Category theory and quantum TGD: generalized Feyman diagrams and generalized planar operads

In the sequel the idea that planar operads [5,6,7,13] or their appropriate generalization might allow to formulate generalized Feynman diagrammatics in zero energy ontology will be considered. Also a description of measurement resolution and arrow of geometric time in terms of operads is discussed.

1. Zeroth order heuristics about zero energy states

Consider now the existing heuristic picture about the zero energy states and coupling constant evolution provided by causal diamonds (CDs), which can be regarded as regions bounded by future directed light-cone and past directed light-cone inside it [2,3].

1. The tentative description for the increase of the measurement resolution in terms CDs is that one inserts to the upper and/or lower light-like boundary of CD smaller CDs by gluing them along light-like radial ray from the tip of CD. It is also possible that the vertices of generalized Feynman diagrams belong inside smaller CDs and it turns out that these CDs must be allowed.

2. The considerations related to the arrow of geometric time suggest that there is asymmetry between upper and lower boundaries of CD. The minimum requirement is that the measurement resolution is better at upper light-like boundary.

3. In zero energy ontology communications to the direction of geometric past are possible and phase conjugate laser photons represent one example of this.

4. Second law of thermodynamics must be generalized in such a manner that it holds with respect to subjective time identified as sequence of quantum jumps. The arrow of geometric time can however vary so that apparent breaking of second law is possible in shorter time scales at least. One must however understand why second law holds true in so good an approximation.

5. One must understand also why the contents of sensory experience is concentrated around a narrow time interval whereas the time scale of memories and anticipation are much longer. The proposed mechanism is that the resolution of conscious experience is higher at the upper boundary of CD. Since zero energy states correspond to light-like 3-surfaces, this could be a result of self-organization rather than a fundamental physical law.
   1. CDs define the perceptive field for self. Selves are curious about the space-time sheets outside their perceptive field in the geometric future of the imbedding space and perform quantum jumps tending to shift the superposition of the space-time sheets to the direction of geometric past (past defined as the direction of shift!). This creates the illusion that there is a time=snapshot front of consciousness moving to geometric future in fixed background space-time as an analog of train illusion.

   2. The fact that news come from the upper boundary of CD implies that self concentrates its attention to this region and improves the resolutions of sensory experience and quantum measurement here. The sub-CDs generated in this manner correspond to mental images with contents about this region. As a consequence, the contents of conscious experience, in particular sensory experience, tend to be about the region near the upper boundary.

   3. This mechanism in principle allows the arrow of the geometric time to vary and depend on p-adic length scale and the level of dark matter hierarchy. The occurrence of phase transitions forcing the arrow of geometric time to be same everywhere are however plausible for the reason that the lower and upper boundaries of given CD must possess the same arrow of geometric time.

   4. If this is the mechanism behind the arrow of time, planar operads can provide a description of the arrow of time but not its explanation.

This picture is certainly not general enough, can be wrong at the level of details, and at best relates to the the whole like single particle wave mechanics to quantum field theory.

2. Planar operads

The geometric definition of planar operads without using the category theoretical jargon goes as follows.

1. There is an external disk and some internal disks and a collection of disjoint lines connecting disk boundaries.

2. To each disk one attaches a non-negative integer k, called the color of disk. The disk with color k has k points at each boundary with the labeling 1,2,...k running clockwise and starting from a distinguished marked point, decorated by '*'. A more restrictive definition is that disk colors are correspond to even numbers so that there are k=2n points lines leaving the disk boundary boundary. The planar tangles with k=2n correspond to inclusions of HFFs.

3. Each curve is either closed (no common points with disk boundaries) or joins a marked point to another marked point. Each marked point is the end point of exactly one curve.

4. The picture is planar meaning that the curves cannot intersect and diks cannot overlap.

5. Disks differing by isotopies preserving *'s are equivalent.

Given a planar k-tangle-one of whose internal disks has color k_i- and a k_i-tangle S, one can define the tangle T _iS by isotoping S so that its boundary, together with the marked points and the '*'s co-indices with that of D_i and after that erase the boundary of D_i. The collection of planar tangle together with the the composition defined in this manner- is called the colored operad of planar tangles.

One can consider also generalizations of planar operads.

1. The composition law is not affected if the lines of operads branch outside the disks. Branching could be allowed even at the boundaries of the disks although this does not correspond to a generic situation. One might call these operads branched operads.

2. The composition law could be generalized to allow additional lines connecting the points at the boundary of the added disk so that each composition would bring in something genuinely new. Zero energy insertion could correspond to this kind of insertions.

3. TGD picture suggests also the replacement of lines with braids. In category theoretical terms this means that besides association one allows also permutations of the points at the boundaries of the disks.

The question is whether planar operads or their appropriate generalizations could allow a characterization of the generalized Feynman diagrams representing the combinatorics of zero energy states in zero energy ontology and whether also the emergence of arrow of time could be described (but probably not explained) in this framework.

3. Planar operads are not enough for zero energy states

Are planar operads sufficiently powerful to code the vision about the geometric correlates for the increase of the measurement resolution and coupling constant evolution formulated in terms of CDs? Or perhaps more realistically, could one improve this formulation by assuming that zero energy states correspond to wave functions in the space of planar tangles or of appropriate modifications of them? It seems that the answer to the first question is almost affirmative.

1. Disks are analogous to the white regions of a map whose details are not visible in the measurement resolution used. Disks correspond to causal diamonds (CDs) in zero energy ontology. Physically the white regions relate to the vertices of the generalized Feynman diagrams and possibly also to the initial and final states (strictly speaking, the initial and final states correspond to the legs of generalized Feynman diagrams rather than their ends).

2. The composition of tangles means addition of previously unknown details to a given white region of the map and thus to an increase of the measurement resolution. This conforms with the interpretation of inclusions of HFFs as a characterization of finite measurement resolution and raises the hope that planar operads or their appropriate generalization could provide the proper language to describe coupling constant evolution and their perhaps even generalized Feynman diagrams.

3. For planar operad there is an asymmetry between the outer disk and inner disks. One might hope that this asymmetry could explain or at least allow to describe the arrow of time. This is not the case. If the disks correspond to causal diamonds (CDs) carrying positive resp. negative energy part of zero energy state at upper resp. lower light-cone boundary, the TGD counterpart of the planar tangle is CD containing smaller CDs inside it. The smaller CDs contain negative energy particles at their upper boundary and positive energy particles at their lower boundary. In the ideal resolution vertices represented 2-dimensional partonic at which light-like 3-surfaces meet become visible. There is no inherent asymmetry between positive and negative energies and no inherent arrow of geometric time at the fundamental level. It is however possible to model the arrow of time by the distribution of sub-CDs. By previous arguments self-organization of selves can lead to zero energy states for which the measurement resolution is better near the upper boundary of the CD.

4. If the lines carry fermion or anti-fermion number, the number of lines entering to a given CD must be even as in the case of planar operads as the following argument shows.
   1. In TGD framework elementary fermions correspond to single wormhole throat associated with topologically condensed CP₂ type extremal and the signature of the induced metric changes at the throat.

   2. Elementary bosons correspond to pairs of wormhole throats associated with wormhole contacts connecting two space-time sheets of opposite time orientation and modellable as a piece of CP₂ type extremal. Each boson therefore corresponds to 2 lines within CP₂ radius.

   3. As a consequence the total number of lines associated with given CD is even and the generalized Feynman diagrams can correspond to a planar algebra associated with an inclusion of HFFs.

5. This picture does not yet describe zero energy insertions.
   1. The addition of zero energy insertions corresponds intuitively to the allowance of new lines inside the smaller CD:s not coming from the exterior. The addition of lines connecting points at the boundary of disk is possible without losing the basic geometric composition of operads. In particular one does not lose the possibility to color the added tangle using two colors (colors correspond to two groups G and H which characterize an inclusion of HFFs [5]).

   2. There is however a problem. One cannot remove the boundaries of sub-CD after the composition of CDs since this would give lines beginning from and ending to the interior of disk and they are invisible only in the original resolution. Physically this is of course what one wants but the inclusion of planar tangles is expected to fail in its original form, and one must generalize the composition of tangles to that of CD:s so that the boundaries of sub-CD:s are not thrown away in the process.

   3. It is easy to see that zero energy insertions are inconsistent with the composition of planar tangles. In the inclusion defining the composition of tangles both sub-tangle and tangle induce a color to a given segment of the inner disk. If these colors are identical, one can forget the presence of the boundary of the added tangle. When zero energy insertions are allowed, situation changes as is easy to see by adding a line connecting points in a segment of given color at the boundary of the included tangle. There exists no consistent coloring of the resulting structure by using only two colors. Coloring is however possible using four colors, which by four-color theorem is the minimum number of colors needed for a coloring of planar map: this however requires that the color can change as one moves through the boundary of the included disk - this is in accordance with the physical picture.

   4. Physical intuition suggests that zero energy insertion as an improvement of measurement resolution maps to an improved color resolution and that the composition of tangles generalizes by requiring that the included disk is colored by using new nuances of the original colors. The role of groups in the definition of inclusions of HFFs is consistent with idea that G and H describe color resolution in the sense that the colors obtained by their action cannot be resolved. If so, the improved resolution means that G and H are replaced by their subgroups G₁ Þ G and H₁ Þ H. Since the elements of a subgroup have interpretation as elements of group, there are good hopes that by representing the inclusion of tangles as inclusion of groups, one can generalize the composition of tangles.

6. Also CDs glued along light-like ray to the upper and lower boundaries of CD are possible in principle and -according the original proposal- correspond to zero energy insertions according. These CDs might be associated with the phase transitions changing the value of (^h/_2p) leading to different pages of the book like structure defined by the generalized imbedding space.

7. p-Adic length scale hypothesis is realized if the hierarchy of CDs corresponds to a hierarchy of temporal distances between tips of CDs given as a=T_n=2^-nT₀ using light-cone proper time.

8. How this description relates to braiding? Each line corresponds to an orbit of a partonic boundary component and in principle one must allow internal states containing arbitrarily high fermion and antifermion numbers. Thus the lines decompose into braids and one must allow also braids of braids hierarchy so that each line corresponds to a braid operad in improved resolution.

4. Relationship to ordinary Feynman diagrammatics

The proposed description is not equivalent with the description based on ordinary Feynman diagrams.

1. In standard physics framework the resolution scale at the level of vertices of Feynman diagrams is something which one is forced to pose in practical calculations but cannot pose at will as opposed to the measurement resolution. Light-like 3-surfaces can be however regarded only locally orbits of partonic 2-surfaces since generalized conformal invariance is true only in 3-D patches of the light-like 3-surface. This means that light-like 3-surfaces are in principle the fundamental objects so that zero energy states can be regarded only locally as a time evolutions. Therefore measurement resolution can be applied also to the distances between vertices of generalized Feynman diagrams and calculational resolution corresponds to physical resolution. Also the resolution can be better towards upper boundary of CD so that the arrow of geometric time can be understood. This is a definite prediction which can in principle kill the proposed scenario.

2. A further counter argument is that generalized Feynman diagrams are identified as light-like 3-surfaces for which Kähler function defined by a preferred extremal of Kähler action is maximum. Therefore one cannot pose any ad hoc rules on the positions of the vertices. One can of course insist that maximum of Kähler function with the constraint that posed by T_n=2ⁿT₀ hierarchy is in question.

It would be too optimistic to believe that the details of the proposal are correct. However, if the proposal is on correct track, zero energy states could be seen as wave functions in the operad of generalized tangles (zero energy insertions and braiding) as far as combinatorics is involved and the coherence rules for these operads would give strong constraints on the zero energy state and fix the general structure of coupling constant evolution.

References

[1] The chapter Nuclear String Model of "p-Adic length scale Hypothesis and Hierarchy of Planck constants".

[2] M. Pitkänen (2008), Quantum TGD: What Might be the General Principles?.

[3] M. Pitkänen (2008), About the Nature of Time.

[4] C. Rogers and A. Hoffnung (2008), Categorified symplectic geometry and the classical string.

[5] Operad theory.

[6] Planar algebra.

[7] D. Bisch, P. Das, and S. K. Gosh (2008), Planar algebra of group-type subfactors.

[8] Multicategories.

[9] J. Baez (2007), Quantum Quandaries.

[10] J. Baez and M. Stay (2008), Physics, topology, logic and computation: a Rosetta Stone.

[11] J. Baez (2008), Categorifying Fundamental Physics.

[12]Planar Algebras, TFTs with Defects.

[13] M. D. Sheppeard (2007), Gluon Phenomenology and a Linear Topos, thesis.

The article Category Theory and Quantum TGD gives a summary of the most recent ideas about applications of category theory in TGD framework. See also the new chapter Category Theory and TGD of "Towards S-matrix".

Category theory and quantum TGD: 2-plectic structures and generalized Feynman diagrams as categories

TGD predicts several hierarchical structures involving a lot of new physics. These structures look frustratingly complex and category theoretical thinking might help to build a bird's eye view about the situation. I have already earlier considered the question how category theory might be applied in TGD. Besides the far from complete understanding of the basic mathematical structure of TGD also my own limited understanding of category theoretical ideas have been a serious limitation. During last years considerable programs in the understanding of quantum TGD proper has taken place and the recent formulation of TGD is in terms of light-like 3-surfaces, zero energy ontology and number theoretic braids. There exist also rather detailed formulations for the fusion of p-adic and real physics and for the dark matter hierarchy. This motivates a fresh look to how category theory might help to understand quantum TGD. In the first posting I will discuss how so called 2-plectic structures emerge in TGD and how generalized Feynman diagrams can be regarded as categories. Second posting will be devoted to the description of generalized Feynman diagrammatics in terms of generalization of planar operads. In the third posting possible other applications will be briefly considered.

1. 2-plectic structures and TGD

Chris Rogers and Alex Hoffnung have demonstrated [1] that the notion of symplectic structure generalizes to n-plectic structure and in n=2 case leads to a categorification of Lie algebra to 2-Lie-algebra. In this case the generalization replaces the closed symplectic 2-form with a closed 3-form w and assigns to a subset of one-forms defining generalized Hamiltonians vector field leaving the 3-form invariant.

There are two equivalent definitions of the Poisson bracket in the sense that these Poisson brackets differ only by a gradient, which does not affect the vector field assignable to the Hamiltonian one-form. The first bracket is simply the Lie-derivate of Hamiltonian one form G with respect to vector field assigned to F. Second bracket is contraction of Hamiltonian one-forms with the three-form w. For the first variant Jacobi identities hold true but Poisson bracket is antisymmetric only modulo gradient. For the second variant Jacobi identities hold true only modulo gradient but Poisson bracket is antisymmetric. This modulo property is in accordance with category theoretic thinking in which commutativity, associativity, antisymmetry,... hold true only up to isomorphism.

For 3-dimensional manifolds n=2-plectic structure has the very nice property that all one-forms give rise to Hamiltonian vector field. In this case any 3-form is automatically closed so that a large variety of 2-plectic structures exists. In TGD framework the natural choice for the 3-form w is as Chern-Simons 3-form defined by the projection of the Kähler gauge potential to the light-like 3-surface. Despite the fact the induced metric is degenerate, one can deduce the Hamiltonian vector field associated with the one-form using the general defining conditions

ivFw = dF  (1)

since the vanishing of the metric determinant appearing in the formal definition cancels out in the expression of the Hamiltonian vector field. The explicit formula is obtained by writing w as

w = Keabg×emndAmJndÖg = eabg×C-S ,

C-S = KEabg AaJbg .

Here E^abg=e_abg holds true numerically and metric determinant, which vanishes for light-like 3-surfaces, has disappeared.

The Hamiltonian vector field is the curl of F divided by the Chern-Simons action density C-S:

vFa = Eabg(¶bFg-¶gFb)/2C-S .

The Hamiltonian vector field multiplied by the dual of 3-form multiplied by the metric determinant has a vanishing divergence and is analogous to a vector field generating volume preserving flow and the value of Chern Simons 3-form defines the analog of the metric determinant for light-like 3-surfaces. The generalized Poisson bracket for Hamiltonian 1-forms defined by the contraction of the Hamiltonian 1-forms with w is Hamiltonian 1-form and unique apart from gradient and the corresponding vector field is the commutator of the corresponding Hamiltonian vector fields.

The objection is that gauge invariance is broken since the expression for the vector field assignal to Hamiltonian one-form depends on gauge. In TGD framework there is no need to worry since Kähler gauge potential has unique natural expression and the U(1) gauge transformations of Kähler gauge potential induced by symplectic transformations of CP₂ are not genuine gauge transformations but dynamical symmetries since the induced metric changes and space-time surface is deformed. Another important point is that Kähler gauge potential for a given CD has M⁴ part which is "pure gauge" constant Lorentz invariant vector and proportional to the inverse of gravitational constant G. Its ratio to CP₂ radius squared is determined from electron mass by p-adic mass calculations and mathematically by quantum criticality fixing also the value of Kähler coupling strength.

2. Operads, number theoretical braids, and inclusions of HFFs

The description of braids leads naturally to category theory and quantum groups when the braiding operation, which can be regarded as a functor, is not a mere permutation. Discreteness is a natural notion in the category theoretical context. To me the most natural manner to interpret discreteness is - not something emerging in Planck scale- but as a correlate for a finite measurement resolution and quantum measurement theory with finite measurement resolution leads naturally to number theoretical braids as fundamental discrete structures so that category theoretic approach becomes well-motivated. Discreteness is also implied by the number theoretic approach to quantum TGD from number theoretic associativity condition central also for category theoretical thinking as well as from the realization of number theoretical universality by the fusion of real and p-adic physics to single coherent whole.

Operads are formally single object multi-categories [3]. This object consist of an infinite sequence of sets of n-ary operations. These operations can be composed and the composition are associative (operations themselves need not be associative) in the sense that the is natural isomorphism (symmetries) mapping differently bracketed compositions to each other. The coherence laws for operads formulate the effect of permutations and bracketing (association) as functors acting as natural isomorphisms. A simple manner to visualize the composition is as an addition of n₁, ...n_k leaves to the leaves 1, ...,k of k-leaved tree.

An interesting example of operad is the braid operad formulating the combinatorics for a hierarchy of braids formed from braids by grouping subsets of braids having n₁,...n_k strands and defining the strands of a k-braid. In TGD framework this grouping can be identified in terms of the formation bound states of particles topologically condensed at larger space-time sheet and coherence laws allow to deduce information about scattering amplitudes. In conformal theories braided categories indeed allow to understand duality of stringy amplitudes in terms of associativity condition.

Planar operads [3] define an especially interesting class of operads. The reason is that the inclusions of HFFs give rise to a special kind of planar operad [4]. The object of this multi-category [5] consists of planar k-tangles. Planar operads are accompanied by planar algebras. It will be found that planar operads allow a generalization which could provide a description for the combinatorics of the generalized Feynman diagrams and also rigorous formulation for how the arrow of time emerges in TGD framework and related heuristic ideas challenging the standard views.

3. Generalized Feynman diagram as category?

John Baez has proposed a category theoretical formulation of quantum field theory as a functor from the category of n-cobordisms to the category of Hilbert spaces [6,7]. The attempt to generalize this formulation looks well motivated in TGD framework because TGD can be regarded as almost topological quantum field theory in a well defined sense and braids appear as fundamental structures. It however seems that formulation as a functor from nCob to Hilb is not general enough. In zero energy ontology events of ordinary ontology become quantum states with positive and negative energy parts of quantum states localizable to the upper and lower light-like boundaries of causal diamond (CD).

1. Generalized Feynman diagrams associated with a given CD involve quantum superposition of light-like 3-surfaces corresponding to given generalized Feynman diagram. These superpositions could be seen as categories with 3-D light-like surfaces containing braids as arrows and 2-D vertices as objects. Zero energy states would represent quantum superposition of categories (different topologies of generalized Feynman diagram) and M-matrix defined as Connes tensor product would define a functor from this category to the Hilbert space of zero energy states for given CD (tensor product defines quite generally a functor).

2. What is new from the point of view of physics that the sequences of generalized lines would define compositions of arrows and morphisms having identification in terms of braids which replicate in vertices. The possible interpretation of the replication is in terms of copying of information in classical sense so that even elementary particles would be information carrying and processing structures. This structure would be more general than the proposal of John Baez that S-matrix corresponds to a function from the category of n-dimensional cobordisms to the category Hilb.

3. p-Adic length scale hypothesis follows if the temporal distance between the tips of CD measured as light-cone proper time comes as an octave of CP₂ time scale: T=2ⁿT₀. This assumption implies that the p-adic length scale resolution interpreted in terms of a hierarchy of increasing measurement resolutions comes as octaves of time scale.

This preliminary picture is of course not far complete since it applies only to single CD. There are several questions. Can one allow CDs within CDs and is every vertex of generalized Feynman diagram surrounded by this kind of CD. Can one form unions of CDs freely?

1. Since light-like 3-surfaces in 8-D imbedding space have no intersections in the generic position, one could argue that the overlap must be allowed and makes possible the interaction of between zero energy states belonging to different CDs. This interaction would be something new and present also for sub-CDs of a given CD.

2. The simplest guess is that the unrestricted union of CDs defines the counterpart of tensor product at geometric level and that extended M-matrix is a functor from this category to the tensor product of zero energy state spaces. For non-overlapping CDs ordinary tensor product could be in question and for overlapping CDs tensor product would be non-trivial. One could interpret this M-matrix as an arrow between M-matrices of zero energy states at different CDs: the analog of natural transformation mapping two functors to each other. This hierarchy could be continued ad infinitum and would correspond to the hierarchy of n-categories.

This rough heuristics represents of course only one possibility among many since the notion of category is extremely general and the only limits are posed by the imagination of the mathematician. Also the view about zero energy states is still rather primitive.

References

[1] The chapter Nuclear String Model of "p-Adic length scale Hypothesis and Hierarchy of Planck constants".

[2] C. Rogers and A. Hoffnung (2008), Categorified symplectic geometry and the classical string.

[3] Operad theory.

[4] Planar algebra.

[5] D. Bisch, P. Das, and S. K. Gosh (2008), Planar algebra of group-type subfactors.

[6] Multicategories.

[7] J. Baez (2007), Quantum Quandaries.

[8] J. Baez and M. Stay (2008), Physics, topology, logic and computation: a Rosetta Stone.

[9] J. Baez (2008), Categorifying Fundamental Physics.

[10]Planar Algebras, TFTs with Defects.

[11] M. D. Sheppeard (2007), Gluon Phenomenology and a Linear Topos, thesis.

The article Category Theory and Quantum TGD gives a summary of the most recent ideas about applications of category theory in TGD framework. See also the new chapter Category Theory and TGD of "Towards S-matrix".

Dark black holes, dark flow, and how to avoid heat death?

Lubos made interesting comments about the calculation of black hole entropy in his blog. I have absolutely nothing to say about this branch of science as far as technicalities are considered. The formulas for blackhole entropy however inspire new visions about black holes if one accepts the hierarchy of Planck constants and the notion of relative darkness in the sense that particles at the different pages of the book like structure, whose pages are labelled by the values of Planck constant are dark relative to each other. I glue below a slightly edited comment in Kea's blog.

1. Black hole entropy and dark black holes

Lubos made in his posting explicit the 1/hbar proportionality of formulas for black hole entropy. This proportionality reflects the basic thermodynamical implication of quantization: the phase space of N-dimensional system decomposes into cells of volume hbar^N and entropy is proportional to the phase space volume using this volume as unit. If hbar becomes large and gigantic as it would in the case of dark gravitation (hbar= GM₁M₂/v₀, v₀/c∼ 2^-11 for inner planetary Bohr orbits) this means that blackhole entropy is extremely small. Black is dark;-) as I realized for few years ago, and it would be interesting to consider the consequences.

2. Hierarchy of Planck lengths

It deserves to be noticed that the rough order of magnitude estimate for the gravitational Planck constant of Sun can be written as hbar_gr=x4GM². This gives for the Planck length the expression

L_P= (Ghbar)^1/2 = x^1/2 2GM .

For x=1 one Planck length would be just Schwartshild radius. This makes sense since these two lengths play rather similar role. Quite generally, one would have a hierarchy of Planck lengths.

3. Dark flow

Second comment is related to the earlier posting of Lubos about the observed dark flow in length scales larger than horizon size towards an attractor outside horizon. The presence of the attractor outside the visible universe conforms with the notion of manysheeted space-time predicting also a manysheeted cosmology.

Many-sheeted cosmology means a hierarchy of space-time sheets obeying their own Robertson-Walker type cosmologies: those with varying p-adic length scale and those labelled by various values of Planck constants at pages of book like structure obtained by gluing together singular coverings and factor spaces of 8-D imbedding space (roughly). Particles at different pages are dark relative to each other in the sense that there are no local interaction vertices: classical interactions and those by exchanges of say photons are possible. Each sheet in many-sheeted cosmology has different horizon size.

The attractor would correspond to a different value of Planck constant and have larger horizon size than our sheet. Dark energy would be dark matter and the phase transitions increasing Planck constant would induce phases of accelerated expansion. In average sense these periods would give ordinary cosmology without accelerated expansion.

4. How to avoid heat death?

Third comment relates to the dark flow and implications of the hierarchy of Planck constants to future prospects of intelligent life. Heat death is believed by standard physicists to be waiting for all forms of life. We would live in the silliest possible Universe. I cannot believe this. I am ready to admit that some of our theories about the Universe are really silly, but entire Universe?--No!

The hierarchy of Planck constants would allow to avoid heat death. For instance, if the rate for the reduction of temperature is proportional to 1/hbar -as looks natural- then there is always an infinite number of hierarchy levels for which temperature is above a given temperature since the temperature at these pages is reduced so slowly.

Life can escape to the pages of the Big Book labelled by larger values of Planck constant without breaking second law since the scaling of the size of the system by hbar increases phase space volume and keeps entropy constant. Evolution by quantum leaps increasing hbar increasing the time scale of planned action and long term memory is another manner to say this.

The observed dark flow might be seen as a direct support for this more optimistic view about Life and the Universe and Everything;-).

Flyby anomaly as a relativistic transverse Doppler effect?

For half year ago I discussed a model for the flyby anomaly based on the hypothesis that dark matter ring around the orbit of Earth causes the effect. The model reproduced the formula deduced for the change of the velocity of the space-craft at qualitative level, and contained single free parameter: essentially the linear density of the dark matter at the flux tube.

From Lubos I learned about a new twist in the story of flyby anomaly. September twelfth 2007 Jean-Paul Mbelek proposed an explanation of the flyby anomaly as a relativistic transverse Doppler effect. The model predicts also the functional dependence of the magnitude of the effect on the kinematic parameters and the prediction is consistent with the empirical findings in the example considered. Therefore the story of flyby anomaly might be finished and dark matter at the orbit of Earth could bring in only an additional effect. It is probably too much to hope for this kind of effect to be large enough if present.

For background see the chapter TGD and Astrophysics.

Tritium beta decay anomaly and variations in the rates of radioactive processes

The determination of neutrino mass from the beta decay of tritium leads to a tachyonic mass squared [2,3,4,5]. I have considered several alternative explanations for this long standing anomaly. The first class of models relies on the presence of dark neutrino or antineutrino belt around the orbit of Earth. The second class of models relies on the prediction of nuclear string model that the neutral color bonds connecting nucleons to nuclear string can be also charged. This predicts large number of fake nuclei having only apparently the proton and neutron numbers deduced from the mass number.

1. ³He nucleus resulting in the decay could be fake (tritium nucleus with one positively charged color bond making it to look like ³He). The idea that slightly smaller mass of the fake ³He might explain the anomaly: it however turned out that the model cannot explain the variation of the anomaly from experiment to experiment.

2. Later (yesterday evenening!) I realized that also the initial ³H nucleus could be fake (³He nucleus with one negatively charged color bond). It turned out that fake tritium option has the potential to explain all aspects of the anomaly and also other anomalies related to radioactive and alpha decays of nuclei.

3. Just one day ago I still believed on the alternative based on the assumption of dark neutrino or antineutrino belt surrounding Earth's orbit. This model has the potential to explain satisfactorily several aspects of the anomaly but fails in its simplest form to explain the dependence of the anomaly on experiment. Since the fake tritium scenario is based only on the basic assumptions of the nuclear string model and brings in only new values of kinematical parameters it is definitely favored.

In the following I shall describe only the models based on the decay of tritium to fake Helium and the decay of fake tritium to Helium.

1. Fake ³He option

Consider first the fake ³He option. Tritium (pnn) would decay with some rate to a fake ³He, call it ³He_f, which is actually tritium nucleus containing one positively charged color bond and possessing mass slightly different than that of ³He (ppn).

1. In this kind of situation the expression for the function K(E,k) differs from K(stand) since the upper bound E₀ for the maximal electron energy is modified:
   

   E0 ® E1=M(3H)-M(3Hef)-mm = M(3H)-M(3He)+DM-mm ,

   
   

   DM = M(3He)-M(3Hef) .

   Depending on whether ³He_f is heavier/lighter than ³He E₀ decreases/decreases. From V_b Î [5-100] eV and from the TGD based prediction order m([`(n)]) ~ .27 eV one can conclude that DM should be in the range 5-100 eV.

2. In the lowest approximation K(E) can be written as
   
   

   K(E) = K0(E,E1)q(E1-E) @ (E1-E)q(E1-E).

   Here q(x) denotes step function and K₀(E,E₁) corresponds to the massless antineutrino.

3. If the fraction p of the final state nuclei correspond to a fake ³He the function K(E) deduced from data is a linear combination of functions K(E,³He) and K(E,³He_f) and given by
   
   

   K(E) = (1-p)K(E,3He)+ pK(E,3Hef)

   
   

   @ (1-p)(E0-E)q(E0-E)+ p(E1-E)q(E1-E)

   in the approximation m_n=0.

   For m(³He_f) < m(³He) one has E₁ > E₀ giving

   
   

   K(E) = (E0-E)q(E0-E)+ p(E1-E0)q(E1-E)q(E-E0).

   K(E,E₀) is shifted upwards by a constant term (1-p)DM in the region E₀ > E. At E=E₀ the derivative of K(E) is infinite which corresponds to the divergence of the derivative of square root function in the simpler parametrization using tachyonic mass. The prediction of the model is the presence of a tail corresponding to the region E₀ < E < E₁.

4. The model does not as such explain the bump near the end point of the spectrum. The decay ³H® ³He_f can be interpreted in terms of an exotic weak decay d® u+W^- of the exotic d quark at the end of color bond connecting nucleons inside ³H. The rate for these interactions cannot differ too much from that for ordinary weak interactions and W boson must transform to its ordinary variant before the decay W® e+`n. Either the weak decay at quark level or the phase transition could take place with a considerable rate only for low enough virtual W boson energies, say for energies for which the Compton length of massless W boson correspond to the size scale of color flux tubes predicted to be much longer than nuclear size. Is so the anomaly would be absent for higher energies and a bump would result.

5. The value of K(E) at E=E₀ is V_b º p(E₁-E₀). The variation of the fraction p could explain the observed dependence of V_b on experiment as well as its time variation. It is however difficult to understand how p could vary.

2. Fake ³H option

Assume that a fraction p of the tritium nuclei are fake and correspond to ³He nuclei with one negatively charged color bond.

1. By repeating the previous calculation exactly the same expression for K(E) in the approximation m_n=0 but with the replacement
   
   

   DM = M(3He)-M(3Hef)® M(3Hf)-M(3H) .

2. In this case it is possible to understand the variations in the shape of K(E) if the fraction of ³H_f varies in time and from experiment to experiment. A possible mechanism inducing this variation is a transition inducing the transformation ³H_f® ³H by an exotic weak decay d+p® u+n, where u and d correspond to the quarks at the ends of color flux tubes. This kind of transition could be induced by the absorption of X-rays, say artificial X-rays or X-rays from Sun. The inverse of this process in Sun could generate X rays which induce this process in resonant manner at the surface of Earth.

3. The well-known poorly understood X-ray bursts from Sun during solar flares in the wavelength range 1-8 A correspond to energies in the range 1.6-12.4 keV, 3 octaves in good approximation. This radiation could be partly due to transitions between ordinary and exotic states of nuclei rather than brehmstrahlung resulting in the acceleration of charged particles to relativistic energies. The energy range suggests the presence of three p-adic length scales: nuclear string model indeed predicts several p-adic length scales for color bonds corresponding to different mass scales for quarks at the ends of the bonds. This energy range is considerably above the energy range 5-100 eV and suggests the range [4×10^-4, 6×10^-2] for the values of p. The existence of these excitations would mean a new branch of low energy nuclear physics, which might be dubbed X-ray nuclear physics.

4. The approximately 1/2 year period of the temporal variation would naturally correspond to the 1/R² dependence of the intensity of X-ray radiation from Sun. There is evidence that the period is few hours longer than 1/2 years which supports the view that the origin of periodicity is not purely geometric but relates to the dynamics of X-ray radiation from Sun. Note that for 2 hours one would have DT/T @ 2^-11, which defines a fundamental constant in TGD Universe and is also near to the electron proton mass ratio.

5. All nuclei could appear as similar anomalous variants. Since both weak and strong decay rates are sensitive to the binding energy, it is possible to test this prediction by finding whether nuclear decay rates show anomalous time variation.

6. The model could explain also other anomalies of radioactive reaction rates including the findings of Shnoll [1] and the unexplained fluctuations in the decay rates of ³²Si and ²²⁶Ra reported quite recently and correlating with 1/R², R distance between Earth and Sun. ²²⁶Ra decays by alpha emission but the sensitive dependence of alpha decay rate on binding energy means that the temporal variation of the fraction of fake ²²⁶Ra isotopes could explain the variation of the decay rates. The intensity of the X-ray radiation from Sun is proportional to 1/R² so that the correlation of the fluctuation with distance would emerge naturally.

7. Also a dip in the decay rates of ⁵⁴Mn coincident with a peak in proton and X-ray fluxes during solar flare has been observed: the proposal is that neutrino flux from Sun is also enhanced during the solar flare and induces the effect. A peak in X-ray flux is a more natural explanation in TGD framework.

8. The model predicts interaction between atomic physics and nuclear physics, which might be of relevance in biology. For instance, the transitions between exotic and ordinary variants of nuclei could yield X-rays inducing atomic transitions or ionization. The wave length range 1-8 Angstroms for anomalous X-rays corresponds to the range Z in the rage [11,30] for ionization energies. The biologically important ions Na⁺, Mg⁺⁺, P^-, Cl^-, K⁺, Ca⁺⁺ have Z= (11,15,17,19,20). I have proposed that Na⁺, Cl^-, K⁺ (fermions) are actually bosonic exotic ions forming Bose-Einstein condensates at magnetic flux tubes (see this). The exchange of W bosons between neutral Ne and A(rgon) atoms (bosons) could yield exotic bosonic variants of Na⁺ (perhaps even Mg⁺⁺, which is boson also as ordinary ion) and Cl^- ions. Similar exchange between A atoms could yield exotic bosonic variants of Cl^- and K⁺ (and even Ca⁺⁺, which is also boson as ordinary variant). This transformation might relate to the paradoxical finding that noble gases can act as narcotics. This hypothesis is testable by measuring the nuclear weights of these ions. X-rays from Sun are not present during night time and this could relate to the night-day cycle of living organisms. Note that the nagnetic bodies are of size scale of Earth and even larger so that the exotic ions inside them could be subject to intense X-ray radiation. X-rays could also be dark X-rays with large Planck constant and thus with much lower frequency than ordinary X-rays so that control could be possible.

References

[1] S. E. Shnoll et al (1998), Realization of discrete states during fluctuations in macroscopic processes, Uspekhi Fisicheskikh Nauk, Vol. 41, No. 10, pp. 1025-1035.

[2]V. M. Lobashev et al(1996), in Neutrino 96 (Ed. K. Enqvist, K. Huitu, J. Maalampi). World Scientific, Singapore.

[3] Ch. Weinheimer et al (1993), Phys. Lett. 300B, 210.

[4] J. I. Collar (1996), Endpoint Structure in Beta Decay from Coherent Weak-Interaction of the Neutrino, hep-ph/9611420. [5]G. J. Stephenson Jr. (1993), Perspectives in Neutrinos, Atomic Physics and Gravitation, ed. J. T. Thanh Van, T. Darmour, E. Hinds and J. Wilkerson (Editions Frontieres, Gif-sur-Yvette), p.31.

For more details see the chapters TGD and Nuclear Physics and Nuclear String Hypothesis of "p-Adic length scale Hypothesis and Dark Matter Hierarchy".

Zero energy ontology, self hierarchy, and the notion of time

In the previous posting I discussed the most recent view about zero energy ontology and p-adicization program. One manner to test the internal consistency of this framework is by formulating the basic notions and problems of TGD inspired quantum theory of consciousness and quantum biology in terms of zero energy ontology. I have discussed these topics already earlier but the more detailed understanding of the role of causal diamonds (CDs) brings many new aspects to the discussion.

In consciousness theory the basic challenges are to understand the asymmetry between positive and negative energies and between two directions of geometric time at the level of conscious experience, the correspondence between experienced and geometric time, and the emergence of the arrow of time. One should also explain why human sensory experience is about a rather narrow time interval of about .1 seconds and why memories are about the interior of much larger CD with time scale of order life time. One should also have a vision about the evolution of consciousness takes place: how quantum leaps leading to an expansion of consciousness take place.

Negative energy signals to geometric past - about which phase conjugate laser light represents an example - provide an attractive tool to realize intentional action as a signal inducing neural activities in the geometric past (this would explain Libet's classical findings), a mechanism of remote metabolism, and the mechanism of declarative memory as communications with the geometric past. One should understand how these signals are realized in zero energy ontology and why their occurrence is so rare.

In the following my intention is to demonstrate that TGD inspired theory of consciousness and quantum TGD proper indeed seem to be in tune and that this process of comparison helps considerably in the attempt to develop the TGD based ontology at the level of details.

1  Causal diamonds as correlates for selves

Quantum jump as a moment of consciousness, self as a sequence of quantum jumps integrating to self, and self hierarchy with sub-selves experienced as mental images, are the basic notion of TGD inspired quantum theory of consciousness. In the most ambitious program self hierarchy reduces to a fractal hierarchy of quantum jumps within quantum jumps.

It is natural to interpret CD:s as correlates of selves. CDs can be interpreted in two manners: as subsets of the generalized imbedding space or as sectors of the world of classical worlds (WCW). Accordingly, selves correspond to CD:s of the generalized imbedding space or sectors of WCW, literally separate interacting quantum Universes. The spiritually oriented reader might speak of Gods. Sub-selves correspond to sub-CD:s geometrically. The contents of consciousness of self is about the interior of the corresponding CD at the level of imbedding space. For sub-selves the wave function for the position of tip of CD brings in the delocalization of sub-WCW.

The fractal hierarchy of CDs within CDs defines the counterpart for the hierarchy of selves: the quantization of the time scale of planned action and memory as T(k) = 2^kT₀ suggest an interpretation for the fact that we experience octaves as equivalent in music experience.

2. Why sensory experience is about so short time interval?

CD picture implies automatically the 4-D character of conscious experience and memories form part of conscious experience even at elementary particle level: in fact, the secondary p-adic time scale of electron is T=1 seconds defining a fundamental time scale in living matter. The problem is to understand why the sensory experience is about a short time interval of geometric time rather than about the entire personal CD with temporal size of order life-time. The obvious explanation would be that sensory input corresponds to sub-selves (mental images) which correspond to CD:s with T(127) @ .1 s (electrons or their Cooper pairs) at the upper light-like boundary of CD assignable to the self. This requires a strong asymmetry between upper and lower light-like boundaries of CD:s.

1. The only reasonable manner to explain the situation seems to be that the addition of CD:s within CD:s in the state construction must always glue them to the upper light-like boundary of CD along light-like radial ray from the tip of the past directed light-cone. This conforms with the classical picture according to which classical sensory data arrives from the geometric past with velocity which is at most light velocity.

2. One must also explain the rare but real occurrence of phase conjugate signals understandable as negative energy signals propagating towards geometric past. The conditions making possible negative energy signals are achieved when the sub-CD is glued to both the past and future directed light-cones at the space-like edge of CD along light-like rays emerging from the edge. This exceptional case gives negative energy signals traveling to the geometric past. The above mentioned basic control mechanism of biology would represent a particular instance of this situation. Negative energy signals as a basic mechanism of intentional action would explain why living matter seems to be so special.

3. Geometric memories would correspond to the lower boundaries of CD:s and would not be in general sharp because only the sub-CD:s glued to both upper and lower light-cone boundary would be present. A temporal sequence of mental images, say the sequence of digits of a phone number, could corresponds to a sequence of sub-CD:s glued to the upper light-cone boundary.

4. Sharing of mental images corresponds to a fusion of sub-selves/mental images to single sub-self by quantum entanglement: the space-time correlate for this could be flux tubes connecting space-time sheets associated with sub-selves represented also by space-time sheets inside their CD:s. It could be that these ëpisodal" memories correspond to CD:s at upper light-cone boundary of CD.

On basis of these arguments it seems that the basic conceptual framework of TGD inspired theory of consciousness can be realized in zero energy ontology. Interesting questions relate to how dynamical selves are.

1. Is self doomed to live inside the same sub-WCW eternally as a lonely god? This question has been already answered: there are interactions between sub-CD:s of given CD, and one can think of selves as quantum superposition of states in CD:s with wave function having as its argument the tips of CD, or rather only the second one since T is assumed to be quantized.

2. Is there a largest CD in the personal CD hierarchy of self in an absolute sense? Or is the largest CD present only in the sense that the contribution to the contents of consciousness coming from very large CD:s is negligible? Long time scales T correspond to low frequencies and thermal noise might indeed mask these contributions very effectively. Here however the hierarchy of Planck constants and generalization of the imbedding space would come in rescue by allowing dark EEG photons to have energies above thermal energy.

3. Can selves evolve in the sense that the size of CD increases in quantum leaps so that the corresponding time scale T=2^kT₀ of memory and planned action increases? Geometrically this kind of leap would mean that CD becomes a sub-CD of a larger CD either at the level of conscious experience or in absolute sense. This leap can occur in two senses: as an increase of the largest p-adic time scale in the personal hierarchy of space-time sheets or as increase of the largest value of Planck constants in the personal dark matter hierarchy. At the level of individual this would mean emergence of increasingly lower frequencies of generalization of EEG and of the levels of dark matter hierarchy with large value of Planck constant.

4. In 2-D illustration of the leap leading to a higher level of self hierarchy would mean simply the continuation of CD to right or left in the 2-D visualization of CD. Since the preferred M² is contained in the tangent space of space-time surfaces, and since preferred M² plays a key role in dark matter hierarchy too, one must ask whether the 2-D illustration might have some deeper truth in it.

3. New view about arrow of time

Perhaps the most fundamental problem related to the notion of time concerns the relationship between experienced time and geometric time. The two notions are definitely different: think only the irreversibility of experienced time and the reversibility of the geometric time and the absence of future of the experienced time. Also the deterministic character of the dynamics in geometric time is in conflict with the notion of free will supported by the direct experience.

In the standard materialistic ontology experienced time and geometric time are identified. In the naivest picture the flow of time is interpreted in terms of the motion of 3-D time=constant surface of space-time towards geometric future without any explanation for why this kind of motion would occur. This identification is plagued by several difficulties. In special relativity the difficulties relate to the impossibility define the notion of simultaneity in a unique manner and the only possible manner to save this notion seems to be the replacement of time=constant 3-surface with past directed light-cone assignable to the world-line of observer. In general relativity additional difficulties are caused by the general coordinate invariance unless one generalizes the picture of special relativity: problems are however caused by the fact that past light-cones make sense only locally. In quantum physics quantum measurement theory leads to a paradoxical situation since the observed localization of the state function reduction to a finite space-time volume is in conflict with the determinism of Schrödinger equation.

TGD forces a new view about the relationship between experienced and geometric time. Although the basic paradox of quantum measurement theory disappears the question about the arrow of geometric time remains.

1. Selves correspond to CD:s the own sub-WCW:s. These sub-WCW:s and their projections to the imbedding space do not move anywhere. Therefore standard explanation for the arrow of geometric time cannot work. Neither can the experience about flow of time correspond to quantum leaps increasing the size of the largest CD contributing to the conscious experience of self.

2. The only plausible interpretation is based on quantum classical correspondence and the fact that space-times are 4-surfaces of the imbedding space. If quantum jump corresponds to a shift of quantum superposition of space-time sheets towards geometric past in the first approximation (as quantum classical correspondence suggests), one can indeed understand the arrow of time. Space-time surfaces simply shift backwards with respect to the geometric time of the imbedding space and therefore to the 8-D perceptive field defined by the CD. This creates in the materialistic mind a kind of temporal variant of train illusion. Space-time as 4-surface and macroscopic and macro-temporal quantum coherence are absolutely essential for this interpretation to make sense.

Why this shifting should always take place to the direction of geometric past of the imbedding space? What seems clear is that the asymmetric construction of zero energy states should correlate with the preferred direction. If question is about probabilities, the basic question would be why the probabilities for shifts in the direction of geometric past are higher. Here some alternative attempts to answer this question are discussed.

1. Cognition and time relate to each other very closely and the required fusion of real physics with various p-adic physics of cognition and intentionality could also have something to do with the asymmetry. Indeed, in the p-adic sectors the transcendental values of p-adic light-cone proper time coordinate correspond to literally infinite values of the real valued light-cone proper time, and one can say that most points of p-adic space-time sheets serving as correlates of thoughts and intentions reside always in the infinite geometric future in the real sense. Therefore cognition and intentionality would break the symmetry between positive and negative energies and geometric past and future, and the breaking of arrow of geometric time could be seen as being induced by intentional action and also due to the basic aspects of cognitive experience.

2. Zero energy ontology suggests also a possible reason for the asymmetry. Standard quantum mechanics encourages the identification of the space of negative energy states as the dual for the space of positive energy states. There are two kinds of duals. Hilbert space dual is identified as the space of continuous linear functionals from Hilbert space to the coefficient field and is isometrically anti-isomorphic with the Hilbert space. This justifies the bra-ket notation. In the case of vector space the relevant notion is algebraic dual. Algebraic dual can be identified as an infinite direct product of the coefficient field identified as a 1-dimensional vector space. Direct product is defined as the set of functions from an infinite index set I to the disjoint union of infinite number of copies of the coefficient field indexed by I. Infinite-dimensional vector space corresponds to infinite direct sum consisting of functions which are non-vanishing for a finite number of indices only. Hence vector space dual in infinite-dimensional case contains much more states than the vector space and does not have enumerable basis.

   If negative energy states correspond to a subspace of vector space dual containing Hilbert space dual, the number of negative energy states is larger than the number of positive energy states. This asymmetry could correspond to better measurement resolution at the upper light-cone cone boundary so that the state space at lower light-cone boundary would be included via inclusion of HFFs to that associated with the upper light-cone boundary. Geometrically this would mean the possibility to glue to the upper light-cone boundary CD which can be smaller than those associated with the lower one.

3. The most convincing candidate for an answer comes from consciousness theory. One must understand also why the contents of sensory experience is concentrated around a narrow time interval whereas the time scale of memories and anticipation are much longer. The proposed mechanism is that the resolution of conscious experience is higher at the upper boundary of CD. Since zero energy states correspond to light-like 3-surfaces, this could be a result of self-organization rather than a fundamental physical law.
   1. The key assumption is that CDs have CDs inside CDs and that the vertices of generalized Feynman diagrams are contained within sub-CDs. It is not assumed that CDs are glued to the upper boundary of CD since the arrow of time results from self organization when the distribution of sub-CDs concentrates around the upper boundary of CD. In a category theoretical formulation for generalized Feynman diagrammatics based on this picture is developed.

   2. CDs define the perceptive field for self. Selves are curious about the space-time sheets outside their perceptive field in the geometric future (relative notion) of the imbedding space and perform quantum jumps tending to shift the superposition of the space-time sheets to the direction of geometric past (past defined as the direction of shift!). This creates the illusion that there is a time=snapshot front of consciousness moving to geometric future in fixed background space-time as an analog of train illusion.

   3. The fact that news come from the upper boundary of CD implies that self concentrates its attention to this region and improves the resolutions of sensory experience and quantum measurement here. The sub-CD:s generated in this manner correspond to mental images with contents about this region. As a consequence, the contents of conscious experience, in particular sensory experience, tend to be about the region near the upper boundary.

   4. This mechanism in principle allows the arrow of the geometric time to vary and depend on p-adic length scale and the level of dark matter hierarchy. The occurrence of phase transitions forcing the arrow of geometric time to be same everywhere are however plausible for the reason that the lower and upper boundaries of given CD must possess the same arrow of geometric time.

For details see chapters TGD as a Generalized Number Theory I: p-Adicization Program.

The most recent vision about zero energy ontology and p-adicization

The generalization of the number concept obtained by fusing real and p-adics along rationals and common algbraics is the basic philosophy behind p-adicization. This however requires that it is possible to speak about rational points of the imbedding space and the basic objection against the notion of rational points of imbedding space common to real and various p-adic variants of the imbedding space is the necessity to fix some special coordinates in turn implying the loss of a manifest general coordinate invariance. The isometries of the imbedding space could save the situation provided one can identify some special coordinate system in which isometry group reduces to its discrete subgroup. The loss of the full isometry group could be compensated by assuming that WCW is union over sub-WCW:s obtained by applying isometries on basic sub-WCW with discrete subgroup of isometries.

The combination of zero energy ontology realized in terms of a hierarchy causal diamonds and hierarchy of Planck constants providing a description of dark matter and leading to a generalization of the notion of imbedding space suggests that it is possible to realize this dream. The article TGD: What Might be the First Principles? provides a brief summary about recent state of quantum TGD helping to understand the big picture behind the following considerations.

1. Zero energy ontology briefly

1. The basic construct in the zero energy ontology is the space CD×CP₂, where the causal diamond CD is defined as an intersection of future and past directed light-cones with time-like separation between their tips regarded as points of the underlying universal Minkowski space M⁴. In zero energy ontology physical states correspond to pairs of positive and negative energy states located at the boundaries of the future and past directed light-cones of a particular CD. CD:s form a fractal hierarchy and one can glue smaller CD:s within larger CD along the upper light-cone boundary along a radial light-like ray: this construction recipe allows to understand the asymmetry between positive and negative energies and why the arrow of experienced time corresponds to the arrow of geometric time and also why the contents of sensory experience is located to so narrow interval of geometric time. One can imagine evolution to occur as quantum leaps in which the size of the largest CD in the hierarchy of personal CD:s increases in such a manner that it becomes sub-CD of a larger CD. p-Adic length scale hypothesis follows if the values of temporal distance T between tips of CD come in powers of 2ⁿ. All conserved quantum numbers for zero energy states have vanishing net values. The interpretation of zero energy states in the framework of positive energy ontology is as physical events, say scattering events with positive and negative energy parts of the state interpreted as initial and final states of the event.

2. In the realization of the hierarchy of Planck constants CD×CP₂ is replaced with a Cartesian product of book like structures formed by almost copies of CD:s and CP₂:s defined by singular coverings and factors spaces of CD and CP₂ with singularities corresponding to intersection M²ÇCD and homologically trivial geodesic sphere S² of CP₂ for which the induced Kähler form vanishes. The coverings and factor spaces of CD:s are glued together along common M²ÇCD. The coverings and factors spaces of CP₂ are glued together along common homologically non-trivial geodesic sphere S². The choice of preferred M² as subspace of tangent space of X⁴ at all its points and having interpretation as space of non-physical polarizations, brings M² into the theory also in different manner. S² in turn defines a subspace of the much larger space of vacuum extremals as surfaces inside M⁴×S².

3. Configuration space (the world of classical worlds, WCW) decomposes into a union of sub-WCW:s corresponding to different choices of M² and S² and also to different choices of the quantization axes of spin and energy and and color isospin and hyper-charge for each choice of this kind. This means breaking down of the isometries to a subgroup. This can be compensated by the fact that the union can be taken over the different choices of this subgroup.

4. p-Adicization requires a further breakdown to discrete subgroups of the resulting sub-groups of the isometry groups but again a union over sub-WCW:s corresponding to different choices of the discrete subgroup can be assumed. Discretization relates also naturally to the notion of number theoretic braid.

Consider now the critical questions.

1. Very naively one could think that center of mass wave functions in the union of sectors could give rise to representations of Poincare group. This does not conform with zero energy ontology, where energy-momentum should be assignable to say positive energy part of the state and where these degrees of freedom are expected to be pure gauge degrees of freedom. If zero energy ontology makes sense, then the states in the union over the various copies corresponding to different choices of M² and S² would give rise to wave functions having no dynamical meaning. This would bring in nothing new so that one could fix the gauge by choosing preferred M² and S² without losing anything. This picture is favored by the interpretation of M² as the space of longitudinal polarizations.

2. The crucial question is whether it is really possible to speak about zero energy states for a given sector defined by generalized imbedding space with fixed M² and S². Classically this is possible and conserved quantities are well defined. In quantal situation the presence of the lightcone boundaries breaks full Poincare invariance although the infinitesimal version of this invariance is preserved. Note that the basic dynamical objects are 3-D light-like "legs" of the generalized Feynman diagrams.

2. Definition of energy inzero energy ontology

Can one then define the notion of energy for positive and negative energy parts of the state? There are two alternative approaches depending on whether one allows or does not allow wave-functions for the positions of tips of light-cones.

Consider first the naive option for which four momenta are assigned to the wave functions assigned to the tips of CD:s.

1. The condition that the tips are at time-like distance does not allow separation to a product but only following kind of wave functions

   Ψ = exp(ip·m)Θ(m²) Θ(m⁰)× Φ(p) , m=m₊-m_-.

   Here m₊ and m_- denote the positions of the light-cones and Q denotes step function. F denotes configuration space spinor field in internal degrees of freedom of 3-surface. One can introduce also the decomposition into particles by introducing sub-CD:s glued to the upper light-cone boundary of CD.

2. The first criticism is that only a local eigen state of 4-momentum operators p_± = (^h/_2p) Ñ/i is in question everywhere except at boundaries and at the tips of the CD with exact translational invariance broken by the two step functions having a natural classical interpretation. The second criticism is that the quantization of the temporal distance between the tips to T = 2^kT₀ is in conflict with translational invariance and reduces it to a discrete scaling invariance.

The less naive approach relies of super conformal structures of quantum TGD assumes fixed value of T and therefore allows the crucial quantization condition T=2^kT₀.

1. Since light-like 3-surfaces assignable to incoming and outgoing legs of the generalized Feynman diagrams are the basic objects, can hope of having enough translational invariance to define the notion of energy. If translations are restricted to time-like translations acting in the direction of the future (past) then one has local translation invariance of dynamics for classical field equations inside dM⁴_± as a kind of semigroup. Also the M⁴ translations leading to interior of X⁴ from the light-like 2-surfaces surfaces act as translations. Classically these restrictions correspond to non-tachyonic momenta defining the allowed directions of translations realizable as particle motions. These two kinds of translations have been assigned to super-canonical conformal symmetries at dM⁴_±×CP₂ and and super Kac-Moody type conformal symmetries at light-like 3-surfaces. Equivalence Principle in TGD framework states that these two conformal symmetries define a structure completely analogous to a coset representation of conformal algebras so that the four-momenta associated with the two representations are identical .

2. The condition selecting preferred extremals of Kähler action is induced by a global selection of M² as a plane belonging to the tangent space of X⁴ at all its points . The M⁴ translations of X⁴ as a whole in general respect the form of this condition in the interior. Furthermore, if M⁴ translations are restricted to M², also the condition itself - rather than only its general form - is respected. This observation, the earlier experience with the p-adic mass calculations, and also the treatment of quarks and gluons in QCD encourage to consider the possibility that translational invariance should be restricted to M² translations so that mass squared, longitudinal momentum and transversal mass squared would be well defined quantum numbers. This would be enough to realize zero energy ontology. Encouragingly, M² appears also in the generalization of the causal diamond to a book-like structure forced by the realization of the hierarchy of Planck constant at the level of the imbedding space.

3. That the cm degrees of freedom for CD would be gauge like degrees of freedom sounds strange. The paradoxical feeling disappears as one realizes that this is not the case for sub-CDs, which indeed can have non-trivial correlation functions with either upper or lower tip of the CD playing a role analogous to that of an argument of n-point function in QFT description. One can also say that largest CD in the hierarchy defines infrared cutoff.

3. p-Adic variants of the imbedding space

Consider now the construction of p-adic variants of the imbedding space.

1. Rational values of p-adic coordinates are non-negative so that light-cone proper time a_4,+=Ö(t²-z²-x²-y²) is the unique Lorentz invariant choice for the p-adic time coordinate near the lower tip of CD. For the upper tip the identification of a₄ would be a_4,-=Ö((t-T)²-z²-x²-y²). In the p-adic context the simultaneous existence of both square roots would pose additional conditions on T. For 2-adic numbers T=2ⁿT₀, n ³ 0 (or more generally T=å_{k ³ n0}b_k 2^k), would allow to satisfy these conditions and this would be one additional reason for T=2ⁿT₀ implying p-adic length scale hypothesis. The remaining coordinates of CD are naturally hyperbolic cosines and sines of the hyperbolic angle h_±,4 and cosines and sines of the spherical coordinates q and f.

2. The existence of the preferred plane M² of un-physical polarizations would suggest that the 2-D light-cone proper times a_2,+ = Ö(t²-z²) a_2,- = Ö((t-T)²-z²) can be also considered. The remaining coordinates would be naturally h_±,2 and cylindrical coordinates (r,f).

3. The transcendental values of a₄ and a₂ are literally infinite as real numbers and could be visualized as points in infinitely distant geometric future so that the arrow of time might be said to emerge number theoretically. For M² option p-adic transcendental values of r are infinite as real numbers so that also spatial infinity could be said to emerge p-adically.

4. The selection of the preferred quantization axes of energy and angular momentum unique apart from a Lorentz transformation of M² would have purely number theoretic meaning in both cases. One must allow a union over sub-WCWs labeled by points of SO(1,1). This suggests a deep connection between number theory, quantum theory, quantum measurement theory, and even quantum theory of mathematical consciousness.

5. In the case of CP₂ there are three real coordinate patches involved . The compactness of CP₂ allows to use cosines and sines of the preferred angle variable for a given coordinate patch.

   ξ¹= tan(u)× cos(Θ/2)× exp(i(Ψ+Φ)/2) ,

   ξ²= tan(u)× sin(Θ/2)× exp(i(Ψ-Φ)/2).

   The ranges of the variables u,Q, F,Y are [0,p/2],[0,p],[0,4p],[0,2p] respectively. Note that u has naturally only the positive values in the allowed range. S² corresponds to the values F = Y = 0 of the angle coordinates.

6. The rational values of the (hyperbolic) cosine and sine correspond to Pythagorean triangles having sides of integer length and thus satisfying m² = n²+r² (m²=n²-r²). These conditions are equivalent and allow the well-known explicit solution . One can construct a p-adic completion for the set of Pythagorean triangles by allowing p-adic integers which are infinite as real integers as solutions of the conditions m²=r²±s². These angles correspond to genuinely p-adic directions having no real counterpart. Hence one obtains p-adic continuum also in the angle degrees of freedom. Algebraic extensions of the p-adic numbers bringing in cosines and sines of the angles p/n lead to a hierarchy increasingly refined algebraic extensions of the generalized imbedding space. Since the different sectors of WCW directly correspond to correlates of selves this means direct correlation with the evolution of the mathematical consciousness. Trigonometric identities allow to construct points which in the real context correspond to sums and differences of angles.

7. Negative rational values of the cosines and sines correspond as p-adic integers to infinite real numbers and it seems that one use several coordinate patches obtained as copies of the octant (x ³ 0,y ³ 0,z ³ 0,). An analogous picture applies in CP₂ degrees of freedom.

8. The expression of the metric tensor and spinor connection of the imbedding in the proposed coordinates makes sense as a p-adic numbers in the algebraic extension considered. The induction of the metric and spinor connection and curvature makes sense provided that the gradients of coordinates with respect to the internal coordinates of the space-time surface belong to the extensions. The most natural choice of the space-time coordinates is as subset of imbedding space-coordinates in a given coordinate patch. If the remaining imbedding space coordinates can be chosen to be rational functions of these preferred coordinates with coefficients in the algebraic extension of p-adic numbers considered for the preferred extremals of Kähler action, then also the gradients satisfy this condition. This is highly non-trivial condition on the extremals and if it works might fix completely the space of exact solutions of field equations. Space-time surfaces are also conjectured to be hyper-quaternionic , this condition might relate to the simultaneous hyper-quaternionicity and Kähler extremal property. Note also that this picture would provide a partial explanation for the decomposition of the imbedding space to sectors dictated also by quantum measurement theory and hierarchy of Planck constants.

4. p-Adic variants for the sectors of WCW

One can also wonder about the most general definition of the p-adic variants of the sectors of the world of classical worlds.

1. The restriction of the surfaces in question to be expressible in terms of rational functions with coefficients which are rational numbers of belong to algebraic extension of rationals means that the world of classical worlds can be regarded as a a discrete set and there would be no difference between real and p-adic worlds of classical worlds: a rather unexpected conclusion.

2. One can of course whether one should perform completion also for WCWs. In real context this would mean completion of the rational number valued coefficients of a rational function to arbitrary real coefficients and perhaps also allowance of Taylor and Laurent series as limits of rational functions. In the p-adic case the integers defining rational could be allowed to become p-adic transcendentals infinite as real numbers. Also now also Laurent series could be considered.

3. In this picture there would be close analogy between the structure of generalized imbedding space and WCW. Different WCW:s could be said to intersect in the space formed by rational functions with coefficients in algebraic extension of rationals just real and p-adic variants of the imbedding space intersect along rational points. In the spirit of algebraic completion one might hope that the expressions for the various physical quantities, say the value of Kähler action, Kähler function, or at least the exponent of Kähler function (at least for the maxima of Kähler function) could be defined by analytic continuation of their values from these sub-WCW to various number fields. The matrix elements for p-adic-to-real phase transitions of zero energy states interpreted as intentional actions could be calculated in the intersection of real and p-adic WCW:s by interpreting everything as real.

For details see chapters TGD as a Generalized Number Theory I: p-Adicization Program.

Dark nuclear strings as analogs of DNA-, RNA- and amino-acid sequences and baryonic realization of genetic code

In the earlier posting I considered the possibility that the evolution of genome might not be random but be controlled by magnetic body and that various DNA sequences might be tested in the virtual world made possible by the virtual counterparts of bio-molecules realized in terms of the homeopathic mechanism as it is understood in TGD framework. The minimal option is that virtual DNA sequences have flux tube connections to the lipids of the cell membrane so that their quality as hardware of tqc can be tested but that there is no virtual variant of transcription and translation machinery. One can however ask whether also virtual amino-acids could be present and whether this could provide deeper insights to the genetic code.

1. Water molecule clusters are not the only candidates for the representatives of linear molecules. An alternative candidate for the virtual variants of linear bio-molecules are dark nuclei consisting of strings of scaled up dark variants of neutral baryons bound together by color bonds having the size scale of atom, which I have introduced in the model of cold fusion and plasma electrolysis both taking place in water environment. Colored flux tubes defining braidings would generalize this picture by allowing transversal color magnetic flux tube connections between these strings.

2. Baryons consist of 3 quarks just as DNA codons consist of three nucleotides. Hence an attractive idea is that codons correspond to baryons obtained as open strings with quarks connected by two color flux tubes. The minimal option is that the flux tubes are neutral. One can also argue that the minimization of Coulomb energy allows only neutral dark baryons. The question is whether the neutral dark baryons constructed as string of 3 quarks using neutral color flux tubes could realize 64 codons and whether 20 aminoacids could be identified as equivalence classes of some equivalence relation between 64 fundamental codons in a natural manner.

The following model indeed reproduces the genetic code directly from a model of dark neutral baryons as strings of 3 quarks connected by color flux tubes.

1. Dark nuclear baryons are considered as a fundamental realization of DNA codons and constructed as open strings of 3 dark quarks connected by two colored neutral flux tubes. DNA sequences would in turn correspond to sequences of dark baryons. It is assumed that the net charge of the dark baryons vanishes so that Coulomb repulsion is minimized.

2. One can classify the states of the open 3-quark string by the total charges and spins associated with 3 quarks and to the two color bonds. Total em charges of quarks vary in the range Z_B Î {2,1,0,-1} and total color bond charges in the range Z_b Î {2,1,0,-1,-2}. Only neutral states are allowed. Total quark spin projection varies in the range J_B=3/2,1/2,-1/2,-3/2 and the total flux tube spin projection in the range J_b = 2,1,-1,-2. If one takes for a given total charge assumed to be vanishing one representative from each class (J_B,J_b), one obtains 4×5=20 states which is the number of amino-acids. Thus genetic code might be realized at the level of baryons by mapping the neutral states with a given spin projection to single representative state with the same spin projection.

3. The states of dark baryons in quark degrees of freedom can be constructed as representations of rotation group and strong isospin group. The tensor product 2Ä2Ä2 is involved in both cases. Physically it is known that only representations with isospin 3/2 and spin 3/2 (D resonance) and isospin 1/2 and spin 1/2 (proton and neutron) are realized. Spin statistics problem forced to introduce quark color (this means that one cannot construct the codons as sequences of 3 nucleons!).

4. Second nucleon spin doublet has wrong parity. Using only 4Å2 for rotation group would give degeneracies (1,2,2,1). One however requires the representations 4Å2Å2 rather than only 4Å2 to get 8 states with a given charge. One should transform the wrong parity doublet to positive parity doublet somehow. Since open string geometry breaks rotational symmetry to a subgroup of rotations acting along the direction of the string, the attractive possible is add a stringy excitation with angular momentum projection L=-1 to the wrong parity doublet so that parity comes out correctly. This would give degeneracies (1,2,3,2).

5. In flux tube degrees of freedom the situation is analogous to construction of mesons from quarks and antiquarks and one obtains pion with spin 0 and r meson with spin 1. States of zero charge correspond to the tensor product 2Ä2=3Å1 for rotation group. Drop the singlet and take only the analog of neutral r meson. The tensor product 3Ä3=5Å3Å1 gives 8+1 states and leaving only spin 2 and spin 1 states gives 8 states. The degeneracies of states with given spin projection for 5Å3 are (1,2,2,2,1). Genetic code means projection of the states of 5Å3 to those of 5 with the same spin projection.

6. Genetic code maps of ( 4Å2Å2)Ä(5Å3) to the states of 4×5. The most natural map maps the states with given spin to state with same spin so that the code is unique. This would give the degeneracies D(k) as products of numbers D_B Î {1,2,3,2} and D_b Î {1,2,2,2,1}. The numbers N(k) of aminoacids coded by D(k) codons would be

   
   

   [N(1),N(2),N(3),N(4),N(6)]=[2,7,2,6,3] .

   The correct numbers for vertebrate nuclear code are (N(1),N(2),N(3),N(4),N(6)) = (2,9,1,5,3). Some kind of symmetry breaking must take place and should relate to the emergence of stopping codons. If one codon in second 3-plet becomes stopping codon, 3-plet becomes doublet. If 2 codons in 4-plet become stopping codons it also becomes doublet and one obtains the correct result (2,9,1,5,3)!

The conclusion is that genetic code can be understand as a map of stringy baryonic states induced by the projection of all states with same spin projection to a representative state with same spin projection. Genetic code would be realized at the level of dark nuclear physics and perhaps also at the level of ordinary nuclear physics and that biochemical representation would be only one particular higher level representation of the code.

For details see chapters Homeopathy in Many-Sheeted Space-time of "Bio-Systems as Conscious Holograms" and The Notion of Wave-Genome and DNA as Topological Quantum Computer of "Genes and Memes"