Sunday, June 29, 2008

Is N=8 super-gravity finite and why this would be the case?

There is an interesting discussion in the blog of Peter Woit about the recent evidence for the finiteness of N=8 super-gravity and perhaps even N<8 super-gravities. I have nothing to say about horrible technicalities of these calculations and I can only admire from distance what the people involved are able to do. Nothing however prevents me to make some uneducated comments from TGD point of view.

Theoretical physicist as a thinker or as a computational virtuoso?

Certainly the cancelation of infinities would be a proof for the magic mathematical power of deep principles like General Coordinate Invariance, Equivalence Principle, and Super Symmetry. That Einstein with his not so fantastic calculational skills discovered the first two principles should make clear that it is conceptual thinking that matters most even in theoretical physics. The return to the study of N=8 super-gravity after 24 years of super-string models and the possible reduction of super string models to a mere auxiliary calculational tool would be a clear-cut answer to the question "Theoretical physics as the art of conceptualization or as mere computational methodology?".

What could be the symmetry behind cancellations of infinities in N=8 supergravities?

One question represented in discussions concerns the underlying symmetry causing the cancelation of the perturbative infinities.

  1. In the latest postings (see this, this,and this) I have told about how the notion symplectic quantum field theory emerges from TGD. These theories are highly analogous to conformal field theories. Fusion rules generalize and N-point functions are proportional to symplectic invariants assignable to the set of points defined by N arguments. The basic symplectic invariants are the symplectic areas of the geodesic triangles defined by 3-point subsets of the arguments of N-point function. Under very general assumptions these invariants vanish when two of N arguments co-incide because the areas of the triangles containing these two points vanish. The behavior of N-point functions is smoothed out so that the artificial UV cutoff is replaced with a dynamical one. Infrared cutoff comes in turn from the size of the causal diamond serving as imbedding space correlate for the zero energy state. One can thus get rid of local divergences as very general arguments based on the absence of local interaction vertices and Kähler geometry of the "world of classical worlds" indeed predicted long time ago.

  2. Very naively, the quantum field theory limit of TGD obtained by replacing the light-like 3-surfaces appearing as "lines" and 2-D partonic surfaces appearing as the vertices of generalized Feynman diagrams are taken to lines and points. The limiting theory could still have symplectic invariance as a hidden symmetry of N-point functions.

The obvious question is whether the N-point functions of N≤8 super-gravities have this kind of symplectic symmetry as a hidden symmetry? Note that N=4 super-conformal symmetry is the most natural candidate for the conformal super-symmetry of TGD and that TGD does not predict super-symmetry at the level of Poincare algebra so that no sparticles are predicted.

Why TGD is needed?

It is very difficult to understand how N=8 super-gravity could be consistent with standard model and my strong conviction is that one cannot avoid TGD if one is interested in physics.

  1. My belief is that it is the generalization of super-conformal invariance provided by the replacement of strings with light-like 3-surfaces as fundamental objects in H=M4×CP2 which is needed. The maximal conformal symmetries are obtained only for 4-D space-time and 4-D Minkowski space.

  2. "Number theoretical compactification" -or more precisely, the duality of pictures based on the identification of 8-D imbedding space as hyper-octonionic space-time HO= M8 or as H=M4×CP2 is absolutely essential for the recent formulation of quantum TGD and most "must-be-trues" of quantum TGD follow from the basic prerequisite of this duality. More precisely: space-time surfaces identified as hyper-quaternionic surfaces of HO contain preferred hyper-complex (commutative) plane HC=M2 defining the plane of non-physical polarizations in their hyper-quaternionic tangent space M4: this kind of tangent spaces are parametrized by CP2 which implies the duality.

  3. One implication of the existence of preferred plane M2 is justification for what I call generalized coset construction (see this) providing an elegant generalization of Einstein's equations replacing them with the condition that the super Virasoro algebras associated with the symplectic analog of Kac-Moody algebra and the ordinary Kac-Moody algebra defined by a symplectic sub-group of isometries of the imbedding space act in the same manner on physical states (the isometric Super Virasoro corresponds to the gravitational four-momentum and symplectic Super Virasoro to the inertial four-momentum). This provides the long sought-for explicit realization of Equivalence Principle in TGD framework.

I dare safely say that there is only a single possible choice of the imbedding space and this choice is consistent with the standard model. The geometry of classical spinor fields in the world of classical worlds and thus also physics is unique just from its mathematical existence, and physics as a generalized number theory vision allows to identify this unique and very special world of classical worlds (as does also the inspection of Particle Data Tables;-)). This motivates my humble suggestion for colleagues: why not to accept the facts and start to transform TGD to mathematical physics. A good place to start would be construction and classification of symplectic quantum field theories.

For a summary of quantum TGD see the article Topological Geometrodynamics: What Might Be the First Principles?.

Wednesday, June 25, 2008

The relationship between super-canonical and Super Kac-Moody algebras, Equivalence Principle, and justification of p-adic thermodynamics

The relationship between super-canonical algebra (SC) acting at light-cone boundary and Super Kac-Moody algebra (SKM) acting on light-like 3-surfaces has remained somewhat enigmatic due to the lack of physical insights. This is not the only problem. The question to precisely what extent Equivalence Principle (EP) remains true in TGD framework and what might be the precise mathematical realization of EP is waiting for an answer. Also the justification of p-adic thermodynamics for the scaling generator L0 of Virasoro algebra -in obvious conflict with the basic wisdom that this generator should annihilate physical states- is lacking. It seems that these three problems could have a common solution.

Before going to describe the proposed solution, some background is necessary. The latest proposal for SC-SKM relationship relies on non-standard and therefore somewhat questionable assumptions.

  1. SKM Virasoro algebra (SKMV) and SC Virasoro algebra (SCV) (anti)commute for physical states.

  2. SC algebra generates states of negative conformal weight annihilated by SCV generators Ln, n < 0, and serving as ground states from which SKM generators create states with non-negative conformal weight.

This picture could make sense for elementary particles. On other hand, the recent model for hadrons [see this] assumes that SC degrees of freedom contribute about 70 per cent to the mass of hadron but at space-time sheet different from those assignable to quarks. The contribution of SC degrees of freedom to the thermal average of the conformal weight would be positive. A contradiction results unless one assumes that there exists also SCV ground states with positive conformal weight annihilated by SCV elements Ln, n < 0, but also this seems implausible.

1. New vision about the relationship between SCV and SKMV

Consider now the new vision about the relationship between SCV and SKMV.

  1. The isometries of H assignable with SKM are also symplectic transformations [see this] (note that I have used the term canonical instead of symplectic previously). Hence might consider the possibility that SKM could be identified as a subalgebra of SC. If this makes sense, a generalization of the coset construction obtained by replacing finite-dimensional Lie group with infinite-dimensional symplectic group suggests itself. The differences of SCV and SKMV elements would annihilate physical states and (anti)commute with SKMV. Also the generators On, n > 0, for both algebras would annihilate the physical states so that the differences of the elements would annihilate automatically physical states for n > 0.

  2. The super-generator G0 contains the Dirac operator D of H. If the action of SCV and SKMV Dirac operators on physical states are identical then cm of degrees of freedom disappear from the differences G0(SCV)-G0(SKMV) and L0(SCV)-L0(SKMV). One could interpret the identical action of the Dirac operators as the long sought-for precise realization of Equivalence Principle (EP) in TGD framework. EP would state that the total inertial four-momentum and color quantum numbers assignable to SC (imbedding space level) are equal to the gravitational four-momentum and color quantum numbers assignable to SKM (space-time level). Note that since super-canonical transformations correspond to the isometries of the "world of classical worlds" the assignment of the attribute "inertial" to them is natural.

  3. The analog of coset construction applies also to SKM and SC algebras which means that physical states can be thought of as being created by an operator of SKM carrying the conformal weight and by a genuine SC operator with vanishing conformal weight. Therefore the situation does not reduce to that encountered in super-string models.

This picture provides also a justification for p-adic thermodynamics.

  1. In physical states the p-adic thermal expectation value of the SKM and SC conformal weights would be non-vanishing and identical and mass squared could be identified to the expectation value of SKM scaling generator L0. There would be no need to give up Super Virasoro conditions for SCV-SKMV.

  2. There is consistency with p-adic mass calculations for hadrons [see this] since the non-perturbative SC contributions and perturbative SKM contributions to the mass correspond to space-time sheets labeled by different p-adic primes. The earlier statement that SC is responsible for the dominating non-perturbative contributions to the hadron mass transforms to a statement reflecting SC-SKM duality. The perturbative quark contributions to hadron masses can be calculated most conveniently by using p-adic thermodynamics for SKM whereas non-perturbative contributions to hadron masses can be calculated most conveniently by using p-adic thermodynamics for SC. Also the proposal that the exotic analogs of baryons resulting when baryon looses its valence quarks [see this] remains intact in this framework.

  3. The results of p-adic mass calculations depend crucially on the number N of tensor factors contributing to the Super-Virasoro algebra. The required number is N=5 and during years I have proposed several explanations for this number. It seems that holonomic contributions that is electro-weak and spin contributions must be regarded as contributions separate from those coming from isometries. SKM algebras in electro-weak degrees and spin degrees of of freedom, would give 2+1=3 tensor factors corresponding to U(2)ew×SU(2). SU(3) and SO(3) (or SO(2) Ì SO(3) leaving the intersection of light-like ray with S2 invariant) would give 2 additional tensor factors. Altogether one would indeed have 5 tensor factors.

2. Can SKM be lifted to a sub-algebra of SC?

A picture introducing only a generalization of coset construction as a new element, realizing mathematically Equivalence Principle, and justifying p-adic thermodynamics is highly attractive but there is a problem. SKM is defined at light-like 3-surfaces X3 whereas SC acts at light-cone boundary dH±=dM4±×CP2. One should be able to lift SKM to imbedding space level somehow. Also SC should be lifted to entire H. This problem was the reason why I gave up the idea about coset construction and SC-SKM duality as it appeared for the first time.

A possible solution of the lifting problem comes from the observation making possible a more rigorous formulation of HO-H duality stating that one can regard space-time surfaces either as surfaces in hyper-octonionic space HO=M8 or in H=M4×CP2 [see this]. Consider first the formulation of HO-H duality.

  1. Associativity also in the number theoretical sense becomes the fundamental dynamical principle if HO-H duality holds true [see this]. For a space-time surface X4 Ì HO=M8 associativity is satisfied at space-time level if the tangent space at each point of X4 is some hyper-quaternionic sub-space HQ=M4 Ì M8. Also partonic 2-surfaces at the boundaries of causal diamonds formed by pairs of future and past directed light-cones defining the basic imbedding space correlate of zero energy state in zero energy ontology and light-like 3-surfaces are assumed to belong to HQ=M4 Ì HO.

  2. HO-H duality requires something more. If the tangent spaces contain the same preferred commutative and thus hyper-complex plane HC=M2, the tangent spaces of X4 are parameterized by the points s of CP2 and X4 Ì HO can be mapped to X4 Ì M4×CP2 by assigning to a point of X4 regarded as point (m,e) of M40×E4=M8 the point (m,s). Note that one must also fix a preferred global hyper-quaternionic subspace M40 Ì M8 containing M2 to be not confused with the local tangent planes M4.

  3. The preferred plane M2 can be interpreted as the plane of non-physical polarizations so that the interpretation as a number theoretic analog of gauge conditions posed in both quantum field theories and string models is possible.

  4. An open question is whether the resulting surface in H is a preferred extremal of Kähler action. This is possible since the tangent spaces at light-like partonic 3-surfaces are fixed to contain M2 so that the boundary values of the normal derivatives of H coordinates are fixed and field equations fix in the ideal case X4 uniquely and one obtains space-time surface as the analog of Bohr orbit.

  5. The light-like "Higgs term" proportional to O=gktk appearing in the generalized eigenvalue equation for the modified Dirac operator [see this] is an essential element of TGD based description of Higgs mechanism. This term can cause complications unless t is a covariantly constant light-like vector. Covariant constancy is achieved if t is constant light-like vector in M2. The interpretation as a space-time correlate for the light-like 4-momentum assignable to the parton might be considered.

  6. Associativity requires that the hyper-octonionic arguments of N-point functions in HO description are restricted to a hyperquaternionic plane HQ=M4 Ì HO required also by the HO-H correspondence. The intersection M4Çint(X4) consists of a discrete set of points in the generic case. Partonic 3-surfaces are assumed to be associative and belong to M4. The set of commutative points at the partonic 2-surface X2 is discrete in the generic case whereas the intersection X3ÇM2 consists of 1-D curves so that the notion of number theoretical braid crucial for the p-adicization of the theory as almost topological QFT is uniquely defined.

  7. The preferred plane M2 Ì M4 Ì HO can be assigned also to the definition of N-point functions in HO picture. It is not clear whether it must be same as the preferred planes assigned to the partonic 2-surfaces. If not, the interpretation would be that it corresponds to a plane containing the over all cm four-momentum whereas partonic planes M2i would contain the partonic four-momenta. M2 is expected to change at wormhole contacts having Euclidian signature of the induced metric representing horizons and connecting space-time sheets with Minkowskian signature of the induced metric.

The presence of globally defined plane M2 and the flexibility provided by the hyper-complex conformal invariance raise the hopes of achieving the lifting of SC and SKM to H. At the light-cone boundary the light-like radial coordinate can be lifted to a hyper-complex coordinate defining coordinate for M2. At X3 one can fix the light-like coordinate varying along the braid strands can be lifted to some hyper-complex coordinate of M2 defined in the interior of X4. The total four-momenta and color quantum numbers assignable to the SC and SKM degrees of freedom are naturally identical since they can be identified as the four-momentum of the partonic 2-surface X2 Ì X3ÇdM4±×CP2. Equivalence Principle would emerge as an identity.

3. Questions

There are still several open questions.

  1. Is it possible to define hyper-quaternionic variants of the superconformal algebras in both H and HO or perhaps only in HO. A positive answer to this question would conform with the conjecture that the geometry of "world of classical worlds" allows Hyper-Kähler property in either or both pictures [see this].

  2. How this picture relates to what is known about the extremals of field equations [see this] characterized by generalized Hamilton-Jacobi structure bringing in mind the selection of preferred M2?

  3. Is this picture consistent with the views about Equivalence Principle and its possible breaking based on the identification of gravitational four-momentum in terms of Einstein tensor is interesting [see this] ?

For more details see the chapter Massless particles and particle massivation.

Wednesday, June 18, 2008

Still more detailed view about the construction of M-matrix elements

After three decades there are excellent hopes of building an explicit recipe for constructing M-matrix elements but the devil is in the details.

1. Elimination of infinities and coupling constant evolution

The elimination of infinities would follow from the symplectic QFT part of the theory. The symplectic contribution to n-point functions vanishes when two arguments co-incide. The UV cancellation mechanism has nothing to do with the finite measurement resolution which corresponds to the size of the causal diamonds inside which the space-time sheets representing radiative corrections are. There is also IR cutoff due to the presence of largest causal diamond.

On can decompose the radiative corrections two two types. First kind of corrections appear both at the level of positive/and negative energy parts of zero energy states. Second kind of corrections appear at the level of interactions between them. This decomposition is standard in quantum field theories and corresponds to the renormalization constants of fields resp. renormalization of coupling constants. The corrections due to the increase of measurement resolution in time comes as very specific corrections to positive and negative energy states involving gluing of smaller causal diamonds to the upper and lower boundaries of causal diamonds along any radial light-like ray. The radiative corresponds to the interactions correspond to the addition of smaller causal diamonds in the interior of the larger causal diamond. Scales for the corrections come as scalings in powers of 2 rather than as continuous scaling of measurement resolution.

2. Conformal symmetries

The basic questions are the following ones. How hyper-octonionic/-quaternionic/-complex super-conformal symmetry relates to the super-canonical conformal symmetry at the imbedding space level and the super Kac-Moody symmetry associated with the light-like 3-surfaces? How do the dual HO=M8 and H=M4×CP2 descriptions (number theoretic compactifcation) relate?

Concerning the understanding of these issues, the earlier construction of physical states poses strong constraints.

  1. The state construction utilizes both super-canonical and super Kac-Moody algebras. Super-canonical algebra has negative conformal weights and creates tachyonic ground states from which Super Kac-Moody algebra generates states with non-negative conformal weight determining the mass squared value of the state. The commutator of these two algebras annihilates the physical states. This requires that both super conformal algebras must allow continuation to hyper-octonionic algebras, which are independent.

  2. The light-like radial coordinate at dM4± can be continued to a hyper-complex coordinate in M2± defined the preferred commutative plane of non-physical polarizations, and also to a hyper-quaternionic coordinate in M4±. Hence it would seem that super-canonical algebra can be continued to an algebra in M2± or perhaps in the entire M4±. This would allow to continue also the operators G, L and other super-canonical operators to operators in hyper-quaternionic M4± needed in stringy perturbation theory.

  3. Also the super KM algebra associated with the light-like 3-surfaces should be continuable to hyper-quaternionic M4±. Here HO-H duality comes in rescue. It requires that the preferred hyper-complex plane M2 is contained in the tangent plane of the space-time sheet at each point, in particular at light-like 3-surfaces. We already know that this allows to assign a unique space-time surface to a given collection of light-like 3-surfaces as hyper-quaternionic 4-surface of HO hypothesized to correspond to (an obviously preferred) extremal of Kähler action. An equally important implication is that the light-like coordinate of X3 can be continued to hyper-complex coordinate M2 coordinate and thus also to hyperquaternionic M4 coordinate.

  4. The four-momentum appears in super generators Gn and Ln. It seems that the formal Fourier transform of four-momentum components to gradient operators to M4± is needed and defines these operators as particular elements of the CH Clifford algebra elements extended to fields in imbedding space.

3. What about stringy perturbation theory?

The analog of stringy perturbation theory does not seems only a highly attractive but also an unavoidable outcome since a generalization of massless fermionic propagator is needed. The inverse for the sum of super Kac-Moody and super-canonical super-Virasoro generators G (L) extended to an operator acting on the difference of the M4 coordinates of the end points of the propagator line connecting two partonic 2-surfaces should appear as fermionic (bosonic) propagator in stringy perturbation theory. Virasoro conditions imply that only G0 and L0 appear as propagators. Momentum eigenstates are not strictly speaking possible since since discretization is present due to the finite measurement resolution. One can however represent these states using Fourier transform as a superposition of momentum eigenstates so that standard formalism can be applied.

Symplectic QFT gives an additional multiplicative contribution to n-point functions and there would be also braiding S-matrices involved with the propagator lines in the case that partonic 2-surface carriers more than 1 point. This leaves still modular degrees of freedom of the partonic 2-surfaces describable in terms of elementary particle vacuum functionals and the proper treatment of these degrees of freedom remains a challenge.

4. What about non-hermiticity of the CH super-generators carrying fermion number?

TGD represents also a rather special challenge, which actually represents the fundamental difference between quantum TGD and super string models. The assignment of fermion number to CH gamma matrices and thus also to the super-generator G is unavoidable. Also M4 and H gamma matrices carry fermion number. This has been a long-standing interpretational problem in quantum TGD and I have been even ready to give up the interpretation of four-momentum operator appearing in Gn and Ln as actual four-momenta. The manner to get rid of this problem would be the assumption of Majorana property but this would force to give up the interpretation of different imbedding space chiralities in terms of conserved lepton and quark numbers and would also lead to super-string theory with critical dimension 10 or 11. A further problem is how to obtain amplitudes which respect fermion number conservation using string perturbation theory if 1/G=G0f/L0 carries fermion number.

The recent picture does not leave many choices so that I was forced to face the truth and see how everything falls down to this single nasty detail! It became as a total surprise that gamma matrices carrying fermion number do not cause any difficulties in zero energy ontology and make sense even in the ordinary Feynman diagrammatics.

  1. Non-hermiticity of G means that the center of mass terms CH gamma matrices must be distinguished from their Hermitian conjugates. In particular, one has g0 ¹ g0f. One can interpret the fermion number carrying M4 gamma matrices of the complexified quaternion space appearing naturally in number theoretical framework.

  2. One might think that M4×CP2 gamma matrices carrying fermion number is a catastrophe but this is not the case in a massless theory. Massless momentum eigen states can be created by the operator pkgkf from a vacuum annihilated by gamma matrices and satisfying massless Dirac equation. For instance, the conserved fermion number defined by the integral of \overline{Y}g0Y over 3-space gives just its standard value. A further experimentation shows that Feynman diagrams with non-hermitian gamma matrices give just the standard results since fermionic propagator and boson-emission vertices give compensating fermion numbers.

  3. If the theory would contain massive fermions or a coupling to a scalar Higgs, a catastrophe would result. Hence ordinary Higgs mechanism is not possible in this framework. Of course, also the quantization of fermions is totally different. In TGD fermion mass is not a scalar in H. Part of it is given by CP2 Dirac operator, part by p-adic thermodynamics for L0, and part by Higgs field which behaves like vector field in CP2 degrees of freedom, so that the catastrophe is avoided.

  4. In zero energy ontology zero energy states are characterized by M-matrix elements constructed by applying the combination of stringy and symplectic Feynman rules and fermionic propagator is replaced with its super-conformal generalization reducing to an ordinary fermionic propagator for massless states. The norm of a single fermion state is given by a propagator connecting positive energy state and its conjugate with the propagator G0/L0 and the standard value of the norm is obtained by using Dirac equation and the fact that Dirac operator appears also in G0.

  5. The hermiticity of super-generators G would require Majorana property and one would end up with superstring theory with critical dimension D=10 or D=11 for the imbedding space. Hence the new interpretation of gamma matrices, proposed already years ago, has very profound consequences and convincingly demonstrates that TGD approach is indeed internally consistent.

For more details see the previous posting, the chapter Construction of Quantum Theory: S-matrix of "Towards S-matrix", and the article Topological Geometrodynamics: What Might Be the First Principles?.

Monday, June 16, 2008

What goes wrong with string models?

Something certainly goes wrong with super-string models and M-theory. But what this something is? One could of course make the usual list involving spontaneous compactification, landscape, non-predictivity, and all that. The point I however want to make relates to the relationship of string models to quantum field theories.

The basic wisdom has been that when Feynman graphs of quantum field theories are replaced by their stringy variants everything is nice and finite. The problem is that stringy diagrams do not describe what elementary particles are doing and quantum field theory limit is required at low energies. This non-renormalizable QFT limit is obtained by the ad hoc procedure called spontaneous compactification, and leads to all this misery that spoils the quality of our life nowadays.

My intention is not to ridicule or accuse string theorists. I feel also myself very very stupid since I realized only now what the relationship between super-conformal and super-symplectic QFTs and generalized Feynman diagrams is in TGD framework: I described this already in the previous posting but did not want to make noise of my stupidity. This discovery (discovery only at level of my own subjective experience) was just becoming aware about something which should have been absolutely obvious for anyone with IQ above 20;-).

A very brief summary goes as follows.

  1. M-matrix elements characterize the time like entanglement between positive and negative energy parts of zero energy states. The positive/negative energy part can be localized to the boundary of past/future directed light-cone and these light-cones form a causal diamond.

  2. M-matrix elements can be expressed in terms of generalized Feynman diagrams with the lines of Feynman diagrams replaced with light-like 3-surfaces glued together along their 2-D ends representing vertices. For a given Feynman diagram of this kind one assigns an n-point function with additional intermediate points coming from the generalized vertices. In hyper-octonionic conformal field theory approach these vertices are fixed uniquely.

  3. The amplitude associated with a given generalized Feynman diagram is calculated by a recursive procedure using the fusion rules of a combination of conformal and symplectic QFT:s as described in the previous posting.

What this means that a fusion of generalizations of stringy conformal QFT to conformal-symplectic QFT and of ordinary QFT gives M-matrix elements. Feynman diagrams are not given up! Only the manner how they are computed is completely new: instead of the iterative approach one uses recursive approach based on fusion rules and involving automatically the cutoff which has interpretation in terms of finite measurement resolution.

For more details see the previous posting, the chapter Construction of Quantum Theory: Symmetries of "Towards S-matrix", and the article Topological Geometrodynamics: What Might Be the First Principles?.

Saturday, June 14, 2008

Could a symplectic analog of conformal field theory be relevant for quantum TGD?

Symplectic (or canonical as I have called them) symmetries of dM4+×CP2 (light-cone boundary briefly) act as isometries of the "world of classical worlds". One can see these symmetries as analogs of Kac-Moody type symmetries with symplectic transformations of S2×CP2, where S2 is rM=constant sphere of lightcone boundary, made local with respect to the light-like radial coordinate rM taking the role of complex coordinate. Thus finite-dimensional Lie group G is replaced with infinite-dimensional group of symplectic transformations. This inspires the question whether a symplectic analog of conformal field theory at dM4+×CP2 could be relevant for the construction of n-point functions in quantum TGD and what general properties these n-point functions would have.

1 Symplectic QFT at sphere

Actually the notion of symplectic QFT emerged as I tried to understand the properties of cosmic microwave background which comes from the sphere of last scattering which corresponds roughly to the age of 5×105 years. In this situation vacuum extremals of Kähler action around almost unique critical Robertson-Walker cosmology imbeddable in M4×S2, where there is homologically trivial geodesic sphere of CP2. Vacuum extremal property is satisfied for any space-time surface which is surface in M4×Y2, Y2 a Lagrangian sub-manifold of CP2 with vanishing induced Kähler form. Symplectic transformations of CP2 and general coordinate transformations of M4 are dynamical symmetries of the vacuum extremals so that the idea of symplectic QFT emerges natural. Therefore I shall consider first symplectic QFT at the sphere S2 of last scattering with temperature fluctuation DT/T proportional to the fluctuation of the metric component gaa in Robertson-Walker coordinates.

  1. In quantum TGD the symplectic transformation of the light-cone boundary would induce action in the "world of classical worlds" (light-like 3-surfaces). In the recent situation it is convenient to regard perturbations of CP2 coordinates as fields at the sphere of last scattering (call it S2) so that symplectic transformations of CP2 would act in the field space whereas those of S2 would act in the coordinate space just like conformal transformations. The deformation of the metric would be a symplectic field in S2. The symplectic dimension would be induced by the tensor properties of R-W metric in R-W coordinates: every S2 coordinate index would correspond to one unit of symplectic dimension. The symplectic invariance in CP2 degrees of freedom is guaranteed if the integration measure over the vacuum deformations is symplectic invariant. This symmetry does not play any role in the sequel.

  2. For a symplectic scalar field n ³ 3-point functions with a vanishing anomalous dimension would be functions of the symplectic invariants defined by the areas of geodesic polygons defined by subsets of the arguments as points of S2. Since n-polygon can be constructed from 3-polygons these invariants can be expressed as sums of the areas of 3-polygons expressible in terms of symplectic form. n-point functions would be constant if arguments are along geodesic circle since the areas of all sub-polygons would vanish in this case. The decomposition of n-polygon to 3-polygons brings in mind the decomposition of the n-point function of conformal field theory to products of 2-point functions by using the fusion algebra of conformal fields (very symbolically FkFl = cklmFm). This intuition seems to be correct.

  3. Fusion rules stating the associativity of the products of fields at different points should generalize. In the recent case it is natural to assume a non-local form of fusion rules given in the case of symplectic scalars by the equation

    Fk(s1)Fl(s2) = ó

    Here the coefficients cklm are constants and A(s1,s2,s3) is the area of the geodesic triangle of S2 defined by the sympletic measure and integration is over S2 with symplectically invariant measure dms defined by symplectic form of S2. Fusion rules pose powerful conditions on n-point functions and one can hope that the coefficients are fixed completely.

  4. The application of fusion rules gives at the last step an expectation value of 1-point function of the product of the fields involves unit operator term òcklf(A(s1,s2,s))Id dms so that one has

    áFk(s1)Fl(s2)ñ = ó

    Hence 2-point function is average of a 3-point function over the third argument. The absence of non-trivial symplectic invariants for 1-point function means that n=1- an are constant, most naturally vanishing, unless some kind of spontaneous symmetry breaking occurs. Since the function f(A(s1,s2,s3)) is arbitrary, 2-point correlation function can have both signs. 2-point correlation function is invariant under rotations and reflections.

2 Symplectic QFT with spontaneous breaking of rotational and reflection symmetries

CMB data suggest breaking of rotational and reflection symmetries of S2. A possible mechanism of spontaneous symmetry breaking is based on the observation that in TGD framework the hierarchy of Planck constants assigns to each sector of the generalized imbedding space a preferred quantization axes. The selection of the quantization axis is coded also to the geometry of "world of classical worlds", and to the quantum fluctuations of the metric in particular. Clearly, symplectic QFT with spontaneous symmetry breaking would provide the sought-for really deep reason for the quantization of Planck constant in the proposed manner.

  1. The coding of angular momentum quantization axis to the generalized imbedding space geometry allows to select South and North poles as preferred points of S2. To the three arguments s1,s2,s3 of the 3-point function one can assign two squares with the added point being either North or South pole. The difference

    DA(s1,s2,s3) º A(s1,s2,s3,N)-A(s1,s2,s3,S)

    of the corresponding areas defines a simple symplectic invariant breaking the reflection symmetry with respect to the equatorial plane. Note that DA vanishes if arguments lie along a geodesic line or if any two arguments co-incide. Quite generally, symplectic QFT differs from conformal QFT in that correlation functions do not possess singularities.

  2. The reduction to 2-point correlation function gives a consistency conditions on the 3-point functions


    = cklr ó

    =cklrcrm ó
    f(DA(s1,s2,s)) f(DA(s,s3,t))dmsdmt.

    Associativity requires that this expression equals to áFk(s1)(Fl(s2)Fm(s3))ñ and this gives additional conditions. Associativity conditions apply to f(DA) and could fix it highly uniquely.

  3. 2-point correlation function would be given by

    áFk(s1)Fl(s2)ñ = ckló
    f(DA(s1,s2,s)) dms

  4. There is a clear difference between n > 3 and n=3 cases: for n > 3 also non-convex polygons are possible: this means that the interior angle associated with some vertices of the polygon is larger than p. n=4 theory is certainly well-defined, but one can argue that so are also n > 4 theories and skeptic would argue that this leads to an inflation of theories. TGD however allows only finite number of preferred points and fusion rules could eliminate the hierarchy of theories.
  5. To sum up, the general predictions are following. Quite generally, for f(0)=0 n-point correlation functions vanish if any two arguments co-incide which conforms with the spectrum of temperature fluctuations. It also implies that symplectic QFT is free of the usual singularities. For symmetry breaking scenario 3-point functions and thus also 2-point functions vanish also if s1 and s2 are at equator. All these are testable predictions using ensemble of CMB spectra.

3 Generalization to quantum TGD

Since number theoretic braids are the basic objects of quantum TGD, one can hope that the n-point functions assignable to them could code the properties of ground states and that one could separate from n-point functions the parts which correspond to the symplectic degrees of freedom acting as symmetries of vacuum extremals and isometries of the 'world of classical worlds'.

  1. This approach indeed seems to generalize also to quantum TGD proper and the n-point functions associated with partonic 2-surfaces can be decomposed in such a manner that one obtains coefficients which are symplectic invariants associated with both S2 and CP2 Kähler form.

  2. Fusion rules imply that the gauge fluxes of respective Kähler forms over geodesic triangles associated with the S2 and CP2 projections of the arguments of 3-point function serve basic building blocks of the correlation functions. The North and South poles of S2 and three poles of CP2 can be used to construct symmetry breaking n-point functions as symplectic invariants. Non-trivial 1-point functions vanish also now.

  3. The important implication is that n-point functions vanish when some of the arguments co-incide. This might play a crucial role in taming of the singularities: the basic general prediction of TGD is that standard infinities of local field theories should be absent and this mechanism might realize this expectation.

Next some more technical but elementary first guesses about what might be involved.

  1. It is natural to introduce the moduli space for n-tuples of points of the symplectic manifold as the space of symplectic equivalence classes of n-tuples.

    i) In the case of sphere S2 convex n-polygon allows n+1 3-sub-polygons and the areas of these provide symplectically invariant coordinates for the moduli space of symplectic equivalence classes of n-polygons (2n-D space of polygons is reduced to n+1-D space). For non-convex polygons the number of 3-sub-polygons is reduced so that they seem to correspond to lower-dimensional sub-space.

    ii) In the case of CP2 n-polygon allows besides the areas of 3-polygons also 4-volumes of 5-polygons as fundamental symplectic invariants. The number of independent 5-polygons for n-polygon can be obtained by using induction: once the numbers N(k,n) of independent k £ n-simplices are known for n-simplex, the numbers of k £ n+1-simplices for n+1-polygon are obtained by adding one vertex so that by little visual gymnastics the numbers N(k,n+1) are given by N(k,n+1) = N(k-1,n)+N(k,n). In the case of CP2 the allowance of 3 analogs {N,S,T} of North and South poles of S2 means that besides the areas of polygons (s1,s2,s3), (s1,s2,s3,X), (s1,s2,s3,X,Y), and (s1,s2,s3,N,S,T) also the 4-volumes of 5-polygons (s1,s2,s3,X,Y), and of 6-polygon (s1,s2,s3,N,S,T), X,Y Î {N,S,T} can appear as additional arguments in the definition of 3-point function.

  2. What one really means with symplectic tensor is not clear since the naive first guess for the n-point function of tensor fields is not manifestly general coordinate invariant. For instance, in the model of CMB, the components of the metric deformation involving S2 indices would be symplectic tensors. Tensorial n-point functions could be reduced to those for scalars obtained as inner products of tensors with Killing vector fields of SO(3) at S2. Again a preferred choice of quantization axis would be introduced and special points would correspond to the singularities of the Killing vector fields.

    The decomposition of Hamiltonians of the "world of classical worlds" expressible in terms of Hamiltonians of S2×CP2 to irreps of SO(3) and SU(3) could define the notion of symplectic tensor as the analog of spherical harmonic at the level of configuration space. Spin and gluon color would have natural interpretation as symplectic spin and color. The infinitesimal action of various Hamiltonians on n-point functions defined by Hamiltonians and their super counterparts is well-defined and group theoretical arguments allow to deduce general form of n-point functions in terms of symplectic invariants.

  3. The need to unify p-adic and real physics by requiring them to be completions of rational physics, and the notion of finite measurement resolution suggest that discretization of also fusion algebra is necessary. The set of points appearing as arguments of n-point functions could be finite in a given resolution so that the p-adically troublesome integrals in the formulas for the fusion rules would be replaced with sums. Perhaps rational/algebraic variants of S2×CP2=SO(3)/SO(2)×SU(3)/U(2) obtained by replacing these groups with their rational/algebraic variants are involved. Tedrahedra, octahedra, and dodecahedra suggest themselves as simplest candidates for these discretized spaces.

    Also the symplectic moduli space would be discretized to contain only n-tuples for which the symplectic invariants are numbers in the allowed algebraic extension of rationals. This would provide an abstract looking but actually very concrete operational approach to the discretization involving only areas of n-tuples as internal coordinates of symplectic equivalence classes of n-tuples. The best that one could achieve would be a formulation involving nothing below measurement resolution.

  4. This picture based on elementary geometry might make sense also in the case of conformal symmetries. The angles associated with the vertices of the S2 projection of n-polygon could define conformal invariants appearing in n-point functions and the algebraization of the corresponding phases would be an operational manner to introduce the space-time correlates for the roots of unity introduced at quantum level. In CP2 degrees of freedom the projections of n-tuples to the homologically trivial geodesic sphere S2 associated with the particular sector of CH would allow to define similar conformal invariants. This framework gives dimensionless areas (unit sphere is considered). p-Adic length scale hypothesis and hierarchy of Planck constants would bring in the fundamental units of length and time in terms of CP2 length.

The recent view about M-matrix described in is something almost unique determined by Connes tensor product providing a formal realization for the statement that complex rays of state space are replaced with N rays where N defines the hyper-finite sub-factor of type II1 defining the measurement resolution. M-matrix defines time-like entanglement coefficients between positive and negative energy parts of the zero energy state and need not be unitary. It is identified as square root of density matrix with real expressible as product of of real and positive square root and unitary S-matrix. This S-matrix is what is measured in laboratory. There is also a general vision about how vertices are realized: they correspond to light-like partonic 3-surfaces obtained by gluing incoming and outgoing partonic 3-surfaces along their ends together just like lines of Feynman diagrams. Note that in string models string world sheets are non-singular as 2-manifolds whereas 1-dimensional vertices are singular as 1-manifolds. These ingredients we should be able to fuse together. So we try once again!

  1. Iteration starting from vertices and propagators is the basic approach in the construction of n-point function in standard QFT. This approach does not work in quantum TGD. Symplectic and conformal field theories suggest that recursion replaces iteration in the construction of given generalized Feynman diagram. One starts from an n-point function and reduces it step by step to a vacuum expectation value of a 2-point function using fusion rules. Associativity becomes the fundamental dynamical principle in this process. Associativity in the sense of classical number fields has already shown its power and led to a hyper-octoninic formulation of quantum TGD promising a unification of various visions about quantum TGD .

  2. Let us start from the representation of a zero energy state in terms of a causal diamond defined by future and past directed light-cones. Zero energy state corresponds to a quantum superposition of light-like partonic 3-surfaces each of them representing possible particle reaction. These 3-surfaces are very much like generalized Feynman diagrams with lines replaced by light-like 3-surfaces coming from the upper and lower light-cone boundaries and glued together along their ends at smooth 2-dimensional surfaces defining the generalized vertices.

  3. It must be emphasized that the generalization of ordinary Feynman diagrammatics arises and conformal and symplectic QFTs appear only in the calculation of single generalized Feynman diagram. Therefore one could still worry about loop corrections. The fact that no integration over loop momenta is involved and there is always finite cutoff due to discretization together with recursive instead of iterative approach gives however good hopes that everything works.

  4. One can actually simplify things by identifying generalized Feynman diagrams as maxima of Kähler function with functional integration carried over perturbations around it. Thus one would have conformal field theory in both fermionic and configuration space degrees of freedom. The light-like time coordinate along light-like 3-surface is analogous to the complex coordinate of conformal field theories restricted to some curve. If it is possible continue the light-like time coordinate to a hyper-complex coordinate in the interior of 4-D space-time sheet, the correspondence with conformal field theories becomes rather concrete. Same applies to the light-like radial coordinates associated with the light-cone boundaries. At light-cone boundaries one can apply fusion rules of a symplectic QFT to the remaining coordinates. Conformal fusion rules are applied only to point pairs which are at different ends of the partonic surface and there are no conformal singularities since arguments of n-point functions do not co-incide. By applying the conformal and symplectic fusion rules one can eventually reduce the n-point function defined by the various fermionic and bosonic operators appearing at the ends of the generalized Feynman diagram to something calculable.

  5. Finite measurement resolution defining the Connes tensor product is realized by the discretization applied to the choice of the arguments of n-point functions so that discretion is not only a space-time correlate of finite resolution but actually defines it. No explicit realization of the measurement resolution algebra N seems to be needed. Everything should boil down to the fusion rules and integration measure over different 3-surfaces defined by exponent of Kähler function and by imaginary exponent of Chern-Simons action. The continuation of the configuration space Clifford algebra for 3-surfaces with cm degrees of freedom fixed to a hyper-octonionic variant of gamma matrix field of super-string models defined in M8 (hyper-octonionic space) and M8« M4×CP2 duality leads to a unique choice of the points, which can contribute to n-point functions as intersection of M4 subspace of M8 with the counterparts of partonic 2-surfaces at the boundaries of light-cones of M8. Therefore there are hopes that the resulting theory is highly unique. Symplectic fusion algebra reduces to a finite algebra for each space-time surface if this picture is correct.
  6. Consider next some of the details of how the light-like 3-surface codes for the fusion rules associated with it. The intermediate partonic 2- surfaces must be involved since otherwise the construction would carry no information about the properties of the light-like 3-surface, and one would not obtain perturbation series in terms of the relevant coupling constants. The natural assumption is that partonic 2-surfaces belong to future/past directed light-cone boundary depending on whether they are on lower/upper half of the causal diamond. Hyper-octonionic conformal field approach fixes the nint points at intermediate partonic two-sphere for a given light-like 3-surface representing generalized Feynman diagram, and this means that the contribution is just N-point function with N=nout+nint+nin calculable by the basic fusion rules. Coupling constant strengths would emerge through the fusion coefficients, and at least in the case of gauge interactions they must be proportional to Kähler coupling strength since n-point functions are obtained by averaging over small deformations with vacuum functional given by the exponent of Kähler function. The first guess is that one can identify the spheres S2 Ì dM4± associated with initial, final and, and intermediate states so that symplectic n-point functions could be calculated using single sphere.

These findings raise the hope that quantum TGD is indeed a solvable theory. Even if one is not willing to swallow any bit of TGD, the classification of the symplectic QFTs remains a fascinating mathematical challenge in itself. A further challenge is the fusion of conformal QFT and symplectic QFT in the construction of n-point functions. One might hope that conformal and symplectic fusion rules can be treated separately.

For details see the chapter Construction of Quantum Theory: Symmetries of "Towards S-matrix" and the article Topological Geometrodynamics: What Might Be the First Principles?.

Friday, June 13, 2008

Could symplectic QFT allow to understand the fluctuations of CMB?

Depending on one's attitudes, the anomalies of the fluctuation spectrum of the cosmic microwave background (CMB) can be seen as a challenge for people analyzing the experiments or that of the inflationary scenario. I do not pretend to be deeply involved with CMB but as I read about one these anomalies in Sean Carrol's blog and next day in Lubos's blog, I felt that I could spend some days by clarifying myself what is involved.
What interests me whether the replacement of inflation with quantum criticality and hbar changing phase transitions could provide fresh insights about fluctuations and anomalies of CMB. In the following I try first to explain to myself what the anomalies are and after that I will consider some TGD inspired crazy (as always) ideas. My motivations to communicate are indeed strong: the consideration of the anomalies led to a generalization of the notion of conformal QFT to what might be called symplectic QFT having very natural place also in quantum TGD proper.
A brief summary about my views is as follows.

  1. There are several types of anomalies (with respect to the expectations motivated by inflation theory). Fluctuation spectrum shows hot and cold spots; there is so called hemispherical asymmetry in the spectrum; rotationally averaged two-point correlation function C(q) is vanishing for angle separations above 60 degrees; the so called cosmic axis of evil means that 3 of the multipole area vectors assignable to the l=2 and l=3 spherical harmonics in the expansion of C(q) are aligned and in the galactic plane (very near to ecliptic) and in roughly the same direction as the dipole corresponding to the motion of the Milky Way with respect to cosmic frame of reference; there is also evidence for non-Gaussianity meaning non-vanishing 3-point functions. Especially strange finding is that the features of the local geometry seems to reflect themselves in CMB at the surface of last scattering. If these findings are not artifacts of the analysis or pure accidents, the consequences for our understanding of the Cosmos would be dramatic.
  2. In TGD framework quantum criticality replaces inflation. This means that the fluctuations of CMB do not correspond to primordial fluctuations of inflaton field evolved into large scale fluctuations during rapid expansion but to long range fluctuations involved with a phase transition increasing Planck constant and occurring at the time of decoupling. The p-adic length scale involved with the sphere of last scattering and the amplitude of the fluctuations provide two dimensionless couplings and allow to make estimates for the scaling of Planck constant in this transition.
  3. I have suggested earlier a conformal field theory defined at the sphere of last scattering as a TGD based model for the anomalies. The analysis of the situation however demonstrates that a more natural approach is based on symplectic variant of conformal QFT at the sphere of last scattering. By combining generalized fusion rules with the knowledge about symplectic invariants associated with 3 or 4 points of sphere, one can deduce surprisingly detailed information about the n-point functions of symplectic QFTs. There are two variants of the theory: the first one is rotation- and reflection symmetric. The fact that the generalization of the notion of imbedding space allows to identify preferred quantization axis, allows to formulate also a variant theory breaking these symmetries. Fusion rules determine n-point functions highly uniquely in terms of 3-point functions expressible as functions of simple symplectic invariants. Symplectic QFT makes special predicions distinguishing it from inflationary models: a sizable non-Gaussianity is predicted and correlation functions vanish when any two arguments are very near to each other. It is certainly possible to reproduce the vanishing of C(q) for large values of q but it is not clear whether fusion rules allow this.
  4. Quite generally, symplectic QFT provides a long sought-for manner to describe the vacuum degeneracy of TGD in terms of n-point functions. What is of special importance is that the n-point functions have no singularities at the limit when some arguments co-incide. This means a profound distinction from quantum field theories and something like this is required by general arguments demonstrating that quantum TGD is free of the standard divergences due to the non-locality of Kähler function as a functional of 3-surface. The classification of symplectic QFTs should be a fascinating challenge for a mathematician. The basic challenge is to determine how uniquely fusion rules determine the 3-point functions generating all other n-point functions.
  5. The possibility of having gigantic values of gravitational Planck constant and zero energy ontology suggests that quantum measurements in cosmological scales are possible. This would mean that time-like entanglement between positive and negative energy parts of zero energy state could correlate galactic geometry with the geometry of fluctuation spectrum so that the hemispherical asymmetry with respect to galactic plane could be produced by this kind of quantum measurement. This would mean a dramatic proof of the notion of participatory Universe introduced by Wheeler.
For details see the chapter Quantum Astrophysics of "Classical Physics in Many-Sheeted Space-Time" or the article Could symplectic quantum field theory allow to model the fluctuations cosmic microwave background?.

Monday, June 09, 2008

About the arrow of psychological time and the notion of self

Quantum classical correspondence predicts that the arrow of subjective time is somehow mapped to that for the geometric time. The detailed mechanism for how the arrow of psychological time emerges has however remained open and I have discussed a handful of alternative explanations. Zero energy ontology and the identification of the space-time correlate of self as causal diamond of imbedding space rather than space-time sheet leads however to a resolution of the problem and precise quantitative predictions.

Also the notion of self has been problematic: the original view identified self as a subsystem able to remain unentangled. It was assumed that the sequence of quantum jumps experienced by self integrates to a stream of consciousness in this kind of situation: self would be for quantum jump what atom is for elementary particles. A more ambitious idea inspired by dark matter hierarchy was that the notion of self hierarchy reduces to a fractal hierarchy of quantum jumps within quantum jumps. It indeed seems that this reduction is consistent with the original definition of self. In the following I shall discuss the new view briefly.

1. About the arrow of psychological time

The explanation for the arrow of psychological time based on zero energy ontology is favored by Occam's razor since it uses only the assumption that space-time sheets are replaced by more evolved ones in each quantum jump. Also the model of topological quantum computation favors it.

  1. In standard picture the attention would gradually shift towards geometric future and space-time in 4-D sense would remain fixed. Now however the fact that quantum state is quantum superposition of space-time surfaces allows to assume that the attention of the conscious observer is directed to a fixed volume of 8-D imbedding space. Quantum classical correspondence is achieved if the evolution in a reasonable approximation means shifting of the space-time sheets and corresponding field patterns backwards backwards in geometric time by some amount per quantum jump so that the perceiver finds the geometric future in 4-D sense to enter to the perceptive field. This makes sense since the shift with respect to M4 time coordinate is an exact symmetry of extremals of Kähler action. It is also an excellent approximate symmetry for the preferred extremals of Kähler action and thus for maxima of Kähler function spoiled only by the presence of light-cone boundaries. This shift occurs for both the space-time sheet that perceiver identifies itself and perceived space-time sheet representing external world: both perceiver and percept change.

  2. Both the landscape and observer space-time sheet remain in the same position in imbedding space but both are modified by this shift in each quantum jump. The perceiver experiences this as a motion in 4-D landscape. Perceiver (Mohammed) would not drift to the geometric future (the mountain) but geometric future (the mountain) would effectively come to the perceiver (Mohammed)!

  3. There is an obvious analogy with Turing machine: what is however new is that the tape effectively comes from the geometric future and Turing machine can modify the entire incoming tape by intentional action. This analogy might be more than accidental and could provide a model for quantum Turing machine operating in TGD Universe. This Turing machine would be able to change its own program as a whole by using the outcomes of the computation already performed.

  4. The concentration of the sensory input and the effects of conscious motor action to a narrow interval of time (.1 seconds typically, secondary p-adic time scale associated with the largest Mersenne M127 defining p-adic length scale which is not completely super-astronomical) can be understood as a concentration of sensory/motor attention to an interval with this duration: the space-time sheet representing sensory "me" would have this temporal length and "me" definitely corresponds to a zero energy state.

  5. The fractal view about topological quantum computation strongly suggests an ensemble of almost copies of sensory "me" scattered along my entire life cycle and each of them experiencing my life as a separate almost copy. My childhood is still sensorily lived but has moved about 57 years backwards in geometric time and would live the year 1897 but enjoy all techno conveniences of the year 1950!

  6. The model of geometric and subjective memories would not be modified in an essential manner: memories would result when "me" is connected with my almost copy in the geometric past by braid strands or massless extremals (MEs) or their combinations (ME parallel to magnetic flux tube is the analog of Alfwen wave in TGD).

This argument leaves many questions open. What is the precise definition for the volume of attention? Is the attention of self doomed to be directed to a fixed volume or can quantum jumps change the volume of attention? What distinguishes between geometric future and past as far as contents of conscious experience are considered? How this picture relates to p-adic and dark matter hierarchies? Does this framework allow to formulate more precisely the notion of self? Zero energy ontology allows to give tentative answers to these questions.

2. About the notion of self

I have considered two different notions of "self" and it is interesting to see whether the new view about time might allow to choose between them or to show that they are actually equivalent.

  1. In the original variant of the theory "self" corresponds to a sequence of quantum jumps. "Self" would result through a binding of quantum jumps to single "selftring" in close analogy and actually in a concrete correspondence with the formation of bound states. Each quantum jump has a fractal structure: unitary process is followed by a sequence of state function reductions and preparations proceeding from long to short scales. Selves can have sub-selves and one has self hierarchy. The questionable assumption is that self remains conscious only as long as it is able to avoid entanglement with environment.

    Even slightest entanglement would destroy self unless on introduces the notion of finite measurement resolution applying also to entanglement. This notion is indeed central for entire quantum TGD also leads to the notion of sharing of mental images: selves unentangled in the given measurement resolution can experience shared mental images resulting as fusion of sub-selves by entanglement not visible in the resolution used.

  2. According to the newer variant of theory, quantum jump has a fractal structure so that there are quantum jumps within quantum jumps: this hierarchy of quantum jumps within quantum jumps would correspond to the hierarchy of dark matters labeled by the values of Planck constant. Each fractal structure of this kind would have highest level (largest Planck constant) and this level would corresponds to the self. What might be called irreducible self would corresponds to a quantum jump without any sub-quantum jumps (no mental images). The quantum jump sequence for lower levels of dark matter hierarchy would create the experience of flow of subjective time.

It would be nice to reduce the original notion of self hierarchy to the hierarchy defined by quantum jumps. There are some objections against this idea. One can argue that fractality is a purely geometric notion and since subjective experience does not reduce to the geometry it might be that the notion of fractal quantum jump does not make sense. It is also not quite clear whether the reasonable looking idea about the role of entanglement as destroyer of self can be kept in the fractal picture.

These objections fail if one can construct a well-defined mathematical scheme allowing to understand what fractality of quantum jump at the level of space-time correlates means and showing that the two views about self are equivalent. The following argument represents such a proposal. Let us start from the causal diamond model as a lowest approximation for a model of zero energy states and for the space-time region defining the contents of sensory experience.

Let us make the following assumptions.

  1. Assume the hierarchy of causal diamonds within causal diamonds in a sense to be specified more precisely below. Causal diamonds would represent the volumes of attention. Assume that the highest level in this hierarchy defines the quantum jump containing sequences of lower level quantum jumps in some sense to be specified. Assume that these quantum jumps integrate to single continuous stream of consciousness as long as the sub...-sub-self in question remains unentangled and that entangling means loss of consciousness or at least that it is not possible to remember anything about contents of consciousness during entangled state.

  2. Assume that the contents of conscious experience come from the interior of the causal diamond. A stronger condition would be that the contents come from the boundaries of the two light-cones involved since physical states are defined at these in the simplest picture. In this case one could identify the lower light-cone boundary as giving rise to memory.

  3. The time span characterizing the contents of conscious experience associated with a given quantum jump would correspond to the temporal distance T between the tips of the causal diamond. T would also characterize the average and approximate shift of the superposition of space-time surfaces backwards in geometric time in single quantum jump at a given level of hierarchy. This time scale naturally scales as Tn=2nTCP2 so that p-adic length scale hypothesis follows as a consequence. T would be essentially the secondary p-adic time scale T2,p=ÖpTp for p @ 2k. This assumption - absolutely essential for the hierarchy of quantum jumps within quantum jumps - would differentiate the model from the model in which T corresponds to either CP2 time scale or p-adic time scale Tp. One would have hierarchy of quantum jumps with increasingly longer time span for memory and with increasing duration of geometric chronon at the highest level of fractal quantum jump. Without additional restrictions, the quantum jump at nth level would contain 2n quantum jumps at the lowest level of hierarchy. Note that in the case of sub-self - and without further assumptions which will be discussed next - one would have just two quantum jumps: mental image appears, disappears or exists all the time. At the level of sub-sub-selves 4 quantum jumps and so on. Maybe this kind of simple predictions might be testable.

  4. We know that that the contents of sensory experience comes from a rather narrow time interval of duration about .1 seconds, which corresponds to the time scale T127 associated with electron. We also know that there is asymmetry between positive and negative energy parts of zero energy states both physically and at the level of conscious experience. This asymmetry must have some space-time correlate. The simplest correlate for the asymmetry between positive and negative energy states would be that the upper light-like boundaries in the structure formed by light-cones within light-cones intersect along light-like radial geodesic. No condition of this kind would be posed on lower light-cone boundaries. The scaling invariance of this condition makes it attractive mathematically and would mean that arbitrarily long time scales Tn can be present in the fractal hierarchy of light cones. At all levels of the hierarchy all contribution from upper boundary of the causal diamond to the conscious experience would come from boundary of same past directed light-cone so that the conscious experience would be sharply localized in time in the manner as we know it to be. The new element would be that content of conscious experience would come from arbitrarily large region of Universe and seing Milky Way would mean direct sensory contact with it.

  5. These assumptions relate the hierarchy of quantum jumps to p-adic hierarchy. One can also include also dark matter hierarchy into the picture. For dark matter hierarchy the time scale hierarchy {Tn} is scaled by the factor r=hbar/hbar0 which can be also rational number. For r=2k the hierarchy of causal diamonds generalizes without difficulty and there is a kind of resonance involved which might relate to the fact that the model of EEG favors the values of k=11n, where k=11 also corresponds in good approximation to proton-electron mass ratio. For more general values of r the generalization is possible assuming that the position of the upper tip of causal diamond is chosen in such a manner that their positions are always the same whereas the position of the lower light-cone boundary would correspond to {rTn} for given value of Planck constant. Geometrically this picture generalizes the original idea about fractal hierarchy of quantum jumps so that it contains both p-adic hierarchy and hierarchy of Planck constants.

The contributions from lower the boundaries identifiable in terms of memories would correspond to different time scales and for a given value of time scale T the net contribution to conscious experience would be much weaker than the sensory input in general. The asymmetry between geometric now and geometric past would be present for all contributions to conscious experience, not only sensory ones. What is nice that the contents of conscious experience would rather literally come from the boundary of the past directed light-cone along which the classical signals arrive. Hence the mystic feeling about telepathic connection with a distant object at distance of billions of light years expressed by an astrophysicist, whose name I have unfortunately forgotten, would not be romantic self deception.

This framework explains also the sharp distinction between geometric future and past (not surprisingly since energy and time are dual): this distinction has also been a long standing problem of TGD inspired theory of consciousness. Precognition is not possible unless one assumes that communications and sharing of mental images between selves inside disjoint causal diamonds is possible. Physically there seems to be no good reason to exclude the interaction between zero energy states associated with disjoint causal diamonds.

3. Feedback to Quantum TGD

The mathematical formulation of this intuition is however a non-trivial challenge and can be used to articulate more precisely the views about what configuration space and configurations space spinor fields actually are mathematically.

  1. Suppose that the causal diamonds with tips at different points of H=M4×CP2 and characterized by distance between tips T define sectors CHi of the full configuration space CH ("world of classical worlds"). Precognition would represent an interaction between zero energy states associated with different sectors CHi in this scheme and tensor factor description is required.

  2. Inside given sector CHi it is not possible to speak about second quantization since every quantum state correspond to a single mode of a classical spinor field defined in that sector.

  3. The question is thus whether the Clifford algebras and zero energy states associated with different sectors CHi combine to form a tensor product so that these zero energy states can interact. Tensor product is required by the vision about zero energy insertions assignable to CHi which correspond to causal diamonds inside causal diamonds. Also the assumption that zero energy states form an ensemble in 4-D sense - crucial for the deduction of scattering rates from M-matrix - requires tensor product.

  4. The argument unifying the two definitions of self requires that the tensor product is restricted when CHi correspond to causal diamonds inside each other. The tensor factors in shorter time scales are restricted to the causal diamonds hanging from a light-like radial ray at the upper end of the common past directed light-cone. If the causal diamonds are disjoint there is no obvious restriction to be posed, and this would mean the possibility of also precognition and sharing of mental images.

This scenario allows also to answers the questions related to a more precise definition of volume of attention. Causal diamond - or rather - the associated light-like boundaries containing positive and negative energy states define the primitive volume of attention. The obvious question whether the attention of a given self is doomed to be fixed to a fixed volume can be also answered. This is not the case. Selves can delocalize in the sense that there is a wave function associated with the position of the causal diamond and quantum jumps changing this position are possible. Also many-particle states assignable to a union of several causal diamonds are possible. Note that the identification of magnetic flux tubes as space-time correlates of directed attention in TGD inspired quantum biology makes sense if these flux tubes connect different causal diamonds. The directedness of attention in this sense should be also understood: it could be induced from the ordering of p-adic primes and Planck constant: directed attention would be always from longer to shorter scale. For a more background see for instance, the chapter Matter,Mind, Quantum of "TGD Inspired Theory of Consciousness".

Cell as gel and the quantum model of nerve pulse

The book Gels and Cells [4] of Pollack should be obligatory reading for anyone seriously interested about the real situation in biology. The book summarizes impressive amount of facts supporting the view that the prevailing view about cytoplasm as water containing molecules dis-solved into it it is badly wrong. These findings force to challenge the notions of channels and pumps and even the notion of continuous cell membrane must be questioned as well as basic view about the generation of action potentials. These findings have served as inspiration in the construction of TGD based view about quantum biology. The solution to various anomalies of living cell proposed by Pollack that cytosplasm is in gel phase [4] and that the phase transitions of gel phase are a universal building brick of various biological functions.

1. Cell as gel

Pollack describes in detail various aspects of cytoplasm as a gel phase and here only short summary can be given.

  1. Cytoplasm can be regarded as a network consisting of cross-linked negatively charged proteins. Water is condensed around the proteins to form structured water. If protein is hydrophilic, water self-organizes around it as a multilayered structure: the number of molecular layers can as high as 600 and the thickness of the layered structure is a considerable fraction of micrometer. If the protein is hydrophobic, water forms another structured phase known as clathrate water: in this case the number of hydrogen bonds between water atoms is large. These phases can be regarded as intermediate between ice and water. Also ordinary ions have this kind of layered structure around them. Chemical cross-links tend to be stable with heat, pH, and solvent composition whereas physical cross-links formed by intermolecular interactions are sensitive to environmental interactions and are of special interest from the point of view of phase transitions.

  2. Pollack proposes that the formation of polymers takes place in an environment containing layered water for the simple reason that monomers cannot diffuse to the layered water so that the probability of association with the end of the growing polymer increases.

  3. Cell interior is populated by micro-tubules, various filamentary structures, and the so called micro-trabecular matrix. Micro-trabecular network divides cell into a compartments in such a manner that the typical distance between two proteins in water is about 5 nm: this corresponds to the p-adic length scale L(149), the thickness of the lipid layer of cell membrane. This is probably not an accident and the micro-trabecular network might be closely involved with the highly folded network of intracellular membranes. There would be a layer of thickness of about 6 water molecules per given protein surface so that a dominating portion of intracellular water could be structured.

  4. The layered water has several tell-tale signatures that have been observed in gels. It freezes at much lower temperature than ordinary water; various relaxation times are shorter since the energy transfer to the water lattice occurs faster than to non-structure water; the diffusion rates of particles into the structured water are much slower than to ordinary water by entropy argument; a simple geometric argument tells that the larger the size of the hydrated ion the lower the diffusion rate; strong gradients of ionic concentrations can form in gel phase as has been observed.

The identification of the cytoplasm as a gel has profound implications for the standard views about cell.

  1. The original motivation for postulating semipermeable cell membrane, channels, and pumps was the need to hinder the diffusion of various ions between cell interior and exterior taking place if cytoplasm is ordinary water into which molecules are dissolved. If cytoplasm is in gel phase, cell membrane need not perform pumping and channeling anymore except perhaps in situations involving the formation of a local sol phase. This raises the question about the proper functions of the cell membrane.

  2. It is possible to drill to cell membrane holes with size of order 1 mm without an appreciable effect on the functioning of the cell and also show that these holes remain as such for long periods of time . It is also possible to splice cells into pieces continuing to function for days. That K+ flux through cell membrane does not change when lipids are partially removed. These findings force to ask whether the assumption about the continuity of the cell membrane might be too strong . Electron micrographs however demonstrate the presence of the bi-layered structure. What is intriguing that this structure is seen even in the absence of lipid layers. In TGD framework this paradoxical finding might be understood in terms of a presence of space-time sheets corresponding to p-adic length scales L(k), k=149,151 as vacuum structures predicted also by TGD inspired model of high Tc super-conductivity [2] .

  3. There is also the strange finding that water flux through cell membrane is much higher than the flux through isolate lipid bi-layer as if some unidentified channels were present. In TGD framework this might be seen as an evidence for the presence of (wormhole) magnetic flux tubes as carriers of water molecules.

  4. The fundamental assumptions about ionic equilibrium must be reconsidered, and the Hodkin-Huxley model for the generation of nerve pulse becomes more or less obsolete. Indeed, it has been found that action potentials can be generated even in absence of Na+ and K+ ions playing a key role in Hodkin-Huxley model. Rather remarkably, the high concentration of K+ ions and low concentration of Na+ ions in cytoplasm could be understood on basis of gel property only. Also new view about cell (note membrane-!) potential emerges. The standard paradigm states that the resting potential is over the cell membrane. Potentials of same order of magnitude have been however seen in de-membraned cells (50 mV in slight excess of action potential and critical potential), colloidal suspensions, and gels which suggest that larger part of cell than mere cell membrane is involved with the generation of the action potential and one should thus speak of cell potential instead of membrane potential.

  5. Pollack suggests that the phase transitions of the gel phase make possible to realize various functions at molecular and cellular level and represents empirical evidence for the phase transition like aspects assigned to these functions including sensitivity to various factors such as pH, temperature, chemical environment, electromagnetic fields, mechanical forces, etc... and the threshold behavior . Also the responses are typical for phase transitions in that they involve dramatic changes in volume, shape, di-electric constant, etc.. With these motivations Pollack discusses phase transition based models for contraction, motility, secretion, transport or molecules, organized flow of particles during cell division, cell locomotion, contraction of muscle, generation of action potentials, etc.. For instance, the transport of bio-molecules along micro-tubule could involve propagating gel-sol-gel phase transition meaning also propagating melting of the layered water around micro-tubule.

  6. Divalent ions, such as Mg++ and Ca++ can act as cross links between negatively charged proteins binding them to form networks. Monovalent ions cannot do this. Peripheral cytoskeleton is this kind of network consisting of micro-tubules and actin molecules cross-linked - according to Pollack- by Ca++ ions. On the other hand, it is known that Mg++ (Ca++) ions dominate in the cell interior (exterior) and that the presence of Ca++ ions in the cell exterior is crucial the for generation of nerve pulse. The influx of Na+ ions having higher affinity to proteins can induce a phase transition to sol-like phase. Pollack suggests a model of nerve pulse based on this mechanism of gel-sol phase transition for peripheral cytoskeleton: this model does not actually explain why Ca++ ions in the exterior of axon are necessary.

2. TGD based vision nerve pulse and its relation to Pollack's model

The vision about dark matter and the model of nerve pulse formulated in terms of Josephson currents brings an additional perspective to the role of pumps and channels and allows to avoid harmony with the standard views about their role.

  1. In long length scales visible matter forms roughly 5 per cent of the total amount of matter. In TGD Universe the dark matter would correspond to matter with large Planck constant including dark variants of ordinary elementary particles. In living matter situation could be the same and visible matter could form only a small part of the living matter. Dark matter would be however visible in the sense that it would interact with visible matter via classical electromagnetic fields and photon exchanges with photons suffering Planck constant changing phase transition. Hence one can consider the possibility that most of the biologically important ions and perhaps even molecules reside at the magnetic flux quanta in large hbar phase.

  2. Bosonic ions could form Bose-Einstein condensates at the flux tubes in which case supra currents flowing without any dissipation would be possible. The model for high Tc super-conductivity suggests that only electronic and protonic super-conductivity are possible at room temperature. If so, Cooper pairs of fermionic ions are excluded. New nuclear physics predicted by TGD could however come in rescue here. The TGD based model for atomic nucleus assumes that nuclei are strings of nucleons connected by color bonds having quark and antiquark at their ends. Also charged color bonds are possible and this means the existence of nuclei with anomalous charge. This makes possible bosonic variants of fermionic ions with different mass number and it would be interesting to check whether biological important ions like Na+,Cl-, and K+ might actually correspond to this kind of exotic ions.

This leads to the following TGD inspired vision about cell as a gel.

  1. DNA as tqc hypothesis and cell membrane as sensory receptor provide possible candidates for the actual functions of the cell membrane and ionic channels and pumps could act as kind of receptors. That standard physics is able to to describe gel phase is of course a mere belief and (wormhole) magnetic flux tubes connecting various molecules (DNA, RNA, aminoacids, biologically important ions) would be "new physics" cross-links could explain the strong correlations between distant molecules of the gel phase.

  2. Dark ionic currents are quantal currents. If the dark ions flow along magnetic or wormhole magnetic flux tubes connecting cell interior and exterior, their currents through cell membrane would be same as through an artificial membrane.

  3. Pumps and channels could serve the role of sensory receptors by allowing to take samples about chemical environment. One cannot exclude the possibility that proteins act as pumps and channels in sol phase if magnetic flux tubes are absent in this phase since also in TGD Universe homeostasis and its control at the level of visible matter in sol phase might requires them. The metabolic energy needed for this purpose would be however dramatically smaller and a reliable estimate for this would allow an estimate of the portion of dark matter in living systems.

  4. Quantum criticality suggests that the phase transitions for the gel phase are induced by quantum phase transitions changing the value of Planck constant for magnetic flux tubes and inducing the change of the length of the flux tube. Macroscopic quantum coherence would explain the observed co-operativity aspect of the phase transitions. Concerning locomotion and transport mountain climbing using pickaxe and rope inspires a guess for a general mechanism. For instance, a packet of molecules moving along actin molecule or a molecule carrying a cargo along micro-tubule could repeat a simple basic step in which a magnetic flux tube with large hbar is shot along the direction of the electric field along micro-tubule and stuck to a rachet followed by a phase transition reducing the value of hbar and shortening the flux tube and forcing the cargo to move forward. The metabolic energy might be provided by the micro-tubule rather than molecular motor.

  5. The reconnection of flux tubes would be a second phase transition of this kind. This phase transition could lead from a phase in phase proteins are unfolded with flux tubes connecting aminoacids to water molecules and thus possessing a large volume of layered water around them to a phase in which they become folded and flux tubes connect aminoacids to each other in the interior of protein. The phase transition could be associated with the contraction of connecting filaments of muscle cell. The phase transitions are also seen in "artificial protein" gels used for drug delivery applications, and are built from polymers arranged in alpha helices, beta sheets and common protein motifs . If wormhole magnetic flux are taken are taken as a basic prerequisite of life, one must ask whether these "rtificial proteins" represent artificial life.

  6. The fact that cytoskeleton rather than only cell membrane is involved with the generation of action potential conforms with the idea that nerve pulse propagating along axon involves also axonal micro-tubules and that Josephson currents between axon and micro-tubules are involved in the process.

  7. Di-valent ions (Ca++ ions according to Pollack) serve as cross links in the peripheral cytoskeleton. The influx of monovalent ions from the exterior of axon induces gel-sol phase transition replacing di-valent ions with monovalent ions. One can consider two models.

    i) The minimal assumption is that this phase transition is induced hbar increasing phase transition the flow of the monovalent ions like Na+ from the cell exterior along the magnetic flux tubes connecting axonal interior and interior. Suppose that in the original situation the flux tubes end to axonal membrane (this is not the only possibility, they could also end to Ca++ ions). The flux tubes extending to the axonal exterior could result by hbar increasing phase transition increasing the length of the flux tubes connecting peripheral cytoskeleton to the axonal membrane so that they extend to the exterior of axon. This option is rather elegant since gel-sol phase transition itself can be understood in terms of ßtandard chemistry". In this model the very slow diffusion rate of the ions to gel phase would have explanation in terms of new physics involving dark matter and (wormhole) magnetic flux tubes.

    ii) One can consider also an option in which divalent ions such as Ca++ or Mg++ are connected by two flux tubes to amino-acids of two negatively charged proteins whereas monovalent biological ions like Na+ would have single flux tube of this kind and could not act as cross links. In the phase transitions removing the cross links the replacement of divalent ion with two monovalent positively charged ions would take place. If one believes in standard chemistry, Na+ ions would flow in automatically. First the increase of Planck constant would induce the lengthening of the magnetic flux tubes and thus the expansion of the gel phase making possible the influx of monovalent ions. If Na+ ions are dark, flux tubes connecting peripheral cytoskeleton to the axonal exterior are required and the mechanism of option i) is also needed.

  8. The mechanisms i) and ii) could be fused to a single one. The hint comes from the presence of Ca++ ions in the exterior of axon is necessary for the generation of action potential. The simplest possibility is that the flux tubes connecting proteins to intracellular Ca++ cross links in gel phase connects them after the length increasing phase transition to extracellular Ca++ ions and Na+ ions flow along these flux tubes.

  9. The increase of the Planck constant would induce the expansion of the peripheral cytoskeleton making possible the inflow of Na+ ions, and divalent ions binding negatively charged actin molecules to a network would be replaced with inflowing Na+ ions. After this a reverse phase transition would occur. Both phase transitions could be induced by a quantal control signal (Josephson current) inducing quantum criticality and a change of Planck constant.

  10. A propagating Ca++ wave inducing the gel-sol-gel phase transition of peripheral cytoskeleton would accompany nerve pulse. Quite generally, Ca++ waves are known to play a fundamental role in living matter as kind of biological rhythms. Irrespective of whether one believes option i) or ii), this might relate to the cross-linking by flux tubes and gel-sol-gel phase transitions induce by phase transitions increasing Planck constant temporarily. The velocities and oscillation periods of Ca++ waves vary in an extremely wide range: this can be understood if the flux tubes involved correspond to a very wide spectrum of Planck constant.

To sum up, the strange discoveries about the behavior of cell membrane provide direct experimental evidence for the presence of dark matter in living systems, for the prediction that it interacts with ordinary matter via classical electromagnetic fields, and for the assumption that it does not dissipate appreciably and could therefore have large value of hbar and form macroscopic quantum phases.

In the model of Pollack for the action potential gel-sol-gel phase transition for the peripheral cytoskeleton accompanies the generation of the action potential. The model allows to understand reasonably well the behavior and the physical role of the ionic currents and explains various anomalies. I have discussed TGD based model of nerve pulse earlier in these blog postings. The Josephson junctions defined by (wormhole) magnetic flux tubes between microtubules and axonal membrane can be modeled as a coupled sequence of analogs of gravitational pendulums and in the continuum idealization Sine-Gordon equation is satisfied. EEG rhythms (actually a fractal hierarchy of EEGs are predicted ) are due to dark photon Josephson radiation associated with sequences of solitons. This corresponds to a situation in which the penduli rotate with a constant phase difference between neighbors. A kick to the rotating pendulum so that it starts to oscillate instead of rotating corresponds to a generation of nerve pulse. This kick would also induce a gel-sol-gel phase transition propagating along the peripheral cytoskeleton.

3. Gel-sol phase transition as quantum critical phase transition

The challenge is to understand how quantum criticality making possible the phase transition is induced.

  1. The primary Josephson currents from the micro-tubuli to the axonal membrane would reduce the magnitude of the cell potential below the critical value (slowing down of the pendulum rotation). This should somehow take the peripheral cytoskeleton near to quantum criticality and induce the increase of Planck constant for the flux tubes connecting peripheral cytoskeleton to the axonal membrane and increasing their length so that they would extend to axonal exterior. This would make possible the flow of monovalent dark ions (say Na+) from the axonal exterior replacing Ca++ acting as cross links between negatively charged proteins and in this manner induce gel-sol phase transition. The reverse phase transition would reduce Planck constant. If ionic currents are non-dissipative they flow back automatically much like oscillating Josephson currents.

  2. There are two forms of quantum criticality corresponding to critical sub-manifolds M2×CP2 and M4×S2, where M2 Ì M4 has interpretation as plane of non-physical polarizations and S2 Ì CP2 is a homologically trivial geodesic sphere of CP2 with vanishing induced Kähler form (see the Appendix of [1]). The latter kind of quantum criticality corresponds to very weak induced Kähler fields and thus to almost vacuum extremals. Given electromagnetic field can be imbedded as a 4-surface in many manners: as a vacuum extremal, as a surface maximizing Kähler electric energy, or something between them.

  3. Quantum criticality suggests that em fields in the cell interior corresponds to nearly vanishing induced Kähler fields and that in the resting state the em fields at cell membrane and peripheral cytoskeleton correspond to strong Kähler fields. The magnitude of the cell potential in the absence of the membrane is about .05 V and slightly below the magnitude of the critical potential . Hence the reduction of the magnitude of the em (-or more precisely- Kähler-) voltage between the inner boundary of the peripheral cytoskeleton and cell exterior to a small enough value could induce quantum criticality making hbar increasing phase transition for the magnetic flux tubes connecting peripheral cytoskeleton to the axonal membrane possible. This framework also allows to understand the paradoxical fact that a reduction of the magnitude of the cell potential induces the action potential rather than its increase as the naive idea about di-electric breakdown would suggest.

  4. The energy of the Josephson photon associated with cell membrane Josephson junction is about .05 eV at criticality for the generation of action potential. This is not too far from the value of the metabolic energy quantum liberated in the dropping of proton Cooper pair from k=139 atomic space-time sheet or of electron Cooper pair from k=151 cell membrane space-time sheet to a much larger space-time sheet. This leads to the idea that phase conjugate IR photons of Josephson radiation couple resonantly to the gel defined by the peripheral cytoskeleton and induce fast dropping of protons to larger space-time sheets and that this in turn induces the increase of Planck constant for magnetic flux tubes inducing gel-to-sol phase transition. This idea has been discussed already earlier and will reconsidered in the section where the relationship of the model with microtubular level is discussed.

  5. A comment relating this picture to DNA as tqc model is in order. The basic difference between TGD and standard model is that color rotations leave invariant the induced Kähler field but affect electro-weak gauge fields. In particular, color rotations change the intensity of em field by transforming em and Z0 fluxes to each other. In the recent case color rotation cannot obviously reduce the value of the electric field. The most elegant variant of the model of DNA as tqc replaces qubit with qutrit (true/false/undefined) presented as color triplet of quarks associated with the (wormhole) magnetic flux tubes connecting nucleotides with lipids . Hence the color rotations representing basic 1-gates would not affect induced Kähler fields and cannot induce phase transitions although they would affect cell potential. For 2-gate represented by the basic braiding operation permuting the ends of the neighboring strands the situation is different. Quantum criticality would make possible the generation of braiding by taking cell membrane to liquid state. The discussion about the effects of anesthetics in the sequel forces however to conclude that in the liquid crystal state action potentials are not possible. Propagating action potentials could however represent tqc programs as time-like braidings if it is microtubular surface that suffer gel-sol-gel transition during the nerve pulse.

4. A model for anesthetic action

The molecular mechanism of the anesthetic action is a fascinating unsolved problem of neurophysiology. Noble gases have very weak chemical interactions. Despite this many noble gas such as Xe, Kr, Ar but to my best knowledge not Ne and He, act as anaesthetics. Also chemically non-inert molecules have quite similar narcotic effect so that chemistry does not seem to matter as Hodgkin-Huxley model would predict.

It is known that the narcotic efficiency of anesthetics correlates with their solubility in lipids . Anesthetics also reduce the melting temperature of the lipid layer. Strong pressure increases the melting temperature and it is known that high pressure brings consciousness back. Thus anesthetic molecules dissolved into the lipid membrane should hinder the generation of the nerve pulse somehow and liquid state of the axonal membrane could be the reason for this. The explanation of the soliton model for the anesthetic action is that the metabolic energy needed to generate an acoustic soliton becomes too high when axon is too high above the critical temperature.

To get a useful perspective note that also the problem why ordinary cell and neuronal soma outside axonal hillock do not allow action potentials is poorly understood. The obvious idea is that anesthetized axonal membrane (or at least axonal hillock) is just like the ordinary cell membrane. The model for DNA-cell membrane system as a topological quantum computer requires the liquid-crystal property of the lipid layers of the ordinary cell membrane and neuronal membrane outside axonal hillock. If this is the case, then liquid phase for axonal membrane implied by the anesthetic action would indeed make it more or less equivalent with the ordinary cell membrane. Therefore the question is why the liquid-crystal property of the ordinary cell membrane prevents the generation of the action potential.

  1. Pollack's model suggests that anesthetics could hinder the occurrence of the gel-sol phase transition for the peripheral cytoskeleton. Suppose that (h/2p) increasing phase transition for the magnetic flux tubes connecting peripheral cytoskeleton to the axon extends them to the axonal exterior and makes possible the influx of monovalent ions inducing gel-sol phase transition.

  2. Suppose that the phase transition increasing (h/2p) is induced by the reduction of the voltage over the axonal membrane (assume to be much smaller than cell potential) inducing almost vacuum property and quantum criticality. Somehow the presence of anesthetics would prevent this. Either the voltage over the membrane is increased in magnitude so that the flow of dark ionic currents to the membrane is not enough to induce quantum criticality or the flow of dark currents is completely prevented. The first option is more economical and could be tested by finding whether the voltage over the axonal membrane (membrane in a solid state) is considerably smaller than that over the ordinary cell membrane (membrane in liquid-crystal state). The first option also predicts that during sleep the increase of cell potential (hyperpolarization) actually corresponds to the increase of the membrane potential.

For background see the chapter TGD Inspired Model for Nerve Pulse of TGD and EEG. References

[1] The chapter DNA as Topological Quantum Computer of Genes and Memes.

[2] The chapter Bio-Systems as Super-Conductors: Part I of Quantum Hardware of Living Matter.

[3] The chapter Quantum Model for Nerve Pulse of TGD and EEG.

[4]G. Pollack (2000), Cells, Gels and the Engines of Life, Ebner and Sons.

[5] Sine-Gordon equation,

[6]K. Graesboll (2006), Function of Nerves-Action of Anesthetics, Gamma 143, An elementary Introduction.