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Genes and Memes

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Year 2013

Negentropic entanglement, NMP, braiding and TQC

Negentropic entanglement for which number theoretic entropy characterized by p-adic prime is negative so that entanglement carries information, is in key role in TGD inspired theory of consciousness and quantum biology.

  1. The key feature of negentropic entanglement is that density matrix is proportional to unit matrix so that the assumption that state function reduction corresponds to the measurement of density matrix does not imply state function reduction to one-dimensional sub-space. This special kind of degenerate density matrix emerges naturally for the hierarchy heff=nh interpreted in terms of a hierarchy of dark matter phases. I have already earlier considered explicit realizations of negentropic entanglement assuming that the entanglement matrix is invariant under the group of unitary or orthogonal transformations (also subgroups of unitary group can be considered - say symplectic group). One can however consider much more general options and this leads to a connection with topological quantum computation (TQC).

  2. Entanglement matrices E equal to n-1/2 factor times unitary matrix U (as a special case to orthogonal matrix O) defines a density matrix given by ρ=UU/n= Idn/n, which is group invariant. One has negentropic entanglement (NE) respected by state function reduction if Negentropy Maximization Principle (NMP) is assumed. This would give huge number of negentropically entangled states providing a representation for some unitary group or its subgroup (such as symplectic group). In principle any unitary representation of any Lie group would allow representation in terms of NE.

  3. In physics as generalized number theory vision, a natural condition is that the matrix elements of E belong to the algebraic extension of p-adic numbers used so that discreted algebraic subgroups of unitary or orthogonal group are selected. This realizes evolutionary hierarchy as a hierarchy of p-adic number fields and their algebraic extensions, and one can imagine that evolution of cognition proceeds by the generation of negentropically entangled systems with increasing algebraic dimensions and increasing dimension reflecting itself as an increase of the largest prime power dividing n and defining the p-adic prime in question.

  4. One fascinating implication is the ability of TGD Universe to emulate itself like Turing machine: unitary S-matrix codes for scattering amplitudes and therefore for physics and negentropically entangled subsystem could represent sub-matrix for S-matrix as rules representing "the laws of physics" in the approximation that the world corresponds to n-dimension Hilbert space. Also the limit n → ∞ makes sense, especially so in the p-adic context were real infinity can correspond to finite number in the sense of p-adic norm. Here also dimensions n given as products of powers of infinite primes can be formally considered.

One can consider various restrictions on E.

  1. In 2-particle case the stronger condition that E is group invariant implies that unitary matrix is identity matrix apart from an overall phase factor: U= exp(iφ)Id. In orthogonal case the phase factor is +/- 1. For n-particle NE one can consider group invariant states by using n-dimensional permutation tensor εi1,...in.
  2. One can give up the group invariance of E and consider only the weaker condition that permutation is represented as transposition of entanglement matrix: Cij→ Cij. Symmetry/antisymmetry under particle exchange would correspond to Cji=ε Cij, ε=+/- 1. This would give in orthogonal case OOT= O2=Id and UU*= Id in the unitary case.

    In the unitary case particle exchange could be also identified as hermitian conjugation Cij→ Cji* and one would have also now U2=Id. Euclidian gamma matrices γi define unitary and hermitian generators of Clifford algebra having dimension 22m for n=2m and n=2m+1. It is relatively easy to verify that the squares of completely anti-symmetrized products of k gamma matrices representing exterior algebra normalized by factor 1/k!1/2 are equal to unit matrix. For k=n the antisymmetrized product gives essentially permutation symbol times the product ∏k γk. In this manner one can construct entanglement matrices representing negentropic bi-partite entanglement.

  3. The possibility of taking tensor products εij..k...nγi⊗ γj..⊗ γk of k gamma matrices means that one can has also co-product of gamma matrices. What is interesting is that quantum groups important in topological quantum computation as well as the Yangian algebra associated with twistor Grassmann approach to scattering amplitudes possess co-algebra structure. TGD leads also to the proposal that this structure plays a central role in the construction of scattering amplitudes. Physically the co-product is time reversal of product representing fusion of particles.
  4. One can go even further. In 2-dimensional QFTs braid statistics replaces ordinary statistics. The natural question is what braid statistics could correspond to at the level of NE. Braiding matrix is unitary so that it defines NE. Braiding as a flow replaces the particle exchange and lifts permutation group to braid group serving as its infinite covering. The allowed unitary matrices representing braiding in tensor product are constructed using braiding matrix R representing the exchange for two braid strands? The well-known Yang-Baxter equation for R defined in tensor product as an invertible element (see this) expresses the associativity of braiding operation. Concretely it states that the two braidings leading from 123 to 321 produce the same result. Entanglement matrices constructed R as basic operation would correspond to unitary matrices providing a representation for braids and each braid would give rise to one particular NE.

    This would give a direct connection with TQC for which the entanglement matrix defines a density matrix proportional to n× n unit matrix: R defines the basic gate (see this). Braids would provide a concrete representation for NE giving rise to "Akashic records". I have indeed proposed the interpretation of braidings as fundamental memory representations much before the vision about Akashic records. This kind of entanglement matrix need not represent only time-like entanglement but can be also associated also with space-like entanglement. The connection with braiding matrices supports the view that magnetic flux tubes are carriers of negentropically entangled matter and also suggests that this kind of entanglement between - say - DNA and nuclear or cell membrane gives rise to TQC.

Some comments concerning the covering space degrees of freedom associated with heff=nh viz. ordinary degrees of freedom are in order.
  1. Negentropic entanglement with n entangled states would correspond naturally to heff=nh and is assigned with "many-particle" states, which can be localized to the sheets of covering but one cannot exclude similar entanglement in other degrees of freedom. Group invariance leaves only group singlets and states which are not singlets are allowed only in special cases. For instance for SU(2) the state kenovert |j,m >kenorangle= | 1,0 > represented as 2-particle state of 2 spin 1/2 particles is negentropically entangled whereas the states | j,m >= |1,+/- 1 > are pure.
  2. Negentropic entanglement associated with heff=nh could factorize as tensor product from other degrees of freedom. Negentropic entanglement would be localised to the covering space degrees of freedom but there would be entropic entanglement in the ordinary degrees of freedom - say spin. The large value of heff would however scale up the quantum coherence time and length also in the ordinary degrees of freedom. For entanglement matrix this would correspond to a direct sum proportional to unitary matrices so that also density matrix would be a direct sum of matrices pn En= pn Idn/n , ∑ pn=1 correspond ing to various values of "other quantum numbers", and state function reduction could take place to any subspace in the decomposition. Also more general entanglement matrices for which the dimensions of direct summands vary, are possible.
  3. One can argue that NMP does not allow halting of quantum computation. The counter argument would be that the halting is not needed if it is indeed possible to deduce the structure of negentropically entangled state by an interaction free quantum measurement replacing the state function reduction with "externalised" state function reduction. One could speak of interaction free TQC. This TQC would be reading of "Akashic records". NE should be able to induce a conscious experience about the outcome of TQC which in the ordinary framework is represented by the reduction probabilities for various possible outcomes.

    One could also counter argue that NMP allows the transfer of NE from the system so that TQC halts. NMP allows this if some another system receives at least the negentropy contained by NE. The interpretation would be as the increase of information obtained by a conscious observer about the outcome of halted quantum computation. It am not able to imagine how this could happen at the level of details.

For details and background see the chapter "DNA as topological quantum computer" and the article "Negentropic entanglement, NMP, braiding and topological quantum computation".

Great Unconformity as a new piece of support for Expanding Earth model

I wrote for some years ago to "Genes and Memes" a chapter entitled Expanding Earth model and Precambrian evolution of continents, climate and life . Single hypothesis - a sudden phase transition increasing the radius of Earth by a factor 2 natural in the many-sheeted space-time of TGD - explains Cambrian explosion in biology (a sudden emergence of huge number of life forms after very slow Precambrian evolution, and also provides a model for Precambrian evolution of continents, climate and life.

Already Darwin realized that the absence of fossils from Precambrian era is a deep problem for his theory and assumed that this is an artefact due to the incomplete fossil record. Fossils of Precambrian origin have been indeed found after Darwin's time but they are simple and very rare, and the conclusion is that Cambrian explosion meaning huge diversification was real. Two mysteries therefore remain. Why the development of life was so slow during Precambrian era? Why the diversification was so incredibly fast during Cambrian explosion? Various explanations have been proposed. Did the oxygen content of the atmosphere reach a critical value and lead to the diversification? Or did predation pose the evolutionary pressure making the pace of evolution dramatically faster?

In New Scientist geologists Robert Gaines and Shanan Peters describe a geological finding perhaps related to the Cambrian Explosion: the mysterious "Great Unconformity", which is a juxtaposition of two different types of rock of very different geological ages along a prominent surface of erosion. This surface represents a very long span of "missing" time. More than 1 billion years of geological record is missing in many places! From the figure of the Wikipedia article about Great Unconformity visible in Grand Canyon the thickness of the missing layer can be estimated to be about 12.6 km. Somehow before the Cambrian the uppermost rocks of the continents were stripped away exposing the underlying crystalline basement rocks. The cause of this gap remains a complete mystery so that we have three mysteries! Plus the mysteries related to the evolution of climate (problems of Snowball Earth model).

The authors suggest that the formation of Great Unconformity relates to the Cambrian explosion. Large scale erosion and chemical weathering of the the exposed crystalline rock caused mineralization of the sea water. The hypothesis is that this led to bio-mineralization: animal groups possessing mineral skeletons - such as silica shells and calcium carbonate shells - emerged. This hypothesis looks rather plausible but does not solve the three great mysteries.

The authors indeed leave open the question about the origin of Great Unconformity and of Cambrian explosion. The TGD based explanation of Cambrian explosion comes from the model realizing the old idea about Expanding Earth in terms of TGD inspired new physics. Already Wegener observed that continents can be fit together nicely and this led to the recent view about plate tectonics. Wegener's model however fits only "half" of the continent boundaries together. One could however do much better: the observation is that the continents would fit nicely to cover the entire surface of Earth if the radius of Earth were 1/2 of its recent value! Expanding Earth model postulates that the radius of Earth grows slowly. Geologists have not taken Expanding Earth model seriously: one good reason is that there is no physics allowing it.

TGD predicts this physics.

  1. At given sheet of the many-sheeted space-time cosmic expansion is predicted to take place as sudden phase transitions in which the size of some space-time sheet suddenly increases. By p-adic length scale hypothesis the preferred scaling factors are powers of 2 and the most favored scaling factor is just two. The proposal is that during the Precambrian era life resided in underground seas being thus shielded from meteor bombardment and cosmic rays. This explains the scarcity of the fossil records and the simplicity of the fossils found. The sudden phase transition was a very violent process increasing the area of the Earth's surface by a factor of 4. The area of continents is 29.1 per cent from the recent area of the Earth's surface - not too far from the naively predicted fraction 1/4.
  2. It is easy to imagine that the uppermost rocks of the continent covering the entire Earth were stripped away and correspond nowadays to 100 km thick continental tectonic plates consisting of mainly silicon and aluminium). This expansion created split first the topmost layer as continental plates and regions between them giving rise to oceans. The magma which was uncovered by the process cooled down and solidifed and the continued expansion gave rise to ocean plates with different composition (mainly silicon and magnesium).
  3. The expansion phase corresponds to criticality so that fractality of the expansion is expected. At least for continental plates this process could have been fractal occurring in various length scales characterizing the thickness and the area of the sub-plates generated in the process. p-Adic length scale hypothesis suggests that the scales involved should appear as powers of 21/2 or 2. Generation of Great Unconformity as a process in which the underlying crystalline basement rocks were uncovered could correspond to a splitting of a layer of the continental plates to pieces. The length scale characterizing the thickness is 12.6 km from the above estimate and with 1 per cent accuracy by a factor 1/8 shorter than 100 km length scale for tectonic plates. This conforms with p-adic fractality. If the process of expansion involved a cascade of scalings by factor 2, one can wonder whether it proceeded from long to short length scales or vice versa. In other words: did continental and oceanic tectonic plates form first and after than the smaller structures such as the Great Unconformity or vice versa?
  4. Note that the p-adic length scale L(237) corresponding p∼ 2237 is 88 km - ten per cent smaller than 100 km. Maybe thermal expansion could account the discrepancy if the original thickness was L(237). There is a nice fractal analogy with cell membrane. The p-adic length scale L(239) defined by the Gaussian Mersenne Mk,G=(1+i)k-1 , k=239- the first one after the Gaussian Mersenne M167,G defining the size of cell nucleus - equals 2× L(237). Remarkably, also M241,G is Gaussian Mersenne. What it interesting that the scale L(127)=88 km would be in the same relation to L(239) associated with Mk=239,G as the thickness L(149) of the lipid layer of cell membrane to the cell membrane thickness L(151) characterized by Gaussian Mersenne M151,G. The two kind tectonic plates (continental and oceanic) would be analogous to the lipid layers of cell membrane. Note that 88 km is rather precisely the thickness of the atmosphere above which begins ionosphere. The thickness of Kennelly–Heaviside layer inside which radiowaves used in terrestrial radio communications propagate, has thickness about 150 km which roughly corresponds to L(239). Also the fact Continental litosphere has typical thickness of 200 km (L(239)) whereas oceanic litosphere is 100 km thick (L(237)) fits nicely with the proposed formation mechanism of continental tectonic plates.
  5. The rapid expansion process could have also brought in daylight the underground seas and the highly developed life in them so that Cambrian diversification would have been only apparent. Skeptic can of course ask whether it is necessary to assume that life resided in underground seas during Precambrian era. Could just the violent geological process be enough to induce extremely fast diversification? This might of course be true.
  6. There is one further argument in favor of the Expanding Earth model. The fact that the solar constant was during proto Earth period only 73 per cent from its recent value, is a problem for the models of the very early evolution of life. If the radius of Earth was 1/2 of its recent value the duration of day and night was from conservation of angular momentum only 1/4:th of the recent value and thus 3 hours. This could have made the environment much more favorable for the evolution of life even at the surface of the Earth since the range for the temperature variation would have been much narrower.

This model relates also to the Snowball Earth model of Precambrian climate in an interesting manner. The reader interested in details should read the chapter Expanding Earth model and Precambrian evolution of continents, climate and life .

Could one find a geometric realization for genetic and memetic codes?

The idea that that icosahedral structures assignable to water clusters could define a geometric representation of some kind of code is very intriguing. Genetic code is of course the code that comes first in mind. The observation that the number of faces of tetrahedron (icosahedron) is 4 (20) raises the question whether genetic code might have a geometric representation. In TGD framework also a second code emerges: I have christened it memetic code. Also memetic code could have a geometric realization. Another purely TGD-based notion is that of dark DNA allowing to assign the states of dark protons with DNA,RNA, tRNA and amino-acids and to predict correctly the numbers of DNA codons coding for a given amino-acid in vertebrate genetic code. A further element is the possibility of strong gravitation in TGD Universe meaning that space-time geometry and topology can be highly non-trivial even in condensed matter length scales. These ingredients allow to imagine geometric representations of genetic and memetic codes.

For background see the chapter Three new physics realizations of the genetic code and the role of dark matter in bio-systems.

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