The last updating

I have been involved with a heavy updating of the books about basic quantum TGD during last three months. This kind of massive cleaning up procedures are unavoidable and seem to become unavoidable with a period of about 5 years. I am now 58 so that I can estimate that not too many updatings are left. Should I be relieved? Or sad for the shortness of the professional life span?

I dare claim that this endless cleaning is not a mere exotic form of cleaning neurosis. My working style is that of a light-hearted jazz musician and this produces a lot of stuff which does not represent eternal truths, and it is better to throw away this stuff away in order to not totally confuse the potential reader (in the beginning of the cleaning operation and seeing what has happened in my household I really hope that no such reader exists! When everything shines again I hope that that my friend might exist after all!.

The progress has been especially jazzy during last five years as several new visions about what TGD might be have seen the daylight. Mention only zero energy ontology, the notion of finite measurement resolution, the role of hyper-finite factors of type II₁, the hierarchy of Planck constants, the construction of configuration space geometry in terms of second quantized induced spinor fields, number theoretic compactification,...). These ideas are now converging to an overall view in which various approaches to quantum TGD (physics as infinite dimensional geometry, physics as generalized number theory, physics from number theoretical universality, physics from finite measurement resolution implying effective discretization, TGD as almost topological QFT) neatly fuse together to single coherent overall view.

What is so fine in this cleaning up process that it forces to read all the stuff written during years and critically estimate the internal consistency or lack of it. I will never get rid of the feeling of deep shame than an age old archeological remnant which should have been destroyed for aeons ago creates in me. It is difficult to tolerate the childish enthusiasm of those older copies of me talking what now seems to me total non-sense.

But there is also a reward from all of this pain and trouble. New beautiful connections emerge and arguments and concepts become more precise. It is also wonderful to feel that you really might have something to give to the human kind. It might not be comparable to the Fantasie Impromptu of Chopin, but it is not totally worthless. May be it reduces just to the message that I did my best.

I also learn how incredibly tortuous the path to truth is and that it is good for ego to learn how fragile the most convincing looking argument is, how many different variants it can evolve to depending on what one means with basic concepts, and that the most difficult part in science is finding the correct interpretation. Without it you cannot write the rules. Some of us have a really good luck, and are able to do it during their lifetime and become heros. They can be however be sure that practically no one bothers to go through the same difficult path to really understand the origin of the rules.

In any case, all this work has not been in vain. I feel that I have good justifications for saying that Quantum TGD is a wonderful child full of vigor and energy and exists more and more intensively also as a mathematical theory. In the following I try to sum up some highlights about what has happened during last months. I hope that I find time to write something also the What's New sections of seven books about quantum TGD.

Number theoretical compactification as a bottle neck notion

The detailed formulation of the notion of "number theoretic compactification" (or M⁸H duality) stating that TGD allows equivalent formulations in terms of 4-surfaces of 8-D Minkowski space M⁸ (hyper-octonions) and H=M⁴× CP₂ is responsible for everything else that has taken place during last months.

Number theoretical compactification makes strong predictions about the structure of preferred extremals of Kähler action consistent with the known extremals. The slicing of preferred extremals by stringy world sheets and their partonic duals is the basic prediction so that dimensional reduction gives string model type theory. A related prediction is a slicing by light-like 3-surfaces parallel to the fundamental light-like 3-surface X³_l at which the signature of the induced metric changes: X³_l carries elementary particle quantum numbers.

Finite measurement resolution replacing effectively light-like 3-surfaces with braids replaces space-time surfaces with collections of string world sheets. Note that the strings connecting braid points at partonic 2-surfaces are like strings connecting branes. The string model in question however differs in many respects from string model and the string tension -essentially density of Kähler action per unit length - does not equal to the inverse of gravitational constant.

Construction of configuration space geometry and spinor structure in terms of second quantized induced spinor fields

Thanks to the input from number theoretical compactification, the construction of the configuration space geometry and spinor structure in terms of second quantized induced spinor fields is now relatively well understood. Second quantization and configuration space geometry are in very intimate relationship and explicit formulas for configuration space Kähler function can be written. Even an explicit formula for Kähler coupling strength revealing its number theoretic anatomy is possible.

1. The key idea is that the Dirac determinant for the modified Dirac operator defined assignable to some action defining the dynamics of space-time surface or perhaps 3-surface. The replacement of induced gamma matrices with modified gamma matrices guarantees super-conformal symmetry. The basic question is "Which action?". For five years ago I would have answered "Kähler action" without hesitation but one of the basic blunders of the last years was the attempt to reduce the entire physics to Chern-Simons action for induced Kähler gauge potential. The motivation was TGD as an almost topological QFT formulated in terms of modified Dirac action associated with C-S action and localized to the light-like 3-surfaces. Step by step I realized that the correct formulation must involve the modified Dirac operator associated with Kähler action, which indeed allows also the almost topological QFT formulation in terms of holography for preferred extremals.

2. For a moment I thought that Kähler action is enough. It however seems that it must be complexified by adding imaginary instanton term for a preferred extremal and defines exponent of Kähler function with a phase defined by instanton term added. By its topological character instanton term does not induce imaginary part to the Kähler metric but induces Chern-Simons action at light-like 3-surfaces representing particles. This would realize the long sought CP breaking at the fundamental level explaing matter antimatter asymmetry and hadronic CP breaking in TGD Universe. Also time reversal asymmetry is implied and becomes quite explicit in the sketched generalized Feynman rules.

3. If instanton term is absent there is only finite number of eigenmodes and the physical interpretation of the situation is very transparent: fermions in electroweak magnetic fields with regions in which induced Kähler form is non-vanishing forming natural units allowing finite number of analogs of cyclotron stats. Second quantization allows to satisfy anticommutation at only finite number of points of partonic 2-surface so that the notion of braid as a correlate for finite measurement resolution would emerge automatically. Dirac determinant reduces to a product of finite number of generalized eigenvalues and everything is nice. This picture is especially attractive from the point of view of number theoretical universality.

4. If instanton term is allowed, infinite number of conformal excitations assignable to the strings connecting braid points are possible. In this case the Dirac determinant can be defined by standard zeta function regularization reducing to that for Riemann Zeta but it is questionable whether this option is number theoretically universal. It is not yet clear whether one must allow conformal excitations in the definition of Dirac determinant or not or whether the two definitions might give rise to same configuration space metric (but not same Kähler function since a real part of a holomorphic function of configuration space coordinates can distinguish between them!). More generally, the independence on conformal cutoff would have interpretation as renormalization group invariance of the configuration space metric.

5. The generalized eigenvalues of the modified Dirac operator relate closely to Higgs mechanism. It however turned out that Higgs vacuum expectation does not cause massivation of gauge bosons. Rather the Higgs expectation value in boson state is expressible in terms of this kind of eigenvalues for gauge boson giving directly the ground state contribution to the mass of fermion (gauge boson is bound state of fermion and antifermion at the opposite throats of a wormhole contact). The ground state contribution to fermion mass would be small since p-adic thermodynamic contribution from the conformal excitations would dominate over the small ground state contribution. This also leads to a formula of Weinberg angle in terms of the generalized eigenvalues. Quite generally, the view about what causes what in particle massivation is drastically modified.

Space-time correlate of quantum criticality and the identification of preferred extremals

The geometric properties of preferred extremals are fixed to a high degree by number theoretic compactification. This is not quite enough. A good candidate for the additional field equations satisfied by the preferred extremals of Kähler action is revealed by the study of the modified Dirac equation (this result could have been deduced for more than decade ago!).

1. The Noether currents associated with Kähler action involve modified gamma matrices, which are contractions of the vector field associated with the first variation of Kähler action with ordinary gamma matrices. These currents are conserved only if the second variation of Kähler action vanishes. This is quite a strong condition. It is satisfied trivially by vacuum extremals but might be too strong for general extremals.

2. A weaker condition is that only the second variations associated with the dynamical symmetries vanish. This would give a hierarchy of criticalities beginning from that for vacuum extremals in which case all second variations vanish identically. Thus field equations alone would imply the basic vision that TGD Universe is a Universe at the edge: it would not be needed as an additional postulate. A generalization of Thom's catastrophe theory would be in question: systems would live only at the edges of catastrophe graph defined by the V shaped boundary of cusp in the simplest situation.

3. This at-the-edge property has also several other aspects. There would be also criticality with respect to phase transitions changing Planck constant very important in TGD inspired quantum biology. As also the number theoretical criticality with respect to quantum jumps transforming p-adic and real space-time sheets to each other and assigned with the formation of cognitive representations and realization of intentional actions in TGD inspired theory of consciousness. Number theoretical would be distinct from number theoretical universality. Only those surfaces whose mathematical representations can be interpret both in terms of real and p-adic numbers would be analogous to rationals common to all number fields and would represent number theoretical criticality.

Finite measurement resolution and number theoretic braids

Finite measurement resolution has the notion of number theoretic braid as a space-time correlate. This concept is now rather well-understood.

1. The basic assumption is that the braids must be definable in purely physical terms: one cannot pick up braid points just randomly as a mathematician armed with selection axiom would do. For instance, braid points could be identified as points of partonic two-surface at which induced Kähler field strength has extremum, as intersections of M⁴ and CP₂ projections with with 2-D critical manifolds associated with the criticality with respect to the phase transition changing Planck constant. There is also a much more general definition inspired by the hierarchy of symplectic triangulations which can be realized in terms of quantization of Kähler magnetic fluxes and extrema of induced Kähler field strength. What is the precise rule characterizing allowed rules defining braids is not quite clear yet. This definition would allow an infinite hierarchy of conformal cutoffs in terms of symplectic triangulations with the resulting cutoff conformal algebras realized in terms of finite number of fermionic oscillator operators assignable to the braid points.

2. Finite measurement resolution reduces the light-like 3-surfaces to braids and space-time surfaces to strings and the infinite-dimensional world of classical worlds reduces to a finite-dimensional space (δ M⁴_+/-× CP₂)ⁿ/S_n consisting of n braid points at partonic 2-surface. In the similar manner the space of configuration space spinor fields modulo finite measurement resolution reduces to a finite-dimensional space. This means enormous simplification at the mathematical level. There is a strong temptation to believe that the Clifford algebra in question can be regarded as a coset space of infinite-dimensional hyper-finite factors of type II₁ N and M, where N subset M defines the measurement resolution, and that this algebra could be regarded as a quantum Clifford algebra for the nonstandard values of Planck constant.

Super-conformal symmetries and the structure of the world of classical worlds

The understanding of super-conformal symmetries is now much more detailed than before and I have deleted an impressive collection of wrong and not-even-wrong statements.

1. It seems now clear that the coset construction for super-symplectic Virasoro algebra and Kac-Moody algebra realizes Equivalence Principle at quantum level. The space-time correlate for Equivalence Principle follows from the stringy picture. General Relativistic form of Equivalence Principle holds only in long length scales, not for a cosmic string like objects for instance. This resolves the basic poorly understood issues which have plagued the understanding of GRT-TGD correspondence and allows to throw away a lot of trash.

2. The understanding of the detailed structure of the configuration space has improved considerably. Configuration space is union of symmetric spaces over zero modes identified as coset spaces and the challenge is to understand what this statement might mean.
   1. The values of the induced Kähler field strength for the partonic 2-surface defines the most important zero modes meaning that dynamics of induced Kähler field is completely classical. Coset construction has its counterpart at the level of configuration space as a union of coset spaces. The symmetric space associated with a given induced Kähler form correspond to the orbit of a symplectic group.
   2. Symplectic group can be made local with respect to the partonic 2-surface - or rather with the coordinate defined by the value of induced Kähler field strength taking the role of complex coordinate in conformal field theories. Kac-Moody sub-algebra defined at light-like 3-surface, whose elements vanish at the partonic 2-surfaces defining its ends, acts as a gauge algebra defining further zero modes.
   3. Quantum fluctuating degrees of freedom correspond to the coset space defined by the symplectic algebra and by the sub-Kac-Moody algebra. Note that the entire Kac-Moody algebra appears in the coset construction and p-adic mass calculations whereas only the sub-algebra appeas in the coset space construction.
3. The identification of induced Kähler form of X² as purely classical field means that configuration space functional integral is only over the fluctuations contributing to the induced metric metric of X². Therefore at the configuration space level the only quantum fluctuating degrees of freedom are purely gravitational. Besides this present are fermionic degrees of freedom, modular degrees of freedom, other zero modes (Kac-Moody algebra), and topological degrees of freedom.

About the construction of M-matrix

The toughest challenge of TGD has been the construction of TGD counterpart of S-matrix - M-matrix as I call it. The understanding of the generalized Feynman rules is now rather detailed and the notion of finite measurement resolution gives excellent hopes about calculational rules making possible practical calculations.

1. The first fundamental element is zero energy ontology allowing to identify M-matrix as time-like entanglement coefficients between positive and negative energy parts of zero energy state (counterpart of physical event) assignable to the light-like boundaries of causal diamond identified as intersection of future and past directed light-cones defining the basic piece of the world of classical worlds. There is entire hierarchy of CDs within CDs and this allow to understand p-adic coupling constant evolution in terms of finite measurement resolution defined by the size of smallest CD included.

2. Second basic notion is generalized Feynman diagram identified as light-like 3-surface or equivalently as region of space-time with Euclidian signature of metric accompanying the light-like 3-surface. Euclidian regions would represent particles and Minkowskian regions classical fields. The conformal symmetries and stringy picture implied by the finite measurement resolution suggest strongly stringy Feynman rules.

3. A very powerful form of General Coordinate Invariance would be the condition that one can deduce configuration space metric by using any light-like 3-surface in the slicing of space-time surface to light-like 3-surfaces parallel to the surface X³_l at which the signature of the induced metric changes. Invariance would not mean invariance of Kähler function but only that of Kähler metric. This condition should pose extremely powerful constraints on the form of various expressions appearing in generalized Feynman diagrammatics.

4. Vertices correspond to partonic 2-surfaces and n-points functions of an N=4 conformal field theory in which second quantized induced spinor fields are the fundamental fields. The TGD based interpretation of N=4 for algebra is now well-understood and reflects directly the basic symmetries of TGD. Discretization implied by the number theoretic braids implies a huge simplification of the situation and mean stringy theory at space-time level.

5. Propagators assigned with light-like 3-surfaces connecting vertices should be stringy. The problem is how to obtain conformal excitations propagating along strings connecting braid points as zero modes of the modified Dirac operator. These excitations with non-vanishing conformal weight necessarily break the effective 2-dimensionality of 3-surfaces and thus holography. In the proposed - and yet admittedly speculative - picture about the properties of preferred extremals the only possible manner to obtain this breaking seems to be complexification of Kähler action by adding to it as imaginary part the CP breaking instanton action. The "only" in the preceding sentence should be taken with a grain of salt since the implications of number theoretical compactification for the geometry of preferred extremals are not completely understood.

6. Besides implying CP breaking and the breaking of time reversal symmetry, the instanton term would break the effective 2-dimensionality of 3-surfaces (holography) and would give rise to stringy propagation of fermions whereas at the space-time level effective 2-dimensionality seem to prevail apart from the non-determinism of Kähler action. One can speak of a radiative generation of kinetic and mass terms in stringy propagator. The classical non-determinism of Kähler action would be responsible for generating the analogs of self energy vertices and break the effective 2-dimensionality of 3-surfaces. This conforms with what one might expect. Note that only the conformal excitations of induced spinor field would break the exact holography.

Einstein's equations and second variation of volume element

Lubos had an interesting posting about how Jacobsen has derived Einstein's equations from thermodynamical considerations as kind of equations of state. This has been actually one the basic ideas of quantum TGD, where Einstein's equations do not make sense as microscopic field equations. The argument involves approximate Poincare invariance, Equivalence principle, and proportionality of entropy to area (dS = kdA) so that the result is perhaps not a complete surprise.

One starts from an expression for the variation of the area element dA for certain kind of variations in direction of light-like Killing vector field and ends up with Einstein's equations. Ricci tensor creeps in via the variation of dA expressible in terms of the analog of geodesic deviation involving curvature tensor in its expression. Since geodesic equation involves first variation of metric, the equation of geodesic deviation involves its second variation expressible in terms of curvature tensor.

The result raises the question whether it makes sense to quantize Einstein Hilbert action and in light of quantum TGD the worry is justified. In TGD (and also in string models) Einstein's equations result in long length scale approximation whereas in short length scales stringy description provides the space-time correlate for Equivalence Principle. In fact in TGD framework Equivalence Principle at fundamental level reduces to a coset construction for two super-conformal algebras: super-symplectic and super Kac-Moody. The four-momenta associated with these algebras correspond to inertial and gravitational four-momenta.

In the following I will consider different -more than 10 year old - argument implying that empty space vacuum equations state the vanishing of first and second variation of the volume element in freely falling coordinate system and will show how the argument implies empty space vacuum equations in the "world of classical worlds". I also show that empty space Einstein equations at space-time level allow interpretation in terms of criticality of volume element - perhaps serving as a correlate for vacuum criticality of TGD Universe. I also demonstrate how one can derive non-empty space Einstein equations in TGD Universe and consider the interpretation.

1. Vacuum Einstein's equations from the vanishing of the second variation of volume element in freely falling frame

The argument of Jacobsen leads to interesting considerations related to the second variation of the metric given in terms of Ricci tensor. In TGD framework the challenge is to deduce a good argument for why Einstein's equations hold true in long length scales and reading the posting of Lubos led to an idea how one might understand the content of these equations geometrically.

1. The first variation of the metric determinant gives rise to

   δ g^1/2 = ∂_μg^1/2dx^μ propto g^1/2 C^ρ_ρμdx^μ.

   Here C^ρ_μν denotes Christoffel symbol.

   The possibility to find coordinates for which this variation vanishes at given point of space-time realizes Equivalence Principle locally.

2. Second variation of the metric determinant gives rise to the quantity

   δ² g^1/2= ∂_μ∂_νg^1/2dx^μdx^ν = g^1/2R_μνdx^μdx^ν.

   The vanishing of the second variation gives Einstein's equations in empty space. Einstein's empty space equations state that the second variation of the metric determinant vanishes in freely moving frame. The 4-volume element is critical in this frame.

2. The world of classical worlds satisfies vacuum Einstein equations

In quantum TGD this observation about second variation of metric led for two decades ago to Einstein's vacuum equations for the Kähler metric for the space of light-like 3-surfaces ("world of classical worlds"), which is deduced to be a union of constant curvature spaces labeled by zero modes of the metric. The argument is very simple. The functional integration over configuration space degrees of freedom (union of constant curvature spaces a priori: R_ij=kg_ij) involves second variation of the metric determinant. The functional integral over small deformations of 3-surface involves also second variation of the volume element Ög. The propagator for small deformations around 3-surface is contravariant metric for Kähler metric and is contracted with R_ij = lg_ij to give the infinite-dimensional trace g^ijR_ij = lD=l×∞. The result is infinite unless R_ij=0 holds. Vacuum Einstein's equations must therefore hold true in the world of classical worlds.

4. Non-vacuum Einstein's equations: light-like projection of four-momentum projection is proportional to second variation of four-volume in that direction

An interesting question is whether Einstein's equations in non-empty space-time could be obtained by generalizing this argument. The question is what interpretation one should give to the quantity

g₄^1/2T_μνdx^μdx^ν

at a given point of space-time.

1. If one restricts the consideration to variations for which dx^m is of form k^me, where k is light-like vector, one obtains a situation similar to used by Jacobsen in his argument. In this case one can consider the component dP_k of four-momentum in direction of k associated with 3-dimensional coordinate volume element dV₃=d³x. It is given by dP_k= g₄^1/2T_μνk^μk^νdV₃ .

2. Assume that dP_k is proportional to the second variation of the volume element in the deformation dx^m =εk^m, which means pushing of the volume element in the direction of k in second order approximation:

   (d²g₄^1/2/dε²)dV₃= (∂²g₄^1/2/∂ x^μ∂ x^ν) k^μk^νg₄^1/2dV₃= R_μνk^μk^νg₄^1/2 dV₃ .

   By light-likeness of k^μ one can replace R_μν by G_μν and add also g_μν for light-like vector k^μ to obtain covariant conservation of four-momentum. Einstein's equations with cosmological term are obtained.

That light-like vectors play a key role in these arguments is interesting from TGD point of view since light-like 3-surfaces are fundamental objects of TGD Universe.

5. The interpretation of non-vacuum Einstein's equations as breaking of maximal quantum criticality in TGD framework

What could be the interpretation of the result in TGD framework.

1. In TGD one assigns to the small deformations of vacuum extremals average four-momentum densities (over ensemble of small deformations), which satisfy Einstein's equations. It looks rather natural to assume that statistical quantities are expressible in terms of the purely geometric gravitational energy momentum tensor of vacuum extremal (which as such is not physical). The question why the projections of four-momentum to light-like directions should be proportional to the second variation of 4-D metric determinant.

2. A possible explanation is the quantum criticality of quantum TGD. For induced spinor fields the modified Dirac equation gives rise to conserved Noether currents only if the second variation of Kähler action vanishes. The reason is that the modified gamma matrices are contractions of the first variation of Kähler action with ordinary gamma matrices.

3. A weaker condition is that the vanishing occurs only for a subset of deformations representing dynamical symmetries. This would give rise to an infinite hierarchy of increasingly critical systems and generalization of Thom's catastrophe theory would result. The simplest system would live at the V shaped graph of cusp catastrophe: just at the verge of phase transition between the two phases.

4. Vacuum extremals are maximally quantum critical since both the first and second variation of Kähler action vanishes identically. For the small deformations second variation could be non-vanishing and probably is. Could it be that vacuum Einstein equations would give gravitational correlate of the quantum criticality as the criticality of the four-volume element in the local freely falling frame. Non-vacuum Einstein equations would characterize the reduction of the criticality due to the presence of matter implying also the breaking of dynamical symmetries (symplectic transformations of CP₂ and diffeomorphisms of M⁴ for vacuum extremals).

For the recent updated view about the relationship between General Relativity and TGD see the chapter TGD and GRT of "Physics in Many-Sheeted Space-time".

A comment about thermodynamics of dark black holes

Lubos Motl had an excellent posting about thermodynamics of black holes. Unfortunately I am too busy with the updatings for a detailed response. Just a hasty comment about thermodynamics of dark black holes inspired by the vision about dark matter as a hierarchy of phases with non-standard value of Planck constant realized in terms of a book like structure of the generalized imbedding space (generalization of H=M⁴×CP₂) with pages labeled by the values of Planck constant and phase transitions changing Planck constant interpreted as a leakage between different pages of the Big Book.

Suppose we accept the identification of dark matter in astrophysical length scales as matter with a gigantic gravitational Planck constant suggested by Bohr orbitology of planetary orbits. For instance, hbar =GM²/v₀, v₀=1/4, would hold true for an ideal black hole with Planck length (hbarG)^1/2 equal to Schwartshild radius 2GM. Since black hole entropy is inversely proportional to hbar, this would predict black hole entropy to be of order single bit. This of course looks totally non-sensible if one believes in standard thermodynamics. For the star with mass equal to 10⁴⁰ Planck masses discussed in the example of Lubos the entropy associated with the initial state of the star would be roughly the number of atoms in star equal to about 10⁶⁰. Black hole entropy proportional to GM²/hbar would be of order 10⁸⁰ provided the standard value of hbar is used as unit.

This stimulates some questions.

1. Does second law pose an upper bound on the value of hbar of dark black hole from the requirement that black hole has at least the entropy of the initial state. The maximum value of hbar would be given by the ratio of black hole entropy to the entropy of the initial state and about 10²⁰ in the example of Lubos to be compared with GM²/v₀ ≈10⁸⁰.

2. Or should one generalize thermodynamics in a manner suggested by zero energy ontology by making explicit distinction between subjective time (sequence of quantum jumps) and geometric time? The arrow of geometric time would correlate with that of subjective time. One can argue that the geometric time has opposite direction for the positive and negative energy parts of the zero energy state interpreted in standard ontology as initial and final states of quantum event. If second law would hold true with respect to subjective time, the formation of ideal dark black hole would destroy entropy only from the point of view of observer with standard arrow of geometric time. The behavior of phase conjugate laser light would be a more mundane example. Do self assembly processes serve as example of non-standard arrow of geometric time in biological systems? In fact, zero energy state is geometrically analogous to a big bang followed by big crunch. One can however criticize the basic assumption as ad hoc guess. One should really understand the the arrow of geometric time. This is discussed in detail in the article About the Nature of Time.

Important notice about links which do not work anymore!!

Readers have perhaps noticed that the discovery of new longlived particle in CDF predicted by TGD already around 1990 turned out to be one of most fantastic breakthroughs of TGD. The reported findings could be explained and even predictd at quantitative level and a lot of testable predictions follow from the model as one might expect since essentially a leptonic variant of QCD is predicted.

My influental colleagues in Helsinki University who have told last 31 years in every possible occasion that I am totally mad, became understandably very angry. Since they could not punish Nature for behaving according to the predictions of TGD, they decided to punish me. I am not allowed to use university computer for my homepage anymore. This would not be a problem as such but they also refused to redirect visitors to my new homepage. These idiots have reached their holy goal: TGD has more or less disappeared from the web.

I have not yet had time and energy to update links from the blog to my homepage. When the link fails to work the recipe is however very simple: replace http://www.helsinki.fi/~matpitka/

in the link with

http://tgd.wippiespace.com/public_html/

and everything should work. Thank you very much for your kind attention.

Matti Pitkänen

The recent view about the construction of configuration space spinor structure

During the last five years both the mathematical and physical understanding of quantum TGD has developed dramatically. Some ideas have died and large number of conjectures have turned to be un-necessary strong, un-necessary, or simply wrong. The outcome is that the books about basic TGD do not correspond the actual situation in the theory. Therefore I decided to perform a major cleaning operation throwing away the obsolete stuff and making good arguments more precise. Good household is not my only motivation: this kind of process, although it challenges the ego, is always extremely fruitful. The basic goal has been to replace the perspective as it was for five years ago with the one which is outcome of the development of visions and concepts like fundamental description of quantum TGD as almost topological QFT in terms of modified Dirac action for fermions at light-like 3-surfaces identified as the basic objects of the theory, zero energy ontology, finite measurement resolution as a fundamental physical principle realized in terms of Jones inclusions and having number theoretic braids as space-time correlate, generalization of S-matrix to M-matrix, number theoretical universality and number theoretical compactification reducing standard model symmetries to number theory and allowing to solve some basic problems of quantum TGD, realization of the hierarchy of Planck constants in terms of the generalization of imbedding space concept, discovery of a hierarchy of symplectic fusion algebras provided concrete understanding of the super-symplectic conformal invariance, and so on.

I started the cleaning up process from the chapter Configuration Space Spinor Structure and I glue below the abstract.

Quantum TGD should be reducible to the classical spinor geometry of the configuration space. In particular, physical states should correspond to the modes of the configuration space spinor fields. The immediate consequence is that configuration space spinor fields cannot, as one might naively expect, be carriers of a definite spin and unit fermion number. Concerning the construction of the configuration space spinor structure there are some important clues.

1. Geometrization of fermionic statistics in terms of configuration space spinor structure

The great vision has been that the second quantization of the induced spinor fields can be understood geometrically in terms of the configuration space spinor structure in the sense that the anti-commutation relations for configuration space gamma matrices require anti-commutation relations for the oscillator operators for free second quantized induced spinor fields.

1. One must identify the counterparts of second quantized fermion fields as objects closely related to the configuration space spinor structure. Ramond model has as its basic field the anti-commuting field G^k(x), whose Fourier components are analogous to the gamma matrices of the configuration space and which behaves like a spin 3/2 fermionic field rather than a vector field. This suggests that the complexified gamma matrices of the configuration space are analogous to spin 3/2 fields and therefore expressible in terms of the fermionic oscillator operators so that their anti-commutativity naturally derives from the anti-commutativity of the fermionic oscillator operators.

   As a consequence, configuration space spinor fields can have arbitrary fermion number and there would be hopes of describing the whole physics in terms of configuration space spinor field. Clearly, fermionic oscillator operators would act in degrees of freedom analogous to the spin degrees of freedom of the ordinary spinor and bosonic oscillator operators would act in degrees of freedom analogous to the 'orbital' degrees of freedom of the ordinary spinor field.

2. The classical theory for the bosonic fields is an essential part of the configuration space geometry. It would be very nice if the classical theory for the spinor fields would be contained in the definition of the configuration space spinor structure somehow. The properties of the modified massless Dirac operator associated with the induced spinor structure are indeed very physical. The modified massless Dirac equation for the induced spinors predicts a separate conservation of baryon and lepton numbers. The differences between quarks and leptons result from the different couplings to the CP₂ Kähler potential. In fact, these properties are shared by the solutions of massless Dirac equation of the imbedding space.

3. Since TGD should have a close relationship to the ordinary quantum field theories it would be highly desirable that the second quantized free induced spinor field would somehow appear in the definition of the configuration space geometry. This is indeed true if the complexified configuration space gamma matrices are linearly related to the oscillator operators associated with the second quantized induced spinor field on the space-time surface and/or its boundaries. There is actually no deep reason forbidding the gamma matrices of the configuration space to be spin half odd-integer objects whereas in the finite-dimensional case this is not possible in general. In fact, in the finite-dimensional case the equivalence of the spinorial and vectorial vielbeins forces the spinor and vector representations of the vielbein group SO(D) to have same dimension and this is possible for D=8-dimensional Euclidian space only. This coincidence might explain the success of 10-dimensional super string models for which the physical degrees of freedom effectively correspond to an 8-dimensional Euclidian space.

4. It took a long time to realize that the ordinary definition of the gamma matrix algebra in terms of the anti-commutators {g_A,g_B} = 2g_AB must in TGD context be replaced with {g_A^f,g_B} = iJ_AB\per, where J_AB denotes the matrix elements of the Kähler form of the configuration space. The presence of the Hermitian conjugation is necessary because configuration space gamma matrices carry fermion number. This definition is numerically equivalent with the standard one in the complex coordinates. The realization of this delicacy is necessary in order to understand how the square of the configuration space Dirac operator comes out correctly.

5. The only possible option is that second quantized induced spinor fields are defined at 3-D light-like causal determinants associated with 4-D space-time sheet. The unique partonic dynamics is almost topological QFT defined by Chern-Simons action for the induced Kähler gauge potential and by the modified Dirac action constructed from it by requiring super-conformal symmetry. The resulting theory has all the desired super-conformal symmetries and is exactly solvable at parton level. It is 3-dimensional lightlike 3-surfaces rather than generic 3-surfaces which are the fundamental dynamical objects in this approach.

   The classical dynamics of the interior of space-time surface defines a classical correlate for the partonic quantum dynamics and provides a realization of quantum measurement theory. It is determined by the vacuum functional identified as the Dirac determinant. There are good arguments suggesting that it reduces to an exponent of absolute extremum of Kähler action in each region of the space-time sheet where the Kähler action density has a definite sign.

2. Modified Dirac equation for induced classical spinor fields

The identification of the light-like partonic 3-surfaces as carriers of elementary particle quantum numbers inspired by the TGD based quantum measurement theory forces the identification of the modified Dirac action as that associated with the Chern-Simons action for the induced Kähler gauge potential. At the fundamental level TGD would be almost-topological super-conformal QFT in the sense that only the light-likeness condition for the partonic 3-surfaces would involve the induced metric. Chern-Simons dynamics would thus involve the induced metric only via the generalized eigenvalue equation for the modified Dirac operator involving the light-like normal of X³_l subset X⁴. N=4 super-conformal symmetry emerges as a maximal Super-Kac Moody symmetry for this option. The application of D to any generalized eigen-mode gives a zero mode and zero modes and generalized eigen-modes define a cohomology.

The basic idea is that Dirac determinant defined by eigenvalues of D_C-S can be identified as the exponent of Kähler action for a preferred extremal. There are however two problems. Without further conditions the eigenvalues of D_C-S are functions of the transversal coordinates of X³_l and the standard definition of Dirac determinant fails. Second problem is how to feed the information about preferred extremal to the eigenvalue spectrum. The solution of these problems is discussed below.

The eigen modes of the modified Dirac equation are interpreted as generators of exact N=4 super-conformal symmetries in both quark and lepton sectors. These super-symmetries correspond to pure super gauge transformations and no spartners of ordinary particles are predicted: in particular N=2 space-time super-symmetry is generated by the righthanded neutrino is absent contrary to the earliest beliefs. There is no need to emphasize the experimental implications of this finding.

An essential difference with respect to standard super-conformal symmetries is that Majorana condition is not satisfied, the super generators carry quark or lepton number, and the usual super-space formalism does not apply. The situation is saved by the fact that super generators of super-conformal algebras anticommute to Hamiltonians of symplectic transformations rather than vector fields representing the transformations.

Configuration space gamma matrices identified as super generators of super-symplectic or super Kac-Moody algebras (depending on CH coordinates used) are expressible in terms of the oscillator operators associated with the eigen modes of the modified Dirac operator. The number of generalized eigen modes turns out to be finite so that standard canonical quantization does not work unless one restricts the set of points involved defined as intersection of number theoretic braid with the partonic 2-surface. The interpretation is in terms of finite measurement resolution and the surprising thing is that this notion is implied by the vacuum degeneracy of Kähler action.

3. The exponent of Kähler function as Dirac determinant for the modified Dirac action

Although quantum criticality in principle predicts the possible values of Kähler coupling strength, one might hope that there exists even more fundamental approach involving no coupling constants and predicting even quantum criticality and realizing quantum gravitational holography.

1. The Dirac determinant defined by the product of Dirac determinants associated with the light-like partonic 3-surfaces X³_l associated with a given space-time sheet X⁴ is the simplest candidate for vacuum functional identifiable as the exponent of the Kähler function. One can of course worry about the finiteness of the Dirac determinant. p-Adicization requires that the eigenvalues belong to a given algebraic extension of rationals. This restriction would imply a hierarchy of physics corresponding to different extensions and could automatically imply the finiteness and algebraic number property of the Dirac determinants if only finite number of eigenvalues would contribute. The regularization would be performed by physics itself if this were the case.

2. The basic problem has been how to feed in the information about the preferred extremal of Kähler action to the eigenvalue spectrum of C-S Dirac operator D_C-S at light-like 3-surface X³_l. The identification of the preferred extremal came possible via boundary conditions at X³_l dictated by number theoretical compactification. The basic observation is that the Dirac equation associated with the 4-D Dirac operator D_K defined by Kähler action can be seen as a conservation law for a super current. By restricting the super current to flow along X³_l by requiring that its normal component vanishes, one obtains a singular solution of 4-D modified Dirac equation restricted to X³_l. The ënergy" spectrum for the solutions of D_K corresponds to the spectrum of eigenvalues for D_C-S and the product of the eigenvalues defines the Dirac determinant in standard manner. Since the eigenmodes are restricted to those localized to regions of non-vanishing induced Kähler form, the number of eigen modes is finite and therefore also Dirac determinant is finite. The eigenvalues can be also algebraic numbers.

3. It remains to be proven that the product of eigenvalues gives rise to the exponent of Kähler action for the preferred extremal of Kähler action. At this moment the only justification for the conjecture is that this the only thing that one can imagine. The identification of super-symplectic conformal weights as zeros of zeta function defined by the eigenvalues of modified Dirac operator would couple them with the dynamics defined by the Kähler action.

4. A long-standing conjecture has been that the zeros of Riemann Zeta are somehow relevant for quantum TGD. Rieman zeta is however naturally replaced Dirac zeta defined by the eigenvalues of D_C-S and closely related to Riemann Zeta since the spectrum consists essentially for the cyclotron energy spectra for localized solutions region of non-vanishing induced Kähler magnetic field and hence is in good approximation integer valued up to some cutoff integer. In zero energy ontology the Dirac zeta function associated with these eigenvalues defines"square root" of thermodynamics assuming that the energy levels of the system in question are expressible as logarithms of the eigenvalues of the modified Dirac operator defining kind of fundamental constants. Critical points correspond to approximate zeros of Dirac zeta and if Kähler function vanishes at criticality as it ineed should, the thermal energies at critical points are in first order approximation proportional to zeros themselves so that a connection between quantum criticality and approximate zeros of Dirac zeta emerges.

5. The discretization induced by the number theoretic braids reduces the world of classical worlds to effectively finite-dimensional space and configuration space Clifford algebra reduces to a finite-dimensional algebra. The interpretation is in terms of finite measurement resolution represented in terms of Jones inclusion M subset N of HFFs with M taking the role of complex numbers. The finite-D quantum Clifford algebra spanned by fermionic oscillator operators is identified as a representation for the coset space N/M describing physical states modulo measurement resolution. In the sectors of generalized imbedding space corresponding to non-standard values of Planck constant quantum version of Clifford algebra is in question.

4. Super-conformal symmetries

The almost topological QFT property of partonic formulation based on Chern-Simons action and corresponding modified Dirac action allows a rich structure of N=4 super-conformal symmetries. In particular, the generalized Kac-Moody symmetries leave corresponding X³-local isometries respecting the light-likeness condition. A rather detailed view about various aspects of super-conformal symmetries emerge leading to identification of fermionic anti-commutation relations and explicit expressions for configuration space gamma matrices and Kähler metric. This picture is consistent with the conditions posed by p-adic mass calculations.

Number theoretical considerations play a key role and lead to the picture in which effective discretization occurs so that partonic two-surface is effectively replaced by a discrete set of algebraic points belonging to the intersection of the real partonic 2-surface and its p-adic counterpart obeying the same algebraic equations. This implies effective discretization of super-conformal field theory giving N-point functions defining vertices via discrete versions of stringy formulas.

For the updated version of the chapter see Configuration Space Spinor Structure of "Physics as Infinite-Dimensional Geometry".

Vision about quantization of Planck constant

The quantization of Planck constant has been the basic them of TGD since 2005 and the perspective in the earlier version of this chapter reflected the situation for about year and one half after the basic idea stimulated by the finding of Nottale that planetary orbits could be seen as Bohr orbits with enormous value of Planck constant given by hbar_gr = GM₁M₂/v₀, v₀ ≈ 2^-11 for the inner planets. The general form of hbar_gr is dictated by Equivalence Principle. This inspired the ideas that quantization is due to a condensation of ordinary matter around dark matter concentrated near Bohr orbits and that dark matter is in macroscopic quantum phase in astrophysical scales.

The second crucial empirical input were the anomalies associated with living matter. Mention only the effects of ELF radiation at EEG frequencies on vertebrate brain and anomalous behavior of the ionic currents through cell membrane. If the value of Planck constant is large, the energy of EEG photons is above thermal energy and one can understand the effects on both physiology and behavior. If ionic currents through cell membrane have large Planck constant the scale of quantum coherence is large and one can understand the observed low dissipation in terms of quantum coherence.

1. The evolution of mathematical ideas

From the beginning the basic challenge -besides the need to deduce a general formula for the quantized Planck constant- was to understand how the quantization of Planck constant is mathematically possible. From the beginning it was clear that since particles with different values of Planck constant cannot appear in the same vertex, a generalization of space-time concept is needed to achieve this.

During last five years or so many deep ideas -both physical and mathematical- related to the construction of quantum TGD have emerged and this has led to a profound change of perspective in this and also other chapters. The overall view about TGD is described briefly here.

1. For more than five years ago I realized that von Neumann algebras known as hyperfinite factors of type II₁ (HFFs) are highly relevant for quantum TGD since the Clifford algebra of configuration space ("world of classical worlds", WCW) is direct sum over HFFs. Jones inclusions are particular class of inclusions of HFFs and quantum groups are closely related to them. This led to a conviction that Jones inclusions can provide a detailed understanding of what is involved and predict very simple spectrum for Planck constants associated with M⁴ and CP₂ degrees of freedom (later I replaced M⁴ by its light cone M⁴_± and finally with the causal diamond CD defined as intersection of future and past light-cones of M⁴).

2. The notion of zero energy ontology replaces physical states with zero energy states consisting of pairs of positive and negative energy states at the light-like boundaries δM⁴_±×CP₂ of CDs forming a fractal hierarchy containing CDs within CDs. In standard ontology zero energy state corresponds to a physical event, say particle reaction. This led to the generalization of S-matrix to M-matrix identified as Connes tensor product characterizing time like entanglement between positive and negative energy states. M-matrix is product of square root of density matrix and unitary S-matrix just like Schrödinger amplitude is product of modulus and phase, which means that thermodynamics becomes part of quantum theory and thermodynamical ensembles are realized as single particle quantum states. This led also to a solution of long standing problem of understanding how geometric time of the physicist is related to the experienced time identified as a sequence of quantum jumps interpreted as moments of consciousness in TGD inspired theory of consciousness which can be also seen as a generalization of quantum measurement theory (see this) .

3. Another closely related idea was the emergence of measurement resolution as the basic element of quantum theory. Measurement resolution is characterized by inclusion M subset N of HFFs with M characterizing the measurement resolution in the sense that the action of M creates states which cannot be distinguished from each other within measurement resolution used. Hence complex rays of state space are replaced with M rays. One of the basic challenges is to define the nebulous factor space N/M having finite fractional dimension N:M given by the index of inclusion. It was clear that this space should correspond to quantum counterpart of Clifford algebra of world of classical worlds reduced to a finite-quantum dimensional algebra by the finite measurement resolution (see this).

4. The realization that light-like 3-surfaces at which the signature of induced metric of space-time surface changes from Minkowskian to Euclidian are ideal candidates for basic dynamical objects besides light-like boundaries of space-time surface was a further decisive step or progress. This led to vision that quantum TGD is almost topological quantum field theory ("almost" because light-likeness brings in induced metric) characterized by Chern-Simons action for induced Kähler gauge potential of CP₂. Together with zero energy ontology this led to the generalization of the notion of Feynman diagram to a light-like 3-surface for which lines correspond to light-like 3-surfaces and vertices to 2-D partonic surface at which these 3-D surface meet. This means a strong departure from string model picture. The interaction vertices should be given by N-point functions of a conformal field theory with second quantized induced spinor fields defining the basic fields in terms of which also the gamma matrices of world of classical worlds could be constructed as super generators of super conformal symmetries (see this).

5. By quantum classical correspondence finite measurement resolution should have a space-time correlate. The obvious guess was that this correlate is discretization at the level of construction of M-matrix. In almost-TQFT context the effective replacement of light-like 3-surface with braids defining basic objects of TQFTs is the obvious guess. Also number theoretic universality necessary for the p-adicization of quantum TGD by a process analogous to the completion of rationals to reals and various p-adic number fields requires discretization since only rational and possibly some algebraic points of the imbedding space (in suitable preferred coordinates) allow interpretation both as real and p-adic points. It was clear that the construction of M-matrix boils to the precise understanding of number theoretic braids (see this).

6. The interaction with M-theory dualities (see this) led to a handful of speculations about dualities possible in TGD framework, and one of these dualities- M⁸-M⁴×CP₂ duality - eventually led to a unique identification of number theoretic braids. The dimensions of partonic 2-surface, space-time, and imbedding space strongly suggest that classical number fields, or more precisely their complexifications might help to understand quantum TGD. If the choice of imbedding space is unique because of uniqueness of infinite-dimensional Kähler geometric existence of world of classical worlds then standard model symmetries coded by M⁴×CP₂ should have some deeper meaning and the most obvious guess is that M⁴×CP₂ can be understood geometrically. SU(3) belongs to the automorphism group of octonions as well as hyper-octonions M⁸ identified by subspace of complexified octonions with Minkowskian signature of induced metric. This led to the discovery that hyper-quaternionic 4-surfaces in M⁸ can be mapped to M⁴×CP₂ provided their tangent space contains preferred M² subset M⁴ subset M⁴×E⁴. Years later I realized that the map generalizes so that M² can depend on the point of X⁴. The interpretation of M²(x) is both as a preferred hyper-complex (commutative) sub-space of M⁸ and as a local plane of non-physical polarizations so that a purely number theoretic interpretation of gauge conditions emerges in TGD framework. This led to a rapid progress in the construction of the quantum TGD. In particular, the challenge of identifying the preferred extremal of Kähler action associated with a given light-like 3-surface X³_l could be solved and the precise relation between M⁸ and M⁴×CP₂ descriptions was understood (see this).

7. Also the challenge of reducing quantum TGD to the physics of second quantized induced spinor fields found a resolution recently (see this). For years ago it became clear that the vacuum functional of the theory must be the Dirac determinant associated with the induced spinor fields so that the theory would predict all coupling parameters from quantum criticality. Even more, the vacuum functional should correspond to the exponent of Kähler action for a preferred extremal. The problem was that the generalized eigenmodes of Chern-Simons Dirac operator allow a generalized eigenvalues to be arbitrary functions of two coordinates in the directions transversal to the light-like direction of X³_l. The progress in the understanding of number theoretic compactification however allowed to understand how the information about the preferred extremal of Kähler action is coded to the spectrum of eigen modes.

   The basic idea is simple and I actually discovered it for more than half decade ago but forgot! The generalized eigen modes of 3-D Chern-Simons Dirac operator D_C-S correspond to the zero modes of a 4-D modified Dirac operator defined by Kähler action localized to X³_l so that induced spinor fields can be seen as 4-D spinorial shock waves. The led to a concrete interpretation of the eigenvalues as analogous to cyclotron energies of fermion in classical electro-weak magnetic fields defined by the induced spinor connection and a connection with anyon physics emerges by 2-dimensionality of the evolving system. Also it was possible to identify the boundary conditions for the preferred extremal of Kähler action -analog of Bohr orbit- at X³_l and also to the vision about how general coordinate invariance allows to use any light-like 3-surface X³ subset X⁴(X³_l) instead of using only wormhole throat to second quantize induced spinor field.

8. It became as a total surprise that due to the huge vacuum degeneracy of induced spinor fields the number of generalized eigenmodes identified in this manner was finite. The good news was that the theory is manifestly finite and zeta function regularization is not needed to define the Dirac determinant. The manifest finiteness had been actually must-be-true from the beginning. The apparently bad news was that the Clifford algebra of WCW world constructed from the oscillator operators is bound to be finite-dimensional. The resolution of the paradox comes from the realization that this algebra represents the somewhat mysterious coset space N/M so that finite measurement resolution and the notion inclusion are coded by the vacuum degeneracy of Kähler action and the maximally economical description in terms of inclusions emerges automatically.

9. A unique identification of number theoretic braids became also possible and relates to the construction of the generalized imbedding space by gluing together singular coverings and factor spaces of CD\M² and CP₂\S²_I to form a book like structure. Here M² is preferred plane of M⁴ defining quantization axis of energy and angular momentum and S²_I is one of the two geodesic sphere of CP₂. The interpretation of the selection of these sub-manifolds is as a geometric correlate for the selection of quantization axes and CD defining basic sector of world of classical worlds is replaced by a union corresponding to these choices. Number theoretic braids come in too variants dual to each other, and correspond to the intersection of M² and M⁴ projection of X³_l on one hand and S²_I and CP₂ projection of X³_l on the other hand. This is simplest option and would mean that the points of number theoretic braid belong to M² (S²_I) and are thus quantum critical although entire X² at the boundaries of CD belongs to a fixed page of the Big Book. This means solution of a long standing problem of understanding in what sense TGD Universe is quantum critical. The phase transitions changing Planck constant correspond to tunneling represented geometrically by a leakage of partonic 2-surface from a page of Big Book to another one.

10. Many other steps of progress have occurred during the last years. Much earlier it had become clear that the basic difference between TGD and string models is that in TGD framework the super algebra generators are non-hermitian and carry quark or lepton number (see this). Super-space concept is un-necessary because super generators anticommute to Hamiltonians of bosonic symmetries rather than corresponding vector fields. This allows to avoid the Majorana condition of super string models fixing space-time dimension to 10 or 11. During last years a much more precise understanding of super-symplectic and super Kac-Moody symmetries has emerged. The generalized coset representation for these two Super Virasoro algebras generalizes Equivalence Principle and predicts as a special case the equivalence of gravitational and inertial masses. Coset construction also provides justification for p-adic thermodynamics in apparent conflict with super-conformal invariance. The construction of the fusion rules of symplectic QFT as analog of conformal QFT led to the notion of number theoretic braid and to an explicit construction of a hierarchy of algebras realizing symplectic fusion rules and the notion of finite measurement resolution (see this). This approach led to the formulation of generalized Feynman diagrams and coupling constant evolution in terms of operads Taylor made for a mathematical realization of the notion of coupling constant evolution. One of the future challenges is to combine symplectic fusion algebras with the realization for the hierarchy of Planck constants.

2. The evolution of physical ideas

The evolution of physical ideas related to the hierarchy of Planck constants and dark matter as a hierarchy of phases of matter with non-standard value of Planck constants was much faster than the evolution of mathematical ideas and quite a number of applications have been developed during last five years.

1. The basic idea was that ordinary matter condenses around dark matter which is a phase of matter characterized by non-standard value of Planck constant.

2. The realization that non-standard values of Planck constant give rise to charge and spin fractionization and anyonization led to the precise identification of the prerequisites of anyonic phase (see this). If the partonic 2-surface, which can have even astrophysical size, surrounds the tip of CD, the matter at the surface is anyonic and particles are confined at this surface. Dark matter could be confined inside this kind of light-like 3-surfaces around which ordinary matter condenses. If the radii of the basic pieces of these nearly spherical anyonic surfaces - glued to a connected structure by flux tubes mediating gravitational interaction - are given by Bohr rules, the findings of Nottale can be understood. Dark matter would resemble to a high degree matter in black holes replaced in TGD framework by light-like partonic 2-surfaces with minimum size of order Schwarstchild radius r_S of order scaled up Planck length: r_S ~ (hbar G)^1/2. Black hole entropy being inversely proportional to hbar is predicted to be of order unity so that dramatic modification of the picture about black holes is implied.

3. Darkness is a relative concept and due to the fact that particles at different pages of book cannot appear in the same vertex of the generalized Feynman diagram. The phase transitions in which partonic 2-surface X² during its travel along X³_l leaks to different page of book are however possible and change Planck constant so that particle exchanges of this kind allow particles at different pages to interact. The interactions are strongly constrained by charge fractionization and are essentially phase transitions involving many particles. Classical interactions are also possible. This allows to conclude that we are actually observing dark matter via classical fields all the time and perhaps have even photographed it (see this).

4. Perhaps the most fascinating applications are in biology. The anomalous behavior ionic currents through cell membrane (low dissipation, quantal character, no change when the membrane is replaced with artificial one) has a natural explanation in terms of dark supra currents. This leads to a vision about how dark matter and phase transitions changing the value of Planck constant could relate to the basic functions of cell, functioning of DNA and aminoacids, and to the mysteries of bio-catalysis. This leads also a model for EEG interpreted as a communication and control tool of magnetic body containing dark matter and using biological body as motor instrument and sensory receptor. One especially shocking outcome is the emergence of genetic code of vertebrates from the model of dark nuclei as nuclear strings (see this).

3. Brief summary about the generalization of the imbedding space concept

A brief summary of the basic vision in order might help reader to assimilate the more detailed representation about the generalization of imbedding space.

1. The hierarchy of Planck constants cannot be realized without generalizing the notions of imbedding space and space-time since particles with different values of Planck constant cannot appear in the same interaction vertex. This suggests some kind of book like structure for both M⁴ and CP₂ factors of the generalized imbedding space is suggestive.

2. Schrödinger equation suggests that Planck constant corresponds to a scaling factor of M⁴ metric whose value labels different pages of the book. The scaling of M⁴ coordinate so that original metric results in M⁴ factor is possible so that the scaling of hbar corresponds to the scaling of the size of causal diamond CD defined as the intersection of future and past directed light-cones. The light-like 3-surfaces having their 2-D and light-boundaries of CD are in a key role in the realization of zero energy states. The infinite-D spaces formed by these 3-surfaces define the fundamental sectors of the configuration space (world of classical worlds). Since the scaling of CD does not simply scale space-time surfaces, the coding of radiative corrections to the geometry of space-time sheets becomes possible and Kähler action can be seen as expansion in powers of hbar/hbar₀.

3. Quantum criticality of TGD Universe is one of the key postulates of quantum TGD. The most important implication is that Kähler coupling strength is analogous to critical temperature. The exact realization of quantum criticality would be in terms of critical sub-manifolds of M⁴ and CP₂ common to all sectors of the generalized imbedding space. Quantum criticality would mean that the two kinds of number theoretic braids assignable to M⁴ and CP₂ projections of the partonic 2-surface belong by the definition of number theoretic braids to these critical sub-manifolds. At the boundaries of CD associated with positive and negative energy parts of zero energy state in given time scale partonic two-surfaces belong to a fixed page of the Big Book whereas light-like 3-surface decomposes into regions corresponding to different values of Planck constant much like matter decomposes to several phases at thermodynamical criticality.

4. The connection with Jones inclusions was originally a purely heuristic guess based on the observation that the finite groups characterizing Jones inclusion characterize also pages of the Big Book. The key observation is that Jones inclusions are characterized by a finite subgroup G Ì SU(2) and that this group also characterizes the singular covering or factor spaces associated with CD or CP₂ so that the pages of generalized imbedding space could indeed serve as correlates for Jones inclusions. The elements of the included algebra M are invariant under the action of G and M takes the role of complex numbers in the resulting non-commutative quantum theory.

5. The understanding of quantum TGD at parton level led to the realization that the dynamics of Kähler action realizes finite measurement resolution in terms of finite number of modes of the induced spinor field. This automatically implies cutoffs to the representations of various super-conformal algebras typical for the representations of quantum groups closely associated with Jones inclusions. The Clifford algebra spanned by the fermionic oscillator operators would provide a realization for the factor space N/M of hyper-finite factors of type II₁ identified as the infinite-dimensional Clifford algebra N of the configuration space and included algebra M determining the finite measurement resolution. The resulting quantum Clifford algebra has anti-commutation relations dictated by the fractionization of fermion number so that its unit becomes r=hbar/hbar₀. SU(2) Lie algebra transforms to its quantum variant corresponding to the quantum phase q=exp(i2p/r).

6. Jones inclusions appear as two variants corresponding to N:M < 4 and N:M=4. The tentative interpretation is in terms of singular G-factor spaces and G-coverings of M⁴ or CP₂ in some sense. The alternative interpretation in terms of two geodesic spheres of CP₂ would mean asymmetry between M⁴ and CP₂ degrees of freedom.

7. Number theoretic Universality suggests an answer why the hierarchy of Planck constants is necessary. One must be able to define the notion of angle -or at least the notion of phase and of trigonometric functions- also in p-adic context. All that one can achieve naturally is the notion of phase defined as root of unity and introduced by allowing algebraic extension of p-adic number field by introducing the phase if needed. In the framework of TGD inspired theory of consciousness this inspires a vision about cognitive evolution as the gradual emergence of increasingly complex algebraic extensions of p-adic numbers and involving also the emergence of improved angle resolution expressible in terms of phases exp(i2p/n) up to some maximum value of n. The coverings and factor spaces would realize these phases geometrically and quantum phases q naturally assignable to Jones inclusions would realize them algebraically. Besides p-adic coupling constant evolution based on hierarchy of p-adic length scales there would be coupling constant evolution with respect to hbar and associated with angular resolution.

For the updated version of the chapter see Does TGD Predict a Spectrum of Planck Constants? of "Towards S-matrix".

Unidentified spectral noise as a new experimental support for quantum TGD?

The news of yesterday morning came in email from Jack Sarfatti. The news was that gravitational detectors in GEO600 experiment have been plagued by unidentified noise in the frequency range 300-1500 Hz. Craig J. Hogan has proposed an explanation in terms of holographic Universe. By reading the paper I learned that assumptions needed are essentially those of quantum TGD. Light-like 3-surfaces as basic objects, holography, effective 2-dimensionality, are some of the terms appearing repeatedly in the article.

Maybe this means a new discovery giving support for TGD. I hope that it does not make my life even more difficult in Finland. Readers have perhaps noticed that the discovery of new longlived particle in CDF predicted by TGD already around 1990 turned out to be one of most fantastic breakthroughs of TGD since the reported findings could be explained at quantitative level. The side effect was that Helsinki University did not allow me to use the computer for homepage anymore and they also refused to redirect visitors to my new homepage. The goal was achieved: I have more or less disappeared from the web. It seems that TGD is becoming really dangerous and power holds of science are getting nervous.

In any case, I could not resist the temptation to spend the day with this problem although I had firmly decided to use all my available time to the updating of basic chapters of quantum TGD.

1. The experiment

Consider first the graviton detector used in GEO600 experiment. The detector consists of two long arms (the length is 600 meters)- essentially rulers of equal length. The incoming gravitational wave causes a periodic stretch of the arms: the lengths of the rulers vary. The detection of gravitons means that laser beam is used to keep record about the varying length difference. This is achieved by splitting the laser beam into two pieces using a beam splitter. After this the beams travel through the arms and bounce back to interfere in the detector. Interference pattern tells whether the beam spent slightly different times in the arms due to the stretching of arm caused by the incoming gravitational radiation. The problem of experimenters has been the presence of an unidentified noise in the range 100-1500 Hz.

The prediction of Measurement of quantum fluctuations in geometry by Craig Hogan published in Phys. Rev. D 77, 104031 (2008) is that holographic geometry of space-time should induce fluctuations of classical geometry with a spectrum which is completely fixed . Hogan's prediction is very general and - if I have understood correctly - the fluctuations depend only on the duration (or length) of the laser beam using Planck length as a unit. Note that there is no dependence on the length of the arms and the fluctuations characterize only the laser beam. Although Planck length appears in the formula, the fluctuations need not have anything to with gravitons but could be due to the failure of the classical description of laser beams. The great surprise was that the prediction of Hogan for the noise is of same order of magnitude as the unidentified noise bothering experiments in the range 100-700 Hz.

2. Hogan's theory

Let us try to understand Hogan's theory in more detail.

1. The basic quantitative prediction of the theory is very simple. The spectral density of the noise for high frequencies is given by h_H= t_P^1/2, where t_P=(hbar G)^1/2 is Planck time. For low frequencies h_H is proportional to 1/f just like 1/f noise. The power density of the noise is given by t_P and a connection with poorly understood 1/f noise appearing in electronic and other systems is suggestive. The prediction depends only Planck scale so that it should very easy to kill the model if one is able to reduce the noise from other sources below the critical level t_P^1/2. The model predicts also the the distribution characterizing the uncertainty in the direction of arrival for photon in terms of the ratio l_P/L. Here L is the length or beam of equivalently its duration. A further prediction is that the minimal uncertainty in the arrival time of photons is given by Δ t= (t_Pt)^1/2 and increases with the duration of the beam.

2. Both quantum and classical mechanisms are discussed as an explanation of the noise. Gravitational holography is the key assumption behind both models. Gravitational holography states that space-time geometry has two space-time dimensions instead of three at the fundamental level and that third dimension emerges via holography. A further assumption is that light-like (null) 3-surfaces are the fundamental objects. Sounds familiar!

2.1 Heuristic argument

The model starts from an optics inspired heuristic argument.

1. Consider a light ray with length L, which ends to aperture of size D. This gives rise to a diffraction spot of size λL/D. The resulting uncertainty of the transverse position of source is minimized when the size of diffraction spot is same as aperture size. This gives for the transverse uncertainty of the position of source Δ x= (λ L)^1/2. The orientation of the ray can be determined with a precision Δ θ= (λ/L)^1/2. The shorter the wavelength the better the precision. Planck length is believed to pose a fundamental limit to the precision. The conjecture is that the transverse indeterminacy of Planck wave length quantum paths corresponds to the quantum indeterminacy of the metric itself. What this means is not quite clear to me.

2. The basic outcome of the model is that the uncertainty for the arrival times of the photons after reflection is proportional to

   Δ t =t_P^1/2× (sin(θ))^1/2×sin(2θ),

   where θ denotes the angle of incidence on beam splitter. In normal direction Δ t vanishes. The proposed interpretation is in terms of Brownian motion for the distance between beam splitter and detector the interpretation being that each reflection from beam splitter adds uncertainty. This is essentially due to the replacement of light-like surface with a new one orthogonal to it inducing a measurement of distance between detector and bean splitter.

This argument has some aspects which I find questionable.

1. The assumption of Planck wave length waves is certainly questionable. The underlying is that it lead to the classical formula involving the aperture size which is eliminated from the basic formula by requiring optimal angular resolution. One might argue that a special status of waves with Planck wave length breaks Lorentz invariance but since the experimental apparatus defines a preferred coordinate system this ned not be a problem.

2. Unless one is ready to forget the argument leading to the formula for Δ θ, one can argue that the description of the holographic interaction between distant points induced by these Planck wave length waves in terms of aperture with size D= (l_PL)^1/2 should have some more abstract physical counterpart. Could elementary particles as extended 2-D objects (as in TGD) play the role of ideal apertures to which a radiation with Planck wave length arrives? If one gives up the assumption about Planck wave radiation the uncertainty increases as λ. To my opinion one should be able to deduced the basic formula without this kind of argument.

2.2 Argument based on uncertainty principle for waves with Planck wave length

Second argument can do without diffraction but still uses Planck wave length waves.

1. The interactions of Planck wave length radiation at null surface at two different times corresponding to normal coordinates z₁ and z₂ at these times are considered. From the standard uncertainty relation between momentum and position of the incoming particle one deduces uncertainty relation for transverse position operators x(z_i), i=1,2. The uncertainty comes from uncertainty of x(z₂) induced by uncertainty of the transverse momentum p_x(z_i). The uncertainty relation is deduced by assuming that (x(z₂)-x(z₁)/(z₂-z₁) is the ratio of transversal and longitudinal wave vectors. This relates x(z₂) to p_x(z_i) and the uncertainty relation can be deduced. The uncertainty increases linearly with z₂-z₁. Geometric optics is used to describing the propagating between the two points and this should certainly work for a situation in which wavelength is Planck wavelength if the notion of Planck wave length wave makes sense. From this formula the basic predictions follow.

2. Hogan emphasizes that the basic result is obtained also classically by assuming that light-like surfaces describing the propagation of light between ends points of arm describe Brownian like random walk in directions transverse to the direction of propagation. I understand that this means that Planck wave length wave is not absolutely necessary for this approach.

2.3 Description in terms of equivalent gravitonic wave packet

Hogan discusses also an effective description of holographic noise in terms of gravitational wave packet passing through the system.

1. The holographic noise at frequency f has equivalent description in terms of a gravitational wave packet of frequency f and duration T=1/f passing through the system. In this description the variance for the length difference of arms using standard formula for gravitational wave packet

   Δl²/l²= h²f ,

   where h characterizes the spectral density of gravitational wave.

2. For high frequencies one obtains

   h= h_P= (t_P)^1/2 .

3. For low frequencies the model predicts

   h= (f_res/f)(t_P)^1/2 .

   Here f_res characterized the inverse residence time in detector and is estimated to be about 700 Hz in GEO600 experiment.

4. The predictions of the theory are compared to the unidentified noise in the frequency range 100-600 Hz which introduces amplifying factor varying from 7 to 1. The orders of magnitude are same.

3. TGD based model

In TGD based model for the claimed noise on can avoid the assumption about waves with Planck wave length. Rather Planck length corresponds to the transversal cross section of so called massless extremals (MEs) assignable to MEs and orthogonal to the direction of propagation. Further elements are so called number theoretic braids leading to the discretization of quantum TGD at fundamental level. The mechanism inducing the distribution for the travel times of reflected photon is due to the transverse extension of MEs, discretization in terms of number theoretic braids. Note that also in Hogan's model it is essential that one can speak about position of particle in the beam.

3.1 Some background

Consider first the general picture behind the TGD inspired model.

1. What authors emphasize can be condensed to the following statement: The transverse indeterminacy of Planck wave length seems likely to be a feature of 3+1 D space-time emerge is as a dual of quantum theory on a 2+1-D null surface. In TGD light-like 3-surfaces indeed are the fundamental objects and 4-D space-time surface is in a holographic relation to these light-like 3-surfaces. The analog of conformal invariance in light-like radial direction implies that partonic 2-surfaces are actually basic objects in short scales in the sense that one 3-dimensionality only in discretized sense.

2. Both the interpretation as almost topological quantum field theory, the notion of finite measurement resolution, number theoretical universality making possible p-adicization of quantum TGD, and the notion of quantum criticality lead to a fundamental description in terms of discrete points sets. These are defined as intersections of what I call number theoretic braids with partonic 2-surfaces X² at the boundaries of causal diamonds identified as intersections of future and paste directed light-cones forming a fractal hierarchy. These 2-surfaces X² correspond to the ends of light-like three surfaces. Only the data from this discrete point set is used in the definition of M-matrix: there is however continuum of selections of this data set corresponding to different directions of light-like ray at the boundary of light-cone, and in detection one of these direction is selected and corresponds to the direction of beam in the recent case.

3. Fermions correspond to CP₂ vacuum extremal with Euclidian signature of induced metric condensed to space-time sheet with Minkowskian signature and light-like wormhole throat for which 4-metric is degenerate carries the quantum numbers. Bosons correspond to wormhole contacts consisting of a piece of CP₂ vacuum extremal connecting two two space-time sheets with Minkowskian signature of induced metric. The strands of number theoretic braids carry fermionic quantum numbers and discretization is interpreted as a space-time correlate for the finite measurement resolution implying the effective grainy nature of 2-surfaces.

3.2 The model

Consider now the TGD inspired model for a laser beam of fixed duration T.

1. In TGD framework the beams of photons and perhaps also photons themselves would have so called massless extremals as space-time correlates. The identification of gauge bosons as wormhole contacts means that there is a pair of MEs connected by a pieces of CP₂ type vacuum extremal and carrying fermion and antifermion at the wormhole throats defining light-like 3-surfaces. The intersection of ME with light-cone boundary would represent partonic 2-surface and any transverse cross section of the M⁴ projection of ME is possible.

2. The reflection of ME has description in terms of generalized Feynman diagrams for which the incoming lines correspond to the light-like three surfaces and vertices to partonic 2-surfaces at which the MEs are glued together. In this simplest model this surface defines transverse cross section of both incoming and outgoing ME. The incoming and outgoing braid strands end to different points of the cross section because if two points coincide the N-point correlation function vanishes. This means that in the reflection the distribution for the positions of braid points representing exact positions of photon change in non-deterministic manner. This induces a quantum distribution of transverse coordinates associated with braid strands and in the detection state function reduction occurs fixing the position of braid strands.

3. The transversal cross section has maximum area when it is parallel to ME. In this case the area is apart from a numerical constant equal to d×L, L the length defined by the duration of laser beam defining the length of ME and d the diameter of orthogonal cross section of ME. This makes natural the assumption about Gaussian distribution for the positions of points in the cross section as Gaussian with variance equal to d×L. The distribution proposed by Hogan is obtained if d is given by Planck length. This would mean that the minimum area for a cross section of ME is very small, about S=hbar×G. This might make sense if the ME represents laser beam.

4. The assumption susceptible to criticism is that for the primordial ME representing photon the area of cross section orthogonal to the direction of propagation is assumed to be always given by Planck length. This assumption of course replaces Hogan's Planck wave. Note that the classical four-momentum of ME is massless. One could however argue that in quantum situation transverse momentum square is well defined quantum number and of order Planck mass mass squared.

5. In TGD Universe single photon would differ from infinitely narrow ray by having thickness defined by Planck length. There would be just single braid strand and its position would change in the reflection. The most natural interpretation indeed is that the pair of space-time sheets associated with photon consists of MEs with different transversal size scales: larger ME could represent laser beam. The noise would come from the lowest level in the hierarchy. One could argue that the natural size for M⁴ projection of wormhole throat is of order CP₂ size R and therefore roughly 10⁴ Planck lengths. If the cross section has area of order R², where R is CP₂ size, the spectral density would be roughly by a factor 100 larger than for Planck length and this might predict too large holographic noise in GEO600 experiment if the value of f_res is correct. The assumption that the Gaussian characterizing the position distribution of the wormhole throat is very strongly concentrated near the center of ME with transverse size given by R looks un-natural.

6. It is important to notice that single reflection of primordial ME corresponds to a minimum spectral noise. Repeated reflections of ME in different directions gradually increase the transversal size of ME so that the outcome is cylindrical ME with radius of order L =cT, where T is the duration of ME. At this limit the spectral density of noise would be T^1/2 meaning that the uncertainty in the frequency assignable to the arrival time of photons would of same order as the oscillation period f=1/T assignable to the original ME. The interpretation is that the repeated reflections gradually generate noise and destroy the coherence of the laser beam. This would however happen at single particle level rather than for a member of fictive ensemble. Quite literally, photon would get old! This interpretation conforms with the fact that in TGD framework thermodynamics becomes part of quantum theory and thermodynamical ensemble is represented at single particle level in the sense and time like entanglement coefficients between positive and negative energy parts of zero energy state define M-matrix as a product of square root of diagonal density matrix and of S-matrix.

7. The notion of number theoretic braid is essential for the interpretation for what happens in detection. In detection the positions of ends of number theoretic braid are measured and this measurement fixes the exact time spent by photons during the travel. Similar position measurement appears also in Hogan's argument. Thus the overall picture is more or less same as in the popular representation where also the grainy nature of space-time is emphasized.

8. I already mentioned the possible connection with poorly understood 1/f noise appearing in very many systems. The natural interpretation would be in terms of MEs.

3.3 The relationship with hierarchy of Planck constants

It is interesting to combine this picture with the vision about the hierarchy of Planck constants (I am just now developing in detail the representation of the ideas involved from a perspective given by the intense work during last five years).

1. If one accepts that dark matter corresponds to a hierarchy of phases of matter labeled by a hierarchy of Planck constants with arbitrarily large values, one must conclude that Planck length l_P proportional to hbar^1/2, has also spectrum. Primordial photons would have transversal size scalings as hbar^1/2. One can consider the possibility that for large values of hbar the transversal size saturates to CP₂ length R ≈10⁴× l_P. The spectral density of the noise would scale as hbar^1/4 at least up to the critical value hbar_cr=R²/G, which is in the range [2.3683, 2.5262]× 10⁷. The preferred values of hbar number theoretically simple integers expressible as product of distinct Fermat primes and power of 2. hbar_crit/hbar₀=3× 2²³ is integer of this kind and belongs to the allowed range of critical values.

2. The order of magnitude for gravitational Planck constant assignable to the space-time sheets mediating gravitational interaction is gigantic - of order hbar_gr≈ GM² - so that the noise assignable to gravitons would be gigantic in astrophysical scales unless R serves as the upper bound for the transverse size of both primordial gauge bosons and gravitons.

3. If ordinary photonic space-time sheets are in question hbar has its standard value. For dark photons which I have proposed to play a key role in living matter, the situation changes and Δl²/l² would scale like hbar^1/2 at least up to critical value of Planck constant. Above this value of Planck constant spectral density would be given by R and Δl²/l² would scale like R/l and Δ θ like (R/l)^1/2.

For details and background see the updated chapter Quantum Astrophysics of "Physics in Many-Sheeted Space-time".

Antimatter as dark matter?

Intuitively the scaling of Planck constant scales up quantum lengths, in particular the size of causal diamond CD defined as intersection of future and past directd lightcones. This looks trivial but one one must describe precisely what is involved to check internal consistency and also to understand how to model the quantum phase transitions changing Planck constant. It turns out that the back of the Big Book along with CDs is analogous to Josephson junction which in presence of dissipation leads to the separation of charges to different pages. This might relate to the generation of matter antimatter asymmetry of the visible matter.

The first manner to understand the situation is to consider CD with a fixed range of M⁴ coordinates. The scaling up of the covariant Kähler metric of CD by r²=(hbar/hbar₀)² scales up the size of CD by r. Another manner to see the situation is by scaling up the linear M⁴ coordinates by r for the larger CD so that M⁴ metric becomes same for both CDs. The smaller CD is glued to the larger one isometrically together along (M²ÇCD) Ì CD anywhere in the interior of the larger CD. What happens is non-trivial for the following reasons.

1. The singular coverings and factor spaces are different and M⁴ scaling is not a symmetry of the Kähler action so that the preferred extrema in the two cases do not relate by a simple scaling. The interpretation is in terms of the coding of the radiative corrections in powers of hbar to the shape of the preferred extremals. This becomes clear from the representation of Kähler action in which M⁴ coordinates have the same range for two CDs but M⁴ metric differs by r² factor.

2. In common M⁴ coordinates the M⁴ gauge part A_a of CP₂ Kähler potential for the larger CD differs by a factor 1/r from that for the smaller CD. This guarantees the invariance of four-momentum assignable to Chern-Simons action in the phase transition changing hbar. The resulting discontinuity of A_a at M² is analogous to a static voltage difference between the two CDs and M² could be seen as an analog of Josephson junction. In absence of dissipation (expected in quantum criticality) the Kähler voltage could generate oscillatory fermion, em, and Z⁰ Josephson currents between the two CDs. Since Kähler gauge potential couples to quarks and leptons with opposite signs the current would be in opposite directions for quarks and leptons as well as for matter and antimatter. In presence of dissipation the currents would be ohmic and could force quarks and leptons and matter and antimatter to different pages of the Big Book and quarks inside hadrons would have nonstandard value of Planck constant.

3. The discontinuities of A_u and A_f allow to assign electric and magnetic Kähler point charges Q_K^e/m with M¹ Ì M² and having sign opposite to those assignable with dCD×CP₂. It should be possible to identify physically M², the line E¹ representing quantization axis of angular momentum, and the position of Q_K.

For details and background see the updated chapter Quantum Hall effect and Hierarchy of Planck Constants of "Physics in Many-Sheeted Space-time".