Wednesday, June 26, 2019

What particles are in TGD Universe?

Savyasanchi Ghose asked the following question. I answer from TGD point of view differing in some respects from the standard view.

"What are the 'elementary particles'? We know that there are some methods to produce them like bombarding in a nuclear reactor, decay or ionization, but what really a 'particle' is? From Quantum field theory perspective, there aren't any particles, what we call as 'particles' are just the excitations in the respective fields, these excitations are tied up in a little bundles of energy which we call as particles, but this view doesn't explain how we perceive the material world as 'solid' matter rather than energy form'

I have also pondered this question many times. I started from a childish attempt to understand elementary particle masses about 45 years ago but gradually realized that I must understand many other things before answering this question!

  1. Elementary particles in lab show their present quite concretely as particle like entities. Say as tracks in magnetic field. In QFT theory description many particle states have purely algebraic description as Fock states created by creation operators labelled by momenta. There is no apparent connection with the concrete geometric picture. The scattering rates are deduced from QFT by rather highly counter-intuitive procedure. Particles correspond to poles for momentum space Fourier transforms of correlation functions for fields representing particles. In path integral approach one studies essentially correlations of classical fields in an analog of thermodynamical partition function defined by the exponent of action. Classical physics results in stationary phase approximation, which need not be good.

  2. The connection with particles become more concrete when one replaces momentum eigenstates with localized modes which however do not remain such in time development. Gaussian wave packets replace momentum replace momentum eigenstates and one has reasonable localization in ordinary space. The observed particles would correspond to these wave packets. Particles in QFT have no geometric size: they are point-like and the ony size is quantum size defined by the support of the wave function.

    For solitonic states appearing in integrable theories- typically 2-D - solitons represent objects having also geometric size and can be de-localized in center of mass degrees of freedom.

TGD view is different in the sense that particle have also geometric size. This is essential for understanding particle massivation not really understood in QFT approach.
  1. Particles correspond to 3-surfaces and their orbits as 4-D regions of space-time surfaces. Particles have geometric size instead of being point-like. The 4-D orbit of particle is a region carrying induced classical fields determined by the imbedding to 8-D H=M4×CP2 in terms of its geometry and spinor structure. Standard model fields and gravitational field are geometrized so that Einstein's dream is realised.

    What is important is that field-particle duality is realised also classically. Induced fields propagate inside the "wave cavity" defined by the orbit of the particle as 3-surface.

  2. The connection with oscillator operator description of QFT comes from the presence of second quantized induced spinor fields creating quarks and possibly also leptons as opposite chiralities of M^4xCP_2 spinor fields. The situation was clarified dramatically quite recently when I finally understood what SUSY is in TGD framework. The conservation of quark number and the behavior of propagators consistent with statistics fixes the TGD view about SUSY.

    1. Super-coordinates of H make sense and have super-part expressible as local hermitian monomials of quark oscillator operators with vanishing quark number. They create point like particles such as weak bosons and graviton. One has creation and annihilation operators rather than anticommuting theta parameters: this is the big difference with respect to the ordinary SUSY and forced by the fact that Majorana spinors are not possible.

    2. Super-Dirac field is odd monomial of quark creation operators and the action depends on the super-coordinates of H. One obtains directly second quantize theory with quarks as fundamental fermions. Anti-leptons correspond to local 3 quark states so that electron as spartner of an antiquark would have been discovered already 1897! Matter antimatter asymmetry finds a nice solution too. Here I had to give up the earlier assumption that also leptons appear as fundamental fermions.


One finally has a concrete view about S-matrix at fundamental level. There are of course many proposals already such as the construction based on quaternionic generalization of twistor Grassmannian approach but the difference is that now one has really fundamental approach.

  1. All reduces formally to classical partial differential equations for super-space-time surface and super-spinors. One solves preferred extremal and its super-variant which means solving the space-time evolution of multi-spinors defining super-coordinates and in this background super-Dirac equation is solve. This is highly non-trivial but in principle precisely defined procedure. If one gives initial values of spinor modes at the first light-like boundary of causal diamond (CD), one can deduce super-spinor field at opposite boundary of CD and express it as a superposition of its basic modes with well-defined quark number and other quantum numbers. This gives S-matrix.

  2. Vertices at partonic 2-surfaces are super-symmetric but in TGD sense and reduce to points at which quark lines meet: one can say that local multi-quark-antiquark states split to local multi-quark-antiquark states. Vertices are determined as vacuum expectations of the bosonic action and super-Dirac action and analogous to those defined by theta integral in SUSY.

  3. No further quantization is needed since super-symmetrization corresponds to second quantization. This is part of the realization of the dream about geometrizing also quantum theory. This should have been realized long time ago also by colleagues since SUSY algebra is Clifford algebra like also oscillator operator algebra.

  4. Situation simplifies dramatically for discrete cognitive representation replacing space-time surface with the set of points having imbedding space coordinates in extension of rationals defining the adele.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Tuesday, June 25, 2019

Finiteness for the number of non-vanishing Wick contractions, quantum criticality, and coupling constant evolution

The consistency with number theoretic vision requires that the number of terms in the super-Taylor expansion of action is finite - otherwise one is led out from the extension: this applies both to the action determining space-time surfaces and to the corresponding modified Dirac action. There are several options that one can consider.

  1. Normal ordering of the fermionic oscillator operators would be a straightforward manner to handle the situation. One would obtain finite number of terms since the number of quark oscillator operators is d=4+4=8. The maximal degree mmax of multiple partial derivative of action with respect to gradient of H-coordinate h would be mmax= d=8 and correspond to monomial with 4+4 quark oscillator operators. Note that the normal ordering of this term gives rise to c-number.

    It however seems that the natural solution of the problem must involve cancellation of the Wick contractions when the degree m of the multiple partial derivative satisfies m>mmax. Some cancellation mechanism for m≥ mmax should guarantee that Wick-contractions give in this case a vanishing contribution to each of the d= 8 monomials in the super-action.

  2. The strongest condition would be that all Wick contraction terms coming from the normal ordering vanish. The contraction terms are expressible as divergences of currents and the interpretation would be in terms of Noether current associated with some symmetry. Super-symplectic symmetry is the best candidate in this respect. Note that besides these currents also the Noether currents coming from the super-symplectic variations should have a vanishing divergence.

  3. One can consider also a weaker condition. Wick contractions vanish for m>mmax such that mmax>8 is possible. This would give rise to the analog of radiative corrections, and if mmax can vary, one obtains the analog coupling constant evolution and discrete coupling constant evolution corresponds to the variation of mmax.

How the value of mmax could be determined?
  1. M8-H duality requires that M8- and H-pictures are structurally similar. Octonionic polynomials are characterized by their order n and also the super-extremals should be characterized by n and even the individual terms of super-polynomial should have counterparts at H-level.

    One can define super-octonionic polynomials at M8-level and also for these normal ordering terms appear. Ordinary derivatives of P(o) with respect to o replace those of the action with respect to the gradients of H coordinates, and one obtains only finite number of Wick contractions. There is no need to require their vanishing now, and the hierarchy of degrees n=heff/h0 for P defines a discrete coupling constant evolution with each level corresponding to its own values of coupling constants differing by the number of Wick contractions. This gives a connection with the ordinary coupling constant evolution with Wick contractions taking the role of loops.

    This picture should have direct image at H-side. In particular, one should have mmax=n.

  2. The cancellation of Wick contractions for the action containing both Kähler term and cosmological term probably happens only for critical values of cosmological constant determined dynamically from the mechanism of dimensional reduction reducing 6-D surface in the product of twistor spaces T(M4)= M4× S2 and T(CP2)= SU(3)/U(1)× U(1) to S2 bundle over space-time surface representing induced twistor structure. The cancellation condition for the higher terms could fix the value of cosmological constant emerging from the mechanism.

  3. The picture could be interpreted in terms of quantum criticality. The polynomials P(o) characterize quantum critical phases. Also Taylor series can be considered but they would not be critical and infinite amount of information would be required to specify them whereas the specification of critical dynamics requires by its universality only a finite number of parameters coded by the rational coefficients of polynomial.

    Criticality corresponds to the vanishing of not only function but also some of its derivatives at critical point. The criticality would be now infinite in the sense that all derivatives of P(o) higher than n would vanish. This is indeed the view about quantum criticality that I ended up to long time ago. This implies that the parameter space for the functions describing criticality is finite-dimensional.

    In Thom's catastrophe theory which essentially describes a hierarchy of criticalities concretely, the finite-dimension of the space of control parameters is essential. For cusp catastrophe this space is 2-dimensional and catastrophe graph is defined by a fourth order polynomial so that all higher order derivatives vanish identically also now.

  4. At the level of H criticality would mean that m-fold partial derivatives of action only up to m=mmax=n-fold partial derivatives contribute to the radiative corrections. The action would be polynomial of finite order in the multi-spinor components of super-coordinates and discrete coupling constant evolution would be realized. The ordinary variations of the action would be of course non-vanishing to arbitrary high order.

    Coupling constant evolution would reduce to the hierarchy of extensions of rationals since the degree n of P determines the dimension of extension. Evolution in terms of the hierarchy of extensions of rationals would dictate also coupling constant evolution. This evolution would also dictate the preferred p-adic length scales if preferred p-adic primes are identifiable as ramified primes. Ramified primes at the lowest level of hierarchy are ramified primes at higher levels if P(0)=0 condition is true for them. Evolutionary hierarchies correspond to functional composition hierarchies for polynomials with degrees ni such that ni+1 is divisible with ni that is ni+1/ni=ki.

    Remark: Functional composition occurs also in the construction of fractals like Mandelbrot fractal and as a special case one iterates single polynomial to get a hierarchy in powers of integers n1. This interpretation would conform with the interpretation of the symmetries guaranteeing the cancellation of Wick terms as super-symplectic symmetries.

  5. A connection with the inclusion hierarchies for super-symplectic algebra is highly suggestive. The fractal hierarchy of super-symplectic sub-algebras (fractality and conformal symmetry - now in generalized sense - are essential for quantum criticality) with levels labelled by n would naturally give rise to counterparts of the functional composition hierarchies.

    Inclusion hierarchies would correspond to sub-hierarchies of super-symplectic algebras formed by sequences of sub-algebras with weights divisible by integer ni such that ni divides ni+1. ni would correspond to a degree of polynomial in the hierarchy formed by their compositions in accordance with functional composition of polynomials.

  6. The inclusion hierarchies of super-symplectic algebras would have interpretation in terms of inclusions of hyper-finite factors of type II1. The ratios ni+1/ni= ki appearing in the composition hierarchies would correspond to the integers labelling the inclusions of HFFs and defining quantum phases U=exp(iπ/ki) characterizing quantum algebras and quantum spaces
    as analogs of state spaces modulo finite measurement resolution.

    The interpretation of finite measurement resolution as an ability to detect only space-time sheets characterized by polynomials of order n below some fixed integer is natural. n would characterize the measurement resolution.

To sum up, this picture rather neatly fuses together several speculative visions about quantum TGD. The reduction of dynamics to polynomial dynamics at the level of M8 has interpretation in terms of quantum criticality with finite-D space of control parameters implying universal dynamics involving very few coupling parameters, which are fixed points of coupling constant evolution for given value of n. M8-H duality maps M8 dynamics to the level of H, where it is realized in terms of a hierarchy of sub-algebras of super-symplectic algebra and sub-hierarchies correspond to sequences of integers ni dividing ni+1. A connection with the inclusions of HFFs and finite measurement resolution emerges. The notion of discrete coupling constant evolution finds a precise formulation, and the notion of radiation correction is realized in terms of Wick contractions.

See the article SUSY in TGD Universe or the chapter TGD view about McKay Correspondence, ADE Hierarchy, Inclusions of Hyperfinite Factors, M8-H Duality, SUSY, and Twistors of "Towards M-matrix".

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Monday, June 17, 2019

Evidence for 96 GeV pseudoscalar predicted by TGD

Lubos had a second posting mentioning new bump at around 96 GeV very near to the masses of weak bosons and tells that physicists seem to take it very seriously. Lubos of course wants to interpret it as a Higgs predicted by standard SUSY already excluded at the energies considered.

What about TGD interpretation?

  1. TGD predicts besides weak gauge bosons, Higgs, and pseudoscalar: about the prediction of pseudoscalar I became aware only now. This follows taking tensor products for spin-isospin representations formed by quarks but for some reason I had not noticed this. The mass scale of pseudoscalar Higgs is most naturally the same as that of scalar Higgs or of weak bosons and p-adic mass calculations allow to estimate its mass. Higgs mass 125 GeV is very nearly the minimal mass for p-adic prime p≈ 2k, k=89. The minimal mass for k=90 defining also the p-adic mass scale of weak bosons would be 88 GeV so that the interpretation as pseudo-scalar with k=90 might make sense (see this).

  2. This lower bound is somewhat smaller than 96 GeV but the estimate neglects effects related to isospin: doublet and complex doublet are actually predicted (or triplet and singlet when SU(2)w action is by automorphism on the 2× 2 matrix defined by the doublets rather than as left or right action on the doublets appearing as its rows/columns). Mass splitting looks natural and the neutral state might be the heavier one as in case of W,Z splitting.

The situation is extremely interesting, since after decades of efforts I finally managed to formulate and understand SUSY in TGD framework.
  1. First of all, SUSY is there but it is very different from standard N=1 SUSY predicting Majorana fermions. The reason is that due to fermion number conservation theta parameters appearing in super-field must be replaced with fermionic - actually quark-like - oscillator operators. The simplest model predicts that theta parameters and their conjugates appearing in the super-fied correspond to quark oscillator operators in a number theoretic discretization of space-time surface. They thus anticommute non-trivially. Anticommutators are finite for cognitive representations for which space-time surface is replaced with a discrete set of points with preferred imbedding space coordinates in an extension of rationals.

  2. Super-spinor field is odd polynomial of creation operators and its conjugate is odd function of annihilation operators whereas the imbedding space coordinates appearing in bosonic action (Kahler action plus volume term) and modified super-Dirac action are replaced by imbedding space super-coordinates, which are polynomials in which super-monomials have vanishing total quark number and appear as sums of monomial and its conjugate to guarantee the hermiticity of the super-coordinate.

    These assumptions guarantee that super-Dirac field describes local states with odd fermion number and propagators have the behavior required by statistics.

  3. In continuum variant of the theory the bosonic Wick contractions would give rise to infinities: this vanishing conforms with the vanishing of loops required quite generally by the number theoretical vision and implying discrete coupling constant evolution. This simplifies the analogs of Feynman diagrams appearing at the level of discrete "cognitive representations" to mere tree diagrams. In twistor approach the vanishing of loops means enormous simplification and implies behavior analogous to that in dual resonance models which initiated superstring models.

    The vanishing of Wick contractions from super-space-time parts of the modified Dirac action and super-counterpart of classical action gives rise to conserved Noether currents having interpretation in terms of symmetries: the most natural interpretation is in terms of gigantic super-symplectic symmetries predicted by TGD. TGD predicts also their Yangian analogs multi-local symmetries.

  4. Super-symmetric vertices are just vacuum expectations of the action. In this picture leptons would be spartners of quarks as local 3-quarks composites. This little discovery I could have made four decades ago. The allowance of only quarks as fundamental fermions follows from SO(1,7) triality in number theoretic vision: here M8-M4×CP2 duality and the part of number theoretic vision involving classical number fields is needed. We would have been staring at super-symmetries for more than century! My heart bleeds for the unlucky colleagues still trying to find standard SUSY at LHC. I can only pray that these lost lambs of experimental and theoretical physics could find their way back to their shepherd.

  5. The quark numbers or protons and leptons would be opposite and matter antimatter asymmetry would mean preference of antiquarks to arrange into local triplets - leptons- whereas quarks would arrange to non-local triplets- baryons. Both (quark) matter and antimatter would have been in front of eyes all the time we have been producing literature about mechanisms possibly explaining the absence of antimatter.

    CP breaking is necessary for this picture and twistor lift of TGD indeed predicts CP breaking term which would be due to the Kähler structure of Minkowski space required by twistor lift of TGD - also non-commutative quantum field theories predict it.

  6. What about SUSY breaking. It has been clear for a long time that the mass formulas could be same for the members of super-multiplet but that p-adic length scale could differ. I realized few weeks ago that the breaking of SUSY is universal and has very little to do with the details of dynamics. In the general case zero energy states are superpositions (mixtures) of states with different mass squared eigenvalues and M8-H duality allows to find an imbedding of M4 to M4 making mass squared vanishing for states without this mixing. For mixtures p-adic thermodynamics predicts the masses. That Minkowski space is a relative notion means obviously new view about the notion of mass.

It is fair to say, that as far as particle physics is considered, TGD is done. The simplicity and elegance of the picture is so stunning that it is difficult to imagine alternatives. Already earlier, I realized that the breaking of SUSY is universal and has very little to do with the details of dynamics. Zero energy states are superpositions (mixtures) of states with different mass squared eigenvalues and M8-H duality allows to find an imbedding of M4 to M8 making mass squared vanishing for states without this mixing. For mixtures p-adic thermodynamics predicts the masses.

See the article SUSY in TGD Universe or the chapter New Physics Predicted by TGD: Part I.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Saturday, June 15, 2019

Twistors in TGD Universe

This article was inspired by a longer paper "TGD view about McKay Correspondence, ADE Hierarchy, Inclusions of Hyperfinite Factors, and Twistors". I found it convenient to isolate the part of paper related to twistors. In twistor Grassmannian approach to N=4 SYM twistors are replaced with supertwistors and the extreme elegance of the description of various helicity states using twistor space wave functions suggests that super-twistors are realized at the level of M8 geometry. These supertwistors are realized at the level of momentum space.

In TGD framework M8-H duality allows to geometrize the notion of super-twistor in the sense that different components of super-field correspond to components of super-octonion each of which corresponds to a space-time surfaces satisfying minimal surface equations with string world sheets as singularities - this is geometric counterpart for masslessness.

In TGD particles are massless in 8-D sense and in general massive in 4-D sense but 4-D twistors are needed also now so that a modification of twistor approach is needed. The incidence relation for twistors suggests the replacement of the usual twistors with either non-commutative quantum twistors or with octo-twistors. Quantum twistors could be associated with the space-time level description of massive particles and octo-twistors with the description at imbedding space level. A possible alternative interpretation of quantum spinors is in terms of quantum measurement theory with finite measurement resolution in which precise eigenstates as measurement outcomes are replaced with universal probability distributions defined by quantum group. This has also application in TGD inspired theory of consciousness.

The outcome of octo-twistor approach together with M8-H duality leads to a nice picture view about twistorial description of massive states based on quaternionic generalization of twistor (super-)Grassmannian approach. A radically new view is that descriptions in terms of massive and massless states are alternative options, and correspond to two different alternative twistorial descriptions and leads to the interpretation of p-adic thermodynamics as completely universal massivation mechanism having nothing to do with dynamics.

The basic problem of the ordinary twistor approach is that the states must be massless in 4-D sense. In TGD framework particles would be massless in 8-D sense. The meaning of 8-D twistorialization at space-time level is relatively well understood but at the level of momentum space the situation is not at all so clear.

  1. In TGD particles are massless in 8-D sense. For M4L description particles are massless in 4-D sense and the description at momentum space level would be in terms of products of ordinary M4 twistors and CP2 twistors. For M4T description particles are massive in 4-D sense. How to generalize the twistor description to 8-D case?

    The incidence relation for twistors and the need to have index raising and lowering operation in 8-D situation suggest the replacement of the ordinary l twistors with eitherwith octo-twistors or non-commutative quantum twistors.

  2. Octotwistors can be expressed as pairs of quaternionic twistors. Octotwistor approach suggests a generalization of twistor Grassmannian approach obtained by replacing the bi-spinors with complexified quaternions and complex Grassmannians with their quaternionic counterparts. Although TGD is not a quantum field theory, this proposal makes sense for cognitive representations identified as discrete sets of spacetime points with coordinates in the extension of rationals defining the adele implying effective reduction of particles to point-like particles.

  3. The notion of super-twistor can be geometrized in TGD framework at M8 level but at H level local many-fermion states become non-local but still having collinear light-like momenta. One would have a proposal for a quite concrete formula for scattering amplitudes!

    Even the existence of sparticles have been far from obvious hitherto but now it becomes clear that spartners indeed exist and SUSY breaking would be caused by the same universal mechanism as ordinary massivation of massless states. The mass formulas would be supersymmetric but the choice of p-adic prime identifiable as ramified prime of extension of rationals would depend on the state of super-multiplet. ZEO would make possible symmetry breaking without symmetry breaking as Wheeler might put it.

  4. What about the interpretation of quantum twistors? They could make sense as 4-D space-time description analogous to description at space-time level. Now one can consider generalization of the twistor Grassmannian approach in terms of quantum Grassmannians.

See the article Twistors in TGD Universe.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.




SUSY in TGD Universe

What SUSY is in TGD framework is a longstanding question. In twistor Grassmannian approach to N=4 SYM twistors are replaced with supertwistors and the extreme elegance of the description of various helicity states using twistor space wave functions suggests that super-twistors are realizex at the level of M8 geometry. These supertwistors are realized at the level of momentum space.

In TGD framework M8-H duality allows to geometrize the notion of super-twistor in the sense that different components of super-field correspond to components of super-octonion each of which corresponds to a space-time surfaces satisfying minimal surface equations with string world sheets as singularities - this is geometric counterpart for masslessness.

The progress in understanding of M8-H duality throws also light to the problem whether SUSY is realized in TGD and what SUSY breaking does mean. It is now rather clear that sparticles are predicted and SUSY remains exact but that p-adic thermodynamics causes thermal massivation: unlike Higgs mechanism, this massivation mechanism is universal and has nothing to do with dynamics. This is due to the fact that zero energy states are superpositions of states with different masses. The selection of p-adic prime characterizing the sparticle causes the mass splitting between members of super-multiplets although the mass formula is same for all of them. Super-octonion components of polynomials have different orders so that also the extension of rational assignable to them is different and therefore also the ramified primes so that p-adic prime as one them can be different for the members of SUSY multiplet and mass splitting is obtained.

The question how to realize super-field formalism at the level of H=M4× CP2 led to a dramatic progress in the identification of elementary particles and SUSY dynamics. The most surprising outcome was the possibility to interpret leptons and corresponding neutrinos as local 3-quark composites with quantum numbers of anti-proton and anti-neutron. Leptons belong to the same super-multiplet as quarks and are antiparticles of neutron and proton as far quantum numbers are consided. One implication is the understanding of matter-antimatter asymmetry. Also bosons can be interpreted as local composites of quark and anti-quark.

Hadrons and hadronic gluons would still correspond to the analog of monopole phase in QFTs. Homology charge would appear as space-time correlate for color at space-time level and explain color confinement. Also color octet variants of weak bosons, Higgs, and Higgs like particle and the predicted new pseudo-scalar are predicted. They could explain the successes of conserved vector current hypothesis (CVC) and partially conserved axial current hypothesis (PCAC).

See the article SUSY in TGD Universe.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

New Aspects of M8-H Duality



M8-H duality (H=M4× CP2) has become one of central elements of TGD. M8-H duality implies two descriptons for the states.

  1. M8-H duality assumes that space-time surfaces in M8 have associative tangent- or normal space M4 and that these spaces share a common sub-space M2⊂ M4, which corresponds to complex subspace of octonions (also integrable distribution of M2(x) can be considered). This makes possible the mapping of space-time surfaces X4⊂ M8 to X4⊂ H=M4× CP2) giving rise to M8-H duality.

  2. M8-H duality makes sense also at the level of 8-D momentum space in one-one correspondence with light-like octonions. In M8=M4× E4 picture light-like 8-momenta are projected to a fixed quaternionic M4T⊂ M8. The projections to M4T⊃ M2 momenta are in general massive. The group of symmetries is for E4 parts of momenta is Spin(SO(4))= SU(2)L× SU(2)R and identified as the symmetries of low energy hadron physics.

    M4⊃ M2 can be also chosen so that the light-like 8-momentum is parallel to M4L⊂ M8. Now CP2 codes for the E4 parts of 8-momenta and the choice of M4L and color group SU(3) as a subgroup of automorphism group of octonions acts as symmetries. This correspond to the usual description of quarks and other elementary particles. This leads to an improved understanding of SO(4)-SU(3) duality. A weaker form of this duality S3-CP2 duality: the 3-spheres S3 with various
    radii parameterizing the E4 parts of 8-momenta with various lengths correspond to discrete set of 3-spheres S3 of CP2 having discrete subgroup of U(2) isometries.

  3. The key challenge is to understand why the MacKay graphs in McKay correspondence and principal diagrams for the inclusions of HFFs correspond to ADE Lie groups or their affine variants. It turns out that a possible concrete interpretation for the hierarchy of finite subgroups of SU(2) appears as discretizations of 3-sphere S3 appearing naturally at M8 side of M8-H duality. Second interpretation is as covering of quaternionic Galois group. Also the coordinate patches of CP2 can be regarded as piles of 3-spheres and finite measurement resolution. The discrete groups of SU(2) define in a natural manner a hierarchy of measurement resolutions realized as the set of light-like M8 momenta. Also a concrete interpretation for Jones inclusions as inclusions for these discretizations emerges.

  4. A radically new view is that descriptions in terms of massive and massless states are alternative options leads to the interpretation of p-adic thermodynamics as a completely universal massivation mechanism having nothing to do with dynamics. The problem is the paradoxical looking fact that particles are massive in H picture although they should be massless by definition. The massivation is unavoidable if zero energy states are superposition of massive states with varying masses. The M4L in this case most naturally corresponds to that associated with the dominating part of the state so that higher mass contributions can be described by using p-adic thermodynamics and mass squared can be regarded as thermal mass squared calculable by p-adic thermodynamics.

  5. As a side product emerges a deeper understanding of ZEO based quantum measurement theory and consciousness theory. 4-D space-time surfaces correspond to roots of octonionic polynomials P(o) with real coefficients corresponding to the vanishing of the real or imaginary part of P(o).

    These polynomials however allow universal roots, which are not 4-D but analogs of 6-D branes and having topology of S6. Their M4 projections are time =constant snapshots t= rn,rM≤ rn 3-balls of M4 light-cone (rn is root of P(x)). At each point the ball there is a sphere S3 shrinking to a point about boundaries of the 3-ball.

    What suggests itself is following "braney" picture. 4-D space-time surfaces intersect the 6-spheres at 2-D surfaces identifiable as partonic 2-surfaces serving as generalized vertices at which 4-D space-time surfaces representing particle orbits meet along their ends. Partonic 2-surfacew would define the space-time regions at which one can pose analogs of boundary values fixing the space-time surface by preferred extremal property. This would realize strong form of holography (SH): 3-D holography is implied already by ZEO.

    This picture forces to consider a modification of the recent view about ZEO based theory of consciousness. Should one replace causal diamond (CD) with light-cone, which can be however either future or past directed. "Big" state function reductions (BSR) meaning the death and re-incarnation of self with opposite arrow of time could be still present. An attractive interpretation for the moments t=rn would be as moments assignable to "small" state function reductions (SSR) identifiable as "weak" measurements giving rise to to sensory input of conscious entity in ZEO based theory of consciousness. One might say that conscious entity becomes gradually conscious about its roots in increasing order. The famous question "What it feels to be a bat" would reduce to "What it feels to be a polynomial?"! One must be however very cautious here.

See the article New Aspects of M8-H Duality.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.




TGD view about McKay Correspondence, ADE Hierarchy, and Inclusions of Hyperfinite Factors


There are two mysterious looking correspondences involving ADE groups. McKay correspondence between McKay graphs characterizing tensor products for finite subgroups of SU(2) and Dynkin diagrams of affine ADE groups is the first one. The correspondence between principal diagrams characterizing inclusions of hyper-finite factors of type II1 (HFFs) with Dynkin diagrams for a subset of ADE groups and Dynkin diagrams for affine ADE groups is the second one.

I have considered the interpretation of McKay correspondence in TGD framework already earlier but the decision to look it again led to a discovery of a bundle of new ideas allowing to answer several key questions of TGD.

  1. Asking questions about M8-H duality at the level of 8-D momentum space led to a realization that the notion of mass is relative as already the existence of alternative QFT descriptions in terms of massless and massive fields suggests (electric-magnetic duality). Depending on choice M4⊂ M8, one can describe particles as massless states in M4× CP2 picture (the choice is M4L depending on state) and as massive states (the choice is fixed M4T) in M8 picture. p-Adic thermal massivation of massless states in M4L picture can be seen as a universal dynamics independent mechanism implied by ZEO. Also a revised view about zero energy ontology (ZEO) based quantum measurement theory as theory of consciousness suggests itself.

  2. Hyperfinite factors of type II1 (HFFs) and number theoretic discretization in terms of what I call cognitive representations provide two alternative approaches to the notion of finite measurement resolution in TGD framework. One obtains rather concrete view about how these descriptions relate to each other at the level of 8-D space of light-like momenta. Also ADE hierarchy can be understood concretely.

  3. The description of 8-D twistors at momentum space-level is also a challenge of TGD. 8-D twistorializations
    in terms of octo-twistors (M4T description) and M4× CP2 twistors (M4L description) emerge at imbedding space level. Quantum twistors could serve as a twistor description at the level of space-time surfaces.

McKay correspondence in TGD framework

Consider first McKay correspondence in more detail.

  1. McKay correspondence states that the McKay graphs characterizing the tensor product decomposition rules for representations of discrete and finite sub-groups of SU(2) are Dynkin diagrams for the affine ADE groups obtained by adding one node to the Dynkin diagram of ADE group. Could this correspondence make sense for any finite group G rather than only discrete subgroups of SU(2)? In TGD Galois group of extensions K of rationals can be any finite group G. Could Galois group take the role of G?

  2. Why the subgroups of SU(2) should be in so special role? In TGD framework quaternions and octonions play a fundamental role at M8 side of M8-H duality. Complexified M8 represents complexified octonions and space-time surfaces X4 have quaternionic tangent or normal spaces. SO(3) is the automorphism group of quaternions and for number theoretical discretizations induced by extension K of rationals it reduces to its discrete subgroup SO(3)K having SU(2)K as a covering. In certain special cases corresponding to McKay correspondence this group is finite discrete group acting as symmetries of Platonic solids. Could this make the Platonic groups so special? Could the semi-direct products Gal(K)×L SU(2)K take the role of discrete subgroups of SU(2)?

HFFs and TGD

The notion of measurement resolution is definable in terms of inclusions of HFFs and using number theoretic discretization of X4. These definitions should be closely related.

  1. The inclusions N M of HFFs with index M: N<4 are characterized by Dynkin diagrams for a subset of ADE groups. The TGD inspired conjecture is that the inclusion hierarchies of extensions of rationals and of corresponding Galois groups could correspond to the hierarchies for the inclusions of HFFs. The natural realization would be in terms of HFFs with coefficient field of Hilbert space in extension K of rationals involved.

    Could the physical triviality of the action of unitary operators N define measurement resolution? If so, quantum groups assignable to the inclusion would act in quantum spaces associated with the coset spaces M/ N of operators with quantum dimension d= M: N. The degrees of freedom below measurement resolution would correspond to gauge symmetries assignable to N.

  2. Adelic approach provides an alternative approach to the notion of finite measurement resolution. The cognitive representation identified as a discretization of X4 defined by the set of points with points having H (or at least M8 coordinates) in K would be common to all number fields (reals and extensions of various p-adic number fields induced by K). This approach should be equivalent with that based on inclusions. Therefore the Galois groups of extensions should play a key role in the understanding of the inclusions.

How HFFs could emerge from TGD?
  1. The huge symmetries of "world of classical words" (WCW) could explain why the ADE diagrams appearing as McKay graphs and principal diagrams of inclusions correspond to affine ADE algebras or quantum groups. WCW consists of space-time surfaces X4, which are preferred extremals of the action principle of the theory defining classical TGD connecting the 3-surfaces at the opposite light-like boundaries of causal diamond CD= cd× CP2, where cd is the intersection of future and past directed light-cones of M4 and contain part of δ M4+/-× CP2. The symplectic transformations of δ M4+× CP2 are assumed to act as isometries of WCW. A natural guess is that physical states correspond to the representations of the super-symplectic algebra SSA.

  2. The sub-algebras SSAn of SSA isomorphic to SSA form a fractal hierarchy with conformal weights in sub-algebra being n-multiples of those in SSA. SSAn and the commutator [SSAn,SSA] would act as gauge transformations. Therefore the classical Noether charges for these sub-algebras would vanish. Also the action of these two sub-algebras would annihilate the quantum states. Could the inclusion hierarchies labelled by integers ..<n1<n2<n3.... with ni+1 divisible by ni would correspond hierarchies of HFFs and to the hierarchies of extensions of rationals and corresponding Galois groups? Could n correspond to the dimension of Galois group of K.

  3. Finite measurement resolution defined in terms of cognitive representations suggests a reduction of the symplectic group SG to a discrete subgroup SGK, whose linear action is characterized by matrix elements in the extension K of rationals defining the extension. The representations of discrete subgroup are infinite-D and the infinite value of the trace of unit operator is problematic concerning the definition of characters in terms of traces. One can however replace normal trace with quantum trace equal to one for unit operator. This implies HFFs and the hierarchies of inclusions of HFFs. Could inclusion hierarchies for extensions of rationals correspond to inclusion hierarchies of HFFs and of isomorphic sub-algebras of SSA?

Quantum spinors are central for HFFs. A possible alternative interpretation of quantum spinors is in terms of quantum measurement theory with finite measurement resolution in which precise eigenstates as measurement outcomes are replaced with universal probability distributions defined by quantum group. This has also application in TGD inspired theory of consciousness: the idea is that the truth value of Boolean statement is fuzzy. At the level of quantum measurement theory this would mean that the outcome of quantum measurement is not anymore precise eigenstate but that one obtains only probabilities for the appearance of different eigenstate. One might say that probability of eigenstates becomes a fundamental observable and measures the strength of belief.

See the article TGD view about McKay Correspondence, ADE Hierarchy, and Inclusions of Hyperfinite Factors.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.


Super-symmetry in TGD Universe

It is now clear that SUSY is realized in TGD Universe but that the realization is very different from that in super-symmetric quantum field theories. The question how to realize super-field formalism at the level of H=M4× CP2 led to a dramatic progress in the identification of elementary particles and SUSY dynamics.

This picture simplifies dramatically the view about particle spectrum and scattering amplitudes. The most surprising outcome was the possibility to interpret leptons and corresponding neutrinos as local 3-quark composites with quantum numbers of anti-proton and anti-neutron. Leptons belong to the same super-multiplet as quarks and are antiparticles of neutron and proton as far quantum numbers are consided. One implication is the understanding of matter-antimatter asymmetry.

Also bosons can be interpreted as local composites of quark and anti-quark. Hadrons and hadronic gluons would still correspond to the analog of monopole phase in QFTs. Homology charge would appear as space-time correlate for color at space-time level and explain color confinement. Also color octet variants of weak bosons, Higgs, and Higgs like particle and the predicted new pseudo-scalar are predicted. They could explain the successes of conserved vector current hypothesis (CVC) and partially conserved axial current hypothesis (PCAC).

See the article Super-symmetry in in TGD Universe or the chapter TGD view about McKay Correspondence, ADE Hierarchy, Inclusions of Hyperfinite Factors, SUSY, and Twistors.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Tuesday, June 04, 2019

Copenhagen interpretation is dead: long live ZEO based quantum measurement theory!

I encountered a very interesting ScienceDaily article " Physicists can predict the jumps of Schrödinger's cat (and finally save it)" ). The experimental findings described in the article are extremely interesting from the point of view provide by TGD inspired quantum measurement theory relying on Zero Energy Ontology (ZEO) and provides a test for it.

In standard quantum measurement theory (Copenhagen interpretation) of Bohr quantum jump is random in the sense that it occurs with predictable probabilities to an eigenstate of the measured observables. Their occurrence cannot be predicted and even less prevented - except by monitoring - Zeno effect.

1. Findings

The findings of Minev et al are described in the article "To catch and reverse a quantum jump mid-flight". The outcome of quantum jump is indeed unpredictable but the time of occurrence is to high degree predictable: there is a detectable warning signal!

A curious feature is that the external signal responsible for the quantum jump can be stopped during the "flight" from the initial to final state. As if the quantum jump is analogous to a domino effect. It is also claimed that the jump can be reversed during flight period by a control signal: if jump has already occurred then one might argue that the control signal induces quantum jump in opposite direction when applied at time which is roughly the mid-time of "flight".

If these findings can be replicated is clear that Bohr's theory is dead also officially and one must finally go back to the blackboard and start serious thinking about fundamentals. It took 92 years - almost a century! State function reduction (SR) is definitely more complex phenomenon than predicted by Bohr. What is most intriguing that SR looks smooth, deterministic classical time evolution although the outcome is not predictable. People loving hidden variables might be happy but better to think about this more precisely before jumping to any conclusions.

What is most intriguing that SR looks smooth, deterministic classical time evolution although the outcome is not predictable. People loving hidden variables might be happy but better to think about this more precisely before jumping to any conclusions. Authors apply so called quantum trajectory theory to describe the findings and report that the model is able to predict the parameters of the parameterization with one per cent accuracy.

Zero energy ontology (ZEO) based view about quantum measurement and the relationship between geometric and subjective time explains why state function reduction looks like a deterministic process. Unfortunately, what ZEO is, is not completely clear. This allows to consider two options.

  1. Both options imply that one can apparently anticipate quantum jump. This could be however an illusion: the observed classical time evolution could occur after the quantum jump in opposite direction of time. The fact that the absence of the signal inducing quantum jump does not affect the occurrence of quantum jump suggests that the "flight" period indeed represents the classical evolution after the quantum jump in the reversed direction of time so that the absence of the external signal would not anymore affect the situation. Generalized Zeno effect is essential element ZEO based quantum measurement theory so that SR might be prevented. Perhaps a more plausible interpretation is that the control signal induces the reversal of the quantum jump already occurred. A careful analysis to distinguish between subjective and geometric time and arrows of time for the observer and atom would be needed.

  2. The more conventional option nearer to the interpretation of experimenters is that the observered time evolution occurs before the quantum jump in standard direction. The period before quantum jump consists of a sequence of "small" state function reductions - "weak" measurements. M8-H duality suggests a concrete assignment of the moments of time to them and there would be also the last moment of this kind. After these things proceed to "big" state function reduction in analogy with domino effect. It is not however obvious why the classical time evolution should appear to converge to the final outcome deterministically.

2. First ZEO based based view about the findings

What about TGD and zero energy ontology (ZEO) based quantum measurement theory? Could it explain the revolutionary findings?

  1. The new element is that quantum states are not time= constant snapshots for time evolution but superpositions of entire deterministic time evolutions at the level of space-time surfaces and at the level of induced spinor fields. SR replaces super position of classical time evolutions with a new one. This like selecting and starting new deterministic computer program. Non-determinism is in these choices.

  2. There are two kinds of state function reductions in ZEO.

    1. In "small" SRs (SSRs) the states change at active boundary of causal diamond (CD) (call it A) but remain unchanged at passive boundary (call it P): generalized Zeno effect occurs at the passive boundary and "weak measurements" at A. The observables measured commute with those determining the states at P as their eigenstates. In particular, the location of A is measured localizing it and corresponds to the measurement of time as distance between the tips of CD.

    2. "Big" SRs (BSRs) reverse the arrow of time of zero energy states and the roles of A and P. BSR is preceded by a sequence of SSRs - weak or almost classical measurements. In TGD inspired theory of consciousness this sequences defines the life cycle of conscious entity - self.


What is of crucial importance that BSR creates the illusion that it is an outcome of a continuous process: this realizes quantum classical correspondence (QCC). Standard observer assumes standard arrow of time and the space-time surfaces in the final time reversed state seem to lead to the the 3-surface serving as a correlate for the final state! As if BSR would be outcome of a smooth deterministic process, which it is not! There is actually a superposition of these 3-surfaces at A after BSR but in the resolution used this is not detected. Re-phrasing it more precisely:

  1. The time reversal of time evolution is in good approximation obtained by time reflection symmetry T but not quite since T is slightly broken. This is extremely small effect.

  2. Before BSR one has a distribution of 3-surfaces X3 defining the ends of space-time surfaces X4 at A: 3-surfaces X3 corresponds to different outcomes of BSR and and can differ dramatically. Observer is not conscious of this. This is like a situtation of Schrödinger cat before measurement: it is impossible to be conscious about the superposition of dead and alive cat.

    After BSR one has quantum superposition of space-time surfaces directed to geometric past. Near the end of space-time at A they look like leading to a unique classical counterpart of final state of state function reduction. As if the state function reduction were a smooth, continuous, deterministic process. BSR guarantees this but BSR is not a smooth evolution.

The experimental findings can be understood by applying this general picture.
  1. One can assign to the evolution from initial state G of atom at P to final state E at A a sequence of small reductions, weak measurements and also superposition of classical time evolutions approximated by single evolution in given measurement resolution. The state E is superposition of various measurement outcomes and each of them corresponds to a superposition of space-time surfaces identical in the measurement resolution used.

  2. Then occurs the BSR: atom jumps from state E to state D. This selects from the superposition of space-time surfaces/time only the evolutions apparently leading to D. Or more precisely: the superposition of reversed time evolutions starting from D at A and very similar near A but deviating farther from it. The illusion about continuous, smooth, deterministic time evolution from G to D is created!

  3. Also the possibility to anticipate the reduction would be an illusion due to the different arrows of time for observer and the observed system after BSR. The time reversed time evolution actually starts from the final state. The warning signal (absence of photon emission would be natural consequence of the reduction but in reversed arrow of time. The illusion would be due to the identification of arrows of time of observer and the atom that made state function reduction. This conforms with the observation that one can drop away the periodic signal inducing the quantum jumps during the "flight" period identified as the deterministic process representing the quantum jump.

    The lesson would be that one must always check whether the arrow of time for the target of attention is same as my own. Not a good idea to be on the wrong lane (means death also in ZEO based consciousness theory).

    It is also claimed that one can prevent the quantum jump using a signal during the "flight" period. Generalized Zeno effect is basic element of TGD but the signal forcing the state to remain in P would be present before the quantum jump. This would suggest that the control signal induced quantum jump in opposite direction. To really understand the situation a careful analysis of the relationships between subjective and geometric times of observer and between geometric time of observer and atomic system after and before the quantum jump would be needed.

To sum up, ZEO is fantastic magician. Maybe this magic is necessary for the mental health of observer: a world without this illusion would be like nightmare where one cannot trust anything.

3. Second ZEO based based view about the findings inspired by M8-H duality

I have learned to take experimental findings very seriously and I am ready to aks whether the above described option the only possibility allowed by ZEO or can one think other alternatives? It would be nice to answer "No" but one can consider variants of ZEO inspired by so called M8-H duality.

The sequence of "small" state function reductions (SSRs) should have the last one. Is the "big" state function reduction (BSR) forced by some condition? One idea is that the life cycle of self corresponds to a measurement of all observables assignable to the active boundary A of CD and commuting with those defining the unaffected states at passive boundary P are measured (time as a location of A belongs to these observables measured in each SSR).

I have discussed possible modifications of ZEO inspired by so called M8-H duality. One motivation is that time flow as shifting M4 time t=constant hyper-plane can be argued to be more natural than that for light-cone boundary. Light-cone boundaries are however favored by its huge symmetries essential for the definition of the geometry of "world of classical worlds" (WCW). M8-H duality forces passive light-cone boundary P and the identification of A as boundary of region where sensory signals can arrive to self is natural.

M8-H duality allows to consider variants the original ZEO.

3.1 M8-H duality

Let us first briefly summarize what M8-H duality is.

  1. M8-H duality is one of the key ideas of TGD, and states that one can regard space-times as surfaces in either complexified octonionic M8 or in M4× CP2. The dynamics M8 is purely algebraic and requires that either tangent or normal space of space-time surface is associative (quaternionic).

  2. The algebraic equations for space-time surfaces in M8 state the vanishing of either the real or imaginary part (defined in quaternionic sense) for octonion valued polynomial P(o) with real coefficients. Besides 4-D roots one obtains as universal exceptional roots 6-spheres at boundary of the light-cone of M8 with radii given by the roots rn of the polynomial in question. They correspond to the balls t= rn (t is octonionic real coordinate) inside Minkowski light-cone with each point have as fiber a 3-sphere S3 with radius contracting to zero at the boundary of the light-cone of M4. These 6-spheres are clearly analogous to branes connected by 4-D space-time surfaces.

  3. The intersections of space-time surfaces with 6-spheres would be 2-D and I have interpreted them as partonic 2-surfaces identifiable as topological particle reaction vertices - partonic 2-surfaces - at which incoming and outgoing light-like 3-surfaces meet along their ends. These light-like 3-surfaces - partonic orbits - would represent the boundaries between space-time regions with Euclidian and Minkowskian signatures of the induced metric. Partonic 2-surfaces would be analogs of the vertices of Feynman diagrams. The boundaries of string world sheets predicted as singularities of minimal surfaces defining space-time surfaces would be along the partonic orbits and give rise to QFT type description using cognitive representations and analogs of twistor diagrams consisting of lines.

3.2 M8-H duality and consciousness

One can ask whether M8-H duality and this braney picture has implications for ZEO based theory of consciousness. Certain aspects of M8-H duality indeed challenge the recent view about consciousness based on ZEO (zero energy ontology) and ZEO itself.

  1. The moments t=rn defining the 6-branes correspond classically to special moments for which phase transition like phenomena occur. Could t=rn have a special role in consciousness theory?

    1. For some SSRs the increase of the size of CD reveals new t=rn plane inside CD. One can argue that these SSRS define very special events in the life of self. This would not modify the original ZEO considerably but could give a classical signature for how many ver special moments of consciousness have occurred: the number of the roots of P would be a measure for the lifetime of self and there would be the largest root after which BSR would occur.

    2. Second possibility is more radical. One could one think of replacing CD with single truncated future- or past-directed light-cone containing the 6-D universal roots of P up to some rn defining the upper boundary of the truncated cone? Could t=rn define a sequence of moments of consciousness? To me it looks more natural to assume that they are associated with very special moments of consciousness.

  2. For both options SSRs increase the number of roots rn inside CD/truncated light-one gradually and thus its size? When all roots of P(o) would have been measured - meaning that the largest value rmax of rn is reached -, BSR would be unavoidable.

    BSR could replace P(o) with P1(r1-o): r1 must be real and one should have r1>rmax. The new CD/truncated light-cone would be in opposite direction and time evolution would be reversed. Note that the new CD could have much smaller size
    size if it contains only the smallest root r0. One important modification of ZEO becomes indeed possible. The size of CD after BSR could be much smaller than before it. This would mean that the re-incarnated self would have "childhood" rather than beginning its life at the age of previous self - kind of fresh start wiping the slate clean.

    One can consider also a less radical BSR preserving the arrow of time and replacing the polynomial with a new one, say a polynomial having higher degree (certainly in statistical sense so that algebraic complexity would increase).

3.3 Is a more conservative view possible?

Could this picture allow to build a more conservative picture more akin to that proposed by experimenters?

  1. The interpretation of the detected time evolution as that before the quantum jump would conform with the interpretation of experimentalists that a kind of domino effect is involved and also with the observation that stopping the signal causing the quantum jumps does not anymore affect the situation.

  2. It is however unclear how to understand why the evolution looks like leading to the outcome unless the sequence of rn:s defines a sequence of steps gradually taking the system near the final state.

  3. What about preventing the BSR by external signal and even reversing the quantum jump? This would require an external perturbation of the octonionic polynomial increasing the value of the largest root rmax or even the degree of the polynomial and bringing in additional significant moments of life. Is it possible to speak about external perturbations of the coefficients of polynomials assumed to be rational numbers? The perturbations would come from a higher level in the hierarchy of selves (experimentalist), and one can imagine them in the framework of many-sheeted space-time.

To sum up, to my opinion (which could change) the first option looks more plausible. The introduction of moments t=rn as special moments in the life of self looks highly attractive and also the possibility of wiping the slate clear.

See the article Copenhagen interpretation dead: long live ZEO based quantum measurement theory or the chapter Zero Energy Ontology and Matrices of "Towards M-matrix".

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Gravitation as square of gauge interactions

I encountered in FB a link to an interesting popular article about theoretical physicist Henrik Johansson who has worked with supergravity in Wallenberg Academy. He has found strong mathematical evidence for a new duality. Various variants of super quantum gravity support the view that supersymmetric quantum theories of gravitation can be seen as a double copy of a gauge theory. One could say that spin 2 gravitons are gluons with color charge replaced with spin. Since the information about charges disappears, gluons can be understood very generally as gauge bosons for given gauge theory, not necessarily QCD.

The article of C. D. White entitled "The double copy: gravity from gluons explains in more detail the double copy duality and also shows that it relates in many cases also exact classical solutions of Einsteins equations and YM theories. One starts from L-loop scattering amplitude involving products of kinematical factors ni and color factors ci and replaces color factors with extra kinematical factors ñi. The outcome is an L-loop amplitude for gravitons.

As if gravitation could be regarded as a gauge theory with polarization and/or momenta identified giving rise to effective color charges. This is like taking gauge potential and giving it additional index to get metric tensor. This naive analogy seems to hold true at the level of scattering amplitudes and also for many classical solutions of field equations. Could one think that gravitons as states correspond to gauge singlets formed from two gluons and having spin 2? Also spin 1 and spin 0 states would be obtained and double copies involve also them.

TGD view about elementary particles indeed predicts that gravitons be regarded in certain sense pairs of gauge bosons. Consider now gravitons and assume for simplicity that spartners of fundamental fermions - identifiable as local multi-fermion states allowed by statistics - are not involved: this does not change the situation much. Graviton's spin 2 requires 2 fermions and 2 anti-fermions: fermion or anti-fermion at each throat. For gauge bosons fermion and anti-fermion at two throats is enough. One could therefore formally see gravitons as pairs of two gauge bosons in accordance with the idea about graviton is a square of gauge boson.

The fermion contents of the monopole flux tube associated with elementary particle determines quantum numbers of the flux tube as particle and characterizes corresponding interaction. The interaction depends also on the charges at the ends of the flux tube. This leads to a possible interpretation for the formation of bound states in terms of flux tubes carrying quantum numbers of particles.

  1. These long flux tubes can be arbitrarily long for large values of ℏeff=n× ℏ0 assigned to the flux tube. A plausible guess for for the expression of ℏ in terms of ℏ0 is as ℏ= 6× ℏ0. The length of the flux tube scales like ℏeff.

  2. Nottale proposed that it makes sense to speak about gravitational Planck constant hgr. In TGD this idea is generalized and interpreted in framework of generalized quantum theory. For flux tubes assignable to gravitational bound states along which gravitons propagate, one would have ℏeff= ℏgr= GMm/v0, where v0<c is parameter with dimensions of velocity. One could write interaction strength as

    GMm = v0× ℏgr .

  3. gr obtained from this formula must satisfy ℏgr>hbar. This generalizes to other interactions. For instance, one has one would have

    Z1Z2e2= v0hbarem

    for electromagnetic flux tubes in the case that ones hem>hbar. The interpretation of the velocity parameter v0 is discussed in at here.

    One could even turn the situation around and say that the value of ℏeff fixes the interaction strength. ℏeff would depend on fermion content and thus of virtual particle and also on the masses or other charges at the ends of the flux tube. The longer the range of the interaction, the larger the typical value of ℏeff.

  4. The interpretation could be in terms long length scale quantum fluctuations at quantum criticality. Particles generate U-shaped monopole flux tubes with varying length proportional to ℏgr. If these U-shaped flux tubes from two different particles find each other, they reconnect to flux tube pairs connecting particles and give rise to interaction. What comes in mind is tiny curious and social animals studying their environment.

  5. I have indeed proposed this picture in biology: the U-shaped flux tubes would be tentacles with which bio-molecules (in particular) would be scanning their environment. This scanning would be the basic mechanism behind immune system. It would also make possible for bio-molecules to find each in molecular crowd and provide a mechanism of catalysis. Could this picture apply completely generally? Would even elementary particles be scanning their environment with these tentacles?

  6. Could one interpret the flux tubes as analogs of virtual particles or could they replace virtual particles of quantum field theories? The objection is that flux tubes would have time-like momenta whereas virtual particle analogs would have space-like momenta. The interpretation makes sense only if the associated momenta are between space-like and time-like that is light-like so that flux tube would correspond to mass shell particle. But this is the case in twistor approach to gauge theories also in TGD (see this).

    Perhaps the following interpretation is more appropriate. Flux tubes are accompanied by strings and string world sheets can be interpreted as stringy description of gravitation and other interactions.

See the article More about the construction of scattering amplitudes in TGD framework or the chapter The Recent View about Twistorialization in TGD Framework of "Towards M-theory: Part II".

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Sunday, June 02, 2019

Three shorter articles related to "TGD view about McKay Correspondence, ADE Hierarchy, Inclusions of Hyperfinite Factors, and Twistors"

I decided to isolate from a rather long article TGD view about McKay Correspondence, ADE Hierarchy, Inclusions of Hyperfinite Factors, and Twistors the following three shorter articles.

M8-H Duality and Consciousness.

This article is part of a longer paper "TGD view about McKay Correspondence, ADE Hierarchy, Inclusions of Hyperfinite Factors, and Twistors". I found it convenient to isolate the part of paper related to the possible implications for TGD inspired theory of consciousness. M8-H duality is one of the key ideas of TGD, and one can ask whether it has implications for TGD inspired theory of consciousness. Certain aspects of M8-H duality indeed challenge the recent view about consciousness based on ZEO (zero energy ontology).

The algebraic equations for space-time surfaces in M8 state the vanishing of either the real or imaginary part (defined in quaternionic sense) for octonion valued polynomial with real coefficients. Besides 4-D roots one obtains as universal exceptional roots 6-spheres at boundary of the light-cone of M8 with radii given by the roots rn of the polynomial in question. They correspond to the balls t= rn inside Minkowski light-cone with each point have as fiber a 3-sphere S3 with radius contracting to zero at the boundary of the light-cone of M4. Could these balls have a special role in consciousness theory? For instance, could they serve as correlates for memories. In this article I consider several scenarios involving a modification of the recent form of ZEO. In the following are the abstracts of these articles.

M8-H Duality and the Two Manners to Describe Particles.

This article is part of a longer paper "TGD view about McKay Correspondence, ADE Hierarchy, Inclusions of Hyperfinite Factors, and Twistors". I found it convenient to isolate the part of paper related to the notion of particle mass to a separate article. The basic new result is that M8-H duality allows to see particles in two manners. In M8 picture particles are massive and correspond to a fixed M4 subset M8: in this case symmetry group os $SO(4)$: this could correspond to low energy hadron physics. In H=M4× CP2 picture particles are massless and symmetry group is SU(3): this picture would correspond to high energy hadron physics with massless quarks and gluons. It is shown that p-adic mass calculations performed M4× CP2 picture are consistent with the massless of the particles: in zero energy ontology (ZEO) it is possible to have quantum superpositions of particles with different mass and this is consistent with the description of the situation in terms of p-adic thermodynamics.

Do Supertwistors Make Sense in TGD?.

This article is part of a longer paper "TGD view about McKay Correspondence, ADE Hierarchy, Inclusions of Hyperfinite Factors, and Twistors". I found it convenient to isolate the part of paper related to supersymmetry. In twistor Grassmannian approach to N =4 SYM twistors are replaced with supertwistors and the extreme elegance of the description of various helicity states using twistor space wave functions suggests that super-twistors are realize at the level of M8 geometry. These supertwistors are realized at the level of momentum space.

In TGD framework M8-H duality allows to geometrize the notion of super-twistor in the sense that different components of super-field correspond to components of super-octonion each of which corresponds to a space-time surfaces satisfying minimal surface equations with string world sheets as singularities - this is geometric counterpart for masslessness.

The progress in understanding of M8-H duality throws also light to the problem whether SUSY is realized in TGD and what SUSY breaking does mean. It is now clear that sparticles are predicted and SUSY remains exact but that p-adic thermodynamics causes thermal massivation: unlike Higgs mechanism this massivation mechanism is universal and has nothing to do with dynamics. This is due to the fact that zero energy states are superpositions of states with different masses. The selection of p-adic prime characterizing the sparticle causes the mass splitting between members of super-multiplets although the mass formula is same for all of them. Super-octonion components of polynomials have different orders so that also the extension of rational assignable to them is different and therefore also the ramified primes so that p-adic prime as one them can be different for the members
of SUSY multiplet and mass splitting is obtained.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.


Thursday, May 30, 2019

TGD view about McKay Correspondence, ADE Hierarchy, Inclusions of Hyperfinite Factors, and Twistors

There are two mysterious looking correspondences involving ADE groups. McKay correspondence between McKay graphs characterizing tensor products for finite subgroups of SU(2) and Dynkin diagrams of affine ADE groups is the first one. The correspondence between principal diagrams characterizing inclusions of hyper-finite factors of type II1 (HFFs) with Dynkin diagrams for a subset of ADE groups and Dynkin diagrams for affine ADE groups is the second one.

I have considered the interpretation of McKay correspondence in TGD framework already earlier
but the decision to look it again led to a discovery of a bundle of new ideas allowing to answer several key questions
of TGD.

  1. Asking questions about M8-H duality at the level of 8-D momentum space led to a realization that the notion of mass is relative as already the existence of alternative QFT descriptions in terms of massless and massive fields suggests (electric-magnetic duality). Depending on choice M4⊂ M8, one can describe particles as massless states in M4× CP2 picture (the choice is M4L depending on state) and as massive states (the choice is fixed M4T) in M8 picture. p-Adic thermal massivation of massless states in M4L picture can be seen as a universal dynamics independent mechanism implied by ZEO. Also a revised view about zero energy ontology (ZEO) based quantum measurement theory as theory of consciousness suggests itself.

  2. Hyperfinite factors of type II1 (HFFs) and number theoretic discretization in terms of what I call cognitive representations provide two alternative approaches to the notion of finite measurement resolution in TGD framework. One obtains rather concrete view about how these descriptions relate to each other at the level of 8-D space of light-like momenta. Also ADE hierarchy can be understood concretely.

  3. The description of 8-D twistors at momentum space-level is also a challenge of TGD. 8-D twistorializations
    in terms of octo-twistors (M4T description) and M4× CP2 twistors (M4L description) emerge at imbedding space level. Quantum twistors could serve as a twistor description at the level of space-time surfaces.

McKay correspondence in TGD framework

Consider first McKay correspondence in more detail.

  1. McKay correspondence states that the McKay graphs characterizing the tensor product decomposition rules for representations of discrete and finite sub-groups of SU(2) are Dynkin diagrams for the affine ADE groups obtained by adding one node to the Dynkin diagram of ADE group. Could this correspondence make sense for any finite group G rather than only discrete subgroups of SU(2)? In TGD Galois group of extensions K of rationals can be any finite group G. Could Galois group take the role of G?

  2. Why the subgroups of SU(2) should be in so special role? In TGD framework quaternions and octonions play a fundamental role at M8 side of M8-H duality. Complexified M8 represents complexified octonions and space-time surfaces X4 have quaternionic tangent or normal spaces. SO(3) is the automorphism group of quaternions and for number theoretical discretizations induced by extension K of rationals it reduces to its discrete subgroup SO(3)K having SU(2)K as a covering. In certain special cases corresponding to McKay correspondence this group is finite discrete group acting as symmetries of Platonic solids. Could this make the Platonic groups so special? Could the semi-direct products Gal(K)×L SU(2)K take the role of discrete subgroups of SU(2)?

HFFs and TGD

The notion of measurement resolution is definable in terms of inclusions of HFFs and using number theoretic discretization of X4. These definitions should be closely related.

  1. The inclusions N M of HFFs with index M: N<4 are characterized by Dynkin diagrams for a subset of ADE groups. The TGD inspired conjecture is that the inclusion hierarchies of extensions of rationals and of corresponding Galois groups could correspond to the hierarchies for the inclusions of HFFs. The natural realization would be in terms of HFFs with coefficient field of Hilbert space in extension K of rationals involved.

    Could the physical triviality of the action of unitary operators N define measurement resolution? If so, quantum groups assignable to the inclusion would act in quantum spaces associated with the coset spaces M/ N of operators with quantum dimension d= M: N. The degrees of freedom below measurement resolution would correspond to gauge symmetries assignable to N.

  2. Adelic approach provides an alternative approach to the notion of finite measurement resolution. The cognitive representation identified as a discretization of X4 defined by the set of points with points having H (or at least M8 coordinates) in K would be common to all number fields (reals and extensions of various p-adic number fields induced by K). This approach should be equivalent with that based on inclusions. Therefore the Galois groups of extensions should play a key role in the understanding of the inclusions.

How HFFs could emerge from TGD?
  1. The huge symmetries of "world of classical words" (WCW) could explain why the ADE diagrams appearing as McKay graphs and principal diagrams of inclusions correspond to affine ADE algebras or quantum groups. WCW consists of space-time surfaces X4, which are preferred extremals of the action principle of the theory defining classical TGD connecting the 3-surfaces at the opposite light-like boundaries of causal diamond CD= cd× CP2, where cd is the intersection of future and past directed light-cones of M4 and contain part of δ M4+/-× CP2. The symplectic transformations of δ M4+× CP2 are assumed to act as isometries of WCW. A natural guess is that physical states correspond to the representations of the super-symplectic algebra SSA.

  2. The sub-algebras SSAn of SSA isomorphic to SSA form a fractal hierarchy with conformal weights in sub-algebra being n-multiples of those in SSA. SSAn and the commutator [SSAn,SSA] would act as gauge transformations. Therefore the classical Noether charges for these sub-algebras would vanish. Also the action of these two sub-algebras would annihilate the quantum states. Could the inclusion hierarchies labelled by integers ..<n1<n2<n3.... with ni+1 divisible by ni would correspond hierarchies of HFFs and to the hierarchies of extensions of rationals and corresponding Galois groups? Could n correspond to the dimension of Galois group of K.


  3. Finite measurement resolution defined in terms of cognitive representations suggests a reduction of the symplectic group SG to a discrete subgroup SGK, whose linear action is characterized by matrix elements in the extension K of rationals defining the extension. The representations of discrete subgroup are infinite-D and the infinite value of the trace of unit operator is problematic concerning the definition of characters in terms of traces. One can however replace normal trace with quantum trace equal to one for unit operator. This implies HFFs and the hierarchies of inclusions of HFFs. Could inclusion hierarchies for extensions of rationals correspond to inclusion hierarchies of HFFs and of isomorphic sub-algebras of SSA?

New aspects of M8-H duality

M8-H duality (H=M4× CP2) has become one of central elements of TGD. M8-H duality implies two descriptons for the states.

  1. M8-H duality assumes that space-time surfaces in M8 have associative tangent- or normal space M4 and that these spaces share a common sub-space M2⊂ M4, which corresponds to complex subspace of octonions (also integrable distribution of M2(x) can be considered). This makes possible the mapping of space-time surfaces X4⊂ M8 to X4⊂ H=M4× CP2) giving rise to M8-H duality.

  2. M8-H duality makes sense also at the level of 8-D momentum space in one-one correspondence with light-like octonions. In M8=M4× E4 picture light-like 8-momenta are projected to a fixed quaternionic M4T⊂ M8. The projections to M4T⊃ M2 momenta are in general massive. The group of symmetries is for E4 parts of momenta is Spin(SO(4))= SU(2)L× SU(2)R and identified as the symmetries of low energy hadron physics.

    M4⊃ M2 can be also chosen so that the light-like 8-momentum is parallel to M4L⊂ M8. Now CP2 codes for the E4 parts of 8-momenta and the choice of M4L and color group SU(3) as a subgroup of automorphism group of octonions acts as symmetries. This correspond to the usual description of quarks and other elementary particles. This leads to an improved understanding of SO(4)-SU(3) duality. A weaker form of this duality S3-CP2 duality: the 3-spheres S3 with various
    radii parameterizing the E4 parts of 8-momenta with various lengths correspond to discrete set of 3-spheres S3 of CP2 having discrete subgroup of U(2) isometries.

  3. The key challenge is to understand why the MacKay graphs in McKay correspondence and principal diagrams for the inclusions of HFFs correspond to ADE Lie groups or their affine variants. It turns out that a possible concrete interpretation for the hierarchy of finite subgroups of SU(2) appears as discretizations of 3-sphere S3 appearing naturally at M8 side of M8-H duality. Second interpretation is as covering of quaternionic Galois group. Also the coordinate patches of CP2 can be regarded as piles of 3-spheres and finite measurement resolution. The discrete groups of SU(2) define in a natural manner a hierarchy of measurement resolutions realized as the set of light-like M8 momenta. Also a concrete interpretation for Jones inclusions as inclusions for these discretizations emerges.

  4. A radically new view is that descriptions in terms of massive and massless states are alternative options leads to the interpretation of p-adic thermodynamics as a completely universal massivation mechanism having nothing to do with dynamics. The problem is the paradoxical looking fact that particles are massive in H picture although they should be massless by definition. The massivation is unavoidable if zero energy states are superposition of massive states with varying masses. The M4L in this case most naturally corresponds to that associated with the dominating part of the state so that higher mass contributions can be described by using p-adic thermodynamics and mass squared can be regarded as thermal mass squared calculable by p-adic thermodynamics.

  5. As a side product emerges a deeper understanding of ZEO based quantum measurement theory and consciousness theory. 4-D space-time surfaces correspond to roots of octonionic polynomials P(o) with real coefficients corresponding to the vanishing of the real or imaginary part of P(o).

    These polynomials however allow universal roots, which are not 4-D but analogs of 6-D branes and having topology of S6. Their M4 projections are time =constant snapshots t= rn,rM≤ rn 3-balls of M4 light-cone (rn is root of P(x)). At each point the ball there is a sphere S3 shrinking to a point about boundaries of the 3-ball.

    What suggests itself is following "braney" picture. 4-D space-time surfaces intersect the 6-spheres at 2-D surfaces identifiable as partonic 2-surfaces serving as generalized vertices at which 4-D space-time surfaces representing particle orbits meet along their ends. Partonic 2-surfacew would define the space-time regions at which one can pose analogs of boundary values fixing the space-time surface by preferred extremal property. This would realize strong form of holography (SH): 3-D holography is implied already by ZEO.

    This picture forces to consider a modification of the recent view about ZEO based theory of consciousness. Should one replace causal diamond (CD) with light-cone, which can be however either future or past directed. "Big" state function reductions (BSR) meaning the death and re-incarnation of self with opposite arrow of time could be still present. An attractive interpretation for the moments t=rn would be as moments assignable to "small" state function reductions (SSR) identifiable as "weak" measurements giving rise to to sensory input of conscious entity in ZEO based theory of consciousness. One might say that conscious entity becomes gradually conscious about its roots in increasing order. The famous question "What it feels to be a bat?" would reduce to "What it feels to be a polynomial?"! One must be however very cautious here.

What twistors are in TGD framework?

The basic problem of the ordinary twistor approach is that the states must be massless in 4-D sense. In TGD framework particles would be massless in 8-D sense. The meaning of 8-D twistorialization at space-time level is relatively well understood but at the level of momentum space the situation is not at all so clear.

  1. In TGD particles are massless in 8-D sense. For M4L description particles are massless in 4-D sense and the description at momentum space level would be in terms of products of ordinary M4 twistors and CP2 twistors. For M4T description particles are massive in 4-D sense. How to generalize the twistor description to 8-D case?

    The incidence relation for twistors and the need to have index raising and lowering operation in 8-D situation suggest the replacement of the ordinary l twistors with eitherwith octo-twistors or non-commutative quantum twistors.

  2. Octotwistors can be expressed as pairs of quaternionic twistors. Octotwistor approach suggests a generalization of twistor Grassmannian approach obtained by replacing the bi-spinors with complexified quaternions and complex Grassmannians with their quaternionic counterparts. Although TGD is not a quantum field theory, this proposal makes sense for cognitive representations identified as discrete sets of spacetime points with coordinates in the extension of rationals defining the adele implying effective reduction of particles to point-like particles.

  3. The notion of super-twistor can be geometrized in TGD framework at M8 level but at H level local many-fermion states become non-local but still having collinear light-like momenta. One would have a proposal for a quite concrete formula for scattering amplitudes!

    Even the existence of sparticles have been far from obvious hitherto but now it becomes clear that spartners indeed exist and SUSY breaking would be caused by the same universal mechanism as ordinary massivation of massless states. The mass formulas would be supersymmetric but the choice of p-adic prime identifiable as ramified prime of extension of rationals would depend on the state of super-multiplet. ZEO would make possible symmetry breaking without symmetry breaking as Wheeler might put it.

  4. What about the interpretation of quantum twistors? They could make sense as 4-D space-time description analogous to description at space-time level. Now one can consider generalization of the twistor Grassmannian approach in terms of quantum Grassmannians.

A possible alternative interpretation of quantum spinors is in terms of quantum measurement theory with finite measurement resolution in which precise eigenstates as measurement outcomes are replaced with universal probability distributions defined by quantum group. This has also application in TGD inspired theory of consciousness: the idea is that the truth value of Boolean statement is fuzzy. At the level of quantum measurement theory this would mean that the outcome of quantum measurement is not anymore precise eigenstate but that one obtains only probabilities for the appearance of different eigenstate. One might say that probability of eigenstates becomes a fundamental observable and measures the strength of belief.

See the article TGD view about McKay Correspondence, ADE Hierarchy, Inclusions of Hyperfinite Factors, and Twistors or the chapter of "Hyper-finite factors, p-adic length scale hypothesis, and dark matter hierarchy" with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.