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    The Hilbert Book Model
    By Hans van Leunen | July 25th 2011 12:12 PM | 40 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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    This paper introduces a new model of physics. It is based on logic. It uses the congruence between the logic of quantum physics and a mathematical construct that got its name from David Hilbert. The Hilbert book model extends this construct such that fields and dynamics also fit in the new model.

    Introduction

    Every time when I read an article about the phenomena, which occur far from us in the universe, I'm surprised about the attention that this Farawayistan gets compared to the phenomena in the world of the smallest. Everything that happens there is dismissed with collective names such as “quantum mechanics” and “field theory”. Rarely or never the treatise goes deeper. In this sub-nano-world spectacular images, such as appear in stories about the cosmos are not available.

    What's playing?

    Still, this is part of our environment is at least as interesting and mysterious as the cosmos. What makes it even more interesting is that the fundamentals of physics can largely be found in this area. This gets enforced by the growing awareness that our knowledge of these foundations contains a lot of gaps.

    Quantum Theory

    Quantum mechanics and the corresponding quantum field theory have been developed mainly in the beginning of the last century. This development occurred fairly violently and in many cases, scientists were already happy with a limited understanding that nevertheless brought enough usable formulas so that one could analyze quantum phenomena and could construct useful applications. 

    History

    In the early days of quantum mechanics the approach was based on adapting equations of motion that were in use in classical mechanics. These equations were quantified via an intuitive process. In Schrödinger’s approach the time dependence is placed in the state function of the particle. The operators that act on these state functions are kept constant. In contrast, the approach of Heisenberg positions the time dependence in the operators that act on the (static) state function. This difference in approach ultimately makes no difference for the properties of the physical particles. That means that the state function and the operators only play a background game. In contrast, the properties of the particles play the foreground act. This indifference also means that time does not belong to the properties of a particle. With respect to the state functions and the operators, time only plays the role of a parameter. Apparently it does not matter whether you place this role in the state functions or in the operators. This parameter characterizes the progress of the dynamics[1]. On the other hand, position belongs to the properties of a particle. This indicates a fundamental difference between the role of space and the role of time.


    The biggest confusion arose when it became clear that the smallest things could behave both as a particle and as a wave package. This confusion continues because it also means that nature is unpredictable in the behavior of its smallest parts. Many are unable or unwilling to accept this fundamental property.

    Clarification

    Already early on in the last century some solid explanations were given. Garret Birkhoff and John von Neumann showed that nature is not complying with the laws of classical logic. Instead nature uses a logic in which exactly one of the laws is weakened when it is compared to classical logic. As in all situations where rules are weakened, this leads to a kind of anarchy. In those areas where the behavior of nature differs from classical logic, its composition is a lot more complicated. That area is the site of the very small items. Actually,that area is in its principles a lot more fascinating than the cosmos. The cosmos conforms, as far as we know, nicely to classical logic. In scientific circles the weakened logic that is mentioned here is named traditional quantum logic.

    Hilbert model

    Birkhoff and von Neumann went a step further. They discovered the fact that a mathematical structure, which more than a century earlier was discovered by mathematician David Hilbert, is in many respects similar to the structure of this quantum logic. This structure is a space with infinitely many dimensions. A position in this Hilbert space can be specified by using numbers. For each position that must be done with infinitely many numbers. Fortunately, that what is happening in this infinite dimensional separable Hilbert space can also be specified with functions. Luckierwise a lot was already known about functions that suit this purpose.

    Numbers

    The numbers that can be used, need not be limited to the real numbers,which we use in order to measure our three dimensional living environment. Constantin Piron found that these numbers at least must be members of a so-called division ring. There are only three division rings: the real numbers,the complex numbers and the quaternions. Virtually no one still knows the quaternions. William Rowan Hamilton discovered quaternions already in the nineteenth century. They are hyper complex numbers with a one-dimensional real part and a three-dimensional imaginary part. 

    Hilbert operators

    Here you see appear an immediate reason for our three dimensional world. It also delivers a mystery, because the structure of Einstein's space-time differs from the structure of the quaternions. However, there are more puzzles. Although the Hilbert space has an infinite number of dimensions, this infinity is countable. Countable means that in principle, a label with an ascending integer can be attached to each dimension. The set of real numbers is uncountable, but the set of rational numbers is countable and the set of rational quaternions is that too. So, to each dimension of the Hilbert space a rational quaternion can be attached. Mathematicians use the name operators for the mathematical things that can do this. The real numbers describe a continuum and the set of quaternions does that too. But the set of rational quaternions does not do this. This means that it is impossible to accurately describe smooth phenomena with the model obtained so far.

    Graininess

    The reality is even worse. There is increasing evidence that in its smallest form nature is grainy. So-called Planck units exist. These are unit sizes for time, place, action and entropy. It is basically impossible to measure the corresponding quantities more accurately than these Planck unit sizes indicate. It is as if within these limits the world does not exist or else, that nature steps over these regions. 

    GPS

    Now suppose that we want to design a three-dimensional GPS system for nature by using the three-dimensional part of quaternions. This system would have to take into account the graininess of length. However, this is a great problem. A lattice consisting of a tightly packed collection of grains is afflicted with preferential directions. Such directions appear in nature in solids but they are not omnipresent in the universe. Therefore we need to find a different solution for the customized GPS system. This solution must not use multidimensional collections of grains, because that would pose the same problem. (However, the inner horizons that also occur in the eigenspace of the new operator form an interesting exception to this rule.)

    Grain chains

    A potential solution is a GPS that works with one-dimensional chains of grains. The chains represent paths. Not actual paths, but hypothetical paths. They can freely move in 3D space. There is one grain in the chain that represents the current position on this path. Only the direct environment of this grain corresponds to an actual path. Now remains the problem to give each grain in the chain its own position. 

    In addition to the Hilbert space with countable dimension themathematicians developed a Gelfand triple. As a kind of sandwich the two outer parts of this triple attach to the previously described separable Hilbert space. Because this triple directly associates to the separable Hilbert space, this sandwich is also known as a “rigged Hilbert space”. In fact this name is incorrect because the triple is not a proper Hilbert space. Fortunately, the rigged Hilbert space has an uncountable number of dimensions and can easily deliver a GPS system that can act as a continuum background coordinate system. The grain chains also have an equivalent in this rigged Hilbert space and this fact can be used to attach a position in the background coordinate system to each of the grains of a selected chain.

    Anchor Points

    The grains of the chains that occupy the current position in the chain’s "path"are in fact anchor points of elementary particles. Depending on its type an elementary particle has one or more of these anchor points. (According to Schiller’s strand theory[2], photons use only one anchor point and electrons have three anchor points.)

    Per time step the anchor point can at the utmost take one space step. If it does that, then it lands in the next grain of the chain. That is why the chain represents a kind of path. The ratio between space step and time step is fixed and is equal to a constant c. That number equals the speed of a freely moving light particle. In each time step a photon invariably takes a space-step. It also means that no particle can go faster than such a freely moving light particle.

    Fields

    The chains are not allowed to move arbitrarily. There is something that ensures that the chain keeps its smooth shape. This is provided by a probability distribution that is associated with the anchor point. In fact, it's a hyper complex function whose squared modulus equals the mentioned probability density distribution. This function has quaternions as its function values and accepts quaternions as a parameter. The three-dimensional imaginary part of the hyper complex parameter may indicate a position. In that case, the probability distribution gives the probability that the next grain will be located at the value of the parameter. The form of the probability distribution ensures that only minimal changes occur. The quaternionic function contributes to the local field. It is the part of the field that corresponds to the considered grain chain.

    Private Fields

    An elementary particle can have one or more anchor points. In this way the corresponding hyper complex functions together form the private field of the elementary particle. This private field has the same properties as the wave function of the particle. Quantum mechanical scholars use this wave function in order to describe the behavior and the properties of the elementary particle. 

    Together, all the private fields of particles form a joint covering field that, like the separate private fields, covers the whole Hilbert space. In our model, this joint covering field is part of the physical fields in our environment.

    The private fields overlap and because they are all probability amplitude distributions their superposition causes an interaction between the particles that anchor on these fields.

    Creation and annihilation

    Chains can split and they can merge. The corresponding creation and annihilation occurs during a progression step and is controlled by probability amplitude distributions that are attached to the current granules.

    Field Theory

    According to field theory each static field can be split in a rotation free (longitudinal) part and a solenoidal (divergence free, transverse) part. Due to the configuration of the field, this split may run along curved lines. This defines a local curvature. The curvature value can be used to define a new field. It is derived from the joined covering field. We can call that new field the curvature field. It has all the aspects of the gravitational field. We can take the part of the curvature field that belongs to a particle as its private curvature field. From this private curvature field the mass of the particle can be computed. Physicists usually apply this relationship in the reverse order.

    The field model

    The field model, which is applied here, differs significantly from the common field model. Usually the electromagnetic fields and the gravitational field are assumed to be independent of each other and the gravitational field is assumed to cause a curvature in the coordinate system that must be taken into account in the treatment of the electromagnetic fields.

    In this new model the cause of the local curvature is laid down in the properties and the configuration of the covering field, which consists out of the superposition of all fields except the gravitational field. The covering field also contains the fields that match the wave functions of particles. The curvature field is then derived from the local curvature. In other words, in this new model the gravitational field is a derived field. This approach causes an immediate unification of field theories.

    Hilbert sandwich

    The Hilbert space itself has no place for fields. Each private field covers the whole Hilbert space. However, in the same manner as described above for the Gelfand triple, it is possible to expand the aforementioned sandwich with three additional layers, which respectively represent the two decomposition parts of the covering field and the curvature field. Therefore, all in all, the expanded Hilbert sandwich consists of six layers.

    Hilbert book

    Each sandwich describes a static condition. Thus, this combination can still not describe any dynamics. This lack can be solved by putting a whole series of these sandwiches in an ordered sequence. In this way, a Hilbert book can be formed, in which each page represents a Hilbert sandwich. Glancing through this book then gives a picture of the dynamics of our universe. The page number acts as a progression step counter. This counter is not our common notion of time, but it has certainly something to do with it.

    Other view

    It might seem that the pages of the Hilbert book are mutually independent. However, this is not so. Each page contains the conditions that determine the contents of the following page. These preconditions are contained in another view of the data. That view can be obtained via a so-called Fourier transformation. This transfers the original coordinate system into an associated coordinate system. The function of position is replaced by the function of displacement. This displacement indicates where the relevant position in the next page moves. In jargon this second coordinate system is called the canonical conjugate of the first coordinate system. The original view gives a good picture of the "particle behavior" of the considered subject, while the new view gives a good picture of the wave behavior of the same subject.

    Discussion

    What is described here is only a model. It is not more than a reflection of reality. The events we see in the cosmos are largely determined by the curvature field. The new GPS operator knows an outside horizon beyond which no chains exist. That operator also has internal horizons inside of which no chains exist. We know these internal horizons as the exterior of black holes. This inside horizon is covered with Planck sized granules of which each represents a bit of information.

    The most controversial aspect of the Hilbert book model is the fact that this book consists of pages, each of which represents a static state of the universe. When considering Lorentz transformations, then usually both inertial frames are distributed over a range of Hilbert book pages. It is possible to locate maximally one of the two uniformly moving frames of reference entirely within a single page of the Hilbert book.  Phenomena such as length contraction and time dilation and relativety of simultaneity fall within the Hilbert book model.

    References

    References: http://www.crypts-of-physics.eu/Cracksofphysics.pdf and http://www.motionmountain.net/research.html

    In this book of Christoph Schiller, he uses the name strand for the equivalents of our chains. Please notice, strands are no strings! The strand model as well as the Hilbert book model have little to do with string theory.



    The role of time becomes clear in the paragraph about the Hilbert book.[1] 

    [2] See References

    Comments

    vongehr
    Oh please - not another one who falls for Christoph Schiller's "motion mountain" crap. If you write crackpot stuff, please make it at least your own original crackpottery, that way it is far more amusing.
    fundamentally
    I am not putting a theory that depends on the work of Schiller. I only indicate that the operator that I introduced as strand operator has eigenspaces that show similarity with the strands in Schillers strand model. The Hilbert book model is "self supporting". Schiller does not belief in theories that are based on axioms.

    The only thing that I borrow from Schiller is the idea that particles depending on their type may anchor on more than one chain/strand.
    If you think, think twice
    Sascha, your "two-times theory" (as propagateded by Vongehr) is crap and real crackpottery. Schiller's strand model is not - it is one of the few models that agrees with LHC results. But for crackpots like you that does not count.

    you may for good reasons not agree with the contents of the above article, but does it really conform to your intellectual standards calling people names because of that? - What a pity!

    fundamentally
    I admire Schiller for the fact that he introduced some revolutionary ideas. They may not be correct, but they cannot be neglected either.
    In the Hilbert book model I introduced a new position operator that implements the granular nature of position. I called it "strand operator" in honor to Schiller's strand model, because the eigenspace of the strand operator shows great similarities with the strands in the strand model. This does not say that the two models are similar. There exist large differences. But Schiller at least dares to take unconventional roads. He might have introduced some useful new ideas. It is far from scrap. It is worth to investigate seriously the value of his ideas. At least he guided me to the strand operator.

    Hans
    If you think, think twice
    Halliday
    Hans:

    You state:
    A lattice consisting of a tightly packed collection of grains is afflicted with preferential directions. Such directions appear in nature in solids but they are not omnipresent in the universe.
    However, this is only true of uniform lattices, just as is the case of crystalline solids.  If, on the other hand, one uses a random graph, rather than a uniform lattice, one avoids "preferential directions" in a manner quite analogous to unordered, amorphous solids.

    David
    fundamentally
    How would you adapt a stochatic or fractal like GPS operator with the special graininess that is set by the existence of Planck limits such as the Planck length?
    If you think, think twice
    Halliday
    Hans:

    A random graph is not, necessarily, the same as some stochastic dynamical system (though it could be), nor need it be fractal (the fact is, one can be fractal without being random).

    The trouble I see with your (or whomever's idea it was) "chains" is the question of interactions.  The image in my mind, at least, is of disjoint "chains".  So, where do the interactions come in?

    However, what I envision is a random graph that "builds" spacetime (a discrete spacetime manifold, with no predetermined dimensionality [dimensionality becomes emergent, and even dynamic-like]), not just space, with time being some "tacked on" or otherwise "separate" parameter, stepping from one configuration to another.

    But what do I know.  ;)

    David

    P.S.  The only Quantum Mechanics (QM) that can truly be couched in Hamiltonian terms is non-relativistic QM, based upon Newtonian Physics.  Once one incorporates Special Relativity (SR), obtaining relativistic QM, one can still have Hilbert spaces, but one can no longer have Hamiltonian-like mechanics (space and time no longer separate out so nicely, though I still see many that attempt such things, as if they just can't let such concepts go).  Actually, even the nature of the Hilbert space is reshaped, but that's another matter (i.e.  the spacial integrals become far less rigid, being definable, and invariant with respect to any and all spacial surfaces [and not just flat ones, such as "inertial reference frames", as well—of course this becomes even more critical if one is to handle curved manifolds]).
    "Non-relativistic quantum mechanics" has always seemed to me to be a contradiction in terms. Knowledge of relativity is as old as knowledge of quantum mechanics. The fact that the spacetime interval between an emitter and absorber of a particle moving with instantaneous speed c is zero provides important clues to the relation between the state function of the emitter and the state function of the absorber.

    Halliday
    Don:

    I'm most definitely inclined to agree with you, above.  Both with respect to the somewhat oxymoronic character of "non-relativistic quantum mechanics", as well as with the null intervals/paths within spacetime.

    David
    fundamentally

    The interaction between grains is installed via the probability amplitude distributions that are attached to the"current" granule in each chain. This distribution can both be seen as the wave function of the corresponding particle and as the private field ofthat particle. The superposition of all private fields forms a covering field. The static version of that covering field can be decomposed in a rotation free part and a divergence free part. This decomposition leads to a local curvature that defines a local metric. This local metric is the value of a (derived) curvature field that features all aspects of the gravitation field. That curvature field can be used to locate corresponding local centers of virtual or actual masses.

    Relativity enters the Hilbert book model via the properties of a generalized displacement transform that describes uniform movements. During a given progression step the particles that are attached to the current granules can at the utmost take one space step. The alternative is to stay at rest. Photons always take their space step. That makes their speed the maximum possible speed. The existence of a maximum possible speed specializes the generalized displacement transform into a Lorentz transform. The Lorentz transform concerns two inertial frames. The Hilbert book model restricts the application of the Lorentz transform to the extra condition that maximally one of the inertial frames is completely located in a single Hilbert book page. The other frame must cover a range of Hilbert book pages.
    With this restriction special relativity is perfectly compatible with the Hilbert book model. The Hilbert book model is aimed to represent a model of quantum physics.

    If you think, think twice
    Halliday
    Hans:

    Are your chains not one (1) dimensional?  If not, then why the use of the term "chain"?

    1. "The existence of a maximum possible speed" does not, necessarily, imply Special Relativity or a Lorentz transform.  (See, for example, sound within solids.)
    2. The existence of a Lorentz transform between any given "special" inertial frame and all other inertial frames implies the applicability of the Lorentz transform between any and all pairs of inertial reference frames, by the group nature of the Lorentz transform.  So, having your "Hilbert book model restrict the application of the Lorentz transform to the extra condition that maximally one of the inertial frames is completely located in a single Hilbert book page" is superfluous.
    And that's just off the top of my head, so far.

    David
    fundamentally
    David
    The chains are one dimensional, but only one element is the current granule and that is the only element in the chain that is attached to a private probability amplitude distribution. It is also the only element that is connected to a Hilbert vector that belongs to the current page of the Hilbert book. The other elements of the chain either belong to the past part or they belong to the future part. In this way the chain touches (tangentially) the path of the particle at the current instant. The chain has two purposes. The probability distribution keeps the chain smooth and the step to the next position is a step to the next granule of the chain. The rest of the chain plays hardly any role and could be discarded. However the other granules play their role of current granule in other pages. Light particles always step to the next granule when they travel to the next page. Slower particles may stay longer at the same granule.
    1. I took the derivation of the Lorentz transform and the maximum speed condition from Wikipedia. http://en.wikipedia.org/wiki/Lorentz_transformation#Derivation. Are they wrong?
    2. I took the extra condition in order to circumvent problems with the relativity of simultaneity. When both inertial frames are located completely in single but different Hilbert book pages then the simultaneity is fixed rather than relative, because everything in a single Hilbert book page occurs simultaneous. So both the proper time step and the coordinate time step are then forced to be zero.
    If you think, think twice
    Halliday
    Hans:

    As I expected, from your use of the term "chain", they are one (1) dimensional.  It's good to know that I wasn't mislead by the term.  :)

    However, this one (1) dimensional characteristic is what prompted my original question about interactions:  Where are the interactions between different particles?  (You know, on different "chains"?)  If each "particle" is on its own one (1) dimensional "chain", then how do you explain interactions between particles, and, a fortiori, the creation and annihilation of particles?

    As for the "derivation of the Lorentz transformation", it appears you are misinterpreting the premiss upon which that derivation is based:  It is not just that there is a "maximum speed", but that that speed must be invariant!  While you have a "maximum speed", you have not shown that said "maximum speed" must be invariant.

    In fact, with only one (1) dimensional "chains", and an external time-like parameter "stepping" the transitions within the "chains" (especially since this time-like parameter is treated as universal among all "chains"), why must there be any invariance at all?

    Furthermore, you cannot "circumvent problems with the relativity of simultaneity" via your "extra condition".  In fact, at best, you have created a "hole" in the group property of the transformations (thus destroying the group property, since closure is now violated).  (That is, unless you renounce the "extra condition".)

    David
    fundamentally

    Each chain contains a granule that represents the current state of the chain. It also represents the location of a particle that is attached to the chain. Further that granule is attached to a probability amplitude distrubution. The task of that distribution is to keep the chain smooth. It plays that role when the anchor point of the particle steps from the current granule to the next granule of the chain, which then becomes the new current granule. The probability amplitude distribution also plays the role of the wave function of the particle. Further it is part of the surrounding fields. The fields also influence other current granules. In that way the particles and their anchor points, the granules, influence each other. In the Hilbert book model, all fields except the gravitation fields are made of the same stuff: probability amplitude distributions that have a quaternionic value and an imaginary quaternionic parameter. The gravitation field is a tensor field. It is derived from the other fields.

    The maximum speed is derived from the ratio of the space step and the progression step. It is the ratio of the Planck-length and the Planck-time. So it fulfills the requirements of the Lorentz transformation as long as the mentionned ratio stays constant.

    I did not mean to change the group properties, but I wanted to restrict the way the inertial frames are choosen. I do not have the idea that this changes the group properties.

    If you think, think twice
    Halliday
    Hans:

    So, it appears that while your particles, anchored to these "chains", can interact, you have two issues that don't match with experimental evidence:
    1. Your particle number is always conserved:  Never any creation or annihilation.
    2. Even though you have a system of multiple interacting particles, each particle only changes along a one dimensional path in parameter space (whether Hilbert or otherwise).*
    As for the "maximum speed" issue.  I completely recognize that your "maximum speed is derived from the ratio of the space step and the progression step. It is the ratio of the Planck-length and the Planck-time."  However, this most certainly is not sufficient to "fulfill the requirements of the Lorentz transformation", even so "long as the mentionned ratio stays constant."  Unless this ratio is also, somehow, required to be the exact same value for all possible "reference frames", then this "constant" is insufficient to require a Lorentz transformation between "reference frames".

    You appear to be mixing concepts without full understanding.

    You say
    I did not mean to change the group properties, but I wanted to restrict the way the inertial frames are choosen. I do not have the idea that this changes the group properties.
    Nonetheless, whether you recognize that restricting the applicability of the transformation between arbitrary "reference frames" "changes the group properties" or not, such an exception does have that affect, whether intended or not (since it violates closure, a fundamental group property).  Yet, I see nothing that you have gained from this restriction.

    My advice?  Drop this apparently useless restriction.  (Unfortunately, this will not help the other issues.)

    David

    *  There is one thing you have that may provide a partial reprieve, here.  That is when you have some particles "anchored" to more than one "chain".  This does provide for a higher dimensional progression through parameter space.  However, you still have a problem with quantum "entanglement"/correlation within a Quantum Mechanical system of interacting "particles".  (My physical intuition suggests you will have other, additional issues.)
    fundamentally

    David,

    The article is kept as simple as possible. So creation and annihilation is not treated. This does not mean that it is not supported, or as you state that the particle number is conserved. Usually after each progression step the chain runs through into the next Hilbert page. However it is possible that after the current page a chain has split, or that two chains have become one. This has the greatest impact on the attached probability amplitude distributions. With other words the fields control the creation and annihilation processes. The fields determine what kind of changes are allowable.

    I do not understand your comment: “Unless this ratio is also, somehow, required to be the exact same value for all possible "reference frames", then this "constant" is insufficient to require a Lorentz transformation between "reference frames".”

    When for some reason the ratio changes, then of course the Lorentz transform loses its global validity. I know that there exist many conditions in nature where the speed of light is lower than c and that due to expansion of space the speed of particles globally can surpass c. But locally and in free space the Lorentz transform is a suitable transform. From Susskind I know that for light particles their proper time step is always zero. (That is why their speed near a BH horizon goes to zero). This zero proper time step is the condition that may hold in a single Hilbert book page.

    I did not state that at least one reference frame must fall within one Hilbert book page. But if you take each of the reference frames completely in single but separate Hilbert pages, then time dilatation and relativity of simultaneity are obstructed. By this selection the dilatation is forced to be zero. May be this condition creates no problem and I see ghosts.

    The Hilbert book model keeps the possibility that depending on the type of the particle, it can anchor on more than one chain. But when chains can split and merge this ability is not required in order to support creation and annihilation. The multiplicity of anchor points may be used in order to support the diversity of types and the corresponding special field configurations.

    If you think, think twice
    Halliday
    Hans:
    From your statement:
    ... it is possible that after the current page a chain has split, or that two chains have become one. ...
    It sounds like your "chains" are not so much "chains", but actually graphs (at least from a temporal point of view), as I have previously suggested.

    On the other hand, you say
    I do not understand your comment: “Unless this ratio is also, somehow, required to be the exact same value for all possible "reference frames", then this "constant" is insufficient to require a Lorentz transformation between "reference frames".”
    OK.  However, you then go on with:
    When for some reason the ratio changes, then of course the Lorentz transform loses its global validity. I know that there exist many conditions in nature where the speed of light is lower than c and that due to expansion of space the speed of particles globally can surpass c. But locally and in free space the Lorentz transform is a suitable transform. From Susskind I know that for light particles their proper time step is always zero. (That is why their speed near a BH horizon goes to zero). This zero proper time step is the condition that may hold in a single Hilbert book page.
    I can't believe how far off the mark you are.  Yes, you certainly don't understand.

    What you have, in this last quote, is a "Red Hearing".  This has absolutely nothing to do with the issue I was pointing out.

    Am I not correct in my understanding that you claim to have Special Relativity, and the Lorentz transformation, as a results of your "Hilbert book model", rather than as some separately imposed criteria?

    If so, then what I am trying to point out is that your arguments concerning this supposed derived condition do not hold.  To put it plainly, you have not succeeded at deriving what you claim to have derived, because there are other, additional conditions you must impose in order to derive what you claim to have derived—conditions your model does not impose or require.

    David
    Halliday
    Hans:

    I'm actually quite surprised that Doug Sweetser hasn't been all over your posts, with your enamorment with quaternions.  :)

    David
    fundamentally
    Doug never reacted on my posts. I am also active in LinkedIn (Quantum physics, Theoretical physics) and in ResearchGate (Quantum logic). He is strong in quaternions but may be not so much in quantum logic.
    If you think, think twice
    Halliday
    Hans:

    While I would not expect Doug to be "strong" in quantum logic, I would expect him to be quite interested in your "playing" with quaternions, and your apparent "advocacy" for their connection with the dimensionality of space.

    My suspicion, however, is that Doug simply hasn't seen your posts, just as I had not, before today.

    David
    fundamentally
    The above article is only a low level introduction to a more elaborate treatise. You can find these on my e-print site: http://vixra.org/author/Ir_J_A_J_van_Leunen and on my website: http://www.crypts-of-physics.eu.
     
    If you think, think twice
    Halliday
    Hans:

    I see another issue:  Do you have anything but a cursory understanding of Hilbert spaces?

    Your ignorance shows with paragraphs like the following:

    Hilbert operators

    Here you see appear an immediate reason for our three dimensional world. It also delivers a mystery, because the structure of Einstein's space-time differs from the structure of the quaternions. However, there are more puzzles. Although the Hilbert space has an infinite number of dimensions, this infinity is countable. Countable means that in principle, a label with an ascending integer can be attached to each dimension. The set of real numbers is uncountable, but the set of rational numbers is countable and the set of rational quaternions is that too. So, to each dimension of the Hilbert space a rational quaternion can be attached. Mathematicians use the name operators for the mathematical things that can do this. The real numbers describe a continuum and the set of quaternions does that too. But the set of rational quaternions does not do this. This means that it is impossible to accurately describe smooth phenomena with the model obtained so far.

    First, do you not know that there are many (infinitely many) different Hilbert spaces?

    Second, do you not know that there are Hilbert spaces with uncountably many dimensions?  While it is true that the Hilbert spaces associated with bound phenomena, like the hydrogen atom, have only a countably infinite number of dimensions; unbound systems, like an ionized hydrogen atom (simply the unbound continuum of states past the highest bound energy state of the hydrogen atom), have an uncountably infinite number of dimensions.

    Third, where do you get the idea that having a countably infinite number of dimensions "means that it is impossible to accurately describe smooth phenomena"?  Each of these dimensions is just as continuous as the real numbers (or the quaternions, not just the rational quaternions).  While we could label each of these dimensions with rational numbers (or even rational quaternions), we can even label them with plain old ordinary integers (or even simply counting numbers), and it will make absolutely no difference in the fact that they can most certainly "accurately describe smooth phenomena".

    By your form of argument, a finite dimensional space, like our three or four dimensional space/spacetime cannot "accurately describe smooth phenomena".  After all, if a countably infinite dimensional space cannot, then what hope is there that a finite dimensional space can?

    I'm sorry, but it is strongly apparent that you need to go back to the "drawing board".  (In fact, you need to go to a stage long before the "drawing board", and "bone up" on the concepts you seem to think may have some bearing upon your ideas.)

    David
    fundamentally

    The Hibert book model uses an infinite dimensional separable Hilbert space and its Gelfand triple, which is called rigged Hilbertspace, but is no Hilbert space. Indeed exists a large number of different representations of this separable Hilbert space that are all based on an orthonormal set of functions or on more abstract items, such as the bra-kets that were introduced by Dirac. All these representations are mutually lattice isomorhic with respect to their closed subspaces. This fact is used in the Hilbert book model and it relies on work done by Garret Birkhoff, John von Neuman, Piron and many others. They proved that the set of closed subspaces of the infinite dimensional separable Hilbert space is lattice isomorphic with the set of propositions in a traditional quantum logic. Piron found that the inner productof this Hilbert space must be specified by using numbers that are taken from a division ring. Only three division rings exist: the real numbers, the complex numbers and the real quaternions. Biquaternions do not form a division ring, so they do not fit. This work was finalized in the sixties. Thus, for the Hilbert book model only three kinds of Hilbert spaces fit. The existence of different representations plays no role in the model. The characterization "separable" means that the Hilbert space has a countable number of dimensions. It means that eigenspaces are countable. This does not hold for the Gelfand triple. For that construct the operators have uncountable eigenspaces. That fact is used by the Hilbert book model for introducing a GPS-like coordinate system as the eigenspace of an operator that resides in that rigged Hilbert space. This background coordinate system is used for adding parameters to fields and for adding positions to the granules, which are attached to the eigenvectors of the strand operator.
    The set of real numbers is uncountable. The same holds for the set of complex numbers and the set of quaternions. But the rational numbers have the same cardinality as the integers, the rational complex numbers and the rational quaternions. The continuum has the same cardinality as the real numbers, the complex numbers and the quaternions.
    Thus the separable Hilbert space does not support a continuum as eigenspace of operators. It is even worse, because it appears that some physical quantities are fundamentally granular with respect to differences. Amongst them are position, action and entropy. Also progression occurs in steps.
    So you cannot represent continuous spectra in a separable Hilbert space. You might do that in the Gelfand triple, but then the link to traditional quantum logic is obstructed. This is what the Hilbert book model is about! It keeps this link.
    The only smooth things in the Hilbert book model are the probability amplitude distributions, which constitute the fields. But they are not represented IN the separable Hilbert space. They are attached to a subset of the Hilbert vectors. This subset is formed by the eigenvectors of the strand operator. Each probability amplitude distribution has a parameter domain that covers the whole Hilbert space (all eigen vectors of the position operator).
    The granules are very small. The same holds for the fundamental steps that correspond to Planck limits. On the scale of atoms these steps already look like continuous changes.

    "By your form of argument, a finite dimensional space, like our three or four dimensional space/spacetime cannot "accurately describe smooth phenomena". After all, if a countably infinite dimensional space cannot, then what hope is there that a finite dimensional space can?"
    Here you seem to mix the dimensions of the Hilbert space with the dimensions of the division ring or the dimensions of eigenspaces.

    "cannot accurately describe smooth phenomena" means that the Hilbert space offers no continuum eigenspaces in order to do that. It can only approximate a continuum. The faults occur at Planck scale.

    I do not think that your comments on the Hilbert space will lead me to a redesign, but I adapted the text of the article a bit based on your previous comments.
    Hans

    If you think, think twice
    Halliday
    Hans:

    First, I never consider isomorphic systems to be different, just as I do not consider different choices in coordinate systems (even general coordinate transformations, such as within General Relativity) to be different systems, even though they may look different.  So, when I said there are an infinite number of Hilbert spaces, I was referring to non-isomorphic classes of Hilbert spaces (I consider all isomorphic elements—together referred to as a "class"—to be the "same" [strictly, of the same class]).

    Second, your "separable" Hilbert spaces (so, Hilbert spaces of only a countable infinite number of dimensions) is only a small subset of the totality of Hilbert spaces.  In fact, it misses important classes of Hilbert spaces, such as ionized atoms, (free) photons, etc., which all have an uncountably infinite number of dimensions.

    Third, there are far more than "Only three division rings".  The three division rings you list—"the real numbers, the complex numbers and the real quaternions"—are the only finite-dimensional division rings/algebras over the reals (a far more restricted set).  There are an uncountably infinite number of other division rings, or even mathematical fields (commutative division rings, like the real and complex numbers).

    Forth, unlike finite dimensional vector spaces (where any two vector spaces of the same dimensionality over the same mathematical field/ring/whatever are always isomorphic), two countably infinite dimensional vector spaces (like Hilbert spaces), even over the same division ring, need not be isomorphic.

    Fifth, even Hilbert spaces (over the real, complex, or quaternion numbers) with only a countably infinite number of dimensions have "the same cardinality as the real numbers, the complex numbers and the quaternions."  Just as finite dimensional spaces, over such numbers, do.

    Even though the dimensionality of finite vector spaces, and of the eigenspaces of operators thereon, are finite, the fact that they are vector spaces over the real, complex, or quaternion numbers gives them this cardinality.  A fortiori, such is also true of such Hilbert spaces.

    In fact, each and every single point within a Hilbert space represents a continuous function over the domain space for (or associated with) that Hilbert space.  So, for instance, the probability distribution of the position of the electron in a hydrogen atom is a continuous function, for each point in the Hilbert space of the hydrogen atom.  (Of course, the position operator is not an eigenfunction of the hydrogen atom.  This is why there are not continually localizable positions for the electron within the hydrogen atom.  However, there is a linear combination of the eigenfunctions of the hydrogen atom that do yield the position operator, for any and every (continuous) position one may wish to choose.)

    This, of course, brings up another issue with your use of what you are calling Hilbert spaces (or even the extension of the Gelfand triple):  You are using multiple points simultaneously (within each "Hilbert" "page" of your "Hilbert book").  Since each point within a Hilbert space represents a configuration of the system—so each point represents a Quantum Mechanical "wave function" of the system—then what does it mean to have multiple points within this space simultaneously?

    This makes as little sense as choosing multiple points, simultaneously, within the phase space of a classical system.

    Yes, you appear to be digging yourself quite the hole.

    David
    fundamentally

    David,

    The Hilbert book model uses infinite dimensional separable Hilbert spaces because it uses traditional quantum logic as its foundation. The consequence is that everything where eigenspaces of operators of these Hilbert spaces are involved, the model is countable. A rigged Hilbert space is taken to help where a continuum background coordinate system is required. This coordinate system and its canonical conjugate are used to give positions a value and to parameterize probability amplitude distributions. Thus these distributions are not hampered by the fact that eigenspaces of operators of the separable Hilbert space are countable.
    The fact that other Hilbert spaces exist does not matter for the Hilbert book model. The Hilbert book model also implements lower limits for the differences between eigenvalues of certain operators. The strand operator is one of them. The fact that division rings with infinite dimensions exist is not important for the model. It only uses division rings that can be used for specifying the inner product of the separable Hilbert space. From these only three versions exist.

    You stated: "This, of course, brings up another issue with your use of what you are calling Hilbert spaces (or even the extension of the Gelfand triple): You are using multiple points simultaneously (within each "Hilbert""page" of your "Hilbert book"). Since each point within a Hilbert space represents a configuration of the system—so each point represents a Quantum Mechanical "wave function" of the system—then what does it mean to have multiple points within this space simultaneously?"

    Each page of the Hilbert book contains a separate Hilbert space. No covering space exists. I can only understand the quoted text when you are mixing the concepts of eigenvalues and eigenvectors otherwise I do not understand what you mean by points. 

    May be it will help when I use the characterization "separable" more early in the article and indicate that this means that the dimensions of the Hilbert space are countable.

    I understand your critics in the light that you thought that this model was targeted to replace current physics. The Hilbert book model is just another model. I do not claim that it replaces current physics.

    String theory is also not meant to be a proper model of physical reality. It is a nice means to understand more of physics. That is also the target of the Hilbert book model. For that reason it tries to stay as close to reality as the (rather simple) model allows. At the same time it takes some unconventional inroads.

    If you think, think twice
    Halliday
    Hans:

    You say:
    String theory is also not meant to be a proper model of physical reality. It is a nice means to understand more of physics.
    However, while this may be a comforting stance for those of us that do not espouse "String theory" (or Super-String theory), this is certainly not the intent of the practitioners of the string theories.  ;)

    David
    fundamentally
    The lessons of Susskind on string theory start with a boost that brings the environment of the observer close to the speed of light. This reduces the number of dimensions with one. In the rest of his ten lectures he stays with that condition, but adds a series of extra dimensions that have a restricted range. In the eight lesson he answers on a question of a student whether the teached string theory is a proper model of reality. The answer that Susskind formulated was that until that moment there existed no sign that the model that he teached conformed with reality. But that this was not the target of the investigation. The target was to get a much better understanding of the physical methodologies and in that respect the project is a great success. Apart of that string theory brought several deep insights. Not in the least in subjects such as black holes and the expansion of the universe. But for example SUSY is not found in experiments. The same seems to hold for the Higgs boson.
    If you think, think twice
    Halliday
    Hans:

    You state:
    You [meaning me, of course] stated: "This, of course, brings up another issue with your use of what you are calling Hilbert spaces (or even the extension of the Gelfand triple): You are using multiple points simultaneously (within each "Hilbert""page" of your "Hilbert book"). Since each point within a Hilbert space represents a configuration of the system—so each point represents a Quantum Mechanical "wave function" of the system—then what does it mean to have multiple points within this space simultaneously?"

    Each page of the Hilbert book contains a separate Hilbert space. No covering space exists. I can only understand the quoted text when you are mixing the concepts of eigenvalues and eigenvectors otherwise I do not understand what you mean by points.
    Hans, from your article's use of the plural "chains" (each with multiple "grains", though "There is one grain in the chain that represents the current position on this path") it appeared that you have more than one chain within "Each page of the Hilbert book", so each "separate Hilbert space".

    Is this not your intent?  Is there only a single chain, with only "one grain in the chain that represents the current position on this path", in each "separate Hilbert space"?

    Hilbert spaces are vector spaces.  Even your "separable" (countably infinite dimensional) Hilbert spaces are vector spaces.  Any and all vectors within a vector space is represented by a "point" within the space.  Are not each of your "grains" associated with a vector within your Hilbert space?

    David
    fundamentally

    David,
    Each subsequent static status quo is handled by its own separable Hilbert space. The corresponding "current granules" all correspond to a Hilbert vector in that Hilbert space that is an eigenvector of the "strand operator". The current granule is the/an anchor point of a particle and at the same time it is the anchor point of a probability amplitude distribution. The size of the granule corresponds with the minimal extent of that probability amplitude distribution and is of the order of magnitude of the Planck-length. The model leaves the possibility open that a particle can anchor on more than one granule (chain). In that case the superposition of the probability amplitude distributions form the private field of that particle. Both in the single anchor case as in the multiple anchor case the private field can be interpreted as the wave function of the particle. At the same time the superposition of all private fields form a covering field. That covering field is a very complex probability amplitude distribution with very many anchor points. The difference of the attached Hilbert vectors and a more general Hilbert vector is that the attached Hilbert vectors are eigenvectors of the strand operator. On the other hand the granules that form horizons are also attached to eigenvectors of the strand operator. They form densely packed sets that get the form of bubbles. They are also attached to a probability amplitude distribution, but for each of the granules that participate in the horizon that distribution has a minimal extent. For the "free" granules the probability amplitude distribution is "folded out" over a much larger extent.

    The minimal extent of the probability amplitude distribution may be interpreted as a kind of ground state of that distribution. It gives the granules their size. So, the granule has no "hard" size.

    This is due to the fact that the continuum background coordinate system that is taken from a GPS-like operator in rigged Hilbert space is not accurately coupled to the eigenvectors of the strand operator that resides in the separable Hilbert space. This coupling is re-established for each Hilbert book page. It results in a corresponding stochastic fluctuation. The ground state of that fluctuation is the source of the granularity. The wider spread is the source of (private) fields.

    Hans

    If you think, think twice
    Halliday
    Hans:

    What you have said here sheds very little light on the simple question of whether or not you have more than one "chain" within a single Hilbert space.  I'm reasonably certain that you have only a single "current granule" for any single chain, but I cannot even be absolutely certain about that.

    David
    fundamentally
    David,

    A huge number of particles exist in the unuverse of the Hilbert book model. Each of these particles has an anchor point in one or more of its own chains. This holds in every static status quo, thus in every Hilbert book page, or more precisely in every separable Hilbert space that is part of the Hilbert book model. For each chain, each separable Hilbert page in the model contains precisely one current granule. When a chain splits or merges it does that between two subsequent Hilbert pages.

    In Schiller's strand model the full length of the strand is used. In the Hilbert book model only the direct environment of the current granule is relevant. 

    Hans 
    If you think, think twice
    Halliday
    Hans:

    You state:
    The Hilbert book model uses infinite dimensional separable Hilbert spaces because it uses traditional quantum logic as its foundation.
    Am I correct to assume that what you are referring to as "traditional quantum logic" corresponds to systems of bound states, such as standing waives, atoms, harmonic oscillators, etc.?

    However, you go on to state:
    The consequence is that everything where eigenspaces of operators of these Hilbert spaces are involved, the model is countable.
    This is false, prima faicie.  Even the eigenspaces of an operator of such a Hilbert space (over the real, complex, or quaternion numbers) that has all distinct eigenvalues (like 1, 2, 3, ...) are all continuous spaces (having exactly the cardinality of their respective division rings).  Even if one looks at only normalized subsets of such spaces (which are, themselves, no longer vector spaces, of course), the complex and quaternion cases remain continuous (only the case where the division ring is the real numbers becomes discrete, taking on a character isomorphic to the set {-1, 1}).

    David
    fundamentally
    David,

    Classical logic is based on about twenty five axioms. It has the structure of a mathematical orthocomplemented modular lattice. Traditional logic differs from classical logic in that the modular law is weakened. It has the structure of an orthocomplemented weakly modular lattice. (It is also called an orthomodular lattice). Garret Birkhoff and John von Neumann used the lattice isomorhism of the set of closed subspaces of an infinite dimensional separatable Hilbert space with the set of propositions of traditional quantum logic as an argument to represent quantum physics in this structure. Traditional quantum logic is a set of propositions that interrelate via the laws set by the axioms of this logic. Some of these propositions concern everything that can be said about a physical item. In that sense these propositions represent the corresponding item. Other propositions treat properties of that item. They belong to the hierarchy of propositions that together have the item's proposition on its top. It corresponds to a similar hierarchy of closed subspaces of the separable Hilbert space.

    The normal operators of the separable Hilbert space have eigenvectors that together span the Hilbert space. They form an orthonormal base of that Hilbert space. The Hilbert space has a countable set of dimensions. So the normal operator must have a countable number of eigenvalues.

    The division rings that are used to specify the inner products are not the eigenspaces of normal operators. Neither do closed subsets of these division rings form eigenspaces of normal operators. however, the eigenspaces may be formed by countable subsets of the division rings. However, other number types could also be used as eigenvalue. For example it is thinkable to use octonions for that purpose. In any case the set of eigenvalues must be countable.

    Hans
    If you think, think twice
    Halliday
    Hans:

    You state:
    "cannot accurately describe smooth phenomena" means that the Hilbert space offers no continuum eigenspaces in order to do that.
    First, even for a complete set of commuting operators, so there is a unique set of eigenvectors, each of these eigenvectors forms the basis for a one dimensional continuum "eigenspace".  (Of course, we usually normalize, but even then there is an infinite number of such normalized vectors, so long as the division ring is larger than the real numbers.)

    However, this lack of "continuum eigenspaces"* does not, necessarily, imply that one "cannot accurately describe smooth phenomena".  Have you not heard of superposition?  Isn't superposition a rather important aspect of Quantum Mechanics (QM)?  In fact, if we didn't have superposition in QM, we wouldn't have many of the most interesting consequences of QM, such as entanglement.  Would we?

    Superposition in QM provides for one to "accurately describe smooth phenomena" in the same way that linear combinations of basis vectors within a finite dimensional space (like our everyday three space) allows one to "accurately describe smooth phenomena" like the motion of planets.

    Sorry, you still haven't "cut it".

    David

    *  I'm assuming that this is for a complete set of commuting operators, since for any single operator with multiple equal eigenvalues, one has an eigenspace that is most certainly continuous, even imposing normalization.
    fundamentally
    David,

    "First, even for a complete set of commuting operators, so there is a unique set of eigenvectors, each of these eigenvectors forms the basis for a one dimensional continuum "eigenspace". (Of course, we usually normalize, but even then there is an infinite number of such normalized vectors, so long as the division ring is larger than the real numbers.)"
    The Hilbert book model only uses division rings that have the same cardinality as the continuum. Still the eigenspaces of operators of the separable Hilbert space have the cardinality of the integers.  

    Superposition uses the numbers of the division ring as coefficients. So, expectation values can be taken from a continuum. Still eigenvalues do not belong to such a continuum. So mixed states can produce observed values that "fall in between". Still the set of all expectation values in a system that consists of a finite number of states is countable.

    The Hilbert book model does not support continuous spectra and also no mix of discrete and continuous spectra. It might very closely approximate a continuous spectrum.

    In the Hilbert book model the separable Hilbert space only treats countable phenomena, but in some cases these phenomena can approximate continuous phenomena. The probability amplitude distributions may describe discrete as well as continuous phenomena.

    The Hilbert book model is strictly building on its foundation. That foundation is traditional quantum logic. In the Hilbert book model this link is never released.
    If you think, think twice
    Halliday
    Hans:

    Again you state, without proof or even further reasoning:
    ... Still the eigenspaces of operators of the separable Hilbert space have the cardinality of the integers.
    Go see my post, above, for the reality.

    You then go on to state, again without proof or even further reasoning:
    ... Still the set of all expectation values in a system that consists of a finite number of states is countable.
    First off, we are not talking about "system[s] that consists of a finite number of states".  Secondly, even if I reinterpret this to mean "a system that consists of a [countably in]finite number of [eigen]states", as is the case of these "separable" (countably infinite dimensional) Hilbert spaces, "the set of all expectation values" has the cardinality of the division ring, not the dimensionality/number-of-eigenstates.

    On the other hand, I have no problem with your statement that
    The Hilbert book model does not support continuous spectra and also no mix of discrete and continuous spectra. It might very closely approximate a continuous spectrum.
    This follows simply from the countably infinite dimensionality of your "separable" (countably infinite dimensional) Hilbert space.

    The problem appears to be that you may be confusing the discrete nature of the eigenvalues of operators on a "separable" (countably infinite dimensional) Hilbert space, with the nature of the space itself, which is most certainly continuous.

    Your statements are like unto statements such as "because operators on a three dimensional vector space can only have up to three distinct eigenvalues, one cannot describe any continuous phenomena in three dimensional space".

    You see, if your statements reduce to something like the above for finite dimensional spaces, then you can know with certainly that what you have stated is, prima facie, false.  For, you see, finite dimensional Euclidean inner-product spaces are also Hilbert spaces.

    David
    fundamentally
    David,

    Above I already stated:"The normal operators of the separable Hilbert space have eigenvectors that together span that Hilbert space. This is a property of (bounded) normal operators. They form an orthonormal base of this Hilbert space. The separable Hilbert space has a countable set of dimensions.  So this normal operator must have a countable number of eigenvalues."

    Countable means having the cardinality of the set of integer numbers.
    The strand operator is bounded by an outer horizon. This outer horizon has a finite surface. Thus the strand operator has a countable number of eigenvalues in a limited region.

    The Hilbert book model contains a huge but finite number of granules.
    In the Hilbert book model the number of states equals the number of particles and that is finite.

    Hans
    If you think, think twice
    The Arab savant Al Biruni found history a book shorn of its covers, without origin or terminus, and attributed events to will. There are strikingly similar ideas in the traditions of the Jains, who evidently knew about electricity and magnetism a long time ago. And if you parse the three imaginary dimensions of the quaternion as time, you get 3DT, which gives a neat account of all known particles. But as the residues of tradition show, these are windowless monads in the sense of Leibniz, and all you have for dynamics is pre-established harmony in the vibrations of the connecting space.

    Much theory that followed was austerely finitist, following the quite midtaken iudea that the Greeks had no notion of infinity. Even Einstein was affected, for his theory, at least in the final Cartan-Einstein form, is omega-inconsistent. But ions, fields and nerve impulses all have uncountable degrees of freedom.

    In this vein, I am very interested in your recent remark on Quantumn Diaries Survivor concerning the mass term in the Dirac equation. If I am not mistaken, Dirac relativised a wave-equation due properly to Huygens, in which the damping term has parameters of mass and frequency, a mass-vibration, which is not just a pre-established harmony.

    fundamentally
    The Hilbert Book Model is one of the subjects of an e-book that is free accessible on:  http://www.crypts-of-physycs.eu . The concept is also treated in sepate papers on that site. I am preparing a concise version: http://www.crypts-of-physics.eu/ConciseHilbertBookModel.pdf
    The book is also available in print. (See the website)
    If you think, think twice