### Could hyperbolic 3-manifolds and hyperbolic lattices be relevant in zero energy ontology?

In zero energy ontology (ZEO) lattices in the 3-D hyperbolic manifold defined by H

^{3}(t

^{2}-x

^{2}-y

^{2}-z

^{2}=a

^{2}) (and known as hyperbolic space to distinguish it from other hyperbolic manifolds emerge naturally. The interpretation of H

^{3}as a cosmic time=constant slice of space-time of sub-critical Robertson-Walker cosmology (giving future light-cone of M

^{4}at the limit of vanishing mass density) is relevant now. ZEO leads to an argument stating that once the position of the "lower" tip of causal diamond (CD) is fixed and defined as origin, the position of the "upper" tip located at H

^{3}is quantized so that it corresponds to a point of a lattice H

^{3}/G, where G is discrete subgroup of SL(2,C) (so called Kleinian group). There is evidence for the quantization of cosmic redshifts: a possible interpretation is in terms of hyperbolic lattice structures assignable to dark matter and energy. Quantum coherence in cosmological scales could be in question. This inspires several questions. How does the crystallography in H

^{3}relate to the standard crystallography in Eucdlidian 3-space E

^{3}? Are there general results about tesselations H

^{3}? What about hyperbolic counterparts of quasicrystals? In this article standard facts are summarized and some of these questions are briefly discussed.

For details see the article Could hyperbolic 3-manifolds and hyperbolic lattices be relevant in zero energy ontology? or the chapter TGD and Cosmology of "Physiscs n Many-Sheeted Space-time".

## 16 Comments:

I've been playing around with code implementation of the Extended Kalman Filter recently.. Is it possible something like this could solve the measurement/observer problem? Looks like someone has thought of it ? Feedback Quantum Control.

https://docs.google.com/viewer?a=v&q=cache:RkL9hRjmmg4J:soar.wichita.edu/dspace/bitstream/handle/10057/632/grasp0614.pdf%3Fsequence%3D1+&hl=en&gl=us&pid=bl&srcid=ADGEESihU37klU2-SF7OT7qIpX0K-ppRdrheL7DPnylCtBVdCsl81_rSZ83dA4dV71yfFEGrWWH9Csz5z-0W9dhv7bmVzFmVlZ4CyXMpvdke9vMHOGAdOcTd5f15pFHLXkbZk5eqq2r2&sig=AHIEtbTq1cmgAwiR3MtTmSjeZoNXNrTZdA

"In quantum mechanics, when the system has been measured, the system will collapse to one of the eigenstates.

This result is chosen randomly according to probability. The purpose of this research is to find the state feedback control law such that a quantum measurement can be engineered to collapse onto the eigenstate of our choice. The system should be globally stabilizing for any spin. The unscented Kalman filter will be used to estimate the state of the system."

Also, I thought of TGD when I read this

http://phys.org/news/2012-10-real-physicists-method-universe-simulation.html

Dear Matti,

I want to learn about modified gamma matrices and modified Dirac equation. I summary what I understood and ask my questions.

(In the bellow, G is modified gamma matrices and g is generalization of 4 gamma matrices to 8D )

G^alphaD_alpha psi = 0

G^alpha = T_l^alpha.g^l

Psi is the octonion valued 2-spinors. Therefore it contains 16 real components!!!

T_l^alpha is canonical energy momentum tensor.

I see the last formula at the topic “Super-symmetry forces modified Dirac equation”

I can guess the two octonion in the 2-spinor are referred to positive and negative energy states or particle and antiparticle states respectively. This can be seen as analogy of zero energy ontology but in the case of spinors." ?

In the modified dirac equation, there isn’t any projection to space time surface. Why?

In TGD there is modified dirac equation and a field equation deduced from kahler action. These are two basic equations of TGD that I learned until now. As I know there are no primary classical gauge fields in TGD, Therefore there is no room for ordinary electroweak equation or chromodynamics equation of standard model when is translated to TGD framework? It is little hard for me to throw away these beauty equations :). Because every term of the equations has physical meaning.

If I understood correctly, the idea is that non-linearization of Schroedinger equation and random feedback could allow to mimic deterministic evolution by Schroedinger equation and combine it with state function reduction. One can safely say that this is doomed to fail as any other attempt to simulate state function reduction classically.

Kalman filter seems to involve randomness based on classical probability. I do not really get the idea about how to use it to mimic state function reduction. The deep fact however that in quantum theory probability amplitudes replace probabilities. This means that this is just partial mimicry at best.

The state function reduction involves deep problems- basically related to our wrong view about the relationship between experienced time and geometric view about time. Many people recognize that something is wrong (t'Hooft is the latest name doing this) but somehow they fail totally to realize what the source of difficulties is and try all kinds of tricks like reduction of quantum theory to classical deterministic theory. Equally many manage in cheating themselves to see no problem at all.

Dear Hamed,

You say: "I want to learn about modified gamma matrices and modified Dirac equation."

A: Here is a brief summary about modified gamma matrices and their octonionic counterparts.

First about modified gamma matrices without any reference to whether they are "ordinary" or octonionic.

a) Modified gamma matrices are linear combinations of 8-D flat space gamma matrices (M^8). The coefficients are analogous to vierbein coefficients e_k^a. Recall that one has

gamma_k= d_k^agamma_a,

where gamma_a are flat space gamma matrices.

b) Modified gamma matrices are defined as

Gamma^alpha= Pi^alpha_k* gamma_k = Pi^alpha_k*e^a_k *gamma_a,

where gamma_a 8-D flat space gamma matrix (they can be ordinary or octonionic!).

Pi^alpha_k is the canonical momentum current associated with any general coordinate invariant action principle for the space-time surface. In TGD it is Kahler action. String theorist would suggest volume giving minimal surface equations which are actually obtained also in TGD for the general ansatz.

c) The expression for Pi^alpha_k is given by

Pi^alpha_k= partial L/partial (partial_alpha h^k).

For M^4 coordinates these coefficients reduces to components of energy momentum current T^alphabeta hkl h^l_beta coming from the variation of induced metric alone. In CP_2 degrees of freedom these is additional contribution due to variation with respect to the induced Kahler form.

!!!! I might have made in some context a hasty and definitely wrong statement that reduction to energy momentum currents takes place generally. If you remember the exact place, where I have said this, tell me so that I can make a correction!

d) Modified gamma matrices defined via their anticommutator what I call effective metric.

If the action density L is four-volume, one obtains minimal 4-surfaces and modified gamma matrices reduce to induced gamma matrices: Gamma_alpha= partial_alpha h^k gamma_k and effective metric is just the induced metric. For Kahler action this is not the case and effective metric G^alphabeta need not even have inverse as a 4x4-matrix.

There are differences with respect to ordinary 4-D gamma matrices.

a) All 8 gamma matrices contribute to 4-D gamma matrices: this is big difference with respect to ordinary gamma matrices.

b) Modified gamma matrices are not covariantly constant. Only the covariant divergence vanishes: D_alpha Gamma^alpha=0. This is implies by classical field equations for the action principle so that spinor dynamics and space-time dynamics get correlated. This condition in turn implies super symmetries in the sense that there is infinite number of conserved fermion currents contractions ubar_n Gamma^alpha Psi, where Psi is second quantized induced spinro field and u_n any classical solution of modified Dirac equation.

What I have said above actually generalizes to the n-dimensional extremals of any general coordinate invariant variational principle in n+k-dimensional curved space.

To be continued....

Dear Hamed,

consider next the octonionic representation of gammas.

a) For ordinary gamma matrices in D=8 tangent space rotation group SO(1,7) is in the role of gauge group: one can indeed select the local basis of flat space gamma matrices in arbitrary manner.

b) In dimension D=8 one can define also octonionic gamma "matrices" defined as as tensor products of selected 2-D Pauli sigma matrices and octonion units. These objects are not actually matrices since octonion units are not associative. For quaternionic 4-surfaces they give rise to associative induced gamma matrices.

Octonionic gamma matrices correspond to a reduction tangent space vielbein rotation group from SO(1,7) to G_2, which is the automorphism group of octonions. SO(1,7) has dimension 28 and G_2 dimension 14. G_2 acts on the imaginary octonion units only and therefore is subgroup of 21-dimensional SO(7) for a special choice of time axis representing real octonion unit: in zero energy ontology octonion real unit corresponds to the time-line would connecting the tips of causal diamond (CD).

c) Question: Is it induced or modified gamma matrices or modified gamma matrices span quaternionic 4-plane of octonions? It seems that induced. For induced gammas this 4-plane is tangential to space-time surface. For modified it would not be in general. For modified gamma matrices this 4-plane could also degenerate to lower-dimensional plane.

I have *not* been able to decide whether octonionic representation is necessary or not. The notion of quaternionicity can be defined also for ordinary gamma matrices. Maybe the octonionic representation is just a curiosity.

Complexified octonionic spinors have indeed 2*8=16 components. One must introduce quark and lepton octonion spinors separately so that one 16+16 "real" components. Ordionary spinors in D=8 have 8 +8 =16 complex components =32 real components corresponding to two 8-D chiralities (quarks and leptons, recall that color corresponds to CP_2 partial waves- not "spin").

Dear Hamed,

I continue by answering to your question about modified Dirac equation.

You say: "In the modified dirac equation, there isn’t any projection to space time surface. Why?"

A: There is!! Modified gamma matrices Gamma^alpha are labelled by space-time coordinates x^alpha (to be distinguished from imbedding space coordinates h^k).

You say: "In TGD there is modified dirac equation and a field equation deduced from kahler action. These are two basic equations of TGD that I learned until now. As I know there are no primary classical gauge fields in TGD, Therefore there is no room for ordinary electroweak equation or chromodynamics equation of standard model when is translated to TGD framework? It is little hard for me to throw away these beauty equations :). Because every term of the equations has physical meaning."

A: This is true but only to very limited extent.

a) Classical electroweak and color gauge field arise purely geometrically. Classical ew gauge potentials are just projections of spinor connection to space-time surface. Color gauge potentials projections of CP_2 Killing vector fields to space-time surface.

This predicts extremely strong correlations between various classical fields but many-sheeted space-time allows to circumvent the claim that superposition is lost for these fields so that basic assumption of perturbation theory ceases to hold true.

Particle having topological sum contacts on several space-time sheets experiences the sum of gauge and gravitaional forces caused by them. Effects superpose, not the fields.

b) At quantum level QFT description is only a long length scale approximation just as it is in string models. Certainly beautiful but only an approximating treating 3-D particle surfaces as points. Field quanta correspond geometrically to deformations of CP_2 type vacuum extremals defining regions of space-time surface with Euclidian signature of metric identifiable as 4-D lines of generalized Feynman diagrams. This is something totally new: in standard QFT space-time is just Minkowski M^4 and Feynman diagram has no geometric counterpart at space-time level.

Much or the recent work in TGD has been concentrate to attempt to gain understanding about generalized Feynman diagrams and zero energy ontology implying twistor description combined with huge symmetries of TGD, are very powerfull guidelines.

Electroweak gauge symmetries in and color symmetries emerge as dynamically for the solutions of the modified Dirac equation (http://tgdtheory.com/public_html/tgdgeom/tgdgeom.html#dirasvira). It is absolutely essential that the solutions are localized to 2-D surfaces of space-time surface (string world sheets and perhaps also to partonic 2-surfaces. This is due to single condition: em charge defined in spinorial sense is conservedfor the extremals. This gives rise to a very profound connection with string theories but with knot theory and braid theory as additional structure coming from 4-dimensionality of space-time.

Dear Hamed,

thanks for mentioning the formula for modified gamma matrices. As I said, they are expressible in terms of canonical momentum currents and besides term involving energy momentum tensor there is a term coming from the variation with respect to induced Kahler form: for some reason I had neglected it.

The chapter "The recent vision about preferred extremals and solutions of the modified Dirac equation indeed contains some wrong formulas assuming that the CP_2 contribution is absent (subsubsection "Definition of quaternionicity based on gamma matrices) . I am correcting them.

Dear Matti,

Thanks a lot. When I was thinking on your answers, I needed to reconsider the concept of vierbein . as I understood the vierbein is the same as the frame vector field that is consist of n orthogonal vector field? If it is correct, the choice of the frame is arbitrary related to our purpose. in the article “The geometry of CP2 and it’s relationship to the standard model”, you choose a special kind of the Vierbein of CP2 space. What is your purpose on the choice?

You wrote there: s_kl= R^2*sigma(e_k^A*e_k^A)

Is e_k^A, the component of e^A on the spherical coordinates of CP2?

Yes. Vierbein is a choice of 4 vector fields orthogonal in induced metric and giving the components of the induced metric in the flat space inner product. Local rotations respect these conditions and Riemann connection and its various variants (spinor connection for instance) can be seen as gauge potentials associated with this gauge symmetry.

The choice of vierbein in the case of CP_2 (not mine originally!) is dictated by symmetry considerations.

Cognition seems to breaks gauge invariance. There are natural and easy choices and most choices are not easy. In fact, I have been forced to seriously consider the possibility that cognition in p-adic sense breaks the gauge symmetries and also other symmetries - also general coordinate invariance - to their discrete variants.

This comment has been removed by the author.

Dear Matti,

I gained a very introduction to modified dirac equation. But now, I think that I need to learn an other important base of TGD about super conformal algebra and super Virasoro and super Kac-Moody algebras.

please guide me how can start to learn them step by step. Thanks.

Dear Hamed,

you are making progress!

Conformal field theories and super-conformal algebras are mathematically rather heavy notions - at least they still remain so for me and create horror in me! My childish understanding of them is basically physical and TGD centered: I however think that physical understanding is the the best way way to proceed.

1. The original motivation for them came from the attempt to describe 2-D critical systems thermodynamically. The great discovery was that scaling invariance generalizes to 2-D conformal invariance which dictates 2-point functions completely and 3-point functions also albeit not uniquely. Fusion rules allow to construct higher n-point functions. One can say that symmetries are so huge that dynamics reduces almost to dynamics.

In string theory one can reduce the situation to conformal field theory at Euclidianized string world sheet: also the formalism using hypercomplex numbers works.

2. 2-D conformal field theory is the basic notion. Alexander Zamoldchikov et al wrote a classic about these and this is "must" I think. see the reference to Nucl. Phys. B at http://en.wikipedia.org/wiki/Alexander_Zamolodchikov. I suggest that you try to find this reference. The Wikipedia article at

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

gives links to related topics.

2-D conformal field theory has 2-D conformal transformations (analytic transformations of complex variable) as symmetries. Super-conformal symmetry is very much analogous to local gauge symmetry but is different. Often this symmetry is broken by the generation of central extension in conformal/super-conformal algebra or associated Kac-Moody type altebras.

3. Super Virasor algebras

a) (N=1) Super Virasoro algebra is the Lie algebra of conformal transformations with central extension.

See this: http://en.wikipedia.org/wiki/Super_Virasoro_algebra

This algebra involves Majorana fermions and is not relevant to TGD since in TGD B and L are separately conserved and one has at least N=2 super conformal symmetry.

b) N=2 Super Virasoro algebra extends this algebra by bringing in conserved charge - say fermion number. Fermions are not anymore Majorana fermions. I would say that in TGD one has large N super-conformal algebra associated with string world sheets. If I understand correctly, N would be in TGD be the number of fermionic oscillator operator families accompanying spinor modes localized to string world sheets. The oscillator operators within given family would be labelled by conformal weight n=0,1,2...

See this to get some perspective: http://en.wikipedia.org/wiki/N_=_2_superconformal_algebra

4. (Super) Kac-Moody algebras accompany very naturally (Super) conformal algebras.

Unfortunately http://en.wikipedia.org/wiki/Kac-Moody_algebra does not seem very helpful.

One might say that Kac-Moody algebra is much like algebra of infinitesimal gauge transformations but with non-trivial central extension. Super-Virasoro generators can be expressed as quadratic expressions of Kac-Moody generators. The article of Zamolodchikov et al might help here. Some physicist friendly introduction would be of enormous help: at least for me reading of mathematical articles is virtually impossible. Maybe arXiv org could help here.

Dear Hamed,

the application of conformal field theory to TGD involves some special features.

a) The basic objects are light-like 3-surfaces (parton orbits) which are metrically 2-D (one coordinate is light-like and does not contribute to induced metric). This imples that conformal invariance generalizes. This gives generalization of Kac-Moody algebras and one can loosely say that the super-conformal invariance more or less reduces physcs at 3-D lightlike surface to that at partonic 2-surfaces at its ends.

b) The light-like boundary of M^4 projection of CD and of M^4 light-cone is metrically 2-D and one obtains now generalized symplectic structure for delta M^4_+/- xCP_2. Hamiltonians can be though of as functions in delta M^4_+/-xCP_2 and delta M^4_+/- can be written as S^2xR_+, where S^2 is radial coordinate r_M= constant sphere and R_+ is the half line along with r_M varies. The Hamiltonians can be written as products r_M^nxHamiltonian in S^2xR_+. r_M is in the role of complex coordinate for ordinary Kac-Moody algebra and Kac-Moody algebra is replaced with infinite-D symplectic algebra generated bt the Hamiltonians of S^2xCP_2 allowing decomposition to products of S^2 partial waves with well-define spin and CP_2 partial waves with well-define color quantum numbers. This is huge extension of ordinary Kac-Moody algebra.

c) What I have done with these super-conformal symmetries besides applying super-conformal partition functions to p-adic thermodynamic, is basically attempts to demonstrate their plausibility: I have not worked with super-space formalism but have constructed directly the fermionic generators of various super conformal algebras. The recent work with the modified Dirac equation takes this work to rather firm basis and gives also connection with gauge symmetries interpreted as Kac-Moody symmetries and there quite not identical with genuine gauge symmetries.

These two super-conformal symmetries relate to the structure of WCW spinor fields. Spin degrees of freedom of WCW spinor field (fermionic Fock space) corresponds to Kac-Moody in ew, color, stringy degrees of freedom. "Orbital" degrees of freedom in WCW correspond to super-symplectic degrees of freedom.

So thanks, I will try as i can to learn the subject.

http://chandra.harvard.edu/photo/2012/phoenix/media/

George Mussor has written http://www.scientificamerican.com/article.cfm?id=humans-think-like-quantum-particles but nothing about Penrose nor the nobelist Eccles? Kant? etc.

Post a Comment

<< Home