Sunday, December 29, 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,

  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 section "Updates since 2012" of chapter "Negentropy Maximization Principle" and and the article ""Negentropic entanglement, NMP, braiding and topological quantum computation"

Saturday, December 28, 2013

Naturalness, fine-tuning, second law, and love

In previous posting we have had a very nice discussion with Hamed about various aspects of "world of classical worlds" realized as a space of space-time surfaces (very rough statement). I thought that the following comment - intended to be a further response relating to the uniqueness of "world of classical worlds" (WCW) - deserves the status of a separate blog posting.

So called naturalness is the physicist's manner to say "of course" as Bee very neatly expresses the gist of this notion. After the results from LHC we know that standard model is not natural and SUSY in standard form cannot help. What one should think about theories whose predictions change dramatically when some parameter is varied only slightly? Certainly these kind of theories are not very useful. Many people are however ready to accept that theory could be of this kind. I see this as giving up. If theory in unstable under small allowed variations of its parameters it is definitely wrong and this is extremely valuable guideline.

The essential attribute appearing in this claim is " allowed". In some situations one cannot allow any variations. Indeed, classical number fields, groups, etc. are extremely rigid mathematical structures, which do not allow any variations of their structure. In the same manner, in TGD WCW is really something God given (or mathematics given). There are no parameters to be varied and fine-tune.

This is of course the great idea of TGD: physics follows from the mere existence of an infinite-dimensional geometry of WCW - more precisely from the existence of Riemann connection, which forces WCW to be a union of infinite-D symmetric spaces with maximal isometries having interpretation as conformal symmetries or analogs of them (symplectic symmetries of boundary of causal diamond). Among other things, this condition fixes imbedding space uniquely to M4 × CP2 (also the condition that the twistor spaces associated with the Cartesian factors exist and are Kähler manifolds forces the same conclusion) allows to avoid the landscape difficulty of super string models, which is actually much more general problem.

Anthropic principle is second aspect related to fine-tuning. Fine-tuning of certain parameters essential for life seems to be present in physics. The idea behind the anthropic principle is that our own existence allows to deduce the values of fine-tuned parameters. This hypothesis involves however many implicit assumptions. For instance, one assumes that life as we know it recently is the only possible form of life and that there cannot be any other kinds of life forms realized for the other values of the key parameters.

The TDG based explanation for the fine-tuning relies on Negentropy Maximization Principle. NMP implies evolution accompanied by increasing negentropy resources realized in terms of negentropic entanglement. It also means finite-tuning: slowly varying parameters approach quantum jump by quantum jump values, which make possible maximal negentropy resources. The evolutionary self-organization process implied by NMP would lead to highly unique final states. Since NMP is more or less the mirror image of second law, the self-organization process could be mathematically analogous to the approach to thermodynamical equilibrium or thermodynamical non-equilibrium state (in presence of energy feed) polishing out all unessential details and leaving only the gem.

It is often said that life generates huge amounts of entropy. Looks at first just the opposite for what NMP predicts! Of course, the entropy in question is ensemble entropy and has as such nothing do with number theoretic entropy characterizing negentropic entanglement having interpretation as a quantum correlate for rule realized as superposition of its instances. NMP implies second law for ordinary entanglement and this might be enough (NMP however also predicts varying arrow of thermodynamical time, something highly non-trivial although considerable support for this exists in living matter!).

I have also considered the pessimistic conjecture that second law holds true in the following sense. If negentropy associated with negentropic entanglement is generated, it is somehow accompanied by a generation of entropy at least compensating for it. I do not believe this conjecture: the two notions do not simply apply in the same context and I do not have any mechanism for how the entropy would be generated. One can however consider the following natural correlation. Systems with a large number of degenerate states (same energy) in thermodynamical equilibrium have large entropy. The same highly degenerate systems can however entangle negentropically and this gives rise to a large entanglement negentropy. Entropy can transforms to negentropy! Love - for which I have suggested negentropic entanglement to serve as a quantum correlate - makes jewels from dirt!

Monday, December 23, 2013

About the notion of four-momentum in TGD framework

The starting point of TGD was the energy problem of General Relativity (see this). The solution of the problem was proposed in terms of sub-manifold gravity and based on the lifting of the isometries of space-time surface to those of M4× CP2 in which space-times are realized as 4-surfaces so that Poincare transformations act on space-time surface as an 4-D analog of rigid body rather than moving points at space-time surface. It however turned out that the situation is not at all so simple.

There are several conceptual hurdles and I have considered several solutions for them. The basic source of problems has been Equivalence Principle (EP): what does EP mean in TGD framework (see this and this)? A related problem has been the interpretation of gravitational and inertial masses, or more generally the corresponding 4-momenta. In General Relativity based cosmology gravitational mass is not conserved and this seems to be in conflict with the conservation of Noether charges. The resolution is in terms of zero energy ontology (ZEO), which however forces to modify slightly the original view about the action of Poincare transformations.

A further problem has been quantum classical correspondence (QCC): are quantal four-momenta associated with super conformal representations and classical four-momenta associated as Noether charges with Kähler action for preferred extremals identical? Could inertial-gravitational duality - that is EP - be actually equivalent with QCC? Or are EP and QCC independent dualities. A powerful experimental input comes p-adic mass calculations (see this) giving excellent predictions provided the number of tensor factors of super-Virasoro representations is five, and this input together with Occam's razor strongly favors QCC=EP identification.

Twistor Grassmannian approach has meant a technical revolution in quantum field theory (for attempts to understand and generalize the approach in TGD framework (see this and this). This approach seems to be extremely well suited to TGD and I have considered a generalization of this approach from N=4 SUSY to TGD framework by replacing point like particles with string world sheets in TGD sense and super-conformal algebra with its TGD version: the fundamental objects are now massless fermions which can be regarded as on mass shell particles also in internal lines (but with unphysical helicity). The approach solves old problems related to the realization of stringy amplitudes in TGD framework, and avoids some problems of twistorial QFT (IR divergences and the problems due to non-planar diagrams). The Yangian variant of 4-D conformal symmetry is crucial for the approach in N=4 SUSY, and implies the recently introduced notion of amplituhedon (see this). A Yangian generalization of various super-conformal algebras seems more or less a "must" in TGD framework. As a consequence, four-momentum is expected to have characteristic multilocal contributions identifiable as multipart on contributions now and possibly relevant for the understanding of bound states such as hadrons.

1. Scale dependent notion of four-momentum in zero energy ontology

Quite generally, General Relativity does not allow to identify four-momentum as Noether charges but in GRT based cosmology one can speak of non-conserved mass cosmo, which seems to be in conflict with the conservation of four-momentum in TGD framework. The solution of the problem comes in terms of zero energy ontology (ZEO) (see this and this), which transforms four-momentum to a scale dependent notion: to each causal diamond (CD) one can assign four-momentum assigned with say positive energy part of the quantum state defined as a quantum superposition of 4-surfaces inside CD.

ZEO is necessary also for the fusion of real and various p-adic physics to single coherent whole. ZEO also allows maximal "free will" in quantum jump since every zero energy state can be created from vacuum and at the same time allows consistency with the conservation laws. ZEO has rather dramatic implications: in particular the arrow of thermodynamical time is predicted to vary so that second law must be generalized. This has especially important implications in living matter, where this kind of variation is observed.

More precisely, this superposition corresponds to a spinor field in the "world of classical worlds" (WCW) (see this): its components - WCW spinors - correspond to elements of fermionic Fock basis for a given 4-surface - or by holography implied by general coordinate invariance (GCI) - for 3-surface having components at both ends of CD. Strong form of GGI implies strong form of holography (SH) so that partonic 2-surfaces at the ends of space-time surface plus their 4-D tangent space data are enough to fix the quantum state. The classical dynamics in the interior is necessary for the translation of the outcomes of quantum measurements to the language of physics based on classical fields, which in turn is reduced to sub-manifold geometry in the extension of the geometrization program of physics provided by TGD.

Holography is very much reminiscent of QCC suggesting trinity: GCI-holography-QCC. Strong form of holography has strongly stringy flavor: string world sheets connecting the wormhole throats appearing as basic building bricks of particles emerge from the dynamics of induced spinor fields if one requires that the fermionic mode carries well-defined electromagnetic charge (see this).

2. Are the classical and quantal four-momenta identical?

One key question concerns the classical and quantum counterparts of four-momentum. In TGD framework classical theory is an exact part of quantum theory. Classical four-momentum corresponds to Noether charge for preferred extremals of Kähler action. Quantal four-momentum in turn is assigned with the quantum superposition of space-time sheets assigned with CD - actually WCW spinor field analogous to ordinary spinor field carrying fermionic degrees of freedom as analogs of spin. Quantal four-momentum emerges just as it does in super string models - that is as a parameter associated with the representations of super-conformal algebras. The precise action of translations in the representation remains poorly specified. Note that quantal four-momentum does not emerge as Noether charge: at at least it is not at all obvious that this could be the case.

Are these classical and quantal four-momenta identical as QCC would suggest? If so, the Noether four-momentum should be same for all space-time surfaces in the superposition. QCC suggests that also the classical correlation functions for various general coordinate invariant local quantities are same as corresponding quantal correlation functions and thus same for all 4-surfaces in quantum superposition - this at least in the measurement resolution used. This would be an extremely powerful constraint on the quantum states and to a high extend could determined the U-, M-, and S-matrices.

QCC seems to be more or less equivalent with SH stating that in some respects the descriptions based on classical physics defined by Kähler action in the interior of space-time surface and the quantal description in terms of quantum states assignable to the intersections of space-like 3-surfaces at the boundaries of CD and light-like 3-surfaces at which the signature of induced metric changes. SH means effective 2-dimensionality since the four-dimensional tangent space data at partonic 2-surfaces matters. SH could be interpreted as Kac-Mody and symplectic symmetries meaning that apart from central extension they act almost like gauge symmetries in the interiors of space-like 3-surfaces at the ends of CD and in the interiors of light-like 3-surfaces representing orbits of partonic 2-surfaces. Gauge conditions are replaced with Super Virasoro conditions. The word "almost" is of course extremely important.

3. What does Equivalence Principle (EP) mean in TGD?

EP states the equivalence of gravitational and inertial masses in Newtonian theory. A possible generalization would be equivalence of gravitational and inertial four-momenta. In GRT this correspondence cannot be realized in mathematically rigorous manner since these notions are poorly defined and EP reduces to a purely local statement in terms of Einstein's equations. What about TGD? What could EP mean in TGD framework?

  1. Is EP realized at both quantum and space-time level? This option requires the identification of inertial and gravitational four-momenta at both quantum and classical level. QCC would require the identification of quantal and classical counterparts of both gravitational and inertial four-momenta. This would give three independent equivalences, say PI,class=PI,quant, Pgr,class=Pgr,quant, Pgr,class=PI,quant, which imply the remaining ones.

    Consider the condition Pgr,class=PI,class. At classical level the condition that the standard energy momentum tensor associated with Kähler action has a vanishing divergence is guaranteed if Einstein's equations with cosmological term are satisfied. If preferred extremals satisfy this condition they are constant curvature spaces for non-vanishing cosmological constant. A more general solution ansatz involves several functions analogous to cosmological constant corresponding to the decomposition of energy momentum tensor to terms proportional to Einstein tensor and several lower-dimensional projection operators (see this). It must be emphasized that field equations are extremely non-linear and one must also consider preferred extremals (which could be identified in terms of space-time regions having so called Hamilton-Jacobi structure): hence these proposals are guesses motivated by what is known about exact solutions of field equations.

    Consider next Pgr,class=PI,class. At quantum level I have proposed coset representations for the pair of super-symplectic algebras assignable to the light-like boundaries of CD and the Super Kac-Moody algebra assignable to the light-like 3-surfaces defining the orbits of partonic 2-surfaces as realization of Equivalence Principle. For coset representation the differences of super-conformal generators would annihilate the physical states so that one can argue that the corresponding four-momenta are identical. One could even say that one obtains coset representation for the "vibrational" parts of the super-conformal algebras in question. I must admit that the notion of coset representation creates uneasy feeling in my stomach. Note however that coset representations occur naturally for the subalgebras of symplectic algebra and Super Kac-Moody algebra and are naturally induced by finite measurement resolution.

  2. Does EP reduce to one aspect of QCC? This would require that classical Noether four-momentum identified as inertial momentum equals to the quantal four-momentum assignable to the states of super-conformal representations and identifiable as gravitational four-momentum. There would be only one independent condition: Pclass== PI,class=Pgr,quant== Pquant.

    Holography realized as AdS/CFT correspondence states the equivalence of descriptions in terms of gravitation realized in terms of strings in 10-D spacetime and gauge fields at the boundary of AdS. What is disturbing is that this picture is not completely equivalent with the proposed one. In this case the super-conformal algebra would be direct sum of super-symplectic and super Kac-Moody parts.

Which of the options looks more plausible? The success of p-adic mass calculations (see this) have motivated theuse of them as a guideline in attempts to understand TGD. The basic outcome was that elementary particle spectrum can be understood if Super Virasoro algebra has five tensor factors. Can one decide the fate of the two approaches to EP using this number as an input?

  1. For the coset option the situation is unclear. Even the definition of coset representation is problematic. If Super Kac-Moody generators vanish at partonic 2-surfaces one would have just direct sum Super-Virasoro algebras and coset representations would reduce to that for symplectic group containing only single tensor factor. If Super-Kac Moody generators do not vanish at partonic 2-surfaces. one must extend the symplectic generators by making them local with respect to partonic 2-surface in order to get a closed algebra. The imbedding of Kac-Moody to Sympl might be well-defined since isometries form subgroup of symplectic transformations. But is it possible to speak about a direct sum of Super Virasoro algebras in this case? It seems that only the inclusion electroweak part to symplectic part represented in terms of fermionic currents allows this and would bring in two tensor factors so that one would have 3 tensor factors.
    If one counts fermionic tensor factors assignable to 2-D transversal part of Kac-Moody algebra one would have 5 tensor factors. This seems however tricky.

  2. For the Pclass,I= Pquant,gr option the number of tensor factors is naturally five. Four tensor factors come from Super Kac-Moody and correspond to translational Kac-Moody type degrees of freedom in M4, to color degrees of freedom and to electroweak degrees of freedom (SU(2)× U(1)). One tensor factor comes from the symplectic degrees of freedom in Δ CD× CP2 (note that Hamiltonians include also products of δ CD and CP2 Hamiltonians so that one does not have direct sum!). Therefore this option seems to be slightly favored. Note that Kac-Moody generators are local isometries localized with respect to the coordinates of light-like 3-surface and vanish at partonic 2-surface so that one can speak about direct sum of algebras.

Clearly, both experimental input and Occam's razor seem to favor the option reducing Equivalence Principle to Quantum Classical Correspondence.

For this option however the GRT inspired interpretation of Equivalence Principle at space-time level remains to be understood. Is it needed at all? The condition that the energy momentum tensor of Kähler action has a vanishing divergence leads in General Relativity to Einstein equations with cosmological term. In TGD framework preferred extremals satisfying the analogs of Einstein's equations with several cosmological constant like parameters can be considered.

Should one give up this idea, which indeed might be wrong? Could the divergence of of energy momentum tensor vanish only asymptotically as was the original proposal? Or should one try to generalize the interpretation? QCC states that quantum physics has classical correlate at space-time level and implies EP. Could also quantum classical correspondence itself have a correlate at space-time level. If so, space-time surface would able to represent abstractions as statements about statements about.... as the many-sheeted structure and the vision about TGD physics as analog of Turing machine able to mimic any other Turing machine suggest.g machine suggests.

4. How translations are represented at the level of WCW?

The four-momentum components appearing in the formulas of super conformal generators correspond to infinitesimal translations. In TGD framework one must be able to identify these infinitesimal translations precisely. As a matter of fact, finite measurement resolution implies that it is probably too much to assume infinitesimal translations. Rather, finite exponentials of translation generators are involved and translations are discretized. This does not have practical signficance since for optimal resolution the discretization step is about CP2 length scale.

Where and how do these translations act at the level of WCW? ZEO provides a possible answer to this question.

1. Discrete Lorentz transformations and time translations act in the space of CDs: inertial four-momentum

Quantum state corresponds also to wave function in moduli space of CDs. The moduli space is obtained from given CD by making all boosts for its non-fixed boundary: boosts correspond to a discrete subgroup of Lorentz group and define a lattice-like structure at the hyperboloid for which proper time distance from the second tip of CD is fixed to Tn=n× T(CP2). The quantization of cosmic redshift for which there is evidence, could relate to this lattice generalizing ordinary 3-D lattices from Euclidian to hyperbolic space by replacing translations with boosts (velocities).

The additional degree of freedom comes from the fact that the integer n>0 obtains all positive values. One has wave functions in the moduli space defined as a pile of these lattices defined at the hyperboloid with constant value of T(CP2): one can say that the points of this pile of lattices correspond to Lorentz boosts and scalings of CDs defining sub-WCW:s.

The interpretation in terms of group which is product of the group of shifts Tn(CP2)→ Tn+m(CP2) and discrete Lorentz boosts is natural. This group has same Cartesian product structure as Galilean group of Newtonian mechanics. This would give a discrete rest energy and by Lorentz boosts discrete set of four-momenta giving a contribution to the four-momentum appearing in the super-conformal representation.

What is important that each state function reduction would mean localisation of either boundary of CD (that is its tip). This localization is analogous to the localization of particle in position measurement in E3 but now discrete Lorentz boosts and discrete translations Tn-->Tn+m replace translations. Since the second end of CD is necessary del-ocalized in moduli space, one has kind of flip-flop: localization at second end implies de-localization at the second end. Could the localization of the second end (tip) of CD in moduli space correspond to our experience that momentum and position can be measured simultaneously? This apparent classicality would be an illusion made possible by ZEO.

The flip-flop character of state function reduction process implies also the alternation of the direction of the thermodynamical time: the asymmetry between the two ends of CDs would induce the quantum arrow of time. This picture also allows to understand what the experience growth of geometric time means in terms of CDs.

2. The action of translations at space-time sheets

The action of imbedding space translations on space-time surfaces possibly becoming trivial at partonic 2-surfaces or reducing to action at δ CD induces action on space-time sheet which becomes ordinary translation far enough from end end of space-time surface. The four-momentum in question is very naturally that associated with Kähler action and would therefore correspond to inertial momentum for PI,class=Pquant,gr option. Indeed, one cannot assign quantal four-momentum to Kähler action as an operator since canonical quantization badly fails. In finite measurement infinitesimal translations are replaced with their exponentials for PI,class=Pquant,gr option.

What looks like a problem is that ordinary translations in the general case lead out from given CD near its boundaries. In the interior one expects that the translation acts like ordinary translation. The Lie-algebra structure of Poincare algebra including sums of translation generators with positive coefficient for time translation is preserved if only timelike superpositions if generators are allowed also the commutators of time-like translation generators with boost generators give time like translations. This defines a Lie-algebraic formulation for the arrow of geometric time. The action of time translation on preferred etxremal would be ordinary translation plus continuation of the translated preferred extremal backwards in time to the boundary of CD. The transversal space-like translations could be made Kac-Moody algebra by multiplying them with functions which vanish at δ CD.

A possible interpretation would be that Pquant,gr corresponds to the momentum assignable to the moduli degrees of freedom and Pcl,I to that assignable to the time like translations. Pquant,gr=Pcl,I would code for QCC. Geometrically quantum classical correspondence would state that timelike translation shift both the interior of space-time surface and second boundary of CD to the geometric future/past while keeping the second boundary of space-time surface and CD fixed.

5. Yangian and four-momentum

Yangian symmetry implies the marvellous results of twistor Grassmannian approach to N=4 SUSY culminating in the notion of amplituhedron which promises to give a nice projective geometry interpretation for the scattering amplitudes (see this). Yangian symmetry is a multilocal generalization of ordinary symmetry based on the notion of co-product and implies that Lie algebra generates receive also multilocal contributions. I have discussed these topics from slightly different point of view (see this).

1. Yangian symmetry

The notion equivalent to that of Yangian was originally introduced by Faddeev and his group in the study of integrable systems. Yangians are Hopf algebras which can be assigned with Lie algebras as the deformations of their universal enveloping algebras. The elegant but rather cryptic looking definition is in terms of the modification of the relations for generating elements (see this). Besides ordinary product in the enveloping algebra there is co-product Δ which maps the elements of the enveloping algebra to its tensor product with itself. One can visualize product and co-product is in terms of particle reactions. Particle annihilation is analogous to annihilation of two particle so single one and co-product is analogous to the decay of particle to two. Δ allows to construct higher generators of the algebra.

Lie-algebra can mean here ordinary finite-dimensional simple Lie algebra, Kac-Moody algebra or Virasoro algebra. In the case of SUSY it means conformal algebra of M4- or rather its super counterpart. Witten, Nappi and Dolan have described the notion of Yangian for super-conformal algebra in very elegant and and concrete manner in the article kenoem Yangian Symmetry in D=4 superconformal Yang-Mills theory (see this). Also Yangians for gauge groups are discussed.

In the general case Yangian resembles Kac-Moody algebra with discrete index n replaced with a continuous one. Discrete index poses conditions on the Lie group and its representation (adjoint representation in the case of N=4 SUSY). One of the conditions conditions is that the tensor product R⊗ R* for representations involved contains adjoint representation only once. This condition is non-trivial. For SU(n) these conditions are satisfied for any representation. In the case of SU(2) the basic branching rule for the tensor product of representations implies that the condition is satisfied for the product of any representations.

Yangian algebra with a discrete basis is in many respects analogous to Kac-Moody algebra. Now however the generators are labelled by non-negative integers labeling the light-like incoming and outgoing momenta of scattering amplitude whereas in in the case of Kac-Moody algebra also negative values are allowed. Note that only the generators with non-negative conformal weight appear in the construction of states of Kac-Moody and Virasoro representations so that the extension to Yangian makes sense.

The generating elements are labelled by the generators of ordinary conformal transformations acting in M4 and their duals acting in momentum space. These two sets of elements can be labelled by conformal weights n=0 and n=1 and and their mutual commutation relations are same as for Kac-Moody algebra. The commutators of n=1 generators with themselves are however something different for a non-vanishing deformation parameter h. Serre's relations characterize the difference and involve the deformation parameter h. Under repeated commutations the generating elements generate infinite-dimensional symmetric algebra, the Yangian. For h=0 one obtains just one half of the Virasoro algebra or Kac-Moody algebra. The generators with n>0 are n+1-local in the sense that they involve n+1-forms of local generators assignable to the ordered set of incoming particles of the scattering amplitude. This non-locality generalizes the notion of local symmetry and is claimed to be powerful enough to fix the scattering amplitudes completely.

2. How to generalize Yangian symmetry in TGD framework?

As far as concrete calculations are considered, it is not much to say. It is however possible to keep discussion at general level and still say something interesting (as I hope!). The key question is whether it could be possible to generalize the proposed Yangian symmetry and geometric picture behind it to TGD framework.

  1. The first thing to notice is that the Yangian symmetry of N=4 SUSY in question is quite too limited since it allows only single representation of the gauge group and requires massless particles. One must allow all representations and massive particles so that the representation of symmetry algebra must involve states with different
    masses, in principle arbitrary spin and arbitrary internal quantum numbers. The candidates are obvious: Kac-Moody algebras (see this) and Virasoro algebras (see this) and their super counterparts. Yangians indeed exist for arbitrary super Lie algebras. In TGD framework conformal algebra of Minkowski space reduces to Poincare algebra and its extension to Kac-Moody allows to have also massive states.

  2. The formal generalization looks surprisingly straightforward at the formal level. In zero energy ontology one replaces point like particles with partonic two-surfaces appearing at the ends of light-like orbits of wormhole throats located to the future and past light-like boundaries of causal diamond (CD× CP2 or briefly CD). Here CD is defined as the intersection of future and past directed light-cones. The polygon with light-like momenta is naturally replaced with a polygon with more general momenta in zero energy ontology and having partonic surfaces as its vertices. Non-point-likeness forces to replace the finite-dimensional super Lie-algebra with infinite-dimensional Kac-Moody algebras and corresponding super-Virasoro algebras assignable to partonic 2-surfaces.

  3. This description replaces disjoint holomorphic surfaces in twistor space with partonic 2-surfaces at the boundaries of CD×CP2 so that there seems to be a close analogy with Cachazo-Svrcek-Witten picture. These surfaces are connected by either light-like orbits of partonic 2-surface or space-like 3-surfaces at the ends of CD so that one indeed obtains the analog of polygon.

What does this then mean concretely (if this word can be used in this kind of context)?

  1. At least it means that ordinary Super Kac-Moody and Super Virasoro algebras associated with isometries of M4 ×CP2 annihilating the scattering amplitudes must be extended to a co-algebras with a non-trivial deformation parameter. Kac-Moody group is thus the product of Poincare and color groups. This algebra acts as deformations of the light-like 3-surfaces representing the light-like orbits of particles which are extremals of Chern-Simon action with the constraint that weak form of electric-magnetic duality holds true. I know so little about the mathematical side that I cannot tell whether the condition that the product of the representations of Super-Kac-Moody and Super-Virasoro algebras contains adjoint representation only once, holds true in this case. In any case, it would allow all representations of finite-dimensional Lie group in vertices whereas N=4 SUSY would allow only the adjoint.

  2. Besides this ordinary kind of Kac-Moody algebra there is the analog of Super-Kac-Moody algebra associated with the light-cone boundary which is metrically 3-dimensional. The finite-dimensional Lie group is in this case replaced with infinite-dimensional group of symplectomorphisms of δ M4+/- made local with respect to the internal coordinates of the partonic 2-surface. A generalization of the Equivalence Principle is in
    question. This picture also justifies p-adic thermodynamics applied to either symplectic or isometry Super-Virasoro and giving thermal contribution to the vacuum conformal and thus to mass squared.

  3. The construction of TGD leads also to other super-conformal algebras and the natural guess is that the Yangians of all these algebras annihilate the scattering amplitudes.

  4. Obviously, already the starting point symmetries look formidable but they still act on single partonic surface only. The discrete Yangian associated with this algebra associated with the closed polygon defined by the incoming momenta and the negatives of the outgoing momenta acts in multi-local manner on scattering amplitudes. It might make sense to speak about polygons defined also by other conserved quantum numbers so that one would have generalized light-like curves in the sense that state are massless in 8-D sense.

3. Could Yangian symmetry provide a new view about conserved quantum numbers?

The Yangian algebra has some properties which suggest a new kind of description for bound states. The Cartan algebra generators of n=0 and n=1 levels of Yangian algebra commute. Since the co-product Δ maps n=0 generators to n=1 generators and these in turn to generators with high value of n, it seems that they commute also with n≥ 1 generators. This applies to four-momentum, color isospin and color hyper charge, and also to the Virasoro generator L0 acting on Kac-Moody algebra of isometries and defining mass squared operator.

Could one identify total four momentum and Cartan algebra quantum numbers as sum of contributions from various levels? If so, the four momentum and mass squared would involve besides the local term assignable to wormhole throats also n-local contributions. The interpretation in terms of n-parton bound states would be extremely attractive. n-local contribution would involve interaction energy. For instance, string like object would correspond to n=1 level and give n=2-local contribution to the momentum. For baryonic valence quarks one would have 3-local contribution corresponding to n=2 level. The Yangian view about quantum numbers could give a rigorous formulation for the idea that massive particles are bound states of massless particles.

See the article About the Notion of Four-momentum in TGD Framework. For background see the chapter TGD and GRT of "Physics in Many-Sheeted Space-time".

Thursday, December 19, 2013

One Mind theory, Akashic records, and negentropic entanglement

Larry Dossey has published a book with title "One Mind: How Our Individual Mind Is Part of a Greater Consciousness and Why It Matters". There is also an article in "Where Does Creativity Come From?" on The Huffington Post.

The article is an excerpt from the book of Dossey and begins with a nice story about miracle like behavior of a five year old child told by a developmental psychologist Joseph Chilton Pearce. I recommend warmly reading it. I have actually had a similar experience in which my younger daughter - I think at age of 5 or 6 - shocked me by saying something very wise that only a very wise adult could have said. It was really weird. What was also peculiar that very many adults behaved in her company as she would have been an adult. I have also had a couple of personal "great experiences", which transformed my view about the world completely and gave problems to ponder for the rest of life.

The key meassage of the book is that there is something one might call One Mind, which serves as an endless storage of wisdom (more technically information) and creativity.

Does the notion of One Mind have place in TGD framework? In TGD framework I speak about a hierarchy of conscious entities. Or - strictly speaking - hierarchy of experiences since the assumption about experiencers might be too much if one accepts that the basic element of consciousness is act of re-creation of the Universe and that quantum jumps are just ordinary state function reductions. As a matter of fact, I do not believe that the latter assumption is true. So the questions are following.

Questions: The entire universe as conscious entity recreating itself could be called God. Could it be identified also as One Mind or is the latter something different? Does One Mind corresponds to conscious entity or some - not necessarily conscious - information source from which it is possible to draw information via conscious experiences?

I must introduce some aspects of TGD inspired theory of consciousness in an attempt to answer this question inside my own jail of thought.

  1. Quantum jump - basically state function reduction - would correspond to moment of consciousness- re-creation but in different sense as in standard quantum theory. The division to separate conscious entities takes place and state function reduction in the ordinary sense reducing entanglement and splitting system to subsystem and its complement could be the quantum physical correlate for it.

    Remark: The notion of state function is richer in Zero Energy Ontology than in standard quantum theory and - most importantly - free of the standard difficulty of quantum measurement theory. The most radical rethinking relates to the relationship between subjective time and geometric time, in particular the arrow of time.

  2. Standard quantum physics would not give anything more than random sequences of state function reductions: no quantum invariants which be called soul or Mind or One Mind. This is not surprising and is mathematically reflected by the fact that in standard physics there exists no information measure. There is only a measure for entropy applying also to entanglement and measuring lack of information - about the state of Schr&oml;dinger cat - to use the standard illustration.

  3. The p-adic approach to cognition gives rise to a p-adic variant of Shannon entropy. The surprise is that p-adic entanglement entropy can be negative and serve therefore as a measure for conscious or potentially conscious information. The standard interpretation for this information cannot hold true. Rather, the information is about the relationship between the cat and bottle of poisson. Negentropic entanglement carries information as rules such that the instances of the rule correspond to the superposed state pairs - or n-tuples in case of negentropic entanglement between n particles. The form of this entanglement is completely unique and one can write a general formula for it in terms of permutation symbols when one knows the number degenerate states assignable to the sheets of n-fold covering of imbedding space assignable to a system with Planck constant heff =nh. The density matrix associated with any decomposition of negentropically entangled n-particle system to a pair is proportional to unit matrix.

    The negentropic states can be measured without changing them by interaction free measurement. My interpretation of negentropic entanglement is as "Akashic records" storing information about the Universe to the structure of the Universe in the sequence of recreations giving rise to more and more complex quantum Universe. Interaction free measurement means their reading without affecting them at all - this is however an idealization since small damage can occur just as in the reading of ordinary book. Universe is a Big Library in this view.

  4. If Negentropy Maximization Principle defines the variational principle of consciousness as a mathematical analog of second law, negentropic entanglement increases and implies evolution. NMP tells essentially that if the decompositions of a system to pairs of subsystems are such that density matrix is proportional to a unit matrix, nothing happens in the reduction. In ordinary measurement theory one cannot say anything about this situation since one can only say that quantum measurement leads to the eigenspace of measured observables: in TGD the density matrix is the universal observable. The total negentropic entanglement cannot decrease in quantum jump: new items appear to the Akashic library. It can be however transferred between subsystems and this increase would be the new element in quantum consciousness theory.

I am cautious and conclude with a question rather than answer: Could the Akashic records - the Big Library - correspond to the One Mind - to information source from which we can read ideas and insights and whis the source of creativity - or rather, recreativity;-)?

Friday, December 13, 2013

Revolutionizing Solar Energy: Quantum Waves Found at the Heart of Organic Solar Cells

Experimentalists seem to be discovering new aspects of TGD Universe almost everyday. The discovery of today reported in Science News is the finding that in organic nanostructures charges are not only formed rapidly but also separated over long distances. This phenomenon has direct technological implications.

This requires quantum mechanics and perhaps even more. The separation of charges indeed looks mysterious. TGD innspired mechanism is a phase transition increasing the value of Planck constant and scaling up all scales - in particular electronic Compton length - by integer heff/h. Large value of Planck constant would make also possible super-conductivity in long length scales at high temperatures.

Charge separation occurs also in water splitting using strong electric fields or cavitation and the so called Brown's gas is the outcome (see older posting). This phase involves charge separation and one talks about charged water clusters and plasmoids. The TGD based model assumes non-standard value of Planck constant for Brown's gas regarded as electrically expanded water. The reduction of Planck constant - say back to normal one - with simultaneous increase of p-adic length scale so that the volume is not changed liberates energy (as becomes clear what happens for states of particle in box and for cyclotron states): this could define a basic mechanism for the liberation of metabolic energy.

One can only imagine how fast progress could be possible if the experimentalists could use a theory on basis of their search but here we must patiently wait for the part of academic community calling themselves theorists to become mature enough to accept TGD as a scientific theory.

Tuesday, December 10, 2013

New results from PHENIX concerning quark gluon plasma

New results have been published on properties of what is conventionally called quark gluon plasma (QGP) . As a matter fact, this phase does not resemble plasma at all. The decay patterns bring in mind decays of string like objects parallel to the collision axes rather than isotropic blackbody radiation. The initial state looks like a perfect fluid rather than plasma and thus more like a particle like object.

The results of QGP - or color glass condensate (CGC) as it is also called - come from three sources and are very similar. The basic characteristic of the collisions is the cm energy s1/2of nucleon pair. The data sources are Au-Au collisions at RHIC, Brookhaven with s1/2=130 GeV, p-p collisions and p-nucleus collisions at LHC with s1/2=200 GeV and d-Au collisions at RHIC with s1/2=200 GeV studied by PHENIX collaboration.

According to the popular article telling about the findings of PHENIX collaboration the collisions are believed to involve a creation of what is called hot spot. In Au-Au collisions this hot spot has size of order Au nucleus. In d-Au collisions it is reported to be much, much smaller. What does this mean? The size of deuteron nucleus or of nucleon? Or something even much smaller? Hardly so if one believes in QCD picture. If this is however the case, the only reasonable candidate for its size would be the longitudinal size scale of colliding nucleon-nucleon system of order L=hbar/s1/2 if an object with this size is created in the collision. I did my best to find some estimate for the very small size of the hot spot from articles some related to the study but failed (see this, this and this): if I were a paranoid I would see this as a conspiracy to keep this as a state secret;-).

How to understand the findings?

I have already earlier considered the basic characteristics of the collisions. What is called QGP does not behave at all like plasma phase for which one would expect particle distributions mimicking blackbody radiation of quarks and gluons. Strong correlations are found between charged particles created in the collision and the best manner to describe them is in terms of a creation of longitudinal string-like objects parallel to the collision axes.

In TGD framework this observation leads to the proposal that the string like objects could be assigned with M89 hadron physics introduced much earlier to explain strange cosmic ray events like Centauro. The p-adic mass scale assignable to M89 hadron physics is obtained from that of electron (given by p-adic thermodynamics in good approximation by m127= me/51/2) as m89= 2(127-89)/2× me/51/2. This gives m89= 111.8 GeV. This is conveniently below the cm mass of nucleon pair in all the experiments.

In standard approach based on QCD the description is completely different. The basic parameters are now thermodynamical. One assumes that thermalized plasma phase is created and is parametrized by the energy density assignable to gluon fields for which QCD gives the estimate ε ≥ 1 GeV/fm3 and by temperature which is about T=170 GeV and more or less corresponds to QCD Λ. One can think of the collision regions as highly flattened pancake (Lorentz contraction) containing very density gluon phase called color glass condensate, which would be something different from QGP and definitely would not conform with the expectations from perturbative QCD since QGP would be precisely a manifestation of perturbative QGP (see this).

Also a proposal has been made that this phase could be described by AdS/CFT correspondence non-perturbatively - again in conflict with the basic idea that perturbative QCD should work. It has however turned out that this approach does not work even qualitatively as Bee ludicly explains this in her blog article Whatever happened to AdS/CFT and the Quark Gluon Plasma?.

Strangely enough, this failure of QGP and AdS/CFT picture has not created any fuss although one might think that the findings challenging the basic pillars of standard model should be seen as sensational and make happy all those who have publicly told that nothing would be more well-come than the failure of standard model. Maybe particle theorists have enough to do with worrying about the failure of standard SUSY and super string inspired particle phenomenology that they do not want to waste their time to the dirty problems of low energy phenomenology.

A further finding mentioned in the popular article is stronger charm-anticharm suppression in head-on collisions than in peripheral collisions (see this). What is clear that if M89 hadrons are created, they consist of lightest quarks present in the lightest hadrons of M89 hadron physics - that is u and d (and possibly also s) of M89 hadrons, which are scaled variants of ordinary u and d quarks and decay to u and d (and possibly s) quarks of M107 hadron physics. If the probability of creating a hot M89 spot is higher in central than peripheral collisions the charm suppression is stronger. Could a hot M89 spot associated with a nucleon-nucleon pair heat some region around it to M89 hadronic phase so that charm suppression would take place inside larger volume than in periphery?

There is also the question whether the underlying mechanism relies on specks of hot QGP or some inherent property of nuclei themselves. At the first sight, the latter option could not be farther from the TGD inspired vision. However, in nuclear string model inspired by TGD nuclei consists of nucleons connected by color bonds having quark and antiquark at their ends. These bonds are characterized by rather large p-adic prime characterizing current quark mass scale of order 5-20 GeV for u and d quarks (the first rough estimate for the p-adic scales involved is p≈ 2^k, k=121 for 5 MeV and k= 119 for 20 MeV). These color bonds Lorentz contract in the longitudinal direction so that nearly longitudinal color bonds would shorten to M89 scale whereas transversal color bonds would get only thinner. Could they be able to transform to color bonds characterized by M89 and in this manner give rise to M89 mesons decaying to ordinary hadrons?

Flowers to the grave of particle phenomenology

The recent situation in theoretical particle physics and science in general does not raise optimism. Super string gurus are receiving gigantic prizes from a theory that was a failure. SUSY has failed in several fronts and cannot be anymore regarded as a manner to stabilize the mass of Higgs. Although the existence of Higgs is established, the status of Higgs mechanism is challenged by its un-naturality: the assumption that massivation is due to some other mechanism and Higgs has gradient coupling provides a natural explanation for Higgs couplings. The high priests are however talking about "challenges" instead of failures. Even evidence for the failure of even basic QCD is accumulating as explained above. Peter Higgs, a Nobel winner of this year, commented the situation ironically by saying that he would have not got a job in the recent day particle physics community since he is too slow.

The situation is not much better in the other fields of science. Randy Scheckman, also this year's Nobel prize winner in physiology and medicine has declared boycott of top science journals Nature, Cell and Science. Schekman said that the pressure to publish in "luxury" journals encourages researchers to cut corners and pursue trendy fields of science instead of doing more important work. The problem is exacerbated, he said, by editors who were not active scientists but professionals who favoured studies that were likely to make a splash.

Theoretical and experimental particle physics is a marvellous creation of humankind. Perhaps we should bring flowers to the grave of the particle physics phenomenology and have a five minutes' respectful silence. It had to leave us far too early.

For background see the chapter "New particle physics predicted by TGD" of "p-Adic Physics".