A never ending story
    By Hans van Leunen | February 7th 2013 05:03 PM | 18 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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    The history of the cosmos.

    In quaternionic physics one equation plays a major role. It is a mixture of a differential equation, a continuity equation and a coupling equation.
    Φ = ∇ ψ = m φ
    Here Φ, ψ and φ are continuous quaternionic distributions or more specially they are quaternionic probability amplitude distributions. ψ and φ are normalized. m acts as a coupling factor. ∇ is the quaternionic nabla. The real part of it takes the differential with respect to the progression parameter.

    m follows from
    m² = ∫ |Φ|² dV

    Here we focus on
    Φ = ∇ ψ
    It represents not only the definition of the differential, it also specifies a differential continuity equation. This becomes clear when a continuous quaternionic distribution is interpreted as the combination of a scalar field and a 3D vector field.

    The integral form of this equation runs
    ∫ Φ dV = ∫ ∇ ψ dV
    The right part can be split into a volume integral that confines to the variation with respect to progression and a second surface integral that can be written as a surface integral. See the above picture.

    The corresponding balance equation reads:

    Total change within V = flow into V + production inside V

    A zero surface integral renders the surface into an inversion surface.

    The inversion surfaces divide universe into compartments. These universe pockets contain matter. If there is enough matter in the pocket it will form a black hole. In that case the rest of the pocket is will be cleared from its mass content. At the same time the size of the pocket may increase. This represents the expansion of the universe. Inside the pocket the holographic principle governs. The black hole represents the densest packaging mode of entropy. This indicates that in this region nature returns to its natal state.

    The pockets may merge. Thus at last a very large part of the universe may return to its natal state, which is a state of densest packaging of entropy. The mass in the rest of the universe, which is positioned at a huge distance will enforce a uniform attraction. This uniform attraction will install an isotropic extension of the considered package. This will disturb the densest packaging quality of that package. The motor behind this is formed by the combination of the attraction through distant massive particles, which installs an isotropic expansion and the influence of the small scale random re-localization which is present even in the state of densest packaging. This re-localization is described by the quantum state functions of the packed particles.

    The rupture of the package and the departure of its densest packaging state represent a new episode in the history of the compartment. This describes an eternal process that takes place in and between the pockets of an affine space.

    The story described here is part of a much larger project with the name Hilbert Book Model. The Hilbert Book Model (HBM) is a simple model of fundamental physics that is strictly based on traditional quantum logic. This results in a quaternionic approach of quantum physics. On its turn it shows that quantum physics is a kind of fluid dynamics. The project runs since 2009 and its latest full description is treated in 
    and is continuously updated and extended at
    The supporting mathematics are compiled in
    The site is devoted to this project.
    The pictures are taken from a slide presentation: .
    I am aware that the HBM is unconventional and in many aspects controversial.
    The HBM departs in many aspects from contemporary physics. I found good reasons for doing this.
    Its main target is the exploration of the undercrofts of physics.
    I do my best to keep the HBM well founded and self-consistent.
    I am open to proper (preferably well founded) criticism.

    If you read "On the Hierarchy of Objects" then do not forget to read the conclusion.


    In your theories, what are quaternions geometrically?
    For example, how do they transform under arbitrary coordinate systems?
    In what sense is the piece you refer to as a "scalar field" actually a scalar value?

    Your last line is not displaying correctly for me. It ends abruptly after an integral sign. Also, the phrase "and a second surface integral that can be written as a surface integral" conveys no information.


    Quaternions form a 1+3D space. As single objects they can be interpreted as the combination of a real scalar and a 3D (real) vector. Due to its dimensionality the quaternionic number systems exist in 16 different versions. The differences are characterized by the discrete symmetries of the number systems.
    A very special characteristic is the handedness of the external vector product.
    Continuous quaternionic distributions also exist in 16 different versions. The symmetry of the target domain may differ from the symmetry of the (quaternionic) parameter space. Continuous quaternionic distributions may be split into a (real) scalar field and a 3D (real) vector field. These fields may be interpreted as potential fields. In that case the scalar potential represents a density distribution of "charge" carriers. The vector potential then represents the corresponding current density distribution. This makes the quaternionic distribution a quaternionic probability amplitude distribution (QPAD).

    If you think, think twice
    Thank you for adding the graphic, however it is still unclear what your equation is saying. What is rho? What is s?

    Also, you didn't really answer my question regarding coordinate transformations.
    Let's say I'm using a (at least locally) inertial coordinate system in which at some point in spacetime the quaternion ψ = (a0, a1, a2, a3). If I change coordinate systems, by doing a Lorentz boost in the x direction, what are now the values of the components of the quaternion ψ at that same spacetime point?

    You seem to be suggesting that the components of your quaternion would transform like the components of the electric four-current J^\mu. Is that the case?

    Quaternion mathematics is treated and compiled in .
    The physical background is treated in 

    When ψ is a quaternionic probability amplitude distribution (QPAD), then the real part is a "charge" density distribution and the imaginary part is a corresponding current density distribution. The "charge" is an ensemble of discrete  properties of the charge carrier. You still have to decide what the carriers are.
    The paper that describes the HBM explains what these carriers are.

    In the pictures the symbol ρ is used instead of ψ, which is used in the text. The real part of the nabla is represented by d/dτ. The pictures were taken from an existing slide presentation. (I had to act fast with the repair of the text. Sorry for that.)
    If you think, think twice
    I looked at your mathematics document, and I don't see any place where you discuss Lorentz transformations. I do see you at times _equate_ the basis of the quaternion representation with the coordinate system basis. So let's try again, and I'll be as specific as I can:

    Let's say I'm using a (at least locally) inertial coordinate system ct,x,y,z in which at some point P in spacetime the quaternion ψ = [a0, a1, a2, a3]. If I change coordinate systems to cT, X, Y, Z, by doing a Lorentz boost in the x direction, such that:
    cT = gamma ct - beta gamma x
    X = gamma x - beta gamma ct
    Y = y
    Z = z
    where beta is the boost parameter = v/c, and as usual gamma = 1/sqrt( 1 - beta^2)

    Are you saying that the values of the components of the quaternion ψ at that same spacetime point P, but using the new coordinate system is:
    ψ = (A0, A1, A2, A3)
    A0 = gamma a0 - beta gamma a1
    A1 = gamma a1 - beta gamma a0
    A2 = a2
    A3 = a3

    Is this what you are saying?
    If not, then please state here clearly what A0,A1,A2,A3 would be according to your treatment of quaternions. It seems you are trying to treat them like four-vectors, but you seem resistant to answering directly whether you are or not. I would appreciate it if you could clearly answer this question on Lorentz transformations. Thank you.


    Since in the HBM nature is supposed to step with universe wide fixed steps and since progression in the HBM is equivalent to proper time, the HBM is inherently Lorentz invariant.
    See the section on the quaternionic metric in 

    If you think, think twice
    Let's agree on coordinate transformations before discussing the metric. And I disagree that a theory with discrete time steps is "inherently Lorentz invariant", but let's stick to one issue at a time.

    Can you please just answer my question. Let's focus on clearly stating how the components of your quaternions transform with an arbitrary change of coordinate system. I gave a very specific example to make discussion easier. I even wrote out the math explicitly, so there would be no miscommunication and you could provide concrete information with a yes or no. So please just answer the question. I don't understand why you keep side-stepping.

    Do the components of your quaternions transform like the components of a four-vector?
    If not, then please state explicitly what A0,A1,A2,A3 would be in the example I gave above. Thank you.


    Sorry, that I did not reply earlier. I was very busy.
    When I interpret your question properly, then you see ψ and the other symbols as quaternions. In the text they are indicated as quaternionic probability amplitude distributions (QPAD's). In the model quaternions are used as elements of the parameter space of these QPAD's. The QPAD's themselves can be seen as combinations of a scalar potential and a 3D vector potential. On their turn these potentials can be interpreted as the representatives of a density distribution of "charge" carriers and a corresponding current density distribution. The target values of the QPAD's are quaternions that represent re-locations in a reference continuum. The “charges” are ensembles of properties of the carriers. The quaternions that are used as parameter values or as target values can be interpreted as the combination of a real scalar and a 3D (real) vector.

    The QPAD’s can be interpreted as the transformers from the topology of the parameter space to the topology of the target space. This target topology is curved.


    The Hilbert Book Model introduces a quaternionic enumeration function that consists of the convolution of a (sharp) continuous function and a randomizing part that can be considered to be generated by a Poisson process and a subsequent binomial process. The binomial process attenuates the efficiency of the Poisson process by spreading the results in 3D space. The result approaches a local 3D Gaussian distribution. The enumeration has two potential modes. One in configuration space and one in the corresponding canonical conjugated space. This is further complicated by the fact that quaternionic number systems exist in 16 different symmetry sets.

    The HBM treats only the positive real part of the quaternions, because it uses the real part in order to represent progression. With respect to QPAD’s, the HBM only uses the positive charge density distributions. These represent presence of charge carriers. The negative charge density distributions represent absence of charge carriers (or holes).


    For finer details I advise to read the referenced paper(s).
    For some mysterious reason I cannot answer on the next comment

    If you think, think twice
    It is frustrating that I ask a direct question (multiple times now), and you respond without actually answering the question. I don't want to get into the merky details of your theory yet, as I don't think you are even treating quaternions in a consistent way. I have already looked at the papers you pointed me to previously, and already explained they don't answer the question I am asking. So let me try this again:

    Can we at least agree up to here: At each point, ψ is quaternion valued?
    For example, let's say I'm using a (at least locally) inertial coordinate system ct,x,y,z in which at some point P in spacetime the "quaternionic probability amplitude" ψ has a quaternion value which can be represented with four components [a0, a1, a2, a3].
    -1-] YES or NO?
    -1a-] If NO, and it does not have a quaternion value then why are you calling it quaternionic and referring to four components such as a 'scalar' component and a '3-vector' component?

    Okay, I'm really hoping the answer to that was yes, as that is just making sure we are on the same page.
    Now, if I change to a new coordinate system cT,X,Y,Z related to ct,x,y,z by a Lorentz boost in the x direction, in this new coordinate system at point P, ψ has the components [A0, A1, A2, A3].

    I want to know how the quaternion components transform when we change coordinate systems.
    To repeat:
    -2-] How do the components A0,A1,A2,A3 relate to a0,a1,a2,a3?
    -2a-] Are they uneffected by the coordinate transformation? ie. A0=a0,A1=a1,A2=a2,A3=a3. Please just definitively answer YES or NO
    -2b-] Do the components transform the same as the components of a four-vector? (you seem to be hinting it transforms like the four-vector J^\mu from electrodynamics, in which case the answer would be yes). Please just definitively answer YES or NO
    -2c-] Or do the components transform some other way than the two options I mentioned here? If you answer NO to 2a and 2b, then please definitively give the equations relating A0,A1,A2,A3 and a0,a1,a2,a3 for the explicit Lorentz boost coordinate transformation I gave in previous post.

    I'm trying to nail down some definitive math, but you keep responding describing how these ideas will fit into your larger theory. Let's nail down this simple math first. If we can't get a strong common foundation, then trying to move onto larger structure of your theory will not be fruitful.

    So please, PLEASE, answer the numbered questions above. Please don't go off on tangents trying to describe how these quaternions evolve in your theory, etc. Please just answer the direct questions with direct answers. Thank you.

    P.S. David or Doug Sweetser, since we've had long discussions on quaternions in the past, I'd much appreciate if you could jump in and help bridge this communication gap. Maybe help explain in a different way to Hans what I'm asking, because I'm not understanding what is causing this communication gap. Thanks.

    First, the HBM does use proper time rather than coordinate time.
    Your questions can be answered in a general way.
    A continuous quaternionic function maps a set of selected points that are taken from a (quaternionic) reference continuum to a set of target points that are embedded in that same reference continuum. The selected points are all rational quaternions. You can enumerate the corresponding target points with that same rational quaternion. Due to the action of the function the target points are no longer embedded in the reference continuum in the same orderly way as they were in the parameter space.
    Any coordinate transformation of the parameter space does not affect the topology of the set of selected points. Only their description will differ. The same holds for the topology of the target points.
    The target values may also be rational quaternions. These will in general differ from the original parameter value.

    The continuous quaternionic function implements the dynamics of the map. The real part of the rational quaternions can be taken as progression value.
    In the HBM the map function is the convolution of a sharp continuous function and a randomizing spread function. This fact is insignificant for the previous consideration.

    The target values form density distributions, where the targets are the carriers of the characteristics of the target. The ensemble of characteristic is the charge of the carrier. The carriers move and the charge density distribution corresponds to a current density distribution. These two distributions form together a quaternionic probability amplitude distribution (QPAD). The density distributions correspond to potential functions. The charge density distribution corresponds to a scalar potential field. The current density function corresponds to a vector potential field. Together the two potential fields form a quaternionic function.
    If you think, think twice
    I don't understand what is causing the communication gap here. I am asking what I feel are straightfoward questions, that I've even reduced to direct YES/NO questions, and you continue to respond with a bunch of details about your theory which completely avoids my questions. Please stop trying to relate things to a larger picture in your theory, until we can agree on the mathematical objects that are being described here. WIthout that foundation, trying to agree on discussions of the larger theory will only be pure hand-waving. I feel there is a flaw in the foundations, before you even get to the larger structure of the theory. So please just answer the questions directly instead of leaping ahead to something else. It is counter-productive.

    So let's try this again. You state:
    "A continuous quaternionic function maps a set of selected points that are taken from a (quaternionic) reference continuum to a set of target points that are embedded in that same reference continuum."

    It sounds like you are saying that ψ is a function mapping quaternions to quaternions:
    ψ: Q → Q
    -1-] Is that correct, YES/NO?

    It also sounds like you interpret the four component representation of a quaterion, to be equivalent to the choice of coordinate system. In other words, any point in 4 dimensional spacetime can be labelled with a four-tuple of real values, and you _equate_ that to a four-component representation of a quaternion.
    -2-] Is that correct, YES/NO?

    Assuming your answer is yes to the previous questions, then it really does sound like you are saying that your quaternions transform like four-vectors. I currently see no way around this.
    -3-] If ψ at some point P in a chosen coordinate system has components a0,a1,a2,a3, and I change coordinate system choice such that ψ at the same point P now is represented with components A0,A1,A2,A3, are you saying that the components of the quaternions transform under a coordinate change in the same way that the components of a four-vector would transform: YES/NO?

    These are direct, straight forward questions on how you are mathematically treating some of the fundemental objects in your theory. Please just answer the direct questions with similarly direct responses. Please stop telling me about something else in your theory, instead of answering these direct questions.

    In the explanation above TWO functions were mentioned. The first one maps rational quaternions selected from a parameter space into a target space. The second function describes the density distributions that are produced in this way. This second function ψ is a QPAD. It is a continuous function that maps a continuum to another continuum. The first function maps a discrete set into another discrete set.
    So for ψ  [1] => yes

    In the parameter space I specified a set of rational quaternions and the values of these numbers can be considered to be the coordinates of these numbers. If you do that it means that the selected rational quaternions are properly ordered in the quaternionic continuum that embeds them.
    With this choice : [2] => yes

    The selected quaternions transform as specified by the first function. Quaternionic functions exist in 16 different symmetry sets. This means that the transform can take sixteen different conditions.
    So, your idea of a transform is far too simple.
    ψ is the second function. It is a QPAD. It is a continuous quaternionic function. Depending on the discrete symmetry of the first function will ψ have a corresponding symmetry. ψ can be split in a real scalar field and a 3D vector field. It describes the distribution of the target quaternions. However, the components of ψ are not real numbers. They are real functions. Since the targets are no longer distributed in an orderly way, the corresponding "coordinate transformation" is a very complex operation. It might be described for the separate targets, but it is not reasonably possible to describe it in a simple way for the full ψ function.
    Thus for a simple coordinate transformation the answer [3]=>no

    Sorry the first function transforms into a curved space. That makes things complicated. Further for the target space the first function may switch to another symmetry set. This also complicates the situation.
    Your questions seem to relate to a very simple transform of a 4D vector space into another 4D vector space. That description would be simple and would have given a yes to all three questions. 
    If you think, think twice
    Can you please answer my question on coordinate transformations.

    *under arbitrary coordinate changes


    To all
    I tried to install a text with embedded scientific equations. However, this text editor cannot cope with some of the equations. I have revised the text and embedded the formulas in the form of pictures. Sorry for the inconvenience.

    If you think, think twice
    This makes no sense. It is just a stream of logically unconnected statements, presented as if it was deriving or elucidating something. This is just crackpot bunk. THIS SHOULD NOT BE LINKED ON THE FRONT PAGE OF SCIENCE 2.0 Anyone with reasonable deduction capabilities should be able to see that there is no logic connecting this freeflow of claims. Sprinkling in equations doesn't negate this.

    Crackpots are hard to eliminate, but I would hope a site like Science 2.0 would have come up with a way of handling it by now. For example, maybe a way people could downvote certain posters that are repeatedly pushing crackpot nonsense, so that it is not outright censored, but at least pushed way down on any page of article lists. Beyond a certain downvote, a moderator could even decide possibly to take action and just remove the article or even the account. This is basically how it works on stackexchange, and the readers are usually able to push nonsense fairly far down on the page, without burdening moderators much.

    Anyone else?
    What do you guys think?

    Classifying statements as crack stuff requires proper argumentation. Otherwise the criticizer disqualifies himself.
    If you think, think twice
    "Classifying statements as crack stuff requires proper argumentation."

    He explained to the extent that is reasonable for responding to a pet "theory" such as this. Your statements are logically disconnected. You present equations, but they are peripheral to any conclusion you make, and seem to be there only to give the appearance of some legitimacy to the stream of unconnected statements that follow.

    What would you like him to do? Since the statements are logically disconnected, he can't say you made a math error going from step 5 to step 6. The error is the sheer lack of any logic at all.