Quaternionic Equations Of Motion For All Elementary Particles
    By Hans van Leunen | October 24th 2011 08:19 AM | 7 comments | Print | E-mail | Track Comments
    About Hans

    I am a retired Physicist (born in 1941) with experience in chemistry, Fourier optics, image intensifiers, quantum logic, quantum physics, modular...

    View Hans's Profile

    This article eliminates the need for the Higgs.

    The quaternionic equation of motion of an elementary particle is in fact a continuity equation in which another quaternionic flavor ψʸ of the transporting field ψˣ is coupled with that transporting field via a source term. The coupling factor acts as the mass of the corresponding particle.

    This fact comes into the foreground when the Dirac equation is converted from its spinor based form to the much cleaner quaternionic form. After stripping away the matrices and reducing the spinors to a single element, the quaternionic Dirac equation for a free particle runs:

      ∇ψ = m ψ*


      ∇*ψ* = m ψ

    A derived equation is

      ∇(ψ ψ) = m (ψ ψ*) = 2 m|ψ|²

    Another derived equation is

      ∇(ψ ψ*) = 2 m Re(ψ ψ)


      ∫˯ (ψ ψ*) dV=1


      ∫˯∇(ψ ψ) dV =2m

    m is acoupling factor. It couples two flavors ψ and ψ* of the quaternionic field ψ

    Four flavors of field ψ exist; ψ, ψ’, ψ” and ψ*.

    The difference switches the sign of 0, 1, 2, resp. 3 imaginary base vectors.

    The general form of the equation of motion fo relementary particles is:

      ∇ψˣ = m ψʸ

    This leads to 16 equations. The four equations in which ψˣ equals ψʸ correspond to a zero coupling factor m. Just reversing the field flavors does not work. With the conjugate the nabla must be conjugated as well. The resulting equations correspond to as many categories of elementary particle types that have non-zero coupling factors. Each category contains three sets of coupling factors.

    The coupling factor follows from:

      ∫˯∇( ψˣ ψʸ*)dV = 2m ∫˯ ( ψʸ ψʸ*) dV

    This equation also shows that m is a property of the combination ψˣ,ψʸ , which defines the particle. Thus, no Higgs is required in order to give elementary particles their mass.

    See details at:
    The most recent version is supported at 
    It is supported by a slideshow:


    All these quaternion aficionados seem to surface in time for Hamilton's bridge walk over in the UK. Despite their mathy interest, quaternions have never found a physical niche, & are largely ignored by the community. 84 yrs of the Dirac eq. shorn of quats prove that. Similar to Hestene's `space-time algebra' : No new physics !
    Also, Why no arxiv posting ? If U want the community to inspect the gory details, THAT is where to unveil it, not here !

    Since arXiv puts a bockade for retired physicists that have little contact to a community that offers an access support (new authors need an endorsement from existing authors), I put my writings on viXra, which is more open than arXiv. ArXiv is open access to readers, but not to writers. I consider viXra as really two way open acess. Still I keep the most recent version of my papers on my website. In that way viXra is not spoiled with a huge amount of new versions. If you do not know viXra, then you must quickly extend your knowledge.
    ViXra is an alternative archive of 2265 e-prints in Science and Mathematics serving the whole scientific community
    If you think, think twice
    You are not blacklisted from Arxiv, trust me. You must submit your paper under the category of `general physics', rather than `qu-ph', for which an endorsement is still no guarantee. Assuming it is typeset correctly & professionally written, it will be posted. Sorry, but Vixra is a bit of a joke, run out of some guy's apartment in the UK, not really read by the community at large, & in obscura. Give it a shot, & may I suggest the math-type SW, `LyX', just google it for free DnLd.

    I would not call the work of the 2K+ people that put serious articles on viXra a "bit of a joke". May be you do, but I do not understand your reasoning. I also do not understand the reasoning behind "endorsement". It is a false way of community guarding. The same as is hided behind the notion of "notability" that Wikipedia applies. Both measures hamper the publication of fresh ideas and show a sick mentality. It is this mentality that keeps me away from arXiv.
    If you think, think twice
    I did not mean to impugn the people who publish on Vixra, just that it is largely Unknown by the gazillions who pre-print on Arxiv. Au contraire, I cheer Vixra ON, & even know a scientists who has Sued Cornell for being blacklisted, but Lost.
    You are correct that the Arxiv has devolved into a sick society.
    Ultimately, the joke is on them, as string theory, SUSY, large extra dimension, & mini-black holes, radical ideas that have been at the core of theoretical physics for 40 yrs now, are suddenly on Life-Support, & in Critical Condition. One remains: Higgsy, but I disagree with you. He will be found kicking, alive & well at 122.8 Gev.
    Trust me: You'll Like Lyx !

    The Stand-Up Physicist
    Hello Hans:

    My approach to the arXiv issue was to build my own pre-print server :-) It is a joke by Jimbo's criteria, having all of 7 papers from 2010, and none afterwards, all by the same author. Anyway, I am glad they have a nice home even if few visit.

    One of them is on this very topic. I did end up with 16 Dirac equations using the various flavors of conjugates like you did, so your writeup sounded familiar to my odd ear. I will have to look into your claim about not needing the Higgs, a claim I make for a different reason.

    The Klein-Gordon equation is even easier (as you probably know). The "new" physics I get out of this process is there is often a phase term that appears to be perfectly valid, but does not show up in the tensor approach. Have you had that experience too? I don't know what to do with those extra terms, too bad for me.

    Just to keep things odd, I also work to make every equation dimensionless. I have figured out how to animate a few expressions, but don't have enough of an animation of quantum field theory library to brag about, just dreams.

    Good luck in your studies,

    Equation 19 in the referred article treats the quaternionic form of the Dirac equation. The conjugate form of this equation delivers the second Dirac equation. A cannot derive from the other part of your article that you use alternative flavors of the quaternionic field in order to generate new equations of motion, such as the Majorana equation and equations that treat quarks, W bosons and the Z boson.

    I do not consider the Klein Gordon equation as a proper physical equation. Eliyahu Comay gives some good arguments (

    From the equations of motion it is possible to derive an equation for the coupling factor m. That equation can be integrated in a volume integral, such that the factor m results. I do not consider the coupling factor as equivalent to mass, but it may intimately related with it.The fact that m can be computed from the field, means that no external field or particle is required to provide that factor.

    I rather seek the cause of the gravitation field that goes with a massive particle in the presence of a geometric anomaly at the location of the particle. The curvature caused by that anomaly causes the values of gravitation field. This gravitation field does no more and no less than administer the local curvature.


    If you think, think twice