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    Quaternionic Versus Complex Probability Ampitude Distributions
    By Hans van Leunen | March 3rd 2012 05:30 AM | 17 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...

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    Introduction

    It is a mathematical fact that both the real numbers and the rational numbers contain an infinite amount of elements. It is possible to devise a procedure that assigns a label containing a different natural number to every rational number. This is not possible for the real numbers. Technically this means that the set of real numbers has a higher cardinality than the set of rational numbers. In simple words it means that there are far more real numbers than there are rational numbers. Still both sets can densely cover a selected continuum, such as a line. However, the rational numbers leave open places,because infinite many real numbers fit between each pair of rational numbers.

    Complex numbers and quaternions have the same cardinality as the real numbers. They all form a continuum. Rational numbers have the same cardinality as the integers and the natural numbers. They all form (infinite) countable sets.

    Now take the fact that the set of observations covers a continuum and presume that the observed objects form a countable set. This poses a problem when the proper observation must be attached to a selected observed object. The problem is over-determined and in general it is inconsistent. The problem can only be solved when a little inaccuracy is allowed in the value of the observations.

    In quantum physics this inaccuracy is represented by the wave function, which is a probability amplitude distribution. It renders the inaccuracy stochastic.

    Quantum physics uses separable Hilbert spaces as the realm in which it does its mathematics. A separable Hilbert space has only a countable set of dimensions. So, a location-operator can only have a countable set of eigenvectors. Each eigenvector represents a location. As a consequence, only a countable set of relevant locations exists. That is why wave functions must solve the problems that are posed by the availability of a too large variety of observations.

    In the Hilbert Book Model this argument is used in order to explain why physics contains fields, thus it has far reaching consequences. One could apply non-separable Hilbert spaces, such as a rigged Hilbert space, but that takes the argument away why physics must contain fields.

    Probability amplitude distributions

    The wave function of physical particles is a typical example of a probability amplitude distribution. Most physicists use a complex probability amplitude distribution (CPAD) for that purpose. The squared modulus of a CPAD can be interpreted as the distribution of the probability density of the presence of the carrier of the properties of the particle. However, it is equally well possible to use a quaternionic probability amplitude distribution (QPAD) instead of a CPAD. This option has several advantages. A QPAD can be considered as the combination of a real charge density distribution and an imaginary current density distribution. What these charges and currents are, can be left to later investigation. It is appropriate to interpret the charge as the ensemble of the properties of the carrier of the charge. As with the CPAD, the squared modulus of a QPAD can still be interpreted as the distribution of the probability density of the presence of the carrier of the properties of the particle. The main difference is that with QPAD’s fundamental physics can be interpreted as a streaming problem.

    This can be taken one step further. This occurs by deliberating about the carriers. What can they be? A daring and at the same time promising choice would be that they are tiny patches of the parameter space of the QPAD. The parameter space can be seen as the imaginary part of a quaternionic distribution. By transporting tiny patches out of that space to another location the space can be locally compressed and may extend at other locations. In this way a kind of space-atmosphere is created.

    This daring interpretation immediately gives an explanation for the existence of space curvature in the neighborhood of particles.

    Sign flavors

    QPAD’s have still another advantage above CPAD’s. Quaternions have two independent sign selections, where complex numbers have only one. One of the sign selections works isotropic. It is called conjugation and switches the sign of all three imaginary base vectors. The other is a reflection and exists in three independent directions. As a consequence eight different sign selections exist. This fact is hardly known.

    Quaternionic distributions keep the same sign selection through all of their values. Thus quaternionic distributions exist in eight different sign flavors. Usually their parameter is a quaternion or the imaginary part of it. Thus the parameter space is itself a quaternionic distribution. This opens the possibility to use the parameter space as the reference for the characterization of the sign flavor of the quaternionic distribution.

    QPAD’s are just special kinds of quaternionic distributions.

    The Hilbert Book Model

    The Hilbert Book Model (HBM) is a simple Higgsless and self-consistent model of fundamental physics that is strictly based on the axioms of traditional quantum logic. It uses a sequenceof instances of an extension of a quaternionic separable Hilbert space that each represents a static status quo of the whole universe. It widely uses the opportunities that are offered by QPAD’s. The HBM uses the sign flavors of coupled QPAD's in order to construct all known elementary particles that occur in the standard model.

    Due to its reliance on QPAD’s rather than on CPAD’s its methodology is unconventional and rather controversial. Methods that work in QDE and QCD cannot all be applied in the HBM. However, the HBM offers circumventions for this dilemma. It treats linear equations of motion such as the Dirac equation as balance equations. It treats fundamental physics as a space streaming problem.

    See: http://vixra.org/abs/1112.0084 and http://vixra.org/abs/1202.0033

    CONCLUSION:
    The fact that a rather small change in the methodology, such as a switch from CPAD’s to QPAD’s is capable of causing a drastic change in the view on the fundaments of physics,shows that these fundaments are still not well comprehended and are far from well established.

    Comments

    BDOA
    Of course in QM the phase factors of any probability distribution aren't measureable, so in principle one
    might not be able to observe the difference between a complex amplitude and a quaternion amplitude. In gauge theory is made by assuming the phase of any amplitude can be multiplied by an arbitary factor at any point in space. Doing that with a complex amplitude leads to a U(1) gauge theory such as electromagnetism. What would happen with a quaternion amplitude mulitplied by a arbitary quaterion phase at any point in space? Well for pure imaginary quaternions the result would be multiplied by a SU(3) matrix of phase, leading to a theory identical to quantum chromodynamics. So perphaps quaterion gauge theory might have some running. Most physicists use group theory for this, and I can
    see no reason a prior why you'd need a algebraic field field like the quaternions rather than a group for the phase and connections between the points in space. Without some principle requiring a (algebriaic) field rather than a group to be used for the phases, physicists are free to choice any group for the phases of the amplitudes, and produce many different gauge theories. Why SU(3)xSU(2)xU(1)/Z6 is of course the name of the game.
    BDOA Adams, Axitronics
    fundamentally

    Several methodologies such as covariant derivation do not (generally) work in the realm of the HBM. The main reason is that for quaternionic distributions in general hold

    (f g) f g + (f) g
    Quaternions and quaternionic distributions offer opportunities that complex numbers and complex functions cannot deliver. If you stick with gauging and su(n) groups you will certainly not discover these opportunities.

    If you think, think twice
    BDOA


    Actually, using general groups and gauging, leads to many different field theory, it not missing out, it may be necessary to examine all the possiblities.

     I've just read both your Vixra articles, nice work on the presentation PDF, looks very good. I read the hilbert model on the basis of this and wasn't as pleased. The presentation promise a dervivation of gravity (which I think is to unlikely to come from just quaternising the dirac equations). As I throught would happen, you did get color from using quaternions as the phase. In the presentation piece you found shadow as well as anti-matter particles, some wouldn't be happy with this, but i'm actually addicted to minor matter groups, of the form [SU(3)xSU(2)LxU_v(1) ]_m x [ SU(3)xSU(2)_r x U(1_v]_s x U_a_{m-s}(1) x U_a_{m+s}(1) , where m and s mean matter and shadow matter, and the missing SU(2)_Rm + SU(2)_Ls gain a huge mass from a right handed neutrino condensate.

    so shadow matter is plus to me, didn't see any talk of shadow matter in main piece through. In your presentation you found two different photons, but the one which switched signs, is actually a axi-photon, under with the left and right handed guage transformations have the opposite sign between them.

    I believe you correctly understand the problem of time, in quantum gravity, but while you promised a
    solution, I didn't actually see one, i find interesting and possibly constructive your idea that time only exists when two observers exist, where a one observer just has a fixed book of timeless information, but those the math lead anywhere?

    Please try and understand the gauge principle, and also the axial anomally these are the cores
    to building physics. I couldn't understand what you meant by foreground and background QPADs, or how your built quaternion spinors. I think a solid defination of spinors when you use QPADs, complete with how they transformation under relativity is necessary. The scalar, vector and spinor quantity defined in ordinary complex quantum mechanics might generalise in more than one ways under quaternions.

    It would be important to see how your eight conjugations, behavor under CPT transformations, if shadow particle also exist you might have two different charge conjugations one for normal matter and one for shadow matter, a C_m C_s PT symmetry.

    Rembember there are also generations, the lack of up quarks and generations in your model, indicates that you have two small a group. What happens if you use octonions? Could up quarks, appear there.
    BDOA Adams, Axitronics
    BDOA
    As an after throught, you might manage to get up quarks in your model if you super-symmetrise it,  if  electron, neutrino, red green and blue down, red green blue anti-down operators, are available with both spins 1/2 and 1, then binding a spin 1/2 down to the spin (-)1 electron, might generate an anti up-quark, this would require a very strong force, somewhat like gauged R symmetry to bind the preons operators.
    BDOA Adams, Axitronics
    Attention to complex dialectrics and refractive indices are opening new ranges of measurement. Of course those are atomic-level phenomena, but possibly linked by the Berry phase to nuclear internals. Phase physics is just getting under way.

    You seem to admit a micro-curvature of space within the proton (or neutron), or at the energy density of quark-gluon plasma. I was looking for something of this kind in Matti Pitkänen's TGD, but he won't admit "submanifold gravity". There are currents of intuition here running back through the vortices of Descartes into the mists of antiquity; with Huygens they pick up the theme of involutes and evolutes, which is quietly all over current theory, but never thematized.

    vongehr
    Now take the fact that the set of observations covers a continuum
    That's fact? News to me!

    fundamentally

    @Barry

    When you start working with QPAD's instead of CPAD's, then every complex number based technology becomes confined to special one dimensional situations. In full 3D everything becomes a streaming problem. It is a completely different way of looking at fundamental physics. In the HBM the methods that you were used to are to a large extend replaced. Dynamic interactions are replaced by couplings of QPAD's. Such a coupling is characterized by a small set of properties: coupling factor, electric charge,spin and if you wish color charge (in the HBM color charge is connected with an anisotropic direction). These properties become conserved properties of the corresponding particles. If you try to understand the HBM in terms of complex number based methodology then you are asking for disappointments. I see that this is your attitude. It is a fruitless approach.

    The HBM is self-consistent, but it differs considerably from conventional physics. Nevertheless it can boast several achievements.
    If you think, think twice
    what is a "streaming problem"?

    fundamentally
    I use the words "this situation is a streaming problem" in order to indicate that the investigation of flows represents the bulk of the solution.
    If you think, think twice
    fundamentally
    @Sacha
    "Now take the fact that the set of observations covers a continuum" is merely used in contrast to the countable set of particles (each represented by an eigenvector of the location operator, which resides in separable Hilbert space). These particles form the observed objects.

    The situation is similar to the case that a large set of linear equations must be solved, while the set of unknowns is much smaller. In general that set of equations is inconsistent, but it can be solved in an optimized way by accepting an appropriate stochastic inaccuracy in the measured results (=boundary conditions).
    If you think, think twice
    Hans, the real 2nd Law, from Newton's actual text, gives dF/dt = m.dv/dt + v.dm/dt : the corpuscular metaphysics was like that, involving mass transfers. And if you look back through the cosmology of Plato, picked up by Galileo, you come to Heraclitus, panta rhei, all is flux.

    When Hamilton unified optics and wave mechanics, he was driven to quaternions for a concise representation, and the mass flux was implicit in the aether of his time, now just the (complex) dielectric of free space. We don't see this, schooled in the superficial corpuscular/wave theory controversy, and the 2nd Law actually given by Euler.

    And with observers I'm certainly with you: look for "observer space" and you now find it in Wikibin. Binned! With this important concept recovered, you have a genuine phenomenology.

    fundamentally

    @Orwin
    Until the sixties many quantum physicists tried to keep quantum physics consistent with its quantum logical base. However, with the growing success of QED and QCD physicist increasingly forgot those fundaments and accepted a rotten theory that contained working formulas above the understanding of why these formulas actually worked. Only now and then, those same scientists clash against the walls of their fragile building and are astonished why that building falls apart.
    In the sixties Constantin Piron proved that the quaternions formed the most extensive division ring that can be used for specifying the inner product of a separable Hilbert space.
    Only most physicists were fooled by the fact that the displacement group requires a Minkowski signature for the space in which displacements occur. That might have led them to think that spacetime is the most natural physical quantity, while quaternions do not fit. If you think it over, it is the other way around.

    If you think, think twice
    Excuse my silence, I've been looking at the Higgs saga: happen they don't have a particle signature, but an intrinsic uncertainty. So I looked again and found four additional candidate sets of uncertainty relations! Sets of two: apparently the position-momentum uncertainty throws up a relativistic image of energy-time uncertainty. So one had better approach time with some skeptical caution!

    On this point, Hans, do your quaternions avoid the advanced/retarded wave problem? This is due to abandoning the Erlanger program in geometry, which starts out from projective geometry: if you skip this foundation the inner product splits and the problem arises.

    A mass in the projective view appears as a mass-profile, in dimensions of mass x area. Could that be your shadow-mass, in the guise of a patch of space? Just such obstructions throw shadows/cause resistance in current flows.

    You can of course also split Plank' action as obstruction and frequency. Or as mass-flux and area. And if you allow micro-curvature, "pot-holes" in space-time, then obstruction can jam nucleons in them, acting like a strong force!

    fundamentally
    The approach that is taken in the Hilbert Book Model (HBM) has two relations to the Higgs model.
    The switch to quaternionic probability amplitude distributions (QPAD's) converts linear equations of motion, such as the Dirac equation into continuity equations. It converts fundamental physics into a kind of fluid dynamics. The QPAD's describe how space in the parameter space of the QPAD is transported. It means that this parameter space shows features such as charge density distributions, current density distributions, sources, drains, compressed regions and decompressed regions. The compression can be related to (parameter) space curvature.
    I take the Dirac equation (in quaternion format) for the electron as an example. This equation couples two QPAD's. One is the quaternionic conjugate of the other. It means that these two QPAD's are nearly equal, but the currents that are described by the separate QPAD's are reversed! One feeds the other. The wave function QPAD (on the left side of the equation) acts as the description of the drain.
    The QPAD on the right side acts as the description of the source. The real action takes place in the parameter space. The drain causes a local space compression. The source takes its food from the regions of distant wave functions. The coupled QPAD can be interpreted as the superposition of the tails of the wave functions of distant particles. This makes the coupled QPAD a very stable and rather static object. In this way the mechanism of inertia is implemented. The factor m in the Dirac equation determines the strength of the coupling. Therefore, I do not call it mass but instead coupling factor.

    If you want to create a relation with the Higgs then you could interpret the coupled QPAD as a Higgs field. It has been there since Dirac created his famous equation.
    If you think, think twice
    Let me add that I was thinking about isospin. Its right at the horizon of your interest, but also symptomatic of the problems in the Standard Theory. Protons and neutrons are given the same isospin, despite differing masses and radii, because the appear to react similarly to the strong force. Yet the physical variable or dimensions of that similarity are not specified. The Higgs is then introduced as a particle with a different isospin, but all along there is a real variability and uncertinty involved, in dimensions unknown, and how can the Higgs mass acting through isoispin then determine the actually different masses of te proton and neutron?

    For an uncertainty in mass you can split Planck's action into terms in mass and kinematic viscosity, also the constant of motion in Kepler's orbits. Ask what string theory actually adds through the laest M-theory, and the answer is just a relation between viscosity and entropy!

    This is where it gets interesting. Your symmetries for mass form a Poisson bracket, as in the Fourier transform which underlies Heisenberg's two uncertainty relations. These turn on length and inverse frequency or time respectively. To keep the symmetry broken within the set, the mass-viscosity uncertainty can be linked another in action and temperature, taken as in thermodynamics to be a pure number. That points to the Hawking-Bekenstein temperature, black hole horizons, and now Eric Verlinde's approach to gravity from entropy.

    And, yes, even free space, the vacuum, has a viscocity, due to virtual particles:
    http://www.newscientist.com/mobile/article/mg20927994.100

    fundamentally
    The Hilbert Book Model has no proper explanation for the existance of generations. On the other hand the coupling factor m can easily be computed from a manipulation of the elementary coupling equation. For the electron this concerns the Dirac equation. So, its computation is straightforward mathematics. The computation involves two integrals. The trick in case of the electron is the multiplication on both sides of the equation with the quaternionic conjugate of the coupled QPAD before the integration is done. The integral on the right side then reduces to the coupling factor. The integral on the left side is completely determined by the form of the wave function QPAD, which in case of the electron is the conjugate of the coupled QPAD. This means that when three generations of the coupling factor exist, there must be three generations of wave function QPAD's for the electron family.
    If you think, think twice
    Barry, short of algebra and the continuum, you face Godel's Incompleteness Theorem, and you cannot rigorously represent the theory on its own terms. So the whole of group-theoretic physics is riddled with extra-theoretic metaphysics. General relativity does not escape: the sympletic representation assumes centres of gravity within the sympletic neighbourhoods, but they're not represented, so the theory is omega-incomplete. I rate micro-curvature as the only systematic approach to his problem.

    And the algebra is there in Poynting's Theorem for conserved electromagnetic energy, which instansiates the E x H vector product. In the dimensions of power/area or mass and three frequencies, from which the switch to QPAD representation is completely natural!

    With that said, I know one shouldn't talk of Lie algebras, and agree that axial anomalies mark the frontier. If Hans' groups is anisotropic, the proton must be dyamically asymmetric and the axial anomaly substantial. And I see that's your Poisson bracket (m+s)(m-s), so I think we need you in the dialogue.

    From mass as coupling one can approach generations trough Koide's work, which is now another fringe interest that Tommaso featured earlier.