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Topological Geometrodynamics: an Overview

Note: Newest contributions are at the top!

Year 2015

Analogs of quantum matrix groups from finite measurement resolution?

The notion of quantum group replaces ordinary matrices with matrices with non-commutative elements. This notion is physically very interesting, and in TGD framework I have proposed that it should relate to the inclusions of von Neumann algebras allowing to describe mathematically the notion of finite measurement resolution (see this). These ideas have developed slowly through various side tracks.

In it is interesting to consider the notion of quantum matrix inspired by recent view about quantum TGD. It turns out that under some additional conditions this approach provides a concrete representation and physical interpretation of quantum groups in terms of finite measurement resolution.

  1. The basic idea is to replace complex matrix elements with operators, which are products of non-negative hermitian operators and unitary operators analogous to the products of modulus and phase as a representation for complex numbers.
  2. The condition that determinant and sub-determinants exist is crucial for the well-definedness of eigenvalue problem in the generalized sense. The weak definition of determinant meaning its development with respect to a fixed row or column does not pose additional conditions. Strong definition of determinant requires its invariance under permutations of rows and columns. The permutation of rows/columns turns out to have interpretation as braiding for the hermitian operators defined by the moduli of operator valued matrix elements: this should lead to quantum commutativity as realized in braid groups with rows and columns appearing taking the role of braids.
  3. The commutativity of all sub-determinants is essential for the replacement of eigenvalues with eigenvalue spectra of hermitian operators and sub-determinants define mutually commuting set of operators.

Quantum matrices define a more general structure than quantum group but provide a concrete representation for them in terms of finite measurement resolution if q is a root of unity. For q=+/- 1 (Bose-Einstein or Fermi-Dirac statistics) one obtains quantum matrices for which the determinant is apart from possible change by sign factor invariant under the permutations of both rows and columns. One can also understand the recursive fractal structure of inclusion sequences of hyper-finite factors resulting by replacing operators appearing as matrix elements with quantum matrices and a concrete connection with quantum groups emerges.

In Zero Energy Ontology (ZEO) M-matrix serving as the basic building brick of unitary U-matrix and identified as a hermitian square root of density matrix provides a possible application for this vision. Especially fascinating is the possibility of hierarchies of measurement resolutions represented as inclusion sequences realized as recursive construction of M-matrices. Quantization would emerge already at the level of complex numbers appearing as M-matrix elements.

This approach might allow to unify various ideas behind TGD. For instance, Yangian algebras emerging naturally in twistor approach are examples of quantum algebras. The hierarchy of Planck constants should have a close relationship with inclusions and fractal hierarchy of sub-algebras of super-symplectic and other conformal algebras.

See the article Analogs of quantum matrix groups from finite measurement resolution? or the chapter Evolution of Ideas about Hyper-finite Factors in TGD.

Variation of Newston's constant and of length of day

J. D. Anderson et al have published an article discussing the observations suggesting a periodic variation of the measured value of Newton constant and variation of length of day (LOD) (see also this). This article represents TGD based explanation of the observations in terms of a variation of Earth radius. The variation would be due to the pulsations of Earth coupling via gravitational interaction to a dark matter shell with mass about 1.3× 10-4ME introduced to explain Flyby anomaly: the model would predict Δ G/G= 2Δ R/R and Δ LOD/LOD= 2Δ RE/RE with the variations pf G and length of day in opposite phases. The expermental finding Δ RE/RE= MD/ME is natural in this framework but should be deduced from first principles.

The gravitational coupling would be in radial scaling degree of freedom and rigid body rotational degrees of freedom. In rotational degrees of freedom the model is in the lowest order approximation mathematically equivalent with Kepler model. The model for the formation of planets around Sun suggests that the dark matter shell has radius equal to that of Moon's orbit. This leads to a prediction for the oscillation period of Earth radius: the prediction is consistent with the observed 5.9 years period. The dark matter shell would correspond to n=1 Bohr orbit in the earlier model for quantum gravitational bound states based on large value of Planck constant. Also n>1 orbits are suggestive and their existence would provide additional support for TGD view about quantum gravitation.

For details see the chapter Cosmology and Astrophysics in Many-Sheeted Space-Time or the article Variation of Newston's constant and of length of day.

What went wrong with symmetries?

Theoretical physics is in deep crisis. This is not bad at all. Crisis forces eventually to challenge the existing beliefs. Crisis gives also hopes about profound changes. In physical systems criticality means sensitivity, long range fluctuations and long range correlations, and this makes phase transition possible. In TGD framework life emerges at criticality!

The crisis of theoretical physics has many aspects. The crisis relates closely to the sociology of science and to the only game in the town attitude. The prevailing materialistic philosophy of science combined with the naive length scale reductionism form part of the sad story. The seeds of the crisis were sown in birthdays of quantum mechanics. The fathers of quantum theory were well aware that quantum measurement theory is the Achilles heel of the newborn quantum theory but later the pragmatically thinking theoreticians labelled questioning of the basic concepts as "philosophy" not meant for a respectable physicist.

The recent quantum measurement theory is just a collection of rules and observer still remains an outsider. To my view the proper formulation of quantum measurement theory requires making observer a part of systems. This means that physics must be extended to a theory of consciousness.

This raises several fundamental challenges and questions. How to define "self" as a conscious entity? How to resolve the conflict between two causalities: that of field equations and that of "free will"? What is the relationship between the geometric time of physicist and the experienced time? How is the arrow of time determined and is it always the same? The evidence that living matter is macroscopic quantum system is accumulating: is a generalization of quantum theory required to describe quantum systems? What about dark matter: can we understand it in the framework of existing quantum theory? This list could be continued.

In the following I will not consider this aspect more but restrict the consideration to an important key notion of recent day theoretical physics, namely symmetries. Physical theories rely nowadays on postulates about symmetries and there are many who say that quantum theory reduces almost totally group representation theory. There are refined mathematical tools making possible to derive the implications of symmetries in quantum theory such as Noether's theorem. These technical tools are extremely useful but it seems that methodology has replaced critical thought.

By this I mean that the real nature of various symmetries has not been considered seriously enough and that this is one of the basic reasons for the recent dead end. In the following I describe what I see as the mistakes due to sloppy thinking (maybe "sloppying" might be shorthand for it) and discuss briefly the TGD based solution of the problems involved.

This sloppiness manifests itself already in general relativity, in standard model there is no unification of color and electroweak symmetries and their different character is not understood, GUT approach is based on naive extension of gauge group and makes problematic predictions, supersymmetry in its standard form predicted to become visible at LHC energies is now strongly dis-favoured experimentally, and superstring model led to landscape catastrophe what has left is AdS/CFT correspondence which has not led to victories. Could it be that also conformal invariance should be re-considered seriously: a non-trivial generalization to 4-D context is highly desirable so that 10-D bulk would be replaced by 4-D space-time in the counterpart of AdS/CFT duality.

Energy problem of GRT

Energy and momentum are not well-defined notions in General Relativity. The Poincare symmetry of flat Minkowski space is lost and one cannot apply Noether's theorem so that the identification of classical conserved charges is lost and one can talk only about local conservation guaranteed by Einstein's equations realizing Equivalence Principle in weak form.

In quantum theory this kind of situation is highly unsatisfactory since Uncertainty Principle means that momentum eigenstates are delocalized. This is sloppy thinking and the fact that quantization is to high extend representation theory for symmetry groups might well explain the failure of the attempts to quantize general relativity.

TGD was born as a reaction to the challenge of constructing Poincare invariant theory of gravitation. The identification of space-times as 4-surfaces of some higher- dimensional space of form H=M4× S lifts Poincare symmetries from space-time level to the level of imbedding space H.

In this framework GRT space-time is an approximate macroscopic description obtained by replacing the space-time sheets of many-sheeted space-time with single piece of M4, which is slightly curved. Gravitational fields -deviations of induced metric from Minkowski metric- are replaced with their sum for various sheets. Same applies to gauge potentials. Einstein's equations express the remnants of Poincare symmetry for the GRT space-time obtained in this manner.

In superstring models one actually considers 10-D Minkowski space so that the lifting of symmetries is possible. Also the compactification (say Calabi-Yau) to M4× C still have Poincare symmetries. But after that one has 10- D gravitation and the same problems that one wanted to solve by introducing strings! School example about sloppying!

Is color symmetry really understood?

Many colleagues use to think that standard model is a closed chapter of theoretical physics. This is a further example of sloppy thinking.

  1. Standard model gauge group is product of color and electro-weak groups which are totally independent. The analogy with Maxwell's equations is obvious. Only after Maxwell and Einstein they could be seen as parts of single tensor representing gauge field.
  2. QCD and electroweak interactions differ in crucial manner. Color symmetry is exact (no Higgs fields in QCD) whereas electroweak symmetry is broken, and QCD is asymptotically free unlike electroweak interactions. In QCD color confinement takes place at low energies and remains still poorly understood.
Again TGD approach suggests a solution to these problems in terms of induced gauge field concept and a more refined view about QCD color.
  1. S=CP2 has color group SU(3) as isometries and electroweak gauge group as holonomies: hence CP2 unifies these symmetries just like Maxwell's theory unified electric and magnetic fields. Note that the choice of H= M4× CP2 is not adhoc: its factors are the only 4- D spaces allowing twistor spaces with Kähler structure.
  2. One can understand also the different nature of these symmetries. Color group represents exact symmetries so that symmetry breaking should not take place. Holonomies are tangent space symmetries and broken already at the level of CP2 geometry and does not therefore give rise to genuine Noether symmetries. One can however assign broken electroweak gauge symmetries to the holonomies.

    The isometry group defines Kac-Moody algebra in quantum TGD and color group acts as Kac-Moody group rather than gauge group. The differences is very delicate since only the central extension of Kac-Moody algebra distinguishes it from gauge algebra.

  3. Color is not spin-like quantum number as in QCD but colored states correspond to color partial waves in CP2 rather. Both leptons and quarks allow colored excitations which are however expected to be very heavy.

Is Higgs mechanism only a parameterization of particle masses?

The discovery of Higgs at LHC was very important step of progress but did not prove Higgs mechanism as a mechanism of massivation as sloppy thinkers believe. Fermion masses are not a prediction of the theory: they are put in by hand by assuming that Higgs couplings are proportional to the Higgs mass. It might well be that Higgs vacuum expectation value is the unique quantum field theoretic representation of particle massivation but that QFT approach cannot predict the masses and that the understanding of the massivation requires transcending QFT so that one describing particles as extended objects. String models were the first step to this direction but one step was not enough.

In TGD framework more radical generalization is performed. Point-like particle is replaced with a 3-surface and particle massivation is described in terms of p-adic thermodynamics, which relies on very general assumptions such as a non-trivial generalization of 2-D conformal invariance to 4-D context to be discussed later, p-adic thermodynamics, p-adic length scale hypothesis, and mapping of the predictions for p-adic mass squared to real mass squared by what I call anonical identification. In this framework Higgs vacuum expectation value parametrizes the QFT limit already described and is calculable from generalized Feynman diagrammatics.

GUT approach as more sloppy thoughts

After the successes of standard model the naive guess was that theory of everything could be constructed by a simple trick: extend the gauge group to a larger group containing standard model gauge group as sub-group. One can do this and there is a refined machinery allowing to deduce particle multiplets, effective actions, beta functions, etc.. There exists of course an infinite variety of Lie groups and endless variety of GUTs have been proposed.

The view about the Universe provided by GUTs is rather weird looking.

  1. Above weak mass scale there should be a huge desert of 14 orders of magnitudes containing no new physics! This is like claiming that the world ends at my backyard.
  2. Only the sum of baryon and lepton numbers would be conserved and proton would be unstable. The experimental lower limit for proton lifetime has been however steadily increasing and all GUTs derived from superstring models share a fine tuning to keep proton alive.
  3. Standard model gauge group seems to be all that is needed: there are no indications for larger gauge group. Fermion families seem to be copies of each other with different mass scales. Also the mass scales of these fermions differ dramatically and forcing them to multiplets of single gauge group could also be sloppy thinking. One would expect that the masses differ by simple numerical factors but they do not.
From TGD viewpoint the GUT approach is un-necessary.
  1. In TGD quarks and leptons correspond to different chiralities of imbedding space spinors. 8-D chiral invariance implies that quark and lepton numbers are separately conserved so that proton does not decay - at least in the manner predicted by GUTs. CP2 mass scale is of same order of magnitude as the mass scale assigned to the super heavy additional gauge bosons mediating proton decay.
  2. Family replication phenomenon does not require extension of gauge group since fermion families correspond to different topologies for partonic 2-surfaces representing fundamental particles (genus-generation correspondence). Note that the orbits of partonic 2-surfaces correspond to light-like 3-surface at which the induced metric changes its signature from Euclidian to Minkowskian: these surfaces or equivalently the 4-surfaces with Euclidian signature can be regarded as lines of generalized Feynman diagrams. The three lowest genera are special in the sense that they always allow Z2 as global conformal symmetry whereas higher genera allow this symmetry only in case of hyper-elliptic surfaces: this leads to an explanation for the experimental absence of higher genera. Higher genera could be more naturally many particle states with continuum mass spectrum with handles taking the role of particles.
  3. p-Adic length scale hypothesis emerging naturally in TGD framework allows to understand the mass ratios of fermions which are very un-natural if different fermion families are assumed to be related by gauge symmetries.

Supersymmetry in crisis

Supersymmetry is very beautiful generalization of the ordinary symmetry concept by generalizing Lie-algebra by allowing grading such that ordinary Lie algebra generators are accompanied by super-generators transforming in some representation of the Lie algebra for which Lie-algebra commutators are replaced with anti-commutators. In the case of Poincare group the super-generators would transform like spinors. Clifford algebras are actually super-algebras. Gamma matrices anti-commute to metric tensor and transform like vectors under the vielbein group (SO(n) in Euclidian signature). In supersymmetric gauge theories one introduced super translations anti-commuting to ordinary translations.

Supersymmetry algebras defined in this manner are characterized by the number of super-generators and in the simplest situation their number is one: one speaks about N=1 SUSY and minimal super-symmetric extension of standard model (MSSM) in this case. These models are most studied because they are the simplest ones. They have however the strange property that the spinors generating SUSY are Majorana spinors- real in well-defined sense unlike Dirac spinors. This implies that fermion number is conserved only modulo two: this has not been observed experimentally. A second problem is that the proposed mechanisms for the breaking of SUSY do not look feasible.

LHC results suggest MSSM does not become visible at LHC energies. This does not exclude more complex scenarios hiding simplest N=1 to higher energies but the number of real believers is decreasing. Something is definitely wrong and one must be ready to consider more complex options or totally new view abot SUSY.

What is the situation in TGD? Here I must admit that I am still fighting to gain understanding of SUSY in TGD framework. That I can still imagine several scenarios shows that I have not yet completely understood the problem and am working hardly to avoid falling to the sin of sloppying myself. In the following I summarize the situation as it seems just now.

  1. In TGD framework N=1 SUSY is excluded since B and L and conserved separately and imbedding space spinors are not Majorana spinors. The possible analog of space-time SUSY should be a remnant of a much larger super-conformal symmetry in which the Clifford algebra generated by fermionic oscillator operators giving also rise to the Clifford algebra generated by the gamma matrices of the "world of classical worlds" (WCW) and assignable with string world sheets. This algebra is indeed part of infinite-D super-conformal algebra behind quantum TGD. One can construct explicitly the conserved super conformal charges accompanying ordinary charges and one obtains something analogous to N=∞ super algebra. This SUSY is however badly broken by electroweak interactions.
  2. The localization of induced spinors to string world sheets emerges from the condition that electromagnetic charge is well-defined for the modes of induced spinor fields. There is however an exception: covariantly constant right handed neutrino spinor νR: it can be de-localized along entire space-time surface. Right-handed neutrino has no couplings to electroweak fields. It couples however to the left handed neutrino by induced gamma matrices except when it is covariantly constant. Note that standard model does not predict νR but its existence is necessary if neutrinos develop Dirac mass. νR is indeed something which must be considered carefully in any generalization of standard model.

Could covariantly constant right-handed spinors generate exact N=2 SUSY? There are two spin directions for them meaning the analog N=2 Poincare SUSY. Could these spin directions correspond to right-handed neutrino and antineutrino. This SUSY would not look like Poincare SUSY for which anticommutator of super generators would be proportional to four-momentum. The problem is that four-momentum vanishes for covariantly constant spinors! Does this mean that the sparticles generated by covariantly constant νR are zero norm states and represent super gauge degrees of freedom? This might well be the case although I have considered also alternative scenarios.

Both imbedding space spinor harmonics and the modified Dirac equation have also right-handed neutrino spinor modes not constant in M4. If these are responsible for SUSY then SUSY is broken.

  1. Consider first the situation at space-time level. Both induced gamma matrices and their generalizations to modified gamma matrices defined as contractions of imbedding space gamma matrices with the canonical momentum currents for Kähler action are superpositions of M4 and CP2 parts. This gives rise to the mixing of right-handed and left-handed neutrinos. Note that non-covariantly constant right-handed neutrinos must be localized at string world sheets.

    This in turn leads neutrino massivation and SUSY breaking. Given particle would be accompanied by sparticles containing varying number of right-handed neutrinos and antineutrinos localized at partonic 2-surfaces.

  2. One an consider also the SUSY breaking at imbedding space level. The ground states of the representations of extended conformal algebras are constructed in terms of spinor harmonics of the imbedding space and form the addition of right handed neutrino with non-vanishing four-momentum would make sense. But the non-vanishing four-momentum means that the members of the super-multiplet cannot have same masses. This is one manner to state what SUSY breaking is.
  3. The simplest form of massivation would be that all members of the super- multiplet obey the same mass formula but that the p-adic length scales associated with them are different. This could allow very heavy sparticles. What fixes the p-adic mass scales of sparticles? If this scale is CP2 mass scale SUSY would be experimentally unreachable.
  4. One can even consider the possibility that SUSY breaking makes sparticles unstable against phase transition to their dark variants with heff =n× h. Sparticles could have same mass but be non-observable as dark matter not appearing in same vertices as ordinary matter! Geometrically the addition of right-handed neutrino to the state would induce many-sheeted covering in this case with right handed neutrino perhaps associated with different space-time sheet of the covering.

    This idea need not be so outlandish at it looks first. The generation of many.sheeted covering has interpretation in terms of breaking of conformal invariance. The sub-algebra for which conformal weights are n-tuples of integers becomes the algebra of conformal transformations and the remaining conformal generators do not represent gauge degrees of freedom anymore. They could however still represent conserved conformal charges.

    This generalization of conformal symmetry breaking gives rise to infinite number of fractal hierarchies formed by sub-algebras of conformal algebra and is also something new and a fruit of an attempt to avoid sloppy thinking. The breaking of conformal symmetry is indeed expected in massivation related to the SUSY breaking.

Have we been thinking sloppily also about super-conformal symmetries?

Super string models were once seen as the only possible candidate for the TOE. By looking at the proceedings of string theory conferences one sees that the age of super strings is over. Landscape problem and multiverse do not give much hopes about predictive theory and the only defence for super string models is as the only game in the town. Super string gurus do not know about competing scenarion but this is not a wonder given the fact that publishing of competing scenarios has been impossible since superstrings have indeed been the only game in the town! One of the very few almost-predictions of superstring theory was N=1 SUSY at LHC and it seems that it is already now excluded at LHC energies.

AdS/CFT correspondence is a mathematical outcome inspired by super-string models. One of the variants of its variants states that there is duality between conformal theory in M4 appearing as boundary of 5-D AdS and string theory in 10-D space AdS5× S5. A more general duality would be between conformal theory in Mn and 10-D space AdSn+1× S10-n-1. For n=2 the CFT would give conformal theory at 2-D Minkowski space for which conformal symmetries (actually their hypercomplex variant) form an infinite-D group. Duality has interpretation in terms of holography but the notion of holography is much more general than AdS/CFT.

AdS/CFT have been applied to nuclear physics but nothing sensational have been discovered. AdS/CFT have been tried also to explain the finding that what was expected to be QCD plasma behaves very differently. The first findings came from RHIC for heavy ion collisions and LHC has found that the strange effects appear already for proton heavy ion collisions. Essentially a deviation from QCD predictions is in question and in the regime where QCD should be a good description. AdS/CFT has not been a success. AdS/CFT is now applied also to condensed matter physics. At least hitherto no dramatic successes have been reported.

This leads to ask whether sloppy thinking should be blamed again. AdS/CFT is mathematically rather sound and well-tested but is the notion of conformal invariance behind it really the one that applies to real world physics?

  1. In TGD framework the ordinary conformal invariance is generalized so that it becomes 4-D one: of course, the ordinary finite-dimensional conformal group in M4 is not in question. The basic observation is that light-like 3-surfaces are metrically 2-dimensional and that this leads to a generalization of conformal transformations. One can locally express light-like 3-surfaces as X2× R and what happens is that the conformal transformations of X2 are localized with respect to the light-like coordinate of R. Light-like orbits of partonic 2-surfaces carrying elementary particle quantum numbers would have this extended conformal invariance.
  2. This is not all. In zero energy ontology (ZEO) the diamond like intersections of future and past directed light-cones - causal diamonds (CDs) are the basic objects. The space-time surfaces having 3-D ends at the boundaries of CD are the basic dynamical units. The boundaries of CD are pieces of δ M4+/-× CP2. The boundary δ M4+/- = S2×R+ is light-like 3-surface and thus allows a huge extension of conformal symmetries: with complex coordinate of S2 and light-like radial coordinate playing the roles of complex coordinate for ordinary conformal symmetry.

    Besides this there is a further analog of conformal symmetry. The symplectic transformations of δ M4+/-× CP2 can be regarded as symplectic transformations of S2× CP2 localized with respect to the light-like coordinate of R+ defining the analog of the complex coordinate z. In TGD Universe a gigantic extension of the conformal symmetry of superstring models experiences applies.

  3. Even these extended symmetries extend to a multi-local (loci correspond to partonic 2-surfaces at boundaries of CD) Yangian variant. Yangian symmetry is very closely related to quantum groups studied for decades but again without serious consideration of the question "Why quantum groups?". The hazy belief has been that they somehow emerge at Planck length scale, which itself is a hazy notion based solely on dimensional analysis and involving Planck constant and Newton's constant characterizing macroscopic gravitation.

    In TGD framework hyper-finite factors of type II1 emerge naturally at the level of WCW since fermionic Fock space provides a canonical representation for them and their inclusions provide an elegant description for finite measurement resolution: the included algebra generates states which are not experimentally distinguishable from the original state.

  4. Against this it is astonishing that AdS/CFT duality has very simple generalization in TGD framework and emerge from a generalization of General Coordinate Invariance (GCI) implying holography. Strong form of GCI postulates that either the space-like 3-surfaces at the ends of causal diamonds or the light-like orbits of partonic 2-surfaces can be taken as 3-surfaces defining the WCW: this is just gauge fixing for general coordinate invariance. If this is true then partonic 2-surfaces and their 4-D tangent space data at the boundaries of CD must code for physics. One would have strong form of holography. This might be too much to require: string world sheets carrying induced spinor fields are present and it might be that they cannot be reduced to data at partonic 2-surfaces.

    In any case, for this duality the 10-D space of AdS/CFT duality would be replaced with space-time surface. Mn would be replaced with the light-like parton orbits and/or space-like ends of CD. Surprisingly, this holography would be very much like holography in its original form!

See the chapter TGD and M-theory or the article What went wrong with symmetries?.

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