Dear Mr Jafari,

I think the main interest is applications of electromagnetism, so you may compare if that representation can be related to this, attached file. Here, the phenomena could be described using a certain fractal (subdomain) format.

Dear Mehdi,

Simply googling turns up numerous refs. As well as split-octonions you'll also find references to hyperbolic octonions and more.

Regards - Paul 

http://link.springer.com/article/10.1140/epjp/i2012-12069-y

Dear Dr. Jafari,

As you know, Many operations on vectors can be defined in terms of quaternions, and this makes it possible to apply quaternion techniques wherever spatial vectors arise. E lectrodynamics is one of many examples.  J. C. Maxwell described the physical feature of electromagnetic field with the vector as well as the quaternion. In 1843 W. R. Hamilton invented the quaternion, and J. T. Graves invented the octonion.

Quaternion space could be used to  describe the gravitational features. Note that meanwhile the S-quaternion space is proper to depict the electromagnetic features.

Dear scientist;

Thank you for your guidance and inspiration.

Dear Professor Paul G. Ellis

Recently, I define the various kinds of octonions(e.g. semi-octonions, split semi-octonions, ...) and study some of algebraic properties of them.

For example, could be used semi-octonions for Dirac’s operator and Maxwell’s equations?

Dear Mehdi

Google gives me 4 hits for ,. Try that?

(I'm not expert in the applications of octonions, merely interested in the physical situations where division algebras are useful - e.g. I started by being interested in whether the rotations implicit in the algebras of complex numbers and quaternions (when applied to physical systems) necessarily encode rotations in space and/or spacetime, or are the apparent rotations purely formal. I'm moving towards the latter view).

Regards

PS I'm haven't been awarded the position of professor - I'm only a PhD.

At detection with a human eye, light moves in a small twist on iris and then into the pupill. Also at HV transmission there is an inplane twist and then a large 'velocity' in another direction to a new point, where it can materialize. The twist agrees with the rot operation in Maxwell's equations.

Posssibly it is both discrete and continous. For example looking at an eye, there is a spot next to the pupill. And facett eyes are discrete.

The Helmholz coil are circular, and there the magnetic field possibly fulfil rot, maybe not at all points. For a cupper coil, in a magnetic field, it works equally well if it is square or circular. It means there are many solutions, and to reach insight, you should focus on one application, with something that could be measured and put into a formula, shape or context.

The front line in mathematical research, as I interpretate it, is 'propagation of singularities' (we found them discrete into bi-Solars), as in the attached article of previous answer, and 'micro-local analysis' ( the bi-Solars are indeed local, since they occur at Earth, not close to the Sun, and we tried to analyse the shape into some more details)

I do not see how Lorenz invariance agrees with the fact that light propagates with a twist on a surface, or lateral to the surface. Maybe in a first approximation, if it moves with a velocity in one direction, but then it appears to reach an object for example a planet, or the eye. For the eye, we know that it takes some time to establish contact with e.g. a star, and then it is there, and a velocity is not so easily motivated. Lorenz invariance is a very strong constraint to put on everything, so why are so many suggesting that?

Best Regards and Good Luck

Dear Lena,

Thanks for raising the point about Lorentz invariance and " the fact that light propagates with a twist" (I assume that what you mean by "a twist" is the same thing as polarisation).

As I understand it, if you use the Lorenz gauge ∂μAμ = 0, then Lorentz invariance holds (unlike using the the Coulomb gauge ∇ · A = 0 ). See e.g. the 4th page (numbered p. 127) of the attached file. 

So I don't think the polarisation of light (its twist) is incompatible with Lorentz invariance, even though intuition might suggest it be so.

Incidentally, there is much confusion between Ludvig Lorenz (to whom the gauge should be attributed, even tho' it's misnamed in the attached file, and with whom your own references suggest a familiaritiy) and Hendrik Lorentz, after whom Lorentz Invariance is named. Wiki refs linked below.

Best - Paul 

https://en.wikipedia.org/wiki/Ludvig_Lorenz

https://en.wikipedia.org/wiki/Hendrik_Lorentz

Since Dr Ellis recommended my answer, I read the text about octonions that he found on internet. 

I should not recommened this if you want to describe electromagnetism. For example, invoking Planck's constant, tacitly assumes a linear relation between energy and frequency, with the same constant for every material. That is not true. The 4-dimensional space-time is not a basis to rely on. You may google crack pot indeces about this.

http://math.ucr.edu/home/baez/crackpot.html

Good luck, in constructing another better space-time, for a specific application (at least one)!! 

Reading Dr Ellis answer, I think he is right, that the twist may be a second order effect on a subscale, where matter is e.g. planet (e.g. Venus) for sunlight, or coil for electromagnetism 

Dear Dr. Jafari,

I wonder non-associativity of octonions is not evil.

Dear Takayoshi,

Unless your use of the word "evil" is an unintended translation of a Japanese term, it's probably more constructive to recognise that as more constraints are imposed on a mathematical structure (more variables but all obeying and 8 x 8 structure table) so some characteristics (associativity, in this case), are lost.

I wonder whether there might be some sort of meta-mathematical law about mathematical structure and constraints.

It's almost suggestive of a sort of "uncertainty-like" relationship, in the sense that the more constrained in one way, the less flexible is a mathematical structure  in another way.

Something along the lines of:

          Properties x constraints

I derived Tti ( c.f. Technical report on my profile), and the spatial variables in that constraint is not at the same time as the time coordinate but delayed or retarded or how to express it.

If one interpretates the constraint for Minkowski space, the same way, it agrees with what I wrote in the previous answer? :

First light goes an arbitrary suitable length where there is a planet ( or an eye if you consider smart phone and the eye as an example). Then, at that point there is a velocity and a time and the eye adjust somewhat. After this, since Maxwell's equations can be rewritten into wave equation there is another constraint, namely the characteristica x-vt=const, and not the Lorenz invariant.  However, if we also adopt the Lorenz invariant, but assumes that time is almost constant and the velocity varies with x, a new condition is obtained. Differentiating Lorenz inv, and assuming that the variables in the characteristica is on a spatial subscale and substituting, there will be a refined model with a stationary field close to the boundary. I did not check this in detail, and maybe you do not agree at all. Or, it is wellknown already.  If not, one could expand the  B-field or E-field, linear in its argument, and solve for the space. It could be discrete solutions, corresponding to a reflexion that appears to be not exactly on a surface?

I would agree that reformulating vector representations into others (such as quaternions, of which vectors can of course be seen as a subset, and even in octonions) probably adds nothing to the physics, directly.

The quaternions offer more material to review the point about alternative formalisms, before returning to octonions.

As André Waser remarks [1] “The Original Quaternion Form of Maxwell‘s Equations: In his Treatise [16] of 1873 MAXWELL has already modified his original equations of 1865. In addition Maxwell tried to introduce the quaternion notation by writing down his results also in a quaternion form.

It is very interesting that Maxwell‘s first formulation of a magnetic charge density and the related discussion about the possible existence of magnetic monopoles became forgotten for more than half a century until in 1931 Paul André Maurice Dirac again speculated about magnetic monopoles.”   … DIRAC,  P. A. M., Quantised Singularities in the Electromagnetic Field“, Proceedings of the London Royal Society A 133 (1931) 60-72.” [Ref edited for errors.]

Further, Ludwik Silberstein [2]: “In 1907 Silberstein described a bivector approach to the fundamental electromagnetic equations. When E and B represent electric and magnetic vector fields with values in R3, then Silberstein suggested E + i B would have values in C3, consolidating the field description with complexification. This contribution has been described as a crucial step in modernizing Maxwell’s equations,[3] while E + i B is known as the Riemann–Silberstein vector.”

It’s been known for a considerable time that the whole set of Maxwell’s Equations can be represented as a single equation in quaternions. A modern derivation in [3] concludes:

“Maxwell’s equation (33) then reduces to DDA¯ + J = 0, (37) which is readily recognized as a wave equation”

More recently, single equation forms have also been developed using geometric algebras (see Wikipedia article, which lists other formalisms as well, e.g. tensor and differential forms) [4] Whether or not Maxwell’s equations can also be put into spinor form [e.g. 6, but see 5] is not entirely clear.

To return to octonions: D R. Beradze and T. Shengelia [6] present “Dirac and Maxwell equations in Split Octonions” [7].

So I still believe that while they may not lead to new physics directly, alternative formulations are still of interest as they may stimulate theoretical advances by suggesting previously unseen relationships – probably not in electromagnetism as we currently understand it, but Waser’s article shows that different formulations can suggest connections to other areas. Similarly, reformulations in geometric algebra may lead to more compact representations suggestive of useful and more widely applicable generalisations, and new methods, which may themselves lead on to new physics.

https://www.zpenergy.com/downloads/Orig_maxwell_equations.pdf

https://en.wikipedia.org/wiki/Ludwik_Silberstein

http://xxx.lanl.gov/pdf/math-ph/0309061v1

https://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_field

https://arxiv.org/pdf/math-ph/0201053.pdf

http://iopscience.iop.org/article/10.1088/1742-6596/788/1/012025/pdf

https://arxiv.org/pdf/1610.06418.pdf

It is better to apply quaternionic differential calculus. Octonions are not associative.

𝙂𝙀𝙉𝙀𝙍𝘼𝙇𝙄𝙕𝙀𝘿 𝙁𝙄𝙀𝙇𝘿 𝙀𝙌𝙐𝘼𝙏𝙄𝙊𝙉𝙎

Generalized field equations hold for all basic fields. Generalized field equations are best treated in a quaternionic setting.

Quaternions are constituted from a real number valued scalar part and a three-dimensional spatial vector that represents the imaginary part.

The multiplication rule of quaternions indicates that the product is constituted by several independent parts.

In this comment, we use a suffix ᵣ to indicate the scalar real part of a quaternion, and we use boldface to indicate the imaginary vector part.

      c = cᵣ + 𝙘 = a b = (aᵣ + 𝙖) (bᵣ + 𝙗) = aᵣ bᵣ − 〈 𝙖, 𝙗 〉 + aᵣ 𝙗 + 𝙖 bᵣ ± 𝙖 × 𝙗

The ± indicates that quaternions exist in right-handed and left-handed versions.

The formula can be used to check the completeness of a set of equations that follow from the application of the product rule.

The quaternionic conjugate of a is a* = (aᵣ − 𝙖)

From the product rule, follows the formula for the norm |a| of quaternion a.

      |a|² = a a* = (aᵣ + 𝙖) (aᵣ − 𝙖) = aᵣ aᵣ + ⟨ 𝙖, 𝙖 ⟩

The quaternionic nabla ∇ acts like a multiplying operator. The (partial) differential ∇ ψ represents the full first-order change of field ψ. 

      ϕ = ∇ ψ = ϕᵣ + 𝟇 = (∇ᵣ + 𝞩 ) (ψᵣ + 𝟁) = ∇ᵣ ψᵣ − ⟨𝞩,𝟁⟩ + ∇ᵣ 𝟁 + 𝞩 ψᵣ ±𝞩 × 𝟁

The terms at the right side show the components that constitute the full first order change.

They represent subfields of field ϕ and often they get special names and symbols.

𝞩 ψᵣ is the gradient of ψᵣ

⟨𝞩,𝟁⟩ is the divergence of 𝟁.

𝞩 × 𝟁 is the curl of 𝟁

The equation is a quaternionic first order partial differential equation.

      ϕᵣ = ∇ᵣ ψᵣ − ⟨𝞩,𝟁⟩ (This is not part of Maxwell equations!)

      𝟇 = ∇ᵣ 𝟁 + 𝞩 ψᵣ ±𝞩 × 𝟁

      𝜠 = −∇ᵣ 𝟁 − 𝞩 ψᵣ

      𝜝 = 𝞩 × 𝟁

From the above formulas follows that the Maxwell equations do not form a complete set.

Physicists use gauge equations to make Maxwell equations more complete.

      χ = ∇* ∇ ψ = (∇ᵣ − 𝞩 )(∇ᵣ + 𝞩 ) (ψᵣ + 𝟁) = (∇ᵣ ∇ᵣ + ⟨𝞩,𝞩⟩) ψ

and

      ζ = (∇ᵣ ∇ᵣ − ⟨𝞩,𝞩⟩) ψ

are quaternionic second order partial differential equations. 

      χ = ∇* ϕ

and

      ϕ = ∇ ψ

split the first second-order partial differential equation into two first order partial differential equations.

The other second order partial differential equation cannot be split into two quaternionic first order partial differential equations. This equation offers waves as parts of its set of solution. For that reason, it is also called a wave equation.

      ∇ᵣ ∇ᵣ ψ = ⟨𝞩,𝞩⟩ ψ = ω ψ ⟹ f = exp(2π𝕚ωxτ)

In odd numbers of participating dimensions both second-order partial differential equations offer shock fronts as part of its set of solutions.

       f(cτ+x𝕚) + g f(cτ−x𝕚) ; one-dimensional fronts

       f(cτ+r𝕚)/r + g f(cτ−r𝕚)/r ; spherical fronts

After integration over a sufficient period the spherical shock front results in the Green’s function of the field under spherical conditions.

      𝔔 = (∇ᵣ ∇ᵣ − ⟨𝞩,𝞩⟩) is equivalent to d'Alembert's operator.

      ⊡ = ∇* ∇ = ∇ ∇* = (∇ᵣ ∇ᵣ + ⟨𝞩,𝞩⟩ describes the variance of the subject

Maxwell equations must be extended by gauge equations to derive the second order partial wave equation.

Maxwell equations use coordinate time, where quaternionic differential equations use proper time. Regarding quaternions, the norm of the quaternion plays the role of coordinate time. These time values are not used in their absolute versions. Thus, only time intervals are used.

Hilbert spaces can only cope with number systems that are division rings. In a division ring, all non-zero members own a unique inverse. Only three suitable division rings exist. These are the real numbers, the complex numbers, and the quaternions. Thus dynamic geometric data that are characterized by a Minkowski signature must first be dismantled into real numbers before they can be applied in a Hilbert space. Quaternions can be applied without dismantling.

Quantum physicists use Hilbert spaces for the modeling of their theory. Quaternionic quantum mechanics appears to represent a natural choice.

The Poisson equation

      Φ = ⟨𝞩,𝞩⟩ ψ = G ∘φ

describes how the field reacts with its Green’s function G on a distribution φ of point-like triggers.

      G(𝙦−𝙘)=1/| 𝙦−𝙘|

      ⟨𝞩 × 𝞩, 𝟁⟩=0

      𝞩 × (𝞩 × 𝟁) = 𝞩⟨𝞩,𝟁 ⟩ − ⟨𝞩,𝞩⟩ 𝟁

       (𝞩𝞩) ψ = (𝞩 × 𝞩) ψ − ⟨𝞩,𝞩⟩ ψ = (𝞩 × 𝞩) 𝟁 − ⟨𝞩,𝞩⟩ ψ = 𝞩⟨𝞩,𝟁 ⟩ − 2 ⟨𝞩,𝞩⟩ ψ + ⟨𝞩,𝞩⟩ ψᵣ

The term (𝞩 × 𝞩) ψ indicates the curvature of field ψ.

The term ⟨𝞩,𝞩⟩ ψ indicates the stress of the field ψ.

(𝞩 × 𝞩) ψ + ⟨𝞩,𝞩⟩ ψ = 𝞩⟨𝞩,𝟁 ⟩ − ⟨𝞩,𝞩⟩ ψᵣ

With respect to a local part of a closed boundary that is oriented perpendicular to vector 𝙣 the partial differentials relate as

      𝞩 ψ = 𝞩 (ψᵣ + 𝟁) = ⟨𝞩,𝟁⟩ + 𝞩 ψᵣ ±𝞩 × 𝟁 ⇔ 𝙣 ψ = 𝙣 (ψᵣ + 𝟁) = ⟨ 𝙣,𝟁⟩ + 𝙣 ψᵣ ± 𝙣 × 𝟁

This is exploited in the generalized Stokes theorem

      ∭ 𝞩 ψ dV = ∯ 𝙣 ψ dS

This turns the differential continuity equation into an integral balance equation.

It also elucidates what the terms in the continuity equation mean.

𝞩 × (𝞩 × 𝟁) = 𝞩⟨𝞩,𝟁 ⟩ − ⟨𝞩,𝞩⟩ 𝟁 ⇔ 𝙣 × (𝙣 × 𝟁) = 𝙣 ⟨ 𝙣,𝟁 ⟩ − ⟨ 𝙣, 𝙣 ⟩ 𝟁

∭ 𝞩 × (𝞩 × 𝟁) dV = ∯ ⟨ 𝙣,𝟁 ⟩ 𝙣 dS − ∯ 𝟁 dS

From fields and charges, you can derive force raising fields.

See:

https://doc.co/U5dVDZ

Thierry,

The field equations are a mix of scalar and vector equations. All formulas of differential geometry are covered. The same holds for a subset of the fluid dynamics equations.

The Maxwell equations differ in the fact that they apply coordinate time, where the quaternionic equations use proper time. I retrieved most equations from Bo Thidé's  book. look at his Formulas section.

http://www.calvin.edu/~pribeiro/courses/engr315/EMFT_Book.pdf

Hans,

I greatly appreciate your link to Bo Thide's EMFT_book,*  for a passage in section 5.2.1 that I find inspirational, referring to a Lagrangian [my bold]: 

"Notice how we made a transition from a discrete description, in which the mass points were identified by a discrete integer variable i = 1, 2, . . . , N, to a continuous description, where the infinitesimal mass points were instead identified by a continuous real parameter x, namely their position along xˆ.

A consequence of this transition is that the number of degrees of freedom for the system went from the finite number N to infinity! Another consequence is that L has now become dependent also on the partial derivative with respect to x of the ‘field coordinate’ η. But, as we shall see, the transition is well worth the price because it allows us to treat all fields, be it classical scalar or vectorial fields, or wave functions, spinors and other fields that appear in quantum physics, on an equal footing.

We may be straying a bit off-topic, but I think this is a worthwhile digression.

Paul

*  from, coincidentally, an author at Uppsala university where I worked for a couple of years

Paul,

I used the formulas in the EMFT book to generate a complete quaternionic differential calculus that covers the first and second order partial differential calculus. This is straightforward when the fact is used that the quaternionic nabla behaves as a quaternionic multiplying operator. Astonishing, this leads to two different homogeneous second order partial differential equations. One of them is similar to the well-known wave equation, which does not follow directly from Maxwell's equations. Maxwell's equations must be extended with gauge equations to reach that result.

https://doc.co/sRXxXN

http://vixra.org/pdf/1506.0111v6.pdf

Maxwell’s equations which are the cornerstones of classical electromagnetism have been formulated in many forms since their discovery in 1873. Although, in his famous book “Treatise on Electricity and Magnetism”, Maxwell used 3-dimensional vector representation to formulate electromagnetism, he also gave their quaternionic forms in a number of places.

The complex quaternions (biquaternions) were used frequently to reformulate the classical electrodynamics and so Maxwell’s equations were reduced to a simple and compact form. Maxwell’s equations have been rewritten in terms of the dual quaternions and by Demir and Ozdas.

By using the same idea on the construction of the complex quaternions, Demir et al., combined two hyperbolic quaternion to express up to 8-dimensional physical quantities. Maxwell’s equations and relevant field equations are investigated with the hyperbolic quaternions by them.