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Quaternions, Octonions,and Physics 

Some Quaternionic Structure

Some QuaternionicHistory

What about Octonions?


Some Quaternionic History

In 1828, GeorgeGreen (1793-1841) discovered Green'sFunctions, which describe the basic Harmonic Mathematics ofPhysics. Green was a self-taught mathematician who worked in hisfamily mill. Green's functions can be used with respect to anyDivision Algebra. At the time Greendiscovered them, the only Division Algebras known were the Real andComplex Numbers. Green used his Green's Functions to describe thePotential Theory of Electromagnetism.

In 1831, JamesClerkMaxwell(1831-1879) was born.

In 1843, the Quaternion Division Algebra was discovered byWilliamRowan Hamilton (1805-1865).

In 1844, N-dimensional Exterior and Interior GeometricProducts were discovered by HermannGrassmann (1809-1877).

In 1845, the Octonion DivisionAlgebra was discovered by John Graves and ArthurCayley (1821-1895). Cayley,who practiced law to make money sothat he could pursue mathematics, also invented MatrixAlgebra.

Also in 1845, WilliamKingdon Clifford (1845-1879) was born.

In 1858, according to Collective Electrodynamics by CarverMead (MIT 2000), "... Bernhard Riemann deduces the phenomena of theinduction of electric currents from a modified form of Poisson'sequation

d^2 V / dx^2 + d^2 V / dy^2 + D^2 V / dz^2 + 4 pi rho = ( 1 /a^2 ) d^2 V / dt^2

where V is the electrostatic potential, and a a velocity. ... Thefour-vector generalization of Riemann's equation was formualted bySommerfeld shortly after Einstein's1905 paper introduced the special theory of relativity. ...".

In 1864, Maxwell discovered the equations ofElectromagnetism, which he wrote in the form of Complex Numbers andVectors in his paper "A dynamical theory of the electromagneticfield".In the abstract to that 1864 paper (dated 27 October 1864, andreprinted in The Scientific Letters and Papers of James ClerkMaxwell, Volume II 1862-1873, edited by P. M. Harman (Cambridge Un.Press 1995) at pages 189-196), Maxwell says:

"The proposed Theory seeks for the origin of electromagnetic effects in the medium surrounding the electric or magnetic bodies ... The properties attributed to the medium in order to explain the propagation of light are -

1st. That the motion of one part communicates motion to the parts in its neighborhood.

2nd. That this communication is ot instantaneous but progressive, and depends on the elasticity of the medium as compared with its density.

The kind of motion attributed to the medium when transmitting light is that called transverse vibration. An elastic medium capable of such motions must be also capable of a vast variety of other motions, and its elasticity may be called into play in other ways, some of which may be discoverable by their effects. ...

... if we look for the explanation of the force of gravitation in the action of a surrounding medium, the constitution of the medium must be such that, when far from the presence of gross matter, it has immense intrinsic energy, part of which is removed from it wherever we find the signs of gravitating force. ...

... The equations of this paper ... show that transverse disturbances, and transverse disturbances only, will be propagated through the field, and that the number which expresses the velocity of propagation must be the same as that which expresses the number of electrostatic units units of electricity in one electromagnetic unit, the standards of space and time being the same. ... ...the ... theory ... restores the medium [luminiferous ether] ... and certifies that the vibrations are transverse, and that the velocity is that of light. With regard to normal [longitudinal] vibrations, the electromagnetic theory does not allow of their transmission. ...".

 

In 1867, Peter Guthrie Tait wrote his book on Quaternions,and Maxwell and Tait discussed Quaternions in theircorrespondence.

On 7 Nov 1870, Maxwell wrote to Tait a letter discussingQuaternion terminology for things like

saying: "... I want phrases of this kind to make statements inelectromagnetism ...".

Also in November 1870, Maxwell wrote a Manuscript on theApplication of Quaternions to Electromagnetism, which is reprinted inVolume II of Maxwell's Scientific Papers at pages 570-576. In itMaxwell uses the term curl instead of twirl, and he also says:

"... The invention of the Calculus of Quaternions by Hamilton is astep towards the knowledge of quantities related to space which canonly be compared for its importance with the invention of triplecoordinates by Descartes. The limited use which has up to thepresent time been made of Quaternions must be attributed partly tothe repugnance of most mature minds to new methods involving theexpenditure of thought ...".

At this time, Maxwell had a clear idea thatwaves should have Scalar and Vector parts, and used the followingterms in his Quaternionic formulation of the equations ofElectromagnetism:

                                                          __  Slope           = what we call Grad (represented by Nabla \/ )Convergence     = what we call DivCurl            = what we call CurlConcentration   = what we call Laplacian

 

Since Maxwell then had both the concept of waves in an elastic medium and the concepts of Grad, Div, Curl, and Laplacian, he had everything you need to write the equations for Longitudinal Waves in an elastic medium as described, for example (as Jack Sarfatti pointed out) on pages 142-144 of Methods of Theoretical Physics by Morse and Feshbach (McGraw-Hill 1953).

Since (as shown in Morse and Feshbach) the Longitudinal Waves are faster than the Transverse Waves, and the Transverse Waves travel at the Speed of Light, the Longitudinal Waves are Superluminal if the Aether is a general elastic medium (Tohu VaVohu).

The question of the existence or non-existence of Longitudinal/Scalar Waves is then the question of whether or not the Aether, regarded as an elastic medium, is compressible:

  • If not, there are no Longitudinal/Scalar Waves.
  • If so, then there are Superluminal Longitudinal/Scalar Waves.

My personal opinion is that the Aether is compressible, but only at energies around the Vacuum Expectation Value of the Higgs field, around 250 GeV, which is corrresponds to the Superposition Separation of an entire single Tubulin in the Brain.

The Dilation of the 15-dim Conformal Group sets the scale of the Higgs VeV at 250 GeV so that general deformations of SpaceTime can take place only above that energy level, while GraviPhoton Special Conformal (Hopf flow) transformations are useful in Conformal deformations of SpaceTime.

Incompressibility of the Aether below 250 GeV is only with respect to the 6-dim vector space of the Conformal Group Spin(2,4), so that below 250 GeV you can see Conformal phenomena that appear to show compressibility from the point of view of 3-dim space or 4-dim Minkowski spacetime. Such conformal phenomena include the Fock superluminal solutions of Maxwell's equations that are described by R. M. Kiehn.

The 4 GraviPhoton Special Conformal transformations of the 15-dim Conformal group are like the Moebius linear fractional transformations, that do deform Minkowski spacetime but take hyperboloids into hyperboloids and are the symmetries of superluminal solutions of the Maxwell equations. They are incompressible/linear from the point of view of a 6-dimensional SpaceTime, with 4 spatial dimensions and 2 time dimensions, because the conformal group over Minkowski spacetime is just SU(2,2) = Spin(2,4), the covering group of SO(2,4), and therefore the Lie algebra generators look like those of rotations in a 6-dim vector space of signature (2,4). This is the 4-dim space with 2-dim time suggested by Robert Neil Boyd, in which things look linear (even though from our conventional 3-dim spatial or 4-dim Minkowski point of view they might appear, due to our limited conventional perspective, to be nonlinear). If you regard Physical SpaceTime as the 6-dimensional vector space of Spin(2,4), and Internal Symmetry Space as 4-dimensional CP2, then the total space is 6+4=10-dimensional. With respect to tthe D4-D5-E6-E7 model, that 10-dim space corresponds:

to the 10-dim vector space of the D5 Lie Algebra Spin(2,8); and

to the 10-dim element of the decomposition of the 27-dim representation of the E6 Lie Algebra into 10 + 16 + 1 under its D5 subalgebra (see, for example, Lie Algebras in Particle Physics, 2nd edition, by Howard Georgi, Perseus Books (1999), page 308).

Since the unit Quaternions form the Lie Group Sp(1) = SU(2) = Spin(3) = S3, Maxwell's use of Quaternions in Electromagnetism anticipated the SU(2) Weak Force and the SU(2)xU(1) ElectroWeak unification, and Maxwell's consideration of a compressible general elastic Aether medium anticipated the Higgs mechanism and Torsion Physics.

Also in 1870, Clifford showed thatenergy and matter are simply different types of curvature of space,thus anticipating Einstein's theory of Gravitation as CurvedSpacetime. According to the EncyclopaediaBritannica, Clifford also coined the phrase "... "mind-stuff"(the simple elements of which consciousnessis composed) ...".

In terms of the smallest charged Elementary Particle, theFirst-Generation Fermion Electron ComptonRadius Vortex Particle, the HiggsVEV (about 250 GeV = 5 x 10^5 Me (Electron Masses)) givesthe linear compressibility of the Aether,Therefore, the Gravitational VEV should begiven by the 4-volume compressibility of the Aether, so that theGravitational VEV is about ( 5 x 10^5 )^4 Me = 6 x 10^22 Me = 3 x10^22 MeV = 3 x 10^19 GeV. Since theGravitational VEV should correspond to a pair of Planck Mass BlackHoles, the Planck Masscould be derived to be about 1.5 x 10^19 GeV.

In 1871, Maxwell wrote a letter of reference for Clifford,saying:

"... The peculiarities of Mr. Clifford's researches ... is that they tend not to the elaboration of abstruse theorems by ingeneous calculations, but to the elucidation of scientific ideas by the concentration upon them of clear and steady thought. ...".

In 1873, Maxwell published his treatise on Electricity andMagnetism. In the same year, in his Lecture on Faraday's Lines ofForce (reprinted in Volume II of Maxwell's Scientific Papers),Maxwell said:

"... If we propose to account for [the attraction of gravitation] in the same way as we have done for magnetism we must admit that there is pressure instead of tension along the lines of force and tension instead of pressure at right angles to them and that here where we sit the ether is supporting a vertical pressure of more than 37000 tons on the square inch. The strength of steel is nothing to this. ...".

Also in 1873, SophusLie (1842-1899) (who had studied with FelixKlein (1849-1925)) began his study of transformation groups whichover the succeeding years produced LieAlgebras, which were independently introduced and classified byWilhelmKilling (1847-1923) and ElieCartan (1869-1951). The classification of Lie Algebrascorresponds to the classification of the AlternativeReal Division Algebras: Real numbers, Complex numbers,Quaternions, and Octonions.

In 1876, Clifford, who had beendescribing the topology of Riemann(1826-1866) surfaces, discovered CliffordAlgebras., thus anticipating Dirac's rediscovery of CliffordAlgebras when he formulated the Dirac Equation of QuantumElectroDynamics.

Dirac not only rediscoveredClifford algebras, he also:

(in 1938) anticipated the Compton Radius Vortex Model of the electron, as shown in this quote from pages 194-195 of Dirac: A Scientific Biography, by Helge Kragh (Cambridge 1990): "... "... It would appear here that we have a contradiction with elementary ideas of causality. The electron seems to know about he pulse before it arrives and to get up an acceleration (as the equations of motion allow it to do), just sufficient to balance the effect of the pulse when it does arrive." Dirac seemed to accept this pre-acceleration as a matter of fact, necessitated by the equations, and did not discuss it further. However, Dirac explained that the strange behavior of electrons in this theory could be understood if the electron was thought of as an extended particle with a nonlocal interior. He suggested that the point electron, embedded in its own radiation field, be interpreted as a sphere of radius a, where a is the distance within which an incoming pulse must arrive before the electron accelerates appreciably. With this interpretation he showed that it was possible for a signal to be propagated faster than light through the interior of the electron. He wrote: "The finite size of the electron now reappears in a new sense, the interior of the electron being a region of failure, not of the field equations of electromagnetic theory, but of some of the elementary properties of space-time." In spite of the appearance of superluminal velocities, Dirac's theory was Lorentz-invariant. ..." ; and

(in 1951-1954) advocated the reality and utility of the aether, as shown in this quote from pages 202-203 of Dirac: A Scientific Biography, by Helge Kragh (Cambridge 1990): "... "Let us imagine the aether to be in a state for which all values of the velocity of any bit of aether, less than the velocity of light, are equally probable. ... In this way the existence of an aether can be brought into complete harmony with the principle of relativity." Dirac identified the ether velocity with the stream velocity of his classical electron theory ... it was the velocity with which small charges would flow if they were introduced. ... in the spring of 1953, Dirac proposed that absolute time be reconsidered. ... The ether, absolute simultaneity, and absolute time "... can be incorporated into a Lorentz invariant theory with the help of quantum mechanics ..." ... he was unable to work out a satisfactory quantum theory with absolute time and had to rest content with the conclusion that "one can try to build up a more elaborate theory with absolute time involving electron spins ...". Recall that Nelson's non-local stochastic quantum mechanics (which I think can be formulated consistently with Bohm theory) involves (see the paper by Smolin in the book Quantum Concepts in Space and Time (Penrose and Isham, eds), at page 156) a diffusion constant that "... is inversely proportional to the inertial mass of the particle, with the constant of proportionality being a universal constant hbar: v = hbar / m ...". Compare this with Dirac's 1951 suggestion that the electromagnetism U(1) gauge-fixing condition should be A A = k^2 where (see page 199 in Kragh's book I am omitting some sub and superscript mus and nus): "... In order to get agreement with the Lorentz equation, the constant k was indentified with m/e The four-velocity v of a stream of electrons ws found to be related to A by v = (1/k) A ..." which gives for Dirac's theory v = e / m.

 

In 1879, both Maxwell and Clifford died, and Einstein was born.

 


 What Might Have Been in the late1800s:

Since Maxwell's equations for U(1)electromagnetism have ConformalSpin(4,2) = SU(2,2) symmetry, and since Gravitation can bedescribed (through the MacDowell-Mansouri mechanism) by gauging theConformal Group = Spin(4,2) = SU(2,2),Clifford's Gravitation could have been unified with Maxwell'sElectromagnetism, giving a theory that included GraviPhotons;

and

Clifford Algebras could have been combined with Maxwell'sQuaternionic Electromagnetism to produce the Dirac Equation;

and

Maxwell's Quaternionic Electromagnetism and Compressible ElasticAether could have been extended to describe not only U(1)Electromagnetism but also photons with mass and longitudinalpolarization, thus anticipating the massive neutralweak boson with gauge group SU(2), the ElectroWeak U(1)xSU(2) = U(2)model, and the Higgs Mechanism, and TorsionPhysics.

 

Further, you could condsider

Maxwell's geometric imagination (he visualized Vortices in theAether, and made stereoscopes to view images in 3-dimensions andzoetropes to view images in motion), as well as

his mathematical/mystical worldview (he said: "... [theaether] is fitted ... toconstitute the material organism of beings excercising functions oflife and mind as high or higher thanours ..."; and "... cubic surfaces! By threes and nines Drawround his camp your seven-and-twentylines The seal of Solomon inthree dimensions. ... we the form may trace Of himwhose soul, too large for vulgar space, In n dimensions flourishedunrestricted. ..."), and

you could consider that Maxwell's 27-dimensional structure issimiilar to the 27-dimensionalgeometry based on the exceptional Jordan algebra of 3x3 HermitianOctonion matrices and

that the 27-complex-dimensionalsymmetric space E7 / (E6xU(1)) describes, at theNearest Neighbor lowest level of Interconnectedness, theSuper Implicate Order, orMacroSpace, whoseGeometry produces JackSarfatti's nonlinear Back-Reaction Quantum Theory of Matter andMind.

With that in mind, you might conjecture that:

Since the symmetry of the 27-lineconfiguration is the same as that of the Weylgroup of the Octonionic E6 Lie algebra; and

Since Octonions (the next, and final, AlternativeReal Division Algebra beyond the Quaternions) had already beendiscovered; and

Since 8-dimensional Octonions can be a representation space forthe 8-dimensional Lie Group SU(3):

the late 1800s might have seen an Extensionfrom Quaternions to Octonions,

resulting in an extension of Gravity and SU(2)xU(1)

to a Unified Theory of

Gravity plus the SU(3)xSU(2)xU(1) Standard Model

such as the D4-D5-E6-E7Model.

 


However, what might have been was notwhat was:

 

What Did Happen after 1879:

 

In 1887, according to Collective Electrodynamics by CarverMead (MIT 2000), "... W. Voigt published a little-known paper inwhich he showed that ... Maxwell's equations, in space free ofcharges and currents, are not altered by a ... Lorentztransformation. ... This transformation was reinvented in 1892 by H.A. Lorentz ...[who]... derived his result independently...".

 

Maxwell's Quaternions were thrown away fromElectromagnetism by Josiah Willard Gibbs at Yale and OliverHeaviside in England. As Saul-Paul Sirag quotes from thebiography, Sir William Rowan Hamilton, by Thomas L. Hankins, JohnsHopkins Press, 1980, pp. 316 - 319:

"... in 1888, Gibbs explained how reading Maxwell's [1873] Treatise on Electricity and Magnetism led him to devise his system of vector analysis: "My first acquaintance with quaternions was in reading Maxwell's E.&M. where Quaternion notations are considerably used. ... I saw, that although the methods were called quaternionic the idea of the quaternion was quite foreign to the subject. ...

I therefore began to work out ab initio, the algebra of the two kinds of multiplication, the three differential operations [del] applied to a scalar, & the two operations to a vector, & those fuctions or rather integrating operators which (under certain limitations) are the inverse of the said differential operators, & which play the leading roles in many departments of Math. Phys. To these subjects was added that of lin. vec. functions which is also prominent in Maxwell's E. & M."

 

In 1903, according to Collective Electrodynamics by CarverMead (MIT 2000), "... Sommerfeldintroduces the Lorentz-invariant quantity S = J . A which he callsthe Schwarzchild invariant .... K . Schwarzchild, Gottinger Nachr.1903 ... Note the publication 1903! Thus Schwarzchild arrivedintuitively at the correct postulate of the theory of invariants sixyears ahead of Minkowski. ... two years before Einstein's paper onspecial relativity ... The Schwarzchild invariant ... When integratedover all four coordinates of space-time ... has the units of energy xtime; it is called the action of the system. When divided by hbar,this integral is dimensionless. ... From our point of view, theaction is flux x charge, rather than energy x time. ...".

 

Lie Groups remained only a narrowfield of Mathematics until around 1920.

 

Clifford's Curved Space Physics and CliffordAlgebras were ignored until Dirac discovered the DiracEquation in 1932.

Quaternions did not returnto fundamental Physics models until 1962, when Finkelstein,Jauch, Schiminovich, and Speiser wrote a paper titled SomePhysical Consequences of General Q-Covariance, Helvetica PhysicaActa, Volume XXXV (1962) 328-329, in which they showed that thequaternion imaginary degrees of freedom corresponded to the Higgsfield that gives mass to the SU(2) gauge bosons. They extended theresult of Stueckelberg that a vector gauge boson could, byinteracting with a scalar field transforming additively under thegauge group, become massive. Their extension was to use aquaternionic gauge structure that naturally produced a scalar fieldwith nonzero vacuum expectation value that transformedmultiplicatively under the gauge group (effectively being theexponential of Stueckelberg's scalar field).

Theirs was the first paper (as far as I know) that usedQuaternionic SU(2) symmetry to describe the mechanism whereby twocharged SU(2) bosons get mass, and the electromagnetic field isunified with the SU(2) bosons. Their paper effectively did the "HiggsMechanism" before Higgs, and did ElectroWeak Unification beforeGlashow,Salam, and Weinberg (who, for their ElectroWeak work, sharedthe 1979 Nobel Prize).

The eta(x) that they call a "fundamental field" corresponds to theHiggs field. Their paper describes three vector fields: two with massand charge (the W+ and W-); and one massless and neutral (thephoton). They do not in their paper construct the neutral massiveZ0.

They gave more details of their model in theirpaper titled Principle of General Q Covariance, J. Math. Phys.4 (1963) 788-796, (received 10 December 1962).

Finkelstein, Jauch, Schiminovich, and Speiser did not proverenormalizability, but neither did Glashow, Salam, and Weinberg.Renormalizability was proven by 't Hooft, who did not win a NobelPrize until 1999, when he shared it with his former adviserVeltman.

Later in the 20th Century, TorsionPhysics was studied by R. M.Kiehn.

 


What if you extend from Quaternions to Octonions?

Note: JohnBaez has a very nice paper,math.RA/0105155, about Octonions,including some interesting history. He also has easilyupdatable and expandable html, ps, and pdf versions on his personalweb site.

Since the circle U(1) = S1, and the 1-sphere S1 is the unitComplex Numbers, the Complex Numbers have a natural local U(1) gaugegroup, which is the gauge group of Electromagnetism. Therefore, theComplex Numbers naturally produce QED.


Since the (6 - 3 = 3)-dimensional symmetric space Spin(4) / SU(2)= S3, and the 3-sphere S3 is the unit Quaternions, the Quaternionshave a natural local SU(2) gauge group, which is the Weak Force gaugegroup. Since the Complex Numbers are a subalgebra of the Quaternions,the Quaternions naturally unify Electromagnetism and the WeakForce, producing the ElectroWeak SU(2)xU(1) sector of the StandardModel.

Further, since the Spin(4) Lie algebra is the Euclidean version ofthe Lie algebra of the Lorentz Group, the Quaternions naturallyunify the ElectroWeak SU(2)xU(1) sector of the Standard Model withSpecial Relativity in 4-dimensional SpaceTime.


Since the (15 - 8 = 7)-dimensional symmetric space Spin(6) / SU(3)= S7, and the 7-sphere S7 is the unit Octonions, the Octonions have anatural local SU(3) gauge group, which is the Color Force gaugegroup. Since the Quaternions are a subalgebra of the Octonions,the Octonions naturally unify Electromagnetism, the Weak Force, andthe Color Force, producing the SU(3)xSU(2)xU(1) StandardModel.

Further, since the Spin(6) Lie algebra is the Euclidean version ofthe Lie algebra of the Conformal Group, and since the Conformal Groupcan produce Gravity by the MacDowell-MansouriMechanism,

the Octonions naturally unify the Standard Model withGeneral Relativistic Gravity in 4-dimensionalSpaceTime.

 

Just as the Quaternions have the 3-sphere S3 that is SU(2) and canmake an SU(2) gauge group, the Octonions have the 7-sphere S7. S7 isnot a Lie group, but

S7 is parallelizable and canhave Torsion Structure.

If you extend S7 in a natural way

then you get the Lie group Spin(8)of the D4 Lie algebra,

which is the starting point of the D4-D5-E6-E7-E8VoDou Physics model.

 

 

 


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