Carl Brannen and the Koide formula – thinking about what kind of simple physics is needed to derive the Koide formula

This post was originally going to be a comment to Kea’s blog, but she has developed a slightly unappreciative attitude towards my comments, and I can’t put subscripts or superscripts in comments, so I’ll put my idea here instead.

“This might give another explanation of the square root.” – Carl Brannen

Kinetic energy can be expressed as

E = (1/2)p2/m


p2/(Em) = 2
[Eq. 1]

Kinetic energy, E = (1/2)mv2, gives velocity v = (2E/m)1/2, hence:

p = mv = m(2E/m)1/2

= (2Em)1/2
[Eq. 2]

Hence, we have a square-root of mass appear, which may be useful in the theory for the Koide formula! So could the Koide formula be an averaging of the momenta of the three leptons in the situation that they are in thermal equilibrium, with identical kinetic energy, only differing masses? Simply use Eq. 1 to sum the momenta given by Eq. 2 for three particles. Let’s assume we can justify writing Eq. 1 for 3 particles as the following (note that we add up momenta before squaring the sum):

(p1 + p2 + p3)2/[E(m1 + m2 + m3)] = 2

Inserting Eq. 2, pn = (2Emn)1/2, gives:

[(2Em1)1/2 + (2Em2)1/2 + (2Em3)1/2]2/[E(m1 + m2 + m3)] = 2

which will simplify to:

(m11/2 + m21/2 + m31/2)2/(m1 + m2 + m3) = 1

which is close to Koide’s formula, which differs in having the dimensionless numerical factor on the right hand side 3/2 instead of 1. Possibly this may arise from relativistic effects and/or angular momentum (spin) of the leptons. Maybe we should be thinking of all leptons having identical angular momentum (spin) energy and just differing masses; which might be the deeper physical meaning of the Koide formula. In any case, it seems to me to be an average of the momenta of the generations of leptons in thermal or energy equilibrium. I’m not claiming to yet have a complete and rigorous theoretical derivation of Koide’s formula. Maybe by putting out this idea, someone else with more time available will be motivated to investigate further along these or related lines.

Carl has a PDF paper about Koide’s formula here. Yoshio Koide in 1982 proposed that if you square the sum of the square roots of the 3 lepton masses, and then divide that result into the simple sum of the 3 lepton masses, you get the ratio 3/2. This is treated by some as a chance result of random numerology, but Carl and others treat it as a more interesting type of coincidence. After all, Balmer’s formula for the curious distance spacings between discrete atomic lines in light spectra was originally entirely empirical, with no theoretical justification. It was decades before Bohr came up with the explanation in his atomic theory of quantum leaps. Although the Koide formula only predicts one mass after requiring two masses as inputs, it has nevertheless been vindicated ever more precisely as more accurate mass data for the leptons has become available over the last 27 years. In particular, Carl has also found it possible to apply it to the 3 neutrino masses if the lightest neutrino mass square root is given a negative sign instead of the positive sign:

Koide formula

8 thoughts on “Carl Brannen and the Koide formula – thinking about what kind of simple physics is needed to derive the Koide formula

    1. Codder,

      The Pioneer Anomaly is an apparent acceleration towards the sun of two Pioneer space probes of roughly a = Hc towards the sun, at least along the trajectories of their escape from the solar system. It’s not experimental proof that this is a universal increase in gravity, see:

      According to :

      “… gas leaks, including helium from the spacecrafts’ radioisotope thermoelectric generators (RTGs) has been viewed as possible causes. Fourth, a real deceleration not accounted for in the current model could result from radiation pressure of sunlight, or the spacecraft’s radio transmissions. Thermal radiation could account for the anomaly as pressure from the RTGs (See Radioisotope rocket) or as asymmetrical radiation of the heat from the spacecraft electronics. Heat from the electronics reflects from the back of the spacecraft’s dish-like main antenna, causing a recoil … A presentation at the APS April 2008 meeting suggests that differential heating may account for as much of 1/3 rd of the observed acceleration.[7] … Finally, a real deceleration not accounted for in the current model could be from electromagnetic forces due to an electric charge on the spacecraft.”

      In other words, it’s BS experimental error accounting. It’s a bit like a simplistic test to prove whether leukemia is due to radiation by comparing the leukemia rate in Denver (a high altitude city, with double the normal cosmic radiatio dose) to New York. If you do that, you find wow, radiation reduces leukemia rates. Problem is, there are lots of other explanations because it’s not a cleanly controlled experiment: the difference could be due to differences in wealth, smoking and exercise habits, air pressure, etc., etc., etc. This is why you need to build theory and especially to test theory upon hard facts not BS experimental hype and guesswork accountancy for discrepancies, which is what the Pioneer anomaly boils down to.

      The thermal radiation pressure from the hot radioisotope thermo-electricity generators from the BACK of the antenna dish of each Pioneer spacecraft (the dishes pointed towards Earth, naturally) would by recoil have accelerated the Pioneer probes towards Earth, adding to the acceleration of gravity. Sunlight was relatively weak by comparison, so the main radiation pressure is from the hot radioactive thermo-electric power source asymetrically radiating heat radiation in the direction away from the Earth (behind the dish), thus causing the probes to recoil towards the sun.

      “Power for the spacecraft was obtained by four SNAP-19 radioisotope thermonuclear generators (RTG), which were held about 3 m from the center of the spacecraft by two three-rod trusses 120° apart. A third boom extended 6.6 m from the experiment compartment to hold the magnetometer away from the spacecraft. The four RTG’s generated about 155 W at launch and decayed to approximately 140 W by the time the spacecraft reached Jupiter, 21 months after launch.” –

      Since these are about 95% inefficient at converting the heat from radioactive decay into electricity and only 5% gets converted, the statement above has no meaning: it’s ambiguous and could refer to 140 W of electric power, and thus 20 times more thermal power, or it could refer to 140 W as the total thermal power of which only 5% is converted into electricity. The extraordinarily low scientific quality of all the BS writing about the Pioneer anomaly facts on the internet is therefore a deterrent to even attempt a calculation. I think it’s more important to confirm predictions of more fundamental numbers in science like the gravitational parameter G, than to mess around with soft science like the Pioneer anomaly.

  1. ok!i don’t think u gave much thought about “pioneer anomaly”.
    u reason like:it’s a coincidence a = Hc so the experiment must be sloppy.
    i reason the other way:it’s a coincidence so the “sloppy” experiment it’s not too might be valid.
    in either case the question is: SO WHAT?
    in u theory netwon’s 3rd law is: a.principle? or b.consequence?
    cause it seems to me that action and reaction is very much a consequence of differential screening that u embrace eaven in the “pioneer anomaly” case.
    that makes u theory capable of explaining the cause by the effect and backwords.
    SO WHAT?in u theory netwon’s 3rd law is: a.principle? or b.consequence?

    1. Hi Codder,

      I don’t think it’s an anomaly, sure the Pioneer Probes are accelerating, it’s just shoddy science to claim that the acceleration is an “anomaly” without properly analyzing whether heat emissions from the radioactive power sources or gas emissions from the stabilizer jets all on the back of the Earth-pointing radio dish were causing the aceleration.

      It’s not a coincidence, because you are picking it out because the size is roughly the size of the cosmological acceleration. Feynman explained this “coincidences” very nicely when looking out of the window, seeing a car number plate, then calculating the incredibly small chance that he should see that particular car number plate out of all the different number plates in the county!

      If you had predicted the Pioneer Anomaly acceleration in advance, then it would be one hell of a coincidence if your theory was wrong! But you didn’t.

      I predicted the value a = Hc cosmological acceleration in May 1996, and it was found right in 1998. I have the feeling that if I am wrong, it will be one hell of a coincidence that I correctly predicted it in advance, and the prediction agreed with the measurements two years later.

      I don’t give much thought to shoddy experimental science like “Cold Fusion” (holding a heat sensitive neutron counter probe near a beaker and claiming the increased count rate is neutrons, not hand heat, what a coincidence!!!), any more than I do to the Pioneer Anomaly. It’s just a coincidence in a tiny measurement close to all kinds of tiny corrections due to heating effects on the probe, similar to Cold Fusion.

      The supernovae recession acceleration of a = Hc is solid science. There is a long range radial acceleration outwards, of hundreds of supernovae measured so far. Unlike the Pioneer probes, the supernovae are not dishes facing us with units on their backs (facing away from us) full of hot radioactive sources and gas supplies for stabilizer nozzles. So while I think it’s likely the Pioneer accelerations towards the sun have a simple explanation in terms of the geometry of the probes and the effects of the heat and gas pollution they emit in causing a slight acceleration towards the sun in addition to gravity, I think the outward a = Hc acceleration of supernovae is evidence of quantum gravity.

      Newton’s 3rd law is a very interesting question. As it stands, it’s a principle based on the conservation of momentum, but Newton made it up based on hard experimental evidence. When I go swimming tonight, I pull water backwards with my hands, and the recoil makes me go forward. Similarly, a helicopter goes up because it blows air downwards, getting an upward recoil. A space rocket fires gas downwards, and thus recoils upwards.

      Newton defined force F as the rate of change of momentum, F = dp/dt. Momentum p = mv, so force F = d(mv)/dt = (m.dv/dt) + ( For non-relativistic situations, dm/dt = 0, so this simplifies to F = m.dv/dt = ma. This is Newton’s 2nd law.

      Newton’s 3rd law, that action and reaction are equal and opposite, just comes from the conservation of momentum. It’ is a CONSEQUENCE of conservation of momentum.

      Conservation of momentum implies that when two things interact, the sum of momenta before and after are the same:

      (mv)_before interaction = (mv)_after interaction.

      Now imagine a gun. Before it is fired, bullet and gun are still and have zero momentum (relative to the observer anyway).

      After you fire the gun, the bullet is going fast out of the gun. BUT, the conservation of momentum means the sum of momenta must still be ZERO. The only way the total momentum of gun + bullet can be ZERO is if the gun GOES BACKWARDS.

      Yep, that’s why people who hold pistols too close to their eye when shooting get painful black eye. Newton’s 3rd law. Recoil of the gun towards you!

      So Newton’s 3rd law can be obtained from the principle of conservation of momentum before and after firing a gun or any other kind of interaction.

      Originally I came up with the gravity mechanism and published it without Newton’s 3rd law. I just pointed out a fluid analogy to the spacetime fabric, which in quantum field theory was supposed to be a frictionless sea of virtual radiations.

      If you move something underwater, the fluid ends up flowing around the sides and filling in the volume you are vacating as the object moves. This is very simple, and you can prove it using colored dyes in water to show the motion.

      If I move a 1 cubic foot object at 1 foot/second Northwards underwater, it turns out that 1 cubic foot of water ends up moving Southwards at 1 foot/second, simply to fill in the volume being vacated as the object is moved!

      This was what I wrote about in my 2004 CERN paper on gravity, where I didn’t use the principle of conservation of momentum or Newton’s 3rd law at all:

      I just explained that if something moves in an ideal spacetime fluid, you will have an equal volume of the spacetime fluid moving at the same speed (or acceleration!) in the OPPOSITE direction, in order to maintain equilibrium of pressure. This is physically the basis for Newton’s 3rd law or the conservation of momentum in nature. Hope this answers your question.

      1. Here a random (very speculative) thought on the “pioneer anomaly”.
        Since the Heliosphere is like a bubble (oddly shaped as it may be), before entering the bubble wall, the Voyager spacecraft were not affected by the graviton shadowing of the matter in the Heliosphere itself, since they were surrounded by it equally on all sides. It isn’t until they start to pass though the heliosphere and ultimately reaches the outside of the heliosphere that both Voyager will be slowed down because the will be pushed toward the Heliosphere more due to the shielding of gravitons coming from the Heliosphere below.

        Of course the above assumes that the particles that make up the Heliosphere have enough mass to make this acceleration significant, which might be completely wrong.

  2. Recently, at “nks forum applied nks” I have explained why the Fredkin-Wolfram information process together with M-theory might the basis for understanding dark matter and dark energy. If Fredkin-factor is the size that I have suggested, then the Pioneer anomaly is at least partially due to violations of Einstein’s equivalence principle by Fredkin forces from alternate universes. Some formula similar to the Koide formula might yield Fredkin-factor. At least the way I see things, the space roar is strong empirical evidence in favor of M-theory.

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