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    A Completely New Look Onto Fundamental Physics
    By Hans van Leunen | March 31st 2012 03:13 PM | 25 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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    People can already count since several millennia. Arithmetic is improved over time. Usually we count by using integers. If we want to get more accurate results, then we better use rational numbers. Sometimes we also use numbers that are not expressible in a fraction, such as the number π (pi) and the square root of two . These are real numbers. The square root of a negative number delivers more problems. For this task it is necessary that we move to a two-dimensional number system. The better calculators under us can also handle this problem fluently. The so-called complex numbers are discovered in the sixteenth century by the Italian mathematician Gerolamo Cardano and are now used for all kinds of applications. Especially wave movements and other oscillations are easily described by complex numbers. It took until half of the nineteenth century before William Rowan Hamilton discovered the successor number collection. These numbers are not three but four-dimensional and are called quaternions. Quickly the eight dimensional octonions followed. This indicates that the dimension of the number systems increase by a factor of two. As the dimension increases, the computational capabilities decrease. For example, the quaternions can no longer be multiplied in random order.

    A quaternion can be seen as the combination of a real number and a three dimensional vector. With their 1 +3D structure quaternions seem well suited for application in physics. Shortly after the discovery of quaternions many physicists have tried this. Remarkably,this failed miserably. The reason is the less convenient quaternionic product rule. In many cases it much easier to apply complex numbers or vector analysis than quaternions. It quickly made the quaternions redundant. This is intensified by the fact that the special theory of relativity of Einstein claims a role for the space-time combination where quaternions do not fit in. The also 1 + 3D spacetime has a so-called Minkowski signature, whereas quaternions have a Euclidean signature. That meant the actual death blow for the application of quaternions in physics. That this was in fact unjustified recognized only few scientists.

    Quaternions prove very suitable as eigenvalues of operators that deliver these values as observable quantities. This cannot be said for spacetime. Thus spacetime is not a genuine observable. So, there must be more to it. However, this only becomes apparent when we dare to let quaternions play a greater role. Quaternions and especially functions with quaternionic values possess some unexpected properties that sofar have remained virtually unexposed.

    Most physicists use complex functions in order to represent the state of small particles with it. In essence, these state functions are probability amplitude distributions. By using complex state functions, these functions can easily show the wave behavior of these particles. This is why the corresponding part of physics is called wave mechanics. However, it is equally well possible to represent the state of the particle by using a quaternionic function. This pure mathematical measure causes that what happens is no longer seen as a wave problem, but rather as a fluid problem. With respect to physical reality, this measure changes nothing. The only thing that differs is that the mathematical toolkit now provides a very different view about what happens. For most physicists this last method of viewing is utterly strange.

    The unexpected move is due to the fact that quaternionic probability amplitude distributions can be considered to be a combination of a charge density distribution and a current density distribution or can be viewed as a combination of a scalar potential and a vector potential. The result is that when a switch from a complex state function is made to a quaternionic state function, any linear equation of motion is converted into a balance equation.

    Fluid mechanics conventionally belongs to another part of physics than wave mechanics. However, the outlined transition lets wave mechanics convert in a kind of fluid dynamics. This is nothing short of a revolution. Instead of oscillations we see high-and low-pressure areas and currents. That is not to say that the oscillations are not there anymore. The only thing that happens is that with the new tools we perceive less well the oscillations, but we get a much better look at the currents and the high-and low-pressure areas.

    A fact that is still not entirely addressed is formed by the sign choices that quaternionic functions offer. Quaternions offer four mutually independent sign choices. The first is called the conjugation. It changes the sign of the imaginary part. The three others are reflections. They change the sign in one of the three imaginary directions. Together, these sign choices provide eight different sign variants. Separately, the conjugation and the reflections cause a transition of the handedness of the quaternionic product. These properties show remarkable effects. This is reflected in the coupling of state functions as is described by equations of motion such as the Dirac and Majorana equations. This coupling causes a local compression of the space that surrounds the considered particles. It can be explained by fluid dynamics.

    The conclusion may be drawn that ignoring quaternions and quaternionic functions was not justified. Correct use of these ingredients gives a sea of new opportunities for fundamental physics.

    The new look is applied in http://www.crypts-of-physics.eu/Quaternionic_Quantum_Physics.pdf

    Comments

    Thank you professor !

    What would be the value of pi in such a notification system ? :

    0
    1,2,3,4,5,6,7,8,9,a,b,c
    11,12,13,14,15,16,17,18,19,1a,1b,1c,
    21,etc... ?

    Or does it make sense at all ??
    i can't solve that alone, please help :) ...

    Johannes Koelman
    Hi Hans -- are you aware of the early work of Finkelstein et al on this subject? Is your theory different? (Sorry, didn't manage to open the pdf, got an I/O error.)
    fundamentally
    Johannes
    In the sixties there was a short uplift of the quaternionic approach. There was a small group at Cern around Jauch, Piron, Emch and others. Finkelstein also had contacts to that group. In that time Constantin Piron proved that the numbers that can be used for the inner product of a separable Hilbert space need to come from a division ring. There are only three suitable division rings; the real numbers, the complex numbers and quaternions. This does not say that higher dimensional hyper complex numbers cannot be used for eigenvalues or field values, but I never saw proper applications of these. Horwitz tried the octonions but he could only close subspaces after seven multiplications. Thus quaternions seem to be the highest dimensional choice for coefficients in Hilbert spaces.
    In the sixties I was a student and I studied everything that was available on quantum logic, quaternions and quaternionic Hilbert spaces. I certainly studied the papers of Finkelstein. After my study I took a career in industry. In 2009 I picked up the thread of quaternionic Hilbert spaces again. I made it a project that investigated the fundaments of physics. It evolved via several stages into its current format. That differs in several points from its origin in the sixties. The project is still strictly based on quantum logic and via an isomorphism on a quaternionic separable Hilbert space, but it extends this primitive model such that it can implement fields and dynamics. The switch from complex to quaternionic wave functions converts wave mechanics into a kind of quantum fluid dynamics. This differs significantly from the vision of the Cern group, including Finkelstein.
    About a year ago I had email contact with Finkelstein. He may still be active.

    The referred article is not signifantly different from my previous column article; "Quaternionic Versus Complex Probability Ampitude Distributions"
    If you think, think twice
    The prospect has evidently been in the air: it irritated people in continuum mechanics, who found their constitutive relations being read as balance equations.

    Balance equations have a heritage of their own from the discourse of the Physiocrats on the Economy of Nature, motivating a view of physics as a kind of accounting procedure. But the only definite trace I have found is a banker named Olinde Rodrigues, a contemporary of Liouville, and a precursor of Hermite in analysing polynomial solutions to field equations.

    Constitutive relations, of course, apply only within materials with boundaries assumed. And continuum mechanics irritates engineers by demanding a foundation in pure mathematics. There has to be a better way.

    fundamentally
    In this context, the notions "balance equation" and "continuity equation" relate to Reynolds's tranport theorem (http://en.wikipedia.org/wiki/Reynolds_transport_theorem) .
    These equations exist in global format (integral format) and local format (differential format).
    If you think, think twice
    fundamentally
    There were earlier interpretations of quantum physics in terms of quantum fluid dynamics.
    See: http://robertkernodle.hubpages.com/hub/Quantum-Theory-Alternative-Fluid-Dynamics-Interpretation-Of-Objective-Reality
    If you think, think twice
    The Stand-Up Physicist
    Have you spent quality and quantity time with Stephen Adler's work, "Quaternionic Quantum Mechanics and Quantum Fields"?  He is very good, getting a position at the Institute for Advanced Studies at Princeton. Get the book from the library as Amazon is charging $265 for the hardcover.  This is the blurb from Amazon:
    It has been known since the 1930s that quantum mechanics can be formulated in quaternionic as well as complex Hilbert space. But systematic work on the quaternionic extension of standard quantum mechanics has scarcely begun. Authored by a world-renowned theoretical physicist, this book signals a major conceptual advance and gives a detailed development and exposition of quaternionic quantum mechanics for the purpose of determining whether quaternionic Hilbert space is the appropriate arena for the long sought-after unification of the standard model forces with gravitation. Significant results from earlier literature, together with many new results obtained by the author, are integrated to give a coherent picture of the subject. The book also provides an introduction to the problem of formulating quantum field theories in quaternionic Hilbert space. The book concludes with a chapter devoted to discussions on where quaternionic quantum mechanics may fit into the physics of unification, experimental and measurement theory issues, and the many open questions that still challenge the field. This well-written treatise is a very significant contribution to theoretical physics. It will be eagerly read by a wide range of physicists.
    Much of my sense of how quaternions fit in with the reset of the power tools of physics are based on reading Adler. Reading his work may help in your research.  Good luck.
    fundamentally
    Doug,
    I own Adler's book
    May be it is a bold statement, but I am convinced that Dirac understood more of quaternion quantum mechanics than Adler ever did. The problem is that Dirac has hidden the typical quaternionic features in spinors and matrices. He wanted to explore the Pauli matrices. If you convert Dirac's equation into (two) quaternionic equations (one for the electron and one for the positron) as Susskind does in his YouTube lectures then you see that it becomes close to Madelung's equation. So already in the early days of quantum physics there was some awareness that there is a relation between wave mechanics and fluid dynamics. Dirac and Majorana both used the sign flavors of quaternionic distributions. These are encoded in the so called gamma matrices.

    In the sixties a small group of scientists (Jauch, Emch, Piron, Speiser, Finkelstein, a.o.) gathered in Cern or other places and studied quaternionic quantum physics. In that time Piron derived that separable Hilbert spaces must use members of a division ring for their inner products. Only three suitable division rings exist: the real numbers, the complex numbers and the quaternions. Most other physicists stayed with complex quantum physics and after a short while quaternionic quantum physics was forgotten. Nobody had picked up the power of quaternionic probability amplitude distributions (QPAD's) in opening a different view on quantum physics using quantum fluid dynamics. So, in 1929 in the early days of quantum mechanics Madelung was a loner. However, he did not apply quaternions. Still his equation comes close to a quaternionic equation.

    Quaternions were only fashionable in a few short periods. The first occurred directly after the discovery of the quaternions. The second period occurred in the sixties by the Cern group.

    Adler had more negative than positive influence. He just tried to translate complex quantum physics into quaternionic quantum physics, but he did not see the opportunities that QPAD's and quaternionic sign selections can bring.
    If you think, think twice
    Here's a really comprehensive and accurate rack of theories: http://ncatlab.org/nlab/show/HomePage. Look for integration approaches and you see the yawning gap.

    This is related to the contrast of groups and algebras: to get from a Lie symmetry group to a Kac-Moody Lie algebra or similar you must allow it to deform over a continuum, including transcendental numbers. This involves the ground (ring or field) where one can introduce scalars and tensor products. Such is relativity, so this is the level of Grand Unification, which no-one has reached from group theory.

    I'm now thinking that ground in this sense is related to Bolzano's grounds of proof, as he found for differential calculus, so the discussion keeps touching on theorems. With quantum logic on board one should be able to secure the quantum theorems.

    Here's the U(1) group from octonion algebra: http://arxiv.org/abs/1004.2816. But remember there are several types of octonions: these are split. As sources/sinks? This is now the hard question.

    Annoying Precision has really clear group math: http://qchu.wordpress.com/2011/02/12/su2-and-the-quaternions/, Also a max argument about group representations: if you try to represent stuff with groups, and then ask about all the stuff, the universe, the answer is no longer stuff, its Lawere Theory! That's just what happened to the string movement!

    So you have to introduce a Copenhagen functor which runs: SU(2)xSU(3) --> U(1)/Z(6), doing observations, assigning measures and relativities. This forces one to ask after the values, which run: quaternion x quaternion ---> split octonion! Short of Lorentz corrections, you may be able to work with quaternions alone. This is daring to run relativity backwards, general -> special.

    fundamentally
    Orwin
    nLab does not work on the explorer of my PC. So I used Google Chrome. That can open it properly, but I still not understand what your advice means.

    The methodology described in the paper uses quaternions and does not need octonions. It does not use group theory either.

    Most scientists that investigated quaternions and quaternionic functions ignored the importance of their sign selections. Further they neglect to consider the interpretation of quaternionic functions as combinations of charge density distributions and current density distributions. Another unexplored fact is the possibility to use the parameter space of quaternionic distributions as the space where these charges and currents reside.

    This also holds for Adler, Baez, and members of the n-cafe.
    If you think, think twice
    The Stand-Up Physicist
    Hans:
    It does not use group theory either.
    This does not sound like a good long term strategy.  Electric charge sure is conserved and that can be viewed as being due to the group U(1) gauge symmetry.  That idea has been around for quite some time.  In the sixties and seventies, folks figured out the symmetry groups that lead to the weak and strong force conservation laws.  The properties of these groups matches up perfectly with the large collection of particles we have seen generated by atom smashers.  To me, a new outlook must make connections to group theory or it is not worth the time to study.  Sorry to be that harsh, but it is my position.
    fundamentally
    The Hilbert Book Model uses the sign selections of the quaternions. This replaces the functionality of the SU(2) group. With the help of these sign selections the HBM can classify 64 elementary coupling equations that lead to 56 elementary particles and eight oscillating wave types (photons and gluons). The HBM uses (all!) the symmetries directly. The geometry together with the number characteristics give more detail than the group.
    If you think, think twice
    blue-green

    Individuals (theorists, Platonist, cats ...) favour wherever they have spent and invested their time.
    This behavior is as natural and instinctive as adults returning to their Xs (even if it makes no sense to those out of the loop). Why should theoretical physics be any different? Accomplished experts in complex variable analysis will remain satisfied (nay, filled to the brim!) with quantum amplitudes being members of equivalence classes of vectors in a complex plane C … perfectly mapped with associated operators to handle everything under their sun and more.

    Sorry, but there are many objections to this post.

    First, this is not any new. Already in 1984 Girard published his The quaternion group and modern physics.

    Second, I read the recent derivation of Maxwell equations from quaternions in science20. Even if we agree on the derivation, the problem is that Maxwell equations are both inconsistent and incomplete and are already being generalized. See for instance this recent

    http://pre.aps.org/abstract/PRE/v53/i5/p5373_1

    http://pre.aps.org/abstract/PRE/v55/i3/p3793_1

    where the Maxwell equations are generalized by introducing new potentials A^b(R(t)) that complement to the old potentials A^b(r,t). Evidently, quaternions are useless for the new non-local potentials because are not defined in a spacetime but over (R(t)).

    Third, the same formalism can be applied to gravitation, where the spacetime potentials h_{ab}(r,t) are extended by new h_{ab}(R(t)). As shown in

    http://www.rxiv.org/pdf/1112.0075v1.pdf

    the new potentials explain gravitational phenomena that cannot be explained by the previous potentials neither by alternative theories as MOND, TeVeS...

    Fourth, before believing that quaternions have something to see with the unification of the standard model forces with gravitation, we would understand that the current failure to unify gravitation with the rest of interactions is a consequence of old mistakes regarding the nature of gravitation. The general textbook claim that general relativity (a geometric theory) is equivalent to a field theory of gravity has been shown to be wrong in recent years. General relativity is only a geometric approximation to FTG. As I wrote in a recent FQXI post

    http://fqxi.org/community/articles/display/165

    http://fqxi.org/community/forum/topic/1246

    the correct field theory of gravity can be quantized as other ordinary field theory.

    fundamentally
    The treated theory is hardly touching Maxwell equations. However it is using field theory. Especially everything that involves balance equations. The theory does not use GRT either, but it has its own approach how curvature is affected by particles and how these particles are generated by fields.
    Quantum Fluid Dynamics differs from conventional fluid dynamics in the fact that the fields act on their own parameter space instead of on some medium like gas or liquid. This parameter space is shared by all quaternionic wave functions and has very special characteristics. That is why I decided to give it a unique name; Palestra. The name signifies that in this space all activity takes place. It comprises the whole universe.
    Please read http://www.crypts-of-physics.eu/Quaternionic_Quantum_Physics.pdf for more details.
    If you think, think twice
    If you look to the title of the Physical Review E paper cited above, you will find the words "Action at a distance". The authors show that Faraday-Maxwell field theory approach to electromagnetic interactions (FMSI) --characterized by potentials A^b(r,t)-- is incomplete:

    "we conclude that the FMSI concept could not give a complete and adequate description of the great variety of electromagnetic phenomena."

    Then they correct field theory by the introduction of a set of new potentials A^b(R(t)), associated to Newtonian action-at-a-distance (NILI). The new potentials go beyond field theory. Indeed, they show irreducibility.

    The same basic ideas about the recent work on gravitation cited above. The field-theoretic potentials h_{ab}(r,t) are incomplete and have been extended by new NILI potentials h_{ab}(R(t)) (again beyond field theory).

    Quantum Fluid Dynamics is only a crude approximation to more advanced formalisms, which are already at our hands as SHP theory, and that do not use the concept of field. SHP theory is a generalization of quantum field theory. The classical limit of SHP theory is discussed in the classic monograph

    http://order.ph.utexas.edu/mtrump/manybody/

    which gives additional technical details in what is wrong with fields.

    In its turn, the SHP theory can be considered an approximation to a more fundamental quantum theory (far beyond wave mechanics). If you and your readers are interested in this research topic I could write more.

    Hans, I didn't intend advice - more to offer some signposts to others. Baez' introduction to renormalization did not impress me.

    Algebraic theories can draw naturally on the algebra of logic, and develop as formal systems, but then your point of departure is a representation, not a set of axioms. This I think makes your system hard to recognise for what it is.

    Against Juan Ramón González Álvarez I would say reading general relativity as pure geometry is wrong: its a statistical mechanics, giving only an average of curvatures at a point. Similarly the Ocedelet Theorem for Lyapounov exponent (for fractal representation) gives only time average. So space-time as we can observe it is a statistical problem. That's why I think this must lead to generalized uncertainty relations.

    And those would secure a foundation for quantum engineering.

    The standard textbook by Misner, Thorne, and Wheeler, notices that general relativity is also named "geometrodynamics" and presented as "Einstein's geometric theory of gravity". Evidently, the fact that general relativity is a macroscopic theory do not change this interpretation.

    Last four decades attempts to extend this geometric formulation by adding quantum effects leads to the well-known quantum geometries, all of which are technical inconsistent, by reasons given in the works cited above. Once the old geometric viewpoint is generalized by the modern physical viewpoint associated to gravitons -- e.g. see the derivation of Hilbert & Einstein equations from the more general FTG equations-- we can obtain a consistent quantization.

    What you say about the generalized uncertainty equations is also well-known. Precisely the SHP theory cited above uses covariant uncertainty relations (which I will write from memory):

    [x^a, p^b] = -i\hbar \eta^ab

    But unlike quantum geometrodynamics, the SHP theory is free of the famous problem of absence of time: i.e. the quantum (covariant) Hamiltonian does not identically vanish.

    fundamentally

    Orwin
    The HBM bases its main structure, (the structure of a book) on the axioms of traditional quantum logic, which leads to its lattice isomorphic companion, the separable Hilbert space. Via the Schrödinger picture and the Heisenberg picture this leads to the conclusion that both primitive models cannot implement progression and the HBM must use a sequence of these primitive models in order to implement dynamics. The fact that the Hilbert vector coefficients can maximally be quaternions lead to quaternionic quantum physics. Thus, much of the HBM comes straight from the axioms. In fact the representation also follows from the axioms. It is possible to choose a real, a complex or a quaternionic representation. These all give different views on physical reality.

    If you think, think twice
    Hans, here's axioms for your observer problem, in the terms of projective geometry, hence formally consistent with the quantum axioms:

    http://www.mendeley.com/research/optical-axiomatization-of-minkowski-spa...

    On leaving Special Relativity for last (as observer-related):

    Towards Relativistic Atomic Physics and Post-Minkowskian Gravitational Waves:
    http://arxiv.org/abs/0908.0209

    Canonical Gravity and Relativistic Metrology: from Clock Synchronization to Dark Matter as a Relativistic Inertial Effect: http://arxiv.org/abs/1108.3224

    I can't vouch for the details, but the idea is consistent with gravitational thermodynamics, and Verlinde's prize-winning statistics, You could then look to the gravitational noise experiment for a critical test....

    fundamentally
    The Hilbert Book Model uses quaternions as eigenvalues. This means that observables in first instance are quaternions. Progression is counted by a special progression parameter, which is not synonymous to time. The notion of spacetime and the corresponding Lorentz transformation and Minkowski signature are only introduced when an observer and the observed object occupy different reference frames. The HBM does not consider spacetime as a genuine physical observable. Instead it considers it as a convenient construct. The same holds for the notions of proper time and coordinate time.
    If you think, think twice
    One must expect an integral model to act on its own parameter-space. But you don't want to loose what is called strong emergence in quantum theory, systems where the medium of interaction matters . E.g. how do you deal with permittivity and permeability and the classical c^2 = 1/nu.eta ?

    How one adds local detail to the point-particles of classical physics is the big theoretical issue of our time, and the main motivation for string theory. Relativity doesn't suit your representation up front: closer to your concerns is the geometric phase or Berry curvature. Here it gives time symmetry as reflection, Z(2) quantization, and SO(3) Chern topology:

    http://arxiv.org/abs/0909.4831

    That gives Z(2)xSO(3) in place of Z(6), but these are complex quaternions, so again its a parallel representation to your flux quaternions. Which can also pick up spin matrices:

    Quaternion eigenbundles:
    http://arxiv.org/abs/math/0304281

    If you are too rigid about representation you won't capture the complexity of reality. And I don't see physics giving up complex refractions and dielectrics.

    fundamentally
    The HBM takes the speed of light, the permittivity and the permeability of Palestra equal to one.
    The HBM does not apply bi-quaternions. 
    If you think, think twice
    That way you can project it all inside the unit circle like they did in Ancient Egypt, with continued fractions. But there's was a slightly different vision, which Wolfgang Pauli revived in a dream:

    Divine Contenders; Wolgang Pauli and the symmetry of the world: http://www.paricenter.com/library/papers/peat26.php

    We have this now as 3DT, with seven vector bosons above 1TeV/c^2, which the pagans revered as gods.

    Following Gauss' infernal units, you suppress a factor of 4.pi which is there for a geometrical reason related to the turn in SI Ampere-turn, commonly winding number. So of course you can't see any electromagnetism, or chemistry. Or even real geometry. All flow with nowhere to go.

    Of course classical electromagnetic optics is only part of the picture, missing displacement and polarization, so a richer representation could help.

    The effective field theory you assume for curvature is also what throws up naturally the whole range of supergravity solutions. So there is now the very interesting prospect that you have the only workable remnant of this whole approach. But to me that means you have a geometric phase and not a true curvature.

    In any case, one would now think of application to renormalization group flow or AdS/holography problems. Take Perelman's proof of the Poincare conjecture by flow analysis in GR: last I heard, he refused the prizes, so he must think its a junk line of argument even if it seems to work!

    Since the Poincare conjecture is about 3-volumes being topologically simple, this issue connects with Douglas Sweetser's interest in volumes in GR, but there's a telling sense that the topology may well not be simple....

    So here's a fresh input from the frontlines: Relativistic Noise: http://arxiv.org/abs/1201.3405. This hydrodynamic analysis gives space-time as a noisy soliton/acoustic dynamic, all "sound and fury, signifying nothing," as Shakespeare put it.

    Interestingly, the hydrodynamic analysis assumes the polarization dynamic, without which reality is perhaps tranquil but also "void" in the sense that carries to Buddhism.