Two different kinds of multidimensional differential calculus exist that can cope with parameter spaces that contain elements, which are constituted from a scalar and a three dimensional vector. These sets of differential calculus differ mainly in the choice of the scalar part of the parameters.
Quaternionic differential calculus uses a parameter space that has quaternions as its elements. Thus, Its parameter space corresponds to a quaternionic number system. This situation is complicated by the fact that several versions of quaternionic number systems exist, which differ in the ordering of their elements. For example, the ordering can be applied via a Cartesian coordinate system, but a Cartesian coordinate system can be ordered in eight mutually independent ways. A spherical coordinate system usually starts from a selected Cartesian coordinate system. It can then proceed by first ordering the azimuth angle or it can start with ordering the polar angle. These angles can be ordered up or down. The radius has a natural ordering. Half of these orderings correspond to quaternions that feature a right handed external vector product. The other half corresponds to quaternions that feature a left handed external vector product. These orderings appear to influence the behavior of elementary objects.
These ordering choices also appear in the parameter spaces of the other set of differential equations that we will indicate as Maxwell based differential calculus. The Maxwell based equations use a spacetime model that has a Minkowski signature. Where the real parts of quaternionic parameters can be interpreted as representing pure progression, will the Maxwell based equations interpret the scalar part as coordinate time. In comparison to the quaternionic parameter space, the coordinate time plays the role of quaternionic distance. This difference may explain the differences between quaternionic differential calculus and Maxwell based differential calculus, but this is not the fact. Trying to convert the quaternionic differential equations into Maxwell based equations is only partially possible.
The difference between the two sets is far more subtle, than just a change of the scalar part of the parameter space. The difference between the two sets clearly shows in the second order partial differential equations. The Maxwell based second order partial differential equation is a wave equation, while the quaternionic second order partial differential equation does not feature waves as its solutions. Physics is full of waves, thus this fact might classify quaternionic differential calculus as unphysical. That conclusion is not justified. The clue of the difference is hidden in the Dirac equation for the free electron and free positron. That equation couples two sets of equations. The usual formulation of the Dirac equation uses spinors and Dirac matrices. It is also possible to formulate these equations in quaternionic format and then it becomes clear that the Dirac equation splits into two coupled first order partial differential equations. In this way the Dirac equation couples solutions that use different quaternionic parameter spaces. One of these parameter spaces uses right handed quaternions and the parameter space of the coupled solution uses left handed quaternions. For each of the solutions a second order partial differential equation can be derived via the coupling of the two solutions. This equation is a wave equation! Still both solutions separately obey a regular quaternionic second order partial differential equation. That equation does not accept waves as members of its set of solutions.
The quaternionic second order partial differential equation accepts other solutions. For example the homogeneous version of this equation accepts shape keeping fronts as its solutions. Shape keeping fronts operate in odd numbers of spatial dimensions. Thus, a one dimensional shape keeping front can travel along a geodesic in the corresponding field. These objects keep their amplitude. Three dimensional shape keeping fronts diminish their amplitude as 1/r with distance r from the trigger point. The wave equation accepts similar shape keeping solutions, but apart from that, the wave equation also accepts waves as its solutions.
Both sets of equations do not reach further than second order differentials. As a consequence they cannot handle violent disruptions of the continuity of the considered fields. Also the coverage of longer ranges will require higher order differentials.
With other words the considered fields will be the same, but the conditions can require methodology, which covers higher order differentials. GRT covers only slightly higher differentials. It neglects the simpler and more localized causes of disruptions of the continuity by point-like artifacts. That should be the task of quantum physics. However, also quantum physics ignores the mechanisms that generate the point-like artifacts, which cause the discontinuities of the physical fields. The equations only describe the behavior of the fields and Physicists seem to interpret the discontinuities as the artifacts that cause the behavior of the fields.
This outcome shows the need to be able to treat fields independent of the equations that describes their behavior. This is possible by exploiting the fact that Hilbert spaces can store discrete quaternions and quaternionic continuums in the eigenspaces of operators that reside in Hilbert spaces. The reverse bra-ket method can create natural parameter spaces from quaternionic number systems and can relate pairs of functions and their parameter spaces with eigenspaces and eigenvectors of corresponding operators that reside in non-separable Hilbert spaces. This also works for separable Hilbert spaces and the defining functions relate the separable Hilbert space with its non-separable companion.
http://vixra.org/abs/1511.0266
http://vixra.org/abs/1511.0007
the other question I pose is the following : Clifford algebra holds about a so restricted number of axioms. only two. How may we explain that a so restricted number of axioms enables us to describe so large fields of physics from quantum mechanics to relativity ?
Most physicists hardly know anything about the features of quaternions, quaternionic Hilbert spaces and quaternionic differential calculus. That knowledge would give them a better view on what they are doing!
Dear Hans,
there are studies which are describing Gravitation with a biquaternionic structure and are promising for unification with quantum physics. The octave remember the model of the Gell Mann octet. The Maxwell equations were written in quaternionic form originally. The quaternions were used by Pauli for describing the spins and the dirac equation can be written with the quaternions as well. The good part is that there are analitical functions which can be used to describe the effect of the singularites by using the residue theorem in one complex variable, the magnetic field of a wire can be seen also as a analitic function singular at the point where the wire passes, the current in the singularity is seen as the outcome of the closed loop integral of the holomorphic function.
I agree in Physics most use the vector analisys instead of the hypercomplex Numbers.
Stefano,
Hilbert spaces can only cope with number systems that are division rings. This restricts the tolerable number systems to real numbers, complex numbers and quaternions or their rational subsets. Biquaternions are not division rings. So with biquaternions you can only build models that do not use Hilbert spaces. In quantum physics the Hilbert spaces take the role of structured storage spaces. So with biquaternions you must use a different kind of structured storage of geometric data.
I tried to formulate the Dirac equation in quaternionic format and encountered some remarkable facts. The quaternionic Dirac equations that represent electrons and positrons use different parameter spaces. The coupling of these equations produces a second order partial differential equation that acts as a wave equation. The natural quaternionic second order partial differential equation does not offer waves as its solutions.
Geometric analysis uses Green's functions in order to cope with point-like artifacts that cause local discontinuities. However, the function does not cause the discontinuity. The discontinuity is a response of the function on the presence of the artifact. Something must put that artifact at that location. (Here I identify the field as a function. The reverse-bra-ket method enables that.)
Physics misses the description of the mechanisms that generate point-like artifacts and thus indirectly cause a dynamic reaction of the affected continuum.
http://arxiv.org/abs/1101.5690
Describing physical behaviour from fields (which we can measure) and regard the sources as singularities or as excitations or as discontinuities I can see it well with the help of analitical fuctions on the complex plane. The effects of the currents for example can be represented by calculating the residual of the analitical function around the singularity. The analogue line of reasoning for a single charge with complex numbers is something like a Surface integral in a complex space around the charge, instead of a line integral over a complex plane... does it make sense?
what could be the answer to the question that quaternions , pertaining to Clifford akgebra, at the same time enable us to formulate the whole standard quantum mechanics, the relativity , the standard electrodynamics here including in particular the Maxwell equations ?
the other question I pose is the following : Clifford algebra holds about a so restricted number of axioms. only two. How may we explain that a so restricted number of axioms enables us to describe so large fields of physics from quantum mechanics to relativity ?
I am sad that such a work cannot be accepted even from arxiv.org.
What to say, we live in eras where original work is out of interest...
Dear Hans,
Keep working and try to combine theory with experimental facts.
If my opinion matters, please avoid working on relativity theories and keep only quantum ones.
Regards
Quaternions have been studied in this context, cf. http://inspirehep.net/search?ln=fr&p=find+a+adler%2C+s+and+t+quaternionic&of=hb&action_search=Recherche&sf=earliestdate&so=d
Dear Prof. Van Leunen,
Quaternionic Algebra since it is an algebra afterall should have some set inclusion kernel like properties which can dynamically ("Arrow of Time") generate Groups and then Rings resulting in Fields by using Special Relativity to give rise to Higgs Bosonic Tachyon like condensation in D-Branes String couples by Systems properties (please refer to my String Theory and Stock Markets paper on my RG page and also the Hagg's Theorem paper) for the translation to geometry of quantum mechanical stock market fluctuations or waves (expermiental). Once that is done, atleast kinematically , any Quantum Mechanical calculation can be Relativistically scaled in an information and energy field ( I don't know whether it will be able to convert the couple into its duality integration) and be amenable to Homotopy Theory for calculations and integration of the parts in which scalars can be special case of isolated vectors made independant by M-Theoretic Transversality conditions (please refer to my RG page). SKM QC PhD EPS Fellow (Indirect)
the fields should be only those previously mentioned. Why not give also a look at my RG profile
First, to Elio Conte:
remark on Clifford algebra is really nice, and to add, it is really joy to me to analyze just Cl3. So beautiful, so powerful, so rich.
About quaternions:
1. quaternions do not have an elegant and intuitive geometric interpretation
2. quaternions do not work for n > 3
3. it is better to use Cl3, quaternions are in even part of the algebra, namely bivectors; but bivectors have nice geometric interpretation and formalism is easy to expand for n > 3; all "quternionic" problems just disappear
4. geometric calculus is powerful and general
Finally, I am using some code in Mathematica (my code) to calculate in geometric algebra. All I need to include quaternions is to define i, j, k as bivectors basis elements.
Hilbert spaces can only cope with division rings. Quaternions, complex numbers and real numbers are the only suitable division rings. If physical reality uses Hilbert spaces as its preferred storage medium, then it uses quaternions as its preferred number system.
Dirac had to use matrices and spinors because quaternionic functions do not help in splitting the Klein Gordon equation into first order partial differential equations, but quaternionic functions can act as solutions of a second order partial differential equation that acts as a wave equation..
It is far too simple to claim which number systems physical reality uses and what its preference will be. You have to dive deep into the secrets of the Dirac equation in order to understand more.
http://www.e-physics.eu/TheDiracEquation.pdf
http://vixra.org/abs/1511.0007
Stam
I own Adler's book. Reading it, was a disappointment to me. It was one of the reasons that I started my own research into the foundations of physical reality.
I took my physics study in the sixties. In those times quaternionic physics was still fashion. A group around Jauch, Piron, Emch and Finkelstein was exploring the advantages of quaternionic Hilbert spaces. Later the interest in this direction faded away. Applied physics became much more important and did not need a new foundation in order to be able to flourish.
I attempt to give a contribution to tis debate considering the question from diffrenet points of view .
First an historical account . Quantum mechanics. As you know when quantum mechanics started , the first studies adopted matrices. The limit . These founding fathers had only such instrument in their knowledge also if Lord Kelvin had invited the international scientific community to intyroduce , generally speaking , a large of use of the quaternions since in his opinion they would have a great future in physics . For a reason or for another the result was that such founding fathers never attempted to use such Q in physics and quantum mechanics dveloped itself first by matrices and after by proper use of operaors and operators acting in a Hilbert space . Certainly all you rememebre that Dirac sio much was involved in the search of q-numbers with difference respect to c-numbers . And certainly all you remeber that Dirac was involved in the problem to find an Hamiltonian representing his relativistic formulation and also having quaternions or biquaternions as just profoundly explained from Hamilton , never he had the idea that the exact solutionof his problem required such kind of q. numbers. . Conclusion , if the scientific pattern would have been to a dopt ab inition Clifford aklgebra , certainly we would have to day a general formulation of quantum mechanics lightly different from the present status. I do not say a more well formulated quantum mechanics but an ALGEBRAIC formulation more elaborated , more elegant and also self-consistent respect to the present some insufficiences of this theory.
This is the first question
Now we may pass to the second question. If you look at my profile on REG you find that I have given a complete reformulation of quantum mechanics only using Clifford algebra. In fact we should be aware that quaternions represent only a field of Clifford , we have biquaternion algebra , we have dihadreal algebra and still more . Properties of such algebraic structures may be deduced with high level of rigour as you may see in my demonstrations and the crucial point is that by using only few algebraic axiooms and some detailed and of course simple algebraic demonstrations , we arrive to reformulate all the quantum mechanics from Heisenberg indeterminatuion principle, to Schridinger equation, to angular momentum , to Hydrogen atom, to more recent features as locality and non locality, contextuality, Bell and still more. We realize a complete bare bone skeleton of quantum mechanics without using elements of physics but only ALGEBRAIC developments. Quantization is explained and finalkly a proof is given of the basic >Von Neumann postulates on quantum measurement arriving for the first time to give mathematical demonstration of what is currently called the ciollapse of the wave function. This last result gives self consistency to the whole structure of quantum mechanics since all you kbow that presently we have standard quantum mechanics from one side and quantum collapse from the other side as ad hoc atached not giving self. consistency to the tehory. I have furnished a lot of demonstrations that we may realize quantum mechanics using such Clifford algebraic structure . Quantum intrinsic and irreducibkle indetermination , quantum probabilities , quantum interference , quantization results as consequent manifestation of such basic features of the Clifford alkgebra . Is not very surprising all this ?
The relativistic question. You certainly know that using Clifford in literarture you find a disseminate lot of studies obtainiong Maxwell equatuons , as example ... as well as we may arrive to the particular field of the elementary particles . Another result : using Clifford we reobtain the standard Einstein relativity, the Lorents transformations and the relativistic invariant ,,, we have a complete jump from quantum mechanics to relativity and always only using the few axioms of Clifford .. An unified formulation arises
Starting such preliminary results a question arises.
How is it that using only three abstarct algebraic elements as the basic elements of the Clifford algebra are we able to delineate a whole set of physics passing from a complete bare bone skeleton of quantum mechanics and arriving to relativity? What such abstract algebraic elements represent in the structure of our reality? This is the question to which I am not able to answer .. Certainly you will object that you have knwoldge also of other algebraic fields that are able to cover fields of physics. Certainly yes , but Clifford arrives where other fields do not arrive. By Clifford we arrive to give mathematicalk demonstration of von Neumann postulates , as example, and they remain POSTULATES if we do not use Clifford? Why? What such abstract three elements of Clifford really represnet in the basic scheme of our reality?. Certainly you remember that e1,e2,e3 bbasic elements of Clifford algebra are connected to spin. But of course you also know that Pauli was forced ... and I repeat.... was forced to admit that his Pauli matrices represneted spin. Ab initio he informed all us...... " be care .. these are foundations that go on the simple notion of spin" and only some experimental results forced him , nad I repeat , forced him to acknowledge the link Pauli matrices with spin .
In conclusion may we advance some further hypothesis ? Are such abstract elements , mathematical abstracty elements, basic foundations of our reality? If so, al our present vison of reality chhanges. Don't forget that , taking as example biuquaternions, they correctly realize a biquaternion space. Consequently, have we a prespace as of course outlined and suggested from many other authiors, existing and giving as projection the space we feel? This is a question that could be of importance if we reason from an algebraic point of view .
There is finally another question. Clifford algebra include deifefrent algebraic fields and they adimit idempotents . p^2=p . What they represent . Celebrated scientists as von Neumann and Orlov never had doubts . They are logical elemnets . If you give a look at my publications in RG you find that all such my elaborations are based on idempotents . Also quantization is based on idempotents. So we have idempoitents and idempotents means logic statements and logic statements involve cognition and cognition involves living being. So? Clifford algebra illustrates that cognition and physical elaborations no more are distinct features of our reality but they both travel ab initio? Have we logical origins of our reality when interpreted by the framework of quantum mecchanics? In other terms, have we about the logical origins of quantum mechanics? Note that if we use logic statements in Clifford we arrive to obtain quantum interference as well as in the corresponding physicsal case of using material entities. I have given a lot of theretuicla and experimental confirmations. In conclusion , it is not the problem of quaternionic Hilbert space or not .... this is very important but my modest view point is that , under such basic question, we have more elaborated problems that possibly could be of interest to pose .
Typical relativistic features, such as time dilation and length contraction are properties of the mathematical Lorentz transform and not of the underlying parameter space. The ingredients of this Lorentz transform are distance, observed speed, speed of information transfer and a notion of time. The corresponding parameter space can have Minkowski signature, but a valid alternative is a parameter space with a Euclidean signature.
Compared to quaternionic differential calculus, the set of Maxwell based differential equations are incomplete. Gauges are applied in order to generate extra functionality. For example the Lorenz gauge is used in order to generate the wave equation from Maxwell equations.
Quaternionic differential calculus does not lead to a wave equation in a natural way. Its natural equivalent of the second order partial differential equation does not offer solutions that implement waves.
Clifford algebra's do not offer easy answers. Clifford algebra helps in investigating the arithmetic of the algebra but it obscures the handling if function theory and differential calculus. This becomes apparent when Dirac's derivation of his equation for the electron and the positron is investigated.
A significant difference exists between the quaternionic four dimensional nabla operator and Dirac's four dimensional nabla operator. The quaternionic nabla stays within the same division ring of the quaternions. Dirac's nabla does not stay within the division ring of the quaternions.
Hilbert spaces are no more and no less than structured storage places for numeric data. They can only cope with number systems that are division rings. The only suitable division rings are quaternions, complex number and real numbers. This includes their rational versions. The same holds for the Gelfand triples, which can be considered as non-separable Hilbert spaces.
http://vixra.org/abs/1505.0149
http://vixra.org/abs/1506.0111
I do not understand if Hans consideration are related to my exposition. All my derivations are based on the rigour of mathematical demonstrations and are included in my RG profile. Lornetz transformation are obtained by using Clifford algebraic transformations and Imaeda , as example, also studied tachyons as theoretical indication. Hilbert space is not the sky full of stras of the physics,. Only some features are important in quantum mechanics , intrinsic indetermination, quantum interference , probabilty calculus .... wave function is again an obscure object in physics and also in tyhese conditions we have given exact interpretation of its meaning .........
When I started to investigate the lower levels of what we know about physical reality I stumbled upon the fact that most theories start with some kind of dynamic geometric space. The choice of that geometric space was not well underpinned. So I looked for a proper underpinning via a deeper foundation that does not yet incorporate notions of space and progression. Such foundation exists already since the 1936 paper of Garret Birkhoff and John von Neumann in which they introduced the relation between the orthomodular lattice and a separable Hilbert space. The orthomodular lattice does not include or apply number systems. As a consequence it does not include concepts such as space and progression. However, it directly leads to the concept of Hilbert spaces and thus indirectly it leads to division rings that can implement the notions of progression and space.
Due to its resemblance with classical logic, Birkhoff and von Neumann interpreted the orthomodular lattice as a logic system. That is why they called their discovery "quantum logic". A more suitable interpretation is that the orthomodular lattice is part of a recipe for modular system construction. That would classify (some) closed subspaces of a separable Hilbert space as representatives of modules or modular systems. The atoms of the lattice then represent elementary modules.
http://arxiv.org/abs/1101.5690
http://vixra.org/abs/1511.0074
van Leunen said:
"Typical relativistic features, such as time dilation and length contraction are properties of the mathematical Lorentz transform and not of the underlying parameter space."
What does it mean? If somebody likes such a strange statements ok, but it is probably wrong statement. Why? Because Lorentz transformations follow naturally from Cl3. Recall that it is possible to formulate special relativity using hyperbolic numbers, but Cl3 is full of objects with hyperbolic-like properties. Square of basis vector is 1. Basis bivector divided (yes, we can divide in Cl3) by his multivector amplitude is 1. Please, read carefully: to obtain (among many other things) Lorentz transformations one needs one thing only - non-commutative vector multiplication!!! So, our 3D space is "hyperbolic" and Lorentz transformations are simple fact. There is no "special relativity" in physics attached to some abstract space.
Quaternions (two things more):
1. there is no way to connect quaternions with vectors, in geometric algebra it is natural.
2. quaternions are just obsolete mathematical objects with historical importance; there is a practical importance, too, to manage rotations of challenger, for example, but EVERYTHING one can do with quaternions could be done with Cl3 (I could!), but this time with geometrical interpretation, without usual "quaternionic" problems and with possibility to expand theory to higher dimensions.
I suggest to read James Chappell (and mine) articles on Cl3, focusing on complex structure and theorems about Clifford conjugation. It is simple, beautiful and far reaching mathematics.
Hamilton was genius, he managed to see some nature of rotations before others, but he recognized only one important thing: objects which squares to -1 ( their multiplication table). He thought that quaternions are some kind of vectors (unit vectors i, j, k), but they are bivectors (duals of cross product, if you like). Bivectors from Cl3 have two properties: amplitude and direction. You can draw them easily, you can IMAGINE them as small directed areas which defines plane and rotate objects (imaginary unit rotates objects, but doesn't define a plane) . They are intuitive. They are powerful.
And finally, one can be arrogant a bit and suggest others to "dig deep" in physical theories, but fact is: you dive into any theory, try using quaternions, I will do it better with Cl3. Any theory!
Elio
I do no not want to downplay the value of Clifford algebra's, but physical reality in some way puts restrictions on what it prefers to implement as its toolkit.
Number systems, algebra's, differential equations are just aids that help describing the structure and the behavior of what we observe when we investigate physical reality. These aids are the tools that we use. These tools may reflect part of what physical reality also uses. That is the reason that we use these tools in order to formulate our physical theories.
These theories only describe the structures and phenomena that we observe. They do not explain why physical reality behaves as it does. None of the physical theories treat the mechanisms that ensure the spatial and dynamic coherence of these observations. However, without these mechanisms our observations would quickly turn into dynamical chaos. That does not happen, but we do not comprehend why that does not happen.
These mechanisms probably use stochastic processes in order to implement their controlling task. That is probably the reason of existence of the wave function of elementary particles.
No Hans, I do not agree. We have a body called Clifford algebra . It gives some important results in physics . I repeat .. all the "bare bone skeleton " of quantum mechanics. As physicist I MUST pose to myself the following question.... what is the reason because some abstract mathematical elements and so fw little axioms as those of the Clifford algebbra are able to give satisfactoiry reproduction of elaborated physivìcal theories as standard quantum mechanics and standard relativity. But if you want avoid to speak of relativity. Consider only quantum mechanics? Why? What such abstract elements represnet? Have they a POSSIBLE counterpart in some scheme of our physical and possibly mental reality? may I pose such question?
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