There seems to be a path forward to higher dimensional systems by constructing coordinate systems from velocity vectors and velocity base units instead of distance vectors and distance base units.
In a 3D velocity vector system a different but identical clock can be attached to each vector giving a 6D system that can be constructed and tested. This is the first step to higher dimensions. It avoids the transformations from space like to time like coordinates near light speed.
A lesson from Kaluza and Klein is that higher dimensions involve electromagnetism as fundamental components.
In extension of the 6D system a different electroscope and magnetometer can be attached to each of the velocity base units giving a 12D super symmetry. This is not the 10D symmetry, 11D cosmology, or 12D super symmetry other people had in mind, but is a set of coordinate systems all of which can be constructed and tested with no leap of faith, fuzzy logic, or dark science.
We could add accelerometers to each velocity vector also, but they appear to be redundant in the 12D system. How about in higher systems?
So far all the clocks and meters have been attached to only 3 velocity components. It appears higher systems are possible by attaching each clock and meter to a different velocity vector and randomly orienting them in the coordinate system that still is defined by the original three velocity vectors.
In this way a 27D coordinate system can be defined by including the accelerometers. It looks so much like the 26D of original string theory that it is tempting to allow one redundancy in the 27.
Certainly this will not be acceptable to some who specialize in higher dimensions. It does seem to agree with Kaluza and Klein and bring higher dimensions within grasp of the non specialists. Especially for the 6D system there seems to be a practical use for it in not so distant future.
Opinions?
I take a clue from Roger Penrose that the imaginary part of a complex representation contains all the necessary additional dimensions that are missing from the construction without saying how many are missing or what types they represent.
Warren Smith has described an algorithm for designing multidimensional number systems. Cayley offers a similar algorithm that after the octonions delivers different multidimensional numbers. Pia Maria Solèr produced a hard prove that Hilbert spaces can only cope with elements of a division ring. In a division ring all non-zero members have a unique inverse. Only three suitable division rings exist: the real numbers, the complex numbers and the quaternions. These number systems exist in several versions that only differ in the way that they are ordered. They can be ordered by a selected Cartesian coordinate system. Subsequently they can be ordered by a selected polar coordinate system.
The existence of different versions of number systems can be correlated with the electric charges and the color charges in the SM.
It is good to know which restrictions mathematics puts onto your freedom of choices. You better obey these restrictions.
http://www.rangevoting.org/WarrenSmithPages/homepage/nce2.pdf
I think that Kaluza and Klein had a good intuition thinking that higher dimensions would involve electromagnetism as a fundamental component.
Indeed, the de Broglie localized double-particle electromagnetic photon can be represented according to Maxwell's equations only by going beyond 4D space, which was a conclusion that de Broglie himself had reached already in 1936.
This expanded geometry is explained in the following paper:
http://www.omicsonline.com/open-access/on-de-broglies-doubleparticle-photon-hypothesis-2090-0902-1000153.php?aid=70373
Christian: "I am not sure, if I understand what you are actually aiming for."
Maybe I should try to decide now what I am actually aiming for. This topic originally came to me as an idea while I was corresponding with Ulla Mattfolk of Finland about a different topic having to do with the properties of space and how the systems of microbiology are constructed upon and limited by systems of micro physics.
I felt that a lot of reference articles have interesting statements to make that get lost in the abstraction of higher dimensions. At that time I wrote the key parts of the present article in about 20 minutes and returned to the other questions at hand. Abstractions were replaced with physical quantities people can find and touch.
At that time I was most interested in the fact that every one of the proposed higher dimensional systems can be constructed with ordinary technology. Also interesting were the results of Kaluza and Klein, apparently pointing in the same direction. Then there is 6D replacement of the inconvenient 4D transformations from space like to time like coordinates near light speed. No transformation is needed in 6D.
For a few years the writing about higher dimensions circulated among friends with a wide variety of opinions. Not everyone liked it but the comments helped to simplify my statements, clarify the content, and delete the parts I liked best. There was no consensus about the validity or the value of my higher dimensions. So I posted the question to a larger group on RG.
Where does it fit into the greater program of science? The best contribution is that coordinate systems can be built up from velocity components instead of distance components. That opens the way for any type of scalar measurement to be attached to a coordinate system in more than one degree of freedom. Also notable is the fact that a velocity system can have an absolute zero in space by observation of red and blue Doppler shifting of cosmic microwave back ground.
Recently I have been working on concepts of high speed in deep space and effects of prolonged acceleration approaching light speed. Higher dimensions become important in the calculations, and results are suggesting that space eventually becomes altered by the objects moving through it, contrary to an opinion of Einstein.
"In this latter statement absolutum means not only physically real, but independent in its physical properties, having physical effect, but not itself influenced by physical conditions."
The correspondence with Ulla was developing biological concepts at high speed and on the quantum mechanical scale which was influenced by other Einstein quotations.
"The principle of inertia, in particular, seems to compel us to ascribe physically objective properties to the space-time continuum." Also "In order to develop this idea within the limits of modern theory of action through a medium, the properties of space-time continuum which determine inertia, must be regarded as field properties of space, analogous to the electromagnetic field."
In 6D systems it is very helpful that transformation of electromagnetic fields to time like coordinates becomes unnecessary at high speed. Maybe Einstein's absolutum was intended for less severe conditions.
Christian, about momentum in QM,
The momentum operator of QM does make an equally feasible system of higher dimensions depending on what it is operating on. The velocity system doesn't fit so well into QM as it is formulated, but otherwise makes an understandable construction for a large audience. To go from momentum to stress energy, a velocity is required, so I guess some one like Heisenberg or Born will eventually find a way to put it together in acceptable formalism.
Why does the wave function own a Fourier transform. or better why does it own a displacement generator?
Christian,
" any absolutely integrable function "owns" a Fourier transform". I know this fact very well, but it shifts the question only a bit deeper. "Why does the wave function belong to the category of functions that owns a Fourier transform. This has no straightforward reason. In fact the reason must be sought in the fact that the wave function is the result of the activity of a stochastic process and the wave function owns a Fourier transform because that stochastic process owns a characteristic function. This goes much "deeper" than the fact that the wave function corresponds to a location density distribution of a point-like object.
Most probably the stochastic process belongs to the category of the inhomogeneous spatial Poisson point processes. I guess that it is something like a modified Thomas process.
Each type of elementary particle corresponds with a private mechanism that secures the coherence of the swarm of locations that corresponds to the location density distribution. For a point-like object the location density distribution corresponds to a detection probability density distribution. In that case the particle hops along a stochastic hopping path. The mechanisms use a particular stochastic process that generates the subsequent hopping landing locations that from the coherent location swarm. Without the mechanism the hops would land in an arbitrary way and the location swarm would be a complete chaos.
The current physical theories do not dive deeper than the wave function.
http://vixra.org/abs/1606.0028
This is exactly the attitude that prohibits physics from developing more in the direction of its actual foundation. A foundation must have a very simple structure and at the same time it must restrict its extension in the direction of a more complicated structure. The simplicity of the foundation causes that it can easily be comprehended by skilled scientists and that quite probable is already added as a part to the mathematics that are discovered by humans. The difficulty is not located in this structure, but in the fact that its extension is significantly restricted, such that after a set of extension steps the result becomes recognizable as a mathematical model that simulates the structure and the behavior of physical reality.
A suitable candidate for the structure of the foundation was already discovered eighty years ago. Mathematicians call it the orthomodular lattice. It extends naturally into a separable Hilbert space and the Hilbert space restricts its usage of number systems to division rings. In a division ring each non-zero member owns a unique inverse. The members of the most elaborate division ring (the quaternions) are ideally suited for the storage of dynamic geometric data. The Hilbert space is nothing more and nothing less than a storage medium of dynamic geometric data. It can harbor many versions of these number systems in parallel. These versions may differ in the location of their origin and in the way that their members are ordered.
The Hilbert space offers no means for the generation of coherent sets of numbers that differ from parameter spaces. These parameter spaces offer the opportunity to define a unique companion non-separable Hilbert space that can store continuums.
Mechanisms that exist outside the separable Hilbert space can fill eigenspaces of stochastic operators. If these eigenspaces form coherent sets that can be described by location density distributions, which have a Fourier transform, then the stochastic process that generates the locations belongs to the category of inhomogeneous spatial point processes. The characteristic function of this process has much in common with the momentum function that describes the kinematic behavior of the location swarm.
Without the activity of the stochastic processes the described model does not feature any coherent dynamics. The swarm contains a huge number of locations, which are all landing locations in a hopping path. Compared to the violent hopping action, the swarm as a whole is moving in a rather well managed way.
It is the stochastic process that generates the form of the location density distribution of the swarm. It is difficult to imagine that the location density distribution controls the stochastic process.
A Hilbert space is a linear vector space in which for every pair of vectors an inner product is defined. If the vectors are perpendicular then their inner product equals zero. Otherwise the value is a member of a division ring. A linear combination of vectors is again a Hilbert vector. The applied coefficients must be member of a division ring.
A division ring is a number system in which every non-zero member owns a unique inverse. Only three suitable division rings exist: the real numbers, the complex numbers and the quaternions. These number systems are continuums. Each of these division rings contain countable subsets: such as rational numbers, integers and natural numbers. These subsets share the same cardinal number. The multidimensional division rings exist in several versions that differ in the way that they are ordered. Thus rational quaternions can be used to enumerate the orthonormal base of a separable Hilbert space.. In this way it is possible to define reference operators that have base vectors as eigenvectors and the corresponding rational enumerators as eigenvalues. Thus, reference operators have parameter spaces as eigenspaces. The Hilbert space can harbor multiple versions of division rings in parallel. By applying the same eigenvectors and the values of continuous functions as eigenvalues it is possible to define new operators from existing reference operators. This method merges function theory and differential calculus with Hilbert space operator calculus.
The orthomodular lattice does not contain number systems. When it extends to a Hilbert space will number systems emerge in the model. Before that extension, notions such as time and location have no place in the model. The Hilbert space puts a significant restriction on the number systems that can be used.
The foundation must somehow explain why notions such as time and location are applied by physical reality. You cannot just start with these notions as basic concepts. It is far from straightforward that these concepts have values that are restricted to division rings. Pia Maria Solèr has provided a hard prove that Hilbert spaces can only cope with division rings.
You must ask yourself what the physical significance is of a Hilbert space, then in fact it is no more and no less than a structured storage medium for dynamic geometrical data. On itself it possesses no means to ensure the dynamic coherence of physical reality.
http://vixra.org/abs/1605.0101
http://arxiv.org/abs/1101.5690
I often state that the actual foundation of physical reality is not accessible to observation. However, I have come to the conclusion that the signature of this foundation can be recognized everywhere in nature.
Look around and you become easily convinced from the fact that all discrete objects are either modules or modular systems. With other words, the creator of this universe must be a modular designer. His motto is “Construct in a modular way”. However, also non-discrete items exist. Universe contains continuums and these continuums appear to relate to the discrete objects. Further, we as observers of these facts, want to place everything into an appropriate model, such that we can comprehend our environment.
http://vixra.org/abs/1607.0146
I am a long retired physicist and my career ended more than a decade ago. My interest no longer is located in applied physics. Instead my interest is targeted at the creation of my living environment. My most urgent question is: "why is universe representing a dynamically coherent system? In order to get an answer to that question, it is necessary to find the fundamental structure of that environment. That fundamental structure must automatically and inescapably expand into the full environment.
Obviously, Hilbert space is nature's storage medium and it is nothing more than that. It restricts the kind of numbers that it can store. Clifford algebras describe far too exotic number systems and obscure the model more than it elucidates. It is no storage medium. Sets are about the simplest structures that can be specified and are often used as foundations of mathematics. However, these structures doe not restrict their own extension. Lattices are sets for which the relational structure is restricted. Many versions of lattices exist. Classical logic is one of them. The orthomodular lattice is another and this one restricts its extension to one possibility: A separable Hilbert space and that thing can act as a structured storage medium for dynamic geometric data. The Hilbert space restricts its usage of number systems to division rings. But the Hilbert space provides no means that ensure the dynamic coherence of the stored data. On the other hand it provides an excellent mathematically well-defined base model. It is fairly easy to specify what is still missing in the model. What is missing are mechanisms that fill the storage places with coherent sets of data. Obviously these mechanisms apply stochastic processes. Current physical theories ignore these mechanisms. They can only describe what happens. They cannot explain why that happens.
The foundation of physical reality is not directly observable and cannot be investigated via experiments. On the other hand the signature of this foundation is clearly visible in every aspect of the universe. All discrete objects in universe are either modules or they are modular systems. Turning this observation into a mathematical structure is not a simple task. But it can be done.
http://www.e-physics.eu/ThouShaltConstructInAModularWay.pdf
http://www.e-physics.eu/TheHilbertBookTestModel.pdf
As soon as you are aware of the fact that nature uses the Hilbert space as its storage medium, then this opens interesting possibilities. Obviously that storage covers the past, the present status quo and the future. The present status quo walks as a vane over the storage space. Thus inside the Hilbert space all aspects of reality are precisely stored. For observers that travel with the vane the future part of the store is hidden. Thus two views are possible. One takes the chair of the creator and surveys the whole creation. In the other view the observer travels with the vane. In this model the role of the mechanisms that ensure dynamic coherence via the application of stochastic processes becomes interesting. In the first view the creator throws his dices, but he throws them all at once. In the second view according to the observers that travel with the vane the creator is still throwing his dices. Einstein will be confused.
These views also throw an interesting light on the beginning and the end of the travel of the vane. It puts big bang in a completely different scene.
Christian,
I rely on the work of Garret Birkhoff and John von Neumann, which is intensively discussed in several papers, but most comprehensively in the paper of John Baez.
They are much better physicists and mathematicians than I am.
G. Birkhoff and J. von Neumann, The Logic of Quantum Mechanics, Annals of Mathematics, Vol. 37, pp. 823–843
For me it is quite acceptable that physical reality is a structured system. Many parts of mathematics are derived by observing this structure and its behavior. I take for granted that an intimate relation exist between physical reality and mathematics. That does not mean that in any way mathematics controls physical reality. It means that reality controls large parts of the mathematics that is "discovered" by humans..
http://arxiv.org/abs/1101.5690
Christian, I enjoyed this discussion and it forces me to sharpen my thoughts.
I have the advantage that I am retired and I have much time to invest in deepening my understanding of the lower levels of physical reality. There are many areas that I did not yet touch. I consider QED and QCD merely as successful recipes rather than as explaining theories. Especially the binding of quarks via the color confinement process intrigues me. Special quaternions can implement the activity of gluons and that activity must disturb the generation of quaternions as complete elementary particles. The mechanisms respond to this by generating hadrons rather than completed quarks. This process must bind the quarks, but this must be a complicated affair. The LHC does not help much because that machine treats all events as black box behaviors. They investigate what goes in and what comes out and do not observe what happens inside the box.
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