### Quaternions, octonions, and TGD

Quaternions and octonions have been lurking around for decades in hope of getting deeper role in physics but as John Baez put it: "I would not have the courage to give octonions as a research topic for a graduate student". Quaternions are algebraically a 4-D structure and this strongly suggests that space-time could be analogous to complex plane.

Classical continuous number fields reals, complex numbers, quaternions, octonions have dimensions 1, 2, 4, 8 coming in powers of 2. In TGD imbedding space is 8-D structure and brings in mind octonions. Space-time surfaces are 4-D and bring in mind quaternions. String world sheets and partonic 2-surfaces are 2-D and bring in mind complex numbers. The boundaries of string world sheets are 1-D and carry fermions and of course bring in mind real numbers. These dimensions are indeed in key role in TGD and form one part of the number theoretic vision about TGD involving p-adic numbers, classical number fields, and infinite primes.

**What quaternionicity could mean?**

Quaternions are non-commutative: AB is not equal to BA. Octonions are even non-associative: A(BC) is not equal to (AB)C. This is problematic and in TGD problems is turned to a victory if space-time surfaces as 4-surface in 8-D M^{4}× CP_{2} are associative (or co-associative in which case normal space orthogonal to the tangent space is associative). This would be extremely attractive purely number theoretic formulation of classical dynamics.

What one means with quaternionicity of space-time is of course highly non-trivial questions. It seems however that this must be a local notion. The tangent space of space-time should have quaternionic structure in some sense.

- It is known that 4-D manifolds allow so called almost quaternionic structure: to any point of space-time one can assign three quaternionic imaginary units. Since one is speaking about geometry, imaginary quaternionic units must be represented geometrically as antisymmetric tensors and obey quaternionic multiplication table. This gives a close connection with twistors: any orientable space-time indeed allows extension to twistor space which is a structure having as "fiber space" unit sphere representing the 3 quaternionic units.

- A stronger notion is quaternionic Kähler manifold, which is also Kähler manifold - one of the quaternionic imaginary unit serves as global imaginary unit and is covariantly constant.CP
_{2}is example of this kind of manifold. The twistor spaces associated with quaternion-Kähler manifolds are known as Fano spaces and have very nice properties making them strong candidates for the Euclidian regions of space-time surfaces obtained as deformations of so called CP_{2}type vacuum extremals represenging lines of generalized Feynman diagrams.

- The obvious question is whether complex analysis including notions like analytic function, Riemann surface, residue integration crucial in twistor approach to scattering amplitudes, etc... generalises to quaternions. In particular, can one generalize the notion of analytic function as a power series in z to that for quaternions q. I have made attempts but was not happy about the outcome and had given up the idea that this could allow to define associative/ co-associative space-time surface in very practically manner. It was quite a surprise to find just month or so ago that quaternions allow differential calculus and that the notion of analytic function generalises elegantly but in a slightly more general manner than I had proposed. Also the conformal invariance of string models generalises to what one might call quaternion conformal invariance. What is amusing is that the notion of quaternion analyticity had been discovered for aeons ago (see this) and I had managed to not stumble with it earlier! See this.

**Octonionicity and quaternionicity in TGD**

In TGD framework one can consider further notions of quaternionicity and octonionicity relying on sub-manifold geometry and induction procedure. Since the signature of the imbedding space is Minkowskian, one must replace quaternions and octonions with their complexification called often split quaternions and split octonions. For instance, Minkowski space corresponds to 4-D subspace of complexified quaternions but not to an algebra. Its tangent space generates by multiplication complexified quaternions.

The tangent space of 8-D imbedding space allows octonionic structure and one can induced (one of the keywords of TGD) this structure to space-time surface. If the induced structure is quaternionic and thus associative (A(BC)= (AB)C), space-time surface has quaternionic structure. One can consider also the option of co-associativity: now the normal space of space-time surface in M^{4}× CP_{2} would be associative. Minkowskian regions of space-time surface would be associative and Euclidian regions representing elementary particles as lines of generalized Feynman diagrams would be co-associative.

Quaternionicity of space-time surface could provide purely number theoretic formulation of dynamics and the conjecture is that it gives preferred extremals of Kähler action. The reduction of classical dynamics to associativity would of course mean the deepest possible formulation of laws of classical physics that one can imagine. This notion of quaternionicity should be consistent with the quaternion-Kähler property for Euclidian space-time regions which represent lines of generalized Feynman graphs - that is elementary particles.

Also the quaternion analyticity could make sense in TGD framework in the framework provided by the 12-D twistor space of imbedding space, which is Cartesian product of twistor spaces of M^{4} and CP_{2} which are the only twistor spaces with Kähler structure and for which the generalization of complex analysis is natural. Hence it seems that space-time in TGD sense might represent an intersection of various views about quaternionicity.

**What about commutativity?: number theory in fermionic sector**

Quaternions are not commutative (AB is not equal to AB in general) and one can ask could one define commutative and co-commutative surfaces of quaternionic space-time surface and their variants with Minkowski signature. This is possible.

There is also a physical motivation. The generalization of twistors to 8-D twistors starts from generalization in the tangent space M^{8} of CP_{2}. Ordinary twistors are defined in terms of sigma matrices identifiable as complexified quaternionic imaginary units. One should replaced the sigma matrices with 7 sigma matrices and the obvious guess is that they represent octonions. Massless irac operator and Dirac spinors should be replaced by their octonionic variant. A further condition is that this spinor structure is equivalent with the ordinary one. This requires that it is quaternionic so that one must restrict spinors to space-time surfaces.

This is however not enough - the associativity for spinor spinor dynamics forces them to 2-D string world sheets. The reason is that spinor connection consisting of sigma matrices replaced with octonion units brings in additional potential source of non-associativity. If induced gauge fields vanish, one has associativity but not quite: induce spinor connection is still non-associative. The stronger condition that induced spinor connection vanishes requires that the CP_{2} projection of string world sheet is not only 1-D but geodesic circle. String world sheets would be possible only in Minkowskian regions of space-time surface and their orbit would contain naturally a light-like geodesic of imbedding space representing point-like particle.

Spinor modes would thus reside at 2-surfaces 2-D surfaces - string world sheets carrying spinors. String world sheets would in turn emerge as maximal commutative space-time regions: at which induced electroweak gauge fields producing problems with associativity vanish. The gamma matrices at string world sheets would be induced gamma matrices and super-conformal symmetry would require that string world sheets are determined by an action which is string world sheet area just as in string models. It would naturally be proportional to the inverse of Newton's constant (string tension) and the ratio hbar G/R^{2} of Planck length and CP_{2} radius squared would be fixed by quantum criticality fixing the values of all coupling strengths appearing in the action principle to be of order 10^{-7}. String world sheets would be fundamental rather than only emerging.

I have already earlier ended up to a weaker conjecture that spinors are localized to string world sheets from the condition that electromagnetic charge is well-defined quantum number for the induced spinor fields: this requires that induced W gauge fields and perhaps even potentials vanish and in the generic case string world sheets would be 2-D. Now one ends up with a stronger condition of commutativity implying that spinors at string world sheets behave like free particles. They do not act with induce gauge fields at string world sheets but just this avoidance behavior induces this interaction implicitly! Your behavior correlates with the behavior of the person whom you try to avoid! One must add that the TGD view about generalized Feynman graphs indeed allows to have non-trivial scattering matrix based on exchange of gauge bosons although the classical interaction vanishes.

**Number theoretic dimensional hierarchy of dynamics**

Number theoretical vision would imply a dimensional hierarchy of dynamics involving the dimensions of classical number fields. The classical dynamics for both space-times surface and spinors would simplify enormously but would be still consistent with standard model thanks to the topological view about interaction vertices as partonic 2-surfaces representing the ends of light-like 3-surface representing parton orbits and reducing the dynamics at fermion level to braid theory. Partonic 2-surfaces could be co-commutative in the sense that their normal space inside space-time surface is commutative at each point of the partonic 2-surface. The intersections of string world sheets and partonic 2-surfaces would consist of discrete points representing fermions. The light-like lines representing intersections of string world sheets with the light-like orbits of partonic 2-surfaces would correspond to orbits of point-like fermions (tangent vector of the light-like line would correspond to hypercomplex number with vanishing norm). The space-like boundary of string world sheet would correspond to real line. Therefore dimensional hierarchy would be realized.

The dimensional hierarchy would relate closely to both the generalization conformal invariance distinguish TGD from superstring models and to twistorialization. All "must be true" conjectures (physics geometry, physics as generalized number theory, M^{8}-H duality, TGD as almost topological QFT, generalization of twistor approach to 8-D situation and induction of twistor structure, etc...) of TGD seems to converge to single coherent conceptual framework.

## 17 Comments:

Beautiful, must let this soak in

Fringe/Alt-Physics have long proclaimed that the Heaviside tensor normalization of Maxwell's original 20 quaternion-based equations has hidden "new physics". A quick Google search turned up:

http://en.wikipedia.org/wiki/Quaternion

http://www.rexresearch.com/maxwell.htm

http://arxiv.org/abs/math-ph/0307038

http://visualphysics.org/de/node/144

http://www.enterprisemission.com/hyper2.html

http://www.cheniere.org/books/aids/ch4.htm

Of course, the Bearden types could still be wrong and yours more fundamentally correct.

Thank you for links. Quaternions have been lurking around already since Maxwell. The problem with quaternion formulations is that breaking of Lorentz invariance takes place. The selection of quaternion real unit selects preferred time direction.

One should be able to interpret this breaking as only apparent. The preferred time direction could for instance correspond to the time direction in rest frame of the subsystem. In zero energy ontology (ZEO) it corresponds to the time-like line connecting the tips of the causal diamond (CD).

Using the language of mathematicians, the CDs with different time direction correspond to moduli characterising different quaternionic structures and changing in Lorentz transformations. This kind of moduli characterise also different complex structures: for torus topology these structure are labelled by points of torus.

For torus topology, see:

4] viXra:1103.0002

3 Dimensional String Based Alternative Particles Model

Matti, could Lorentz transforms show themselves as a peak in some event time data having a Cauchy (Lorentz ) distribution? http://stats.stackexchange.com/questions/139790/does-this-look-like-a-cauchy-distribution

To Anonymous:

I guess that you refer to a distribution/ wave function for causal diamonds (CDs) defining

the perceptive field of conscious entities selves in ZEO - that is Lorentz transforms defining moduli space for quaternion structures). I can only try to formulate what this distribution/wave function means in the framework provided by zero energy ontology (ZEO).

*Zero energy states are characterised by wave function in the moduli space for CDs (I call it M for simplicity). State function reductions form sequences. During them second boundary of CD remains located at light-cone boundary common to all CDs. That part of any zero energy state in superposition is unaffected just like the quantum state in repeated quantum measurement is not affected after the first measurement (Zeno effect).

*The wave function for the position of the opposite boundary of CD changes and (lower

level wave functions at the opposite boundary). In other words, the wave function in M changes. This sequence gives rise to self/mental image/.. in TGD inspired consciousness theory. Also the average temporal distance between the tips increases during this period and gives rise to experienced flow of time. When the first reduction at the opposite boundary of CD occurs, situation changes and it becomes fixed. Self "reincarnates".

In the first reduction to second boundary the moduli are partially "measured" in the sense that second boundary of CDs is localized to fixed light-cone boundary. The opposite boundary of CD represents degrees of freedom analogous to momenta in the sense that it cannot be localized. The analogy with position-momentum duality can be made much more concrete and is probably much more than only an analogy. This is like measuring position: momentum becomes maximally uncertain. Uncertainty Principle prevents the measurement of the moduli distribution.

This is all I can say. Maybe we can return to this question after century or two;-).

Matti, that data came from the stock market... so, in an indirect sense it does have to do with intention as you say :)

Interesting post here, http://math.stackexchange.com/questions/821881/riemann-zeta-function-quaternions-and-physics

-crow

Thank you for the link. I must admit that I failed to understand the point.

In any case, quaternion holomorphy has been discovered long time ago as I discovered recently. The trick is to define left- and right analytic series consisting of terms a_nq^n reap. q^na_n. This allows to circumvent the problems due to non-commutativity. The definition of quaternion analyticity is not unique. One form gives analyticity in 2 complex variables.

Second form gives what one expects from quaternion analyticity: in the first case one has CR involving on t and radial coordinate r and corresponding unit vector as imaginary unit. Same trick works for octonions too and one avoids complications due to non-associativity.

The continuation to Minkowski signature indeed works since z^n is of same form as z and belongs to the M^4 subspace of complexified quaternions as is easy to verify. Same for octonions.M

Even though its stock market data(time between trades for the s&p500), intentionality and physical effects are there.. aa well as the interesting observation that the empirical distribution has inflection points at about 200ms, and 1 second, corresponding to cognition timeshttp://www.newscientist.com/article/dn27107-confident-your-voice-gives-you-away-in-milliseconds.html the LHC protons speed around the ring at approximately 11khz. . Human audible range is 20hz to 20khz... ?! I don't know if this is pure coincidence or not

http://www.newscientist.com/article/dn27107-confident-your-voice-gives-you-away-in-milliseconds.html

Stephen,

could you elaborate this stock market claim. I am not sure whether I understood. 10 Hz is fundamental biorhythm and in TGD corresponds to the secondary p-adic time scale for electron. The frequency spectra for EEG, sound, etc… are not co-incidences in TGD Universe.

Cyclotron frequencies in the magnetic field of Earth (or in its dark counterpart) are in EEG range and hearing as also other forms of sensory perception relies strongly on magnetic flux tubes and associated cyclotron frequencies. Cyclotron energies for these photons are extremely small unless one has large Planck constant.

The wavelength of 10 Hz dark photon is about size of Earth. One could imagine that these photons could relate very closely to collective levels of consciousness. Maybe they could even give a background rhythm for all these idiocies that stock market people are doing to destroy our civilisation!

I have developed this idea in detail using the h_eff=n*h= h_gr= GMm/v_0 hypothesis. The flux tube connections with magnetic Mother Gaia would be essential for life. Even nutrients- typically biomolecules - could mediate this connections and this would make them nutrients.

Matti, the data is "time between trades" in seconds (fractional real number line) modeled as a jump process and in this formalism it has an associated "stochastic intensity process" which is akin to a wavefunction which randomly jumps , so point process theory has very interesting relation to wave/particle duality i think

http://arxiv.org/abs/1301.5605

on page 22 of the pdf, it looks like the distribution is a mixture of Poissionion (shot 'noise' process) and a Cauchy process , reflected 2d 'brownian motion' aka (Wiener process) at the origin

Reflected stable subordinators for fractional Cauchy problems

Boris Baeumer, Mihály Kovács, Mark M. Meerschaert, René L. Schilling, Peter Straka

(Submitted on 23 Jan 2013)

In a fractional Cauchy problem, the first time derivative is replaced by a Caputo fractional derivative of order less than one. If the original Cauchy problem governs a Markov process, a non-Markovian time change yields a stochastic solution to the fractional Cauchy problem, using the first passage time of a stable subordinator. This paper proves that a spectrally negative stable process reflected at its infimum has the same one dimensional distributions as the inverse stable subordinator. Therefore, this Markov process can also be used as a time change, to produce stochastic solutions to fractional Cauchy problems. The proof uses an extension of the D. Andr\'e reflection principle. The forward equation of the reflected stable process is established, including the appropriate fractional boundary condition, and its transition densities are explicitly computed.

correction, its the "compensator" aka the "dual predictable projection" of a Hawkes process conditioned on the (almost) maximum likelihood estimate of the paramaters to a particular realization of the symbol SPY(S&P 500) on halloween of 2014 .. if the Hawkes process removed all predictibility it should turn the resulting output into a homogeneous unit rate Poisson process (a martingale) but, the leftovers in this case has a Cauchy kernel remaining(unaccounted for), and then i just discovered this.. .and i research it, and its related to .... Lorentz... and brownian motion.. and maybe im off my rocker, but the riemann hypothesis is still involved somehow i am almost sure of it.

Matti, do these non-commutative fractional derivatives come up in TGD?

http://courses2.cit.cornell.edu/pp396/Patie_Simon.pdf

"Intertwining Certain Fractional Derivatives" it seems like it might relate to some of the twistor stuff

--anonymouscrow :)

I know whether little about these things. I wonder how many definitions of fractional derivatives exists or is the definition unique by some god argument.

Two things come however in my mind.

a) p-Adic fractals are obtained by mapping real continuous differentiable functions suchs f=x^2 to its p-adic counterpart by mapping x to p-adic number canonical identification x= SUM x_np^(-n)

-->x_p =SUM x_np^n. Forming the p-adic variant F(x_p) = x_p^2 and mapping its back to the reals by the inverse canonical identification. I have plotted this kind of fractals at my homepage. See

http://www.tgdtheory.fi/figu.html .

The special feature of these fractals is that when p-adic norm of p-add norm changes, the real counterpart develops discontinuity since the numbers (p-1)(1+p) and 1 are mapped to real number p under canonical identification (analogy: .99999..=1 so that decimal expansion is not unique for real number).

One could also form p-adic derivative dF/dx_p and map back to the reals to get what one might call fractal derivative. Left-right asymmetry is characteristic since canonical identification is well-defined only for non-negative reals. I have speculated that number theoretical universality could be behind the positive Grassmannians found in the construction of twistor representation of scattering amplitudes: in this case it relates to projectivity of the amplitudes.

To be continued...

b) Finite measurement resolution leads to hyper-finite factors and quantum groups characterised by quantum phases. One can introduce derivative, which is discretised version of ordinary derivative and approaches it when quantum group parameter q= exp(i2pi/n) approaches unity. What is beautiful is that the theory of group representations generalises and one can define notions like q-special function.

The physical meaning of this mathematics has remained obscure: to my opinion the idea to regard it as Planck length scale exotics is not good: one example of sloppy thinking characterising recent day thinking about physics by theoretical physics that I have been talking about. To my opinion it could relate to the description of finite measurement resolution in all length scales, just as p-adic fractals would do.

To be continued...

There should be a connection between these two since quantum groups and p-adicization are parts of TGD and both indeed relate to finite measurement resolution.

Discretization is the space-time counterpart for the inclusion of hyper finite factors as description of finite measurement resolution and cutoffs. q-derivative might relate to discretized functions of angle variables. p-Adicization forces discretization of angle variables by representing the allowed angles by corresponding phases which are roots of unity exp(ipi/n) up to some maximal n. This would naturally give rise to q-spherical harmonics and their generalizations and group theory would generalise to p-adic context.

"Radial" coordinates can be mapped by discretised version of canonical identification between real and p-adic (cognitive) realms. Finite measurement resolution destroys well-orderedness of real numbers below resolution scale and p-adic numbers are indeed not well-ordered. One would get simpler number field which would not have well-orderedness not possessed by measurement data below resolution. I propose p-adic manifold as formulation of this.

Interesting , it makes sense.. some theorem of Landau says things can only be ordered or unordered, there is no partially ordered states?The physics of clouds..http://phys.org/pdf345300363.pdf

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