper having to do with symmetries of matrices can also be thought of as "symmetries of Hilbert space operators" and hence one can get exactly the same results by switching the symmetry operators from acting on the Hilbert space vectors to acting on the Hilbert space operators.

The natural Hilbert space operators to so abuse first are the Pauli spin matrices, hence my paper "Group Geometric Algebra and the Standard Model" which is unfortunately confused by having its symmetry written in Hestenes' language of geometric algebra. But as far as using symmetry to manipulate Hilbert space operators (the Weyl equations) it's clear enough: https://www.scirp.org/pdf/jmp_2020081416294422.pdf

This sort of thing was overlooked in the origins of symmetry in QM because the group algebras the mathematicians used at the time (1930s) always used the complex numbers. In harmonic analysis the mathematicians soon generalized to rings and in the case of that paper the ring is the geometric algebra, i.e. the 4-dimensional algebra with basis 1, sigma_x, sigma_y, and sigma_z. In terms of relativity, you can think of the "1" as standing for time and then it might make it more obvious that the paper is compatible with special relativity and all that.

So yet another way of interpreting this single idea of Weinberg and density matrices is that you can apply a symmetry group to special relativity itself. And that gets back to what Dirac was doing when he wrote the Dirac equation. He was deriving the particle content from special relativity. So what I'm doing here is making special relativity bigger (okay at least the spatial part of special relativity is getting bigger) and then deriving the particle content from the bigger special relativity.

 

[ohwilleke]

A. C. Kleppe has uploaded a preprint entitled "Quark mass matrices inspired by a numerical relation" that explores how Koide's rule for charged lepton masses can be extended to quarks. This conference paper presentation is Kleppe's first paper on arXiv and it isn't clear that the author has a university affiliation.

The abstract, after stating Koide's 1981 charged lepton mass rule (which still holds to high precision as the inputs have become more accurate over the last 44 years) states that:

Inspired by this relation, we introduce tentative mass matrices, using numerical values, and find matrices that display an underlying democratic texture.

There have been other attempts to make this extension and I've tinkered with these extensions a little myself. The paper does not, however, meaningfully engage with (or even mention) most of the prior literature in this area.

The statement in the paper that:

It should be noted that for the square roots of the running charged lepton masses at MZ around 91 GeV, the results no longer give the exact Koide formula.

is particularly concerning when it comes to understanding, because Koide's rule is a rule about the pole masses of particles and not about the running mass of those particles at a consistent energy scale. And, Koide's rule is, in fact, exquisitely confirmed when applied to pole masses.

This distinction matters because the proper definition of mass to use for light quarks, when extending Koide's rule, is not self-evident.

The conclusion, which I have screenshotted rather than cut and paste from to preserve the integrity of the notation, states:

 

[arivero]

A. C. Kleppe has uploaded a preprint entitled "Quark mass matrices inspired by a numerical relation" that explores how Koide's rule for charged lepton masses can be extended to quarks. This conference paper presentation is Kleppe's first paper on arXiv and it isn't clear that the author has a university affiliation.

She is surely retired (b. 1951), the arxiv only has some collaborations with Nielsen (b. 1941, we invited him in out lab time ago in the nineties). Surely not used to internet forums, because clearly she missed completely this thread. It is more infuriating that her previous paper on democratic textures cites Harari Hauts Willers and does not notice that the masses in that paper match Koide rule.

 

[arivero]

Furthermore, if we now assemble from these, the analogs of the physical yukawa matrices - $m_t$ |1>, $m_c$ |3>, $m_u$ |2> and $m_b$ |2>, $m_s$ |1>, $m_d$ |3> - also turn out to be circulant! Study of a circulant form for the physical yukawas dates back at least to Harrison & Scott 1994.

Hmm @Kea and @CarlB did mention this paper back in 2006 but I had not checked it. The generalised circulant in (3)

still matches Koide, does it? I mean, if a,b defined a Koide circulant.

Mitchell has suggested to look again into circulants and well, I agree they were pretty elegant; the trick is to look not at delta=15 degrees but to the next period, delta=135. I was sort of obsessed with the 15 degrees shape because of the factors of three.

<br /> C(45^\circ) =<br /> \begin{pmatrix}<br /> 2 &amp; 1 + \tfrac{i}{2} &amp; 1 - \tfrac{i}{2} \\<br /> 1 - \tfrac{i}{2} &amp; 2 &amp; 1 + \tfrac{i}{2} \\<br /> 1 + \tfrac{i}{2} &amp; 1 - \tfrac{i}{2} &amp; 2<br /> \end{pmatrix}=<br /> \begin{pmatrix}<br /> 1 &amp; 0 &amp; 0\\<br /> 0 &amp; 1 &amp; 0\\<br /> 0 &amp; 0 &amp; 1<br /> \end{pmatrix}<br /> \;+\;<br /> \begin{pmatrix}<br /> 1 &amp; 1 &amp; 1\\<br /> 1 &amp; 1 &amp; 1\\<br /> 1 &amp; 1 &amp; 1<br /> \end{pmatrix}<br /> \;+\;<br /> \frac{i}{2}\,<br /> \begin{pmatrix}<br /> 0 &amp; -1 &amp; 1\\<br /> 1 &amp; 0 &amp; -1\\<br /> -1 &amp; 1 &amp; 0<br /> \end{pmatrix}<br />

<br /> C(135^\circ) =<br /> \begin{pmatrix}<br /> 2 &amp; -1 - \tfrac{3i}{2} &amp; -1 + \tfrac{3i}{2} \\<br /> -1 + \tfrac{3i}{2} &amp; 2 &amp; -1 - \tfrac{3i}{2} \\<br /> -1 - \tfrac{3i}{2} &amp; -1 + \tfrac{3i}{2} &amp; 2<br /> \end{pmatrix}=<br /> 3 \begin{pmatrix}<br /> 1 &amp; 0 &amp; 0\\<br /> 0 &amp; 1 &amp; 0\\<br /> 0 &amp; 0 &amp; 1<br /> \end{pmatrix}<br /> \;-\;<br /> \begin{pmatrix}<br /> 1 &amp; 1 &amp; 1\\<br /> 1 &amp; 1 &amp; 1\\<br /> 1 &amp; 1 &amp; 1<br /> \end{pmatrix}<br /> \;-\;<br /> \frac{3i}{2}\,<br /> \begin{pmatrix}<br /> 0 &amp; -1 &amp; 1\\<br /> 1 &amp; 0 &amp; -1\\<br /> -1 &amp; 1 &amp; 0<br /> \end{pmatrix}<br />

 

[arivero]

In related news, I have updated the wikipedia page with the new (well, 2024) pdg measure, which was from the Belle measure that @ohwilleke noticed last year. It had not been moved since 2008 So it seems we are at Q = 0.66666446(508)

Today it makes 20 years since the first mention in PhysicsForums https://www.physicsforums.com/threads/all-the-lepton-masses-from-g-pi-e.46055/page-4#post-541835

 

[mitchell porter]

I have not had time or mental space in which to think about it, but I've just noticed that a recent Clifford-algebra GUT paper from Portugal

"Spacetime Grand Unified Theory" by Gonçalo M. Quinta

mentions the Brannen parametrization of Koide! This is still sufficiently rare that we should try to obtain at least a basic understanding of the author's paradigm.

 

[ohwilleke]

A new paper looks at fundamental fermion mass ratios from the perspective of something similar to an extended Koide's rule approach (and discusses Koide's rule in the body text at pages 38-39 within the context of their approach). Its 90 pages are not easy reading, however, and it is more technical than I can easily follow.

We revisit the "three generations" problem and the pattern of charged-fermion masses from the vantage of octonionic and Clifford algebra structures. Working with the exceptional Jordan algebra J3(OC) (right-handed flavor) and the symmetric cube of SU(3) (left-handed charge frame), we show that a single minimal ladder in the symmetric cube, together with the Dynkin Z2 swap (the A2 diagram flip), leads to closed-form expressions for the square-root mass ratios of all three charged families. The universal Jordan spectrum (q - delta, q, q + delta) with a theoretically derived delta squared = 3/8 fixes the endpoint contrasts; fixed Clebsch factors (2, 1, 1) ensure rung cancellation ("edge universality") so that adjacent ratios depend only on which edge is taken. The down ladder determines one step, its Dynkin reflection gives the lepton ladder, and choosing the other outward leg from the middle yields the up sector.

From the same inputs we obtain compact CKM "root-sum rules": with one 1-2 phase and a mild 2-3 cross-family normalization, the framework reproduces the Cabibbo angle and Vcb and provides leading predictions for Vub and Vtd/Vts. We perform apples-to-apples phenomenology (common scheme/scale) and find consistency with current determinations within quoted uncertainties. Conceptually, rank-1 idempotents (points of the octonionic projective plane), fixed symmetric-cube Clebsches, and the Dynkin swap together account for why electric charge is generation-blind while masses follow the observed hierarchies, and they furnish clear, falsifiable mass-ratio relations beyond the Standard Model.

Tejinder P. Singh, "Fermion mass ratios from the exceptional Jordan algebra" arXiv:2508.10131 (August 13, 2025) (90 pages).

The acknowledgments section is exceptional, so I restate it here:

The author gratefully acknowledges extensive assistance from OpenAI’s conversational AI system (ChatGPT; models GPT-5 Thinking, GPT-5 Pro). Under the author’s guidance, the assistant performed the numerical work reported here, including the extraction and fits of CKM and PMNS parameters, the “concurrency” tests against experimental inputs, and the running of fermion masses and gauge couplings at the reference scales used in this paper.

The conceptual ideas of employing the symmetric-cube construction Symm3 for generation structure and of implementing a “Dynkin swap” within the E6-based embedding were original suggestions arising in dialogue with OpenAI’s GPT-3o. Following standard authorship policies, the assistant is not listed as a co-author; nevertheless, the author wishes to explicitly recognize its substantial technical and conceptual contributions. The author alone bears responsibility for the interpretation and for any errors in the results reported here.

I gratefully acknowledge collaboration and useful conversations with Torsten AsselmeyerMaluga, Vivan Bhatt, Felix Finster, Mohammad Furquan, Niels Gresnigt, Jose Isidro, Priyank Kaushik, Rajrupa Mandal, Antonino Marciano, Claudio Paganini, Aditya Ankur Patel, Vatsalya Vaibhav and Samuel Wesley.

 

[mitchell porter]

Tejinder P. Singh, "Fermion mass ratios from the exceptional Jordan algebra" arXiv:2508.10131 (August 13, 2025) (90 pages).

A previous paper by this author was discussed in this thread in posts #251-252, and the author himself appeared at #262. I commented on his research program here.

Of all the ingredients in his calculation of fermion masses, the idea that I find the most credible, is that there might be some relationship between mass ratios and charge ratios in the first generation. This is not high on my list of ideas, but it's there. However, the mechanism proposed here (that the fermions possess a "dark electromagnetic charge" equal to the square root of their mass) does not make sense to me. I do commend the author for trying to creatively build on the chiral graviweak observation that SU(2) can give you both electroweak interactions (in the standard model) and gravity (in Ashtekar variables) - he's supposing that the analogy somehow extends to the way that charge and mass interact with the electroweak gauge field and the Ashtekar gravitational connection, respectively.