Monday, July 09, 2012

How quaternionicity of space-time could be consistent with Hermitian/Hamilton-Jacobi structure?

The recent progress in the understanding of preferred extremals of Kähler action suggests also an interesting connection to the number theoretic vision about field equations. In particular, it might be possible to understand how one can have Hermitian/Harmilton-Jacobi structure simultaneously with quaternionic structure and how quaternionic structure is possible for the Minkowskian signature of the induced metric.

One can imagine two manners of introducing octonionic and quaternionic structures. The first one is based on the introduction of octonionic representation of gamma matrices and second on the notion of octonion real-analycity.

  1. If quaternionic structure is defined in terms of the octonionic representation of the imbedding space gamma matrices, there seems to be no obvious problems since one considers automatically complexification of quaternions represented in terms of gamma matrices. For the approach based on the notion of quaternion real analyticity, one is forced to use Wick rotation to define the quaternionic structure in Minkowskian regions or to introduce what I have called hyper-quaternionic structure by imbedding the space-time surface to a sub-space $M^8$ of complexified octonions. This is admittedly artificial.

  2. The octonionic representation effectively replaces SO(7,1) as tangent space group with G2 and means selection of preferred M2⊂ M4 having interpretation complex plane of octonionic space. A more general condition is that the tangent space of space-time surface at each point contains preferred sub-space M2(x)⊂ M4 forming an integrable distribution. The same condition is involved with the definition of Hamilton-Jacobi structure. What puts bells ringing is that the modified Dirac equation for the octonionic representation of gamma matrices allows the conservation of electromagnetic charge in the proposed sense a observed for years ago. One can ask whether the conditions on the charged part of energy momentum tensor could relate to the reduction of SO(7,1) to G2.

  3. Octonionic gamma matrices appear also in the proposal stating that space-time surfaces are quaternionic in the sense that tangent space of the space-time surface is quaternionic in the sense that induced octonionic gamma matrices generate a quaternionic sub-space at a given point of space-time time. Besides this the already mentioned additional condition stating that the tangent space contains preferred sub-space M2⊂ M4 or integrable distribution of this kind of sub-spaces is required. It must be emphasized that induced rather than modified gamma matrices are natural in these conditions.
The definition of quaternionicity in terms of gamma matrices looks more promising. This however raises two questions.
  1. Could the quaternionicity of the space-time surface together with a preferred distribution of tangent planes M2(x)⊂ M4 or E2(x)⊂ CP2 be equivalent with the reduction of the field equations to the analogs of minimal surface equations stating that certain components of the induced metric in complex/Hamilton-Jacobi coordinates vanish in turn guaranteeing that field equations reduce to algebraic identifies following from the fact that energy momentum tensor and second fundamental form have no common components? This should be the case if one requires that the two solution ansätze are equivalent.

  2. Can the conditions for the modified Dirac equation select complex of co-complex 2-sub-manifold of space-time surface identified as quaternionic or co-quaternionic 4-surface? Could the conditions stating the vanishing of charged energy momentum currents state that the spinor fields are localized to complex or co-complex (hyper-complex or co-hypercomplex) 2-surfaces?
One should assign to the space-time sheets both quaternionic and Hermitian or Hamilton-Jacobi structure. There are two structures involved. Euclidian metric is an essential aspect of what it is to be quaternionic or octonions. It however seems that one can assign to the induced metric only Hermitian or Hamilton-Jacobi structure. This leads to a serious of innocent questions.
  1. Could these two structures be associated with energy momentum tensor and metric respectively? Or perhaps vice versa? Anti-commutators of the modified gamma matrices define an effective metric as

    Gαβ=TαμTμβ .

    This effective metric should have a deep physical and mathematical meaning but this meaning has remained a mystery. Note that iT takes the role of Kähler form for Kähler metric in this expression with imaginary unit plus symmetry replacing antisymmetry. The neutral and charged parts of T would be analogous to quaternionic imaginary units and real part but one would have sum over the squares of all of them rather than only single square as for Kähler metric.

  2. Could G be assigned with the quaternionic structure and induced metric to the Hermitian/Hamilton-Jacobi structures? Could the neutral and charged components of the energy momentum tensor somehow correspond to quaternionic units?
The basic potential problem with the assignment of quaternionic structure to the induced gamma matrices is the signature of the metric in Minkowskian regions.
  1. If quaternionic structrures is defined in terms of the octonionic representation of the imbedding space gamma matrices, there seems to be no obvious problems since one considers automatically complexification of quaternions.
  2. For the approach based on the notion of quaternion real analyticity, one is forced to use Wick rotation to define the quaternionic structure or to introduce hyper-quaternionic structure by imbedding the space-time surface to a sub-space M8 of complexified octonions. This is admittedly artificial.

Could one pose the additional requirement that the signature of the effective metric G defined by the modified gamma marices (and to be distinguished from Einstein tensor) is Euclidian in the sense that all four eigenvalues of this tensor would have same sign.

  1. For the induced metric the projections of gamma matrices are given by

    Γαae , e= eakα hk .

    For the modified gamma matrices their analogs would be given by

    Γαa E , E= eTμα .

    Projection would be followed by multiplication with energy momentum tensor. One cannot induce G from any metric defined in the imbedding space but the notion of tangent space quaternionicity is well-defined.

  2. What quaternionic structure for G could mean? One can imagine several options.

    1. For the ordinary complex structure metric has vanishing diagonal components and the infinitesimal line element ds2=gzz*dzdz*. Could this formula generalize to


      The generalization would be a direct generalization of conformal invariance to 4-D context stating that 4-metric is quaternion-conformally equivalent to flat metric. This would give additional strong condition on energy momentum tensor:

      G= TαμTμβ= T2δαβ .

      The proportionality to Euclidian metric means in Minkowskian realm that the G is of form G= T2(2uαuβ -gαβ. Here u is time-like vector field satisfying uαuα=1 and having interpretation as local four-velocity (in Robertson-Walker cosmology similar situation is encountered). The eigen value problem in the form Gαβxβ= λ xα makes sense and eigenvectors would be u with eigenvalue λ=T2 and three vectors orthogonal to u with eigenvalue -T2. This requires integrable flow defined by u and defining a preferred time coordinate. In number theoretic vision this kind of time coordinate is introduced and corresponds to the direction assignable to the octonionic real unit. Note that the vanishing of charged projections of the energy momentum tensor does not imply a reduction of the rank of T so that this option might work.

    2. Quaternionicity could mean also the structure of hyper-Kähler manifold. Metric and Kähler form for Kähler manifold are generalized to metric representing quaternion real unit and three covariantly constant Kähler forms Ii obeying the multiplication rules for quaternions. The necessary condition is that the holonomy group equals to SU(2) identifiable as automorphism group of quaternions. One can also define quaternionic structure: there would exist three antisymmetric tensors, whose squares give the negative of the metric. CP2 allows quaternionic structure in this sense and only one of these forms is covariantly constant.

      Could space-time surface allow Hyper-Kähler or quaternionic structure somehow induced from that of CP2? This does not work for G. G is quadratic in energy momentum tensor and therefore involves four power of J rather than being square of projection of J or two other quaternionic imaginary units of CP2. One can of course ask whether the induced quaternionic units could obey the multiplication of quaternionic units and have same square given by the projection of CP2 metric. In this case CP2 metric would define the effective metric and would be indeed Euclidian. For the ansatz for preferred extremals with Minkowskian signature CP2 projection is at most 3-dimensional but also in this case the imaginary units might allow a realization as projections.

For more details see the chapter Does the Modified Dirac Equation Define the Fundamental Action Principle? or the article.


At 8:26 AM, Blogger ThePeSla said...


You certainty have a high level post here and it asks questions I and others I have encountered in the reading. I am stating new principles in this sort of supersymmetry or hyper-color theory as more foundational than the quantum terms and notations used here. To a great extent these complex field theories do correspond as some suspect, like Rowlands.

As to your quaint term magnetic body I compliment with the quaint term body electric. I think on this deeper multilevel physics that any result of some particle such as the Higgs is a victory for super symmetry or that of our take on the varieties--- we need to make clearer what happens in the sub and superscripts and why in the physical world the quaternal reduces to the 3+1 signature and there find the source of what hold existents together and keeps them apart- the standard theory cannot explain this.

In the super symmetry along our lines there are connections and conservation laws of hyper-reflection that exceed the formalism of quaternions in the expression of mass or other inertial and energetic values- and yes much of it can seem biased to the more conventional ideas of numbers

The PeSla

At 2:47 PM, Anonymous Orwin O'Dowd said...


Remember I said close to you now is covariant argument from Bethe-Salpeter equation leading to t'Hooft?

T'Hooft finds gravity is dissipative, which echoes Poincare's discovery of chaos in celestial mechanics.

Now I noticed that the residual terms in Stirling formula in free energy expression for thermodynamics contain a phase term 2.pi.N. Take this as your imaginary component, and the energy measure has an entropy measure.

Then you may be able to find geometry for your meson entropy result. Is it like a curvature of charge-parity relation, hence an entropy of micro-curvature that matters at meson orbit, between electron and nucleus?

A meson-aware thermodynamics would be a real improvement, because mesons contribute to pH, and hence chemical reactions.

At 2:16 AM, Anonymous Orwin said...

Around here you also meet Walter Eslasser, who was the established Hegelian theoretical biologist - stressed complexity and thus epistemological emergence - no continuity of theory with physics. Hayeck admired him, but no real following in biology.

Also devised dynamo theory of Earth's core and magnetic field, which has problems with chaos!!!

Elsasser's work on nucleus also provided springboard for Wigner, who then hid behind isospin construct. Now Higgs is isospin - iso- with what? More Higgses? Some massive charge?

Right now you have G as square root of Einstain's cosmic scale statistical mechanics - flat space with no dark matter, consistent with the massless SM. Now SM is massless due to time-reversal symmetry: introduce affine tensors and the inner product splits, breaking the symmetry!

This is the mathematically rigorous approach to Higgs enigma. Equivalently, you have symmetric SU(2), but SU(2)*SU(3) is broken - by a new inner product! Is this your 'mutiplication by quaternionic units'?!?

What about the off-diagonal elements of energy-momentum tensor representing radiation pressure? Is this the non-linear factor, hence neutrinos as radiation pressure-pulses, ie solitons?

At 8:48 AM, Anonymous Orwin said...

Where the non-linear aspect comes into the picture: here's a relativistic generalization of Young's modulus of elasticity in a blackstring model: Not superstring, and real elasticity! I think this is a result worth recovering.

Otherwise your position seems underdetermined, lacking some additional constraint, axiom, or empirical law. These people are also looking at conformal symmetry and hypotheses about the cosmic microwave background, which are in principle testable:

Topological black hole solitons:

Non-linearity from backreaction:

At 1:39 PM, Blogger Ulla said...

At 2:27 PM, Anonymous ◘Fractality◘ said...


Evidence for primary qualia taking place in organs.

...stomach, whose complex work is under the control of what’s sometimes called "the little brain", a network of neurons that line your stomach and your gut.

Surprisingly, there are over 100 million of these cells in your gut, as many as there are in the head of a cat.

The little brain does not do a lot of complex thinking but it does get on with the essential daily grind involved in digesting food - lots of mixing, contracting and absorbing, to help break down our food and begin extracting the nutrients and vitamins we need.

And all those neurons lining our digestive system allow it to keep in close contact with the brain in your skull, via the vagus nerves, which often influence our emotional state.


At 12:15 AM, Blogger Ulla said...

Primary qualia? This is networks. Where does the consciousness enter? I agree it is very important, much more than we usually know of, It is the 'gut' feeling, but primary qualia is not met at this level, I think. You have to go to the actual braidings and computations, and have all 'strings' there. As instance you begin to feel ill if you SEE a special person.

At 5:52 AM, Blogger Ulla said...

I hope your silence not mean bad things, just holiday? I begin to worry again.

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At 8:24 AM, Blogger Ulla said...

Excitement :D
I start see hospitals.

At 12:20 PM, Blogger Stephen said...

A novel optical encryption based on polarization is proposed and applied to a holographic memory system. Original binary data are described as two orthogonal linear polarization states. These input polarization states can be modulated by use of two polarization-modulation masks located at the input and the Fourier planes. Each modulation mask can convert an input polarization state into a random polarization state. Once encrypted, the polarization state is recorded as a hologram. For the decryption, the hologram can generate a vector phase-conjugate beam. When the same polarization-modulation masks are used, the vector phase-conjugate readout can cancel the polarization modulation at each mask, and the original polarization state can be recovered. The encryption of the proposed method is evaluated numerically. We also present experimental results by demonstrating holographic recording in a bacteriorhodopsin film.

At 11:39 PM, Blogger Ulla said...

New Dark Matter Detectors using DNA for Nanometer Tracking

A 1-metre-square sheet of gold foil and a "forest" of single-strand DNA molecules suspended beneath in an ordered array, like the bristles on a toothbrush. When a WIMP strikes a gold atom in the foil, it would dislodge a gold nucleus and send it careening through the array, severing the DNA strands along its path.

A DNA-based detector has other advantages too. It can operate at room temperature, as opposed to near absolute zero for current detectors. And by changing the material of the foil, it can be tuned to look for particles of different energies, including the WIMPs

This requires interaction with dark matter.

At 12:07 PM, Blogger Ulla said...

Living cells as lasers using bioluminescens? One can see the flux tubes in work :)

See also C quantum dots (many atoms) so many properties in a cell? But Si have problems with water. Neutron stars that are superfluidy at very high temp.

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