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Physics in Many-Sheeted Space-Time

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Year 2015



Quantum fluctuations in geometry as a new kind of noise?

Bee told in rather critical tone about an article titled "Search for Space-Time Correlations from the Planck Scale with the Fermilab Holometer" reporting Fermilab experiment. The claim of Craig Hogan, who leads the experimental group, is that that the experiment is able to demonstrate the absence of quantum gravity effects. The claim is based on a dimensional estimate for transversal fluctuations of distances between mirrors reflecting light. The fluctuations of the distances between mirrors would be visible as a variation of interference pattern and the correlations of fluctuations between distant mirrors could be interpreted as correlations forced by gravitational holography. No correlations were detected and the brave conclusion was that predicted quantum gravitational effects are absent.

Although no quantitative theory for the effect exists, the effect is expected to be extremely small and non-detectable. Hogan has however different opinion based on his view about gravitational holography not shared by workers in the field (such as Lenny Susskind). Argument seems to go like follows (I am not a specialist so that there might be inaccuracies).

One has volume size R and the area of of its surface gives bound on entanglement entropy implying that fluctuations must be correlated. A very naive dimensional order of magnitude estimate would suggest that the transversal fluctuation of distance between mirrors (due to the fluctuations of space-time metric) would be given by ⟨ Δ x2 ⟩ ∼ (R/lP) ×lP2. For macroscopic R this could be measurable number. This estimate is of course ad hoc, involves very special view about holography, and also Planck length scale mysticism is involved. There is no theory behind it as Bee correctly emphasizes. Therefore the correct conclusion of the experiments would have been that the formula used is very probably wrong.

Why I saw the trouble of writing about this was that I want to try to understand what is involved and maybe make some progress in understanding TGD based holography to the GRT inspired holography.

  1. The argument of Hogan involves an assumption, which seems to be made routinely by quantum holographists: the 2-D surface involved with holography is outer boundary of macroscopic system and bulk corresponds to its interior. This would make the correlation effect large for large R if one takes seriously the dimensional estimate large for large R. The special role of outer boundaries is natural in AdS/CFT framework.
  2. In TGD framework outer boundaries do not have any special role. For strong form of holography (SH) the surfaces involved are string world sheets and partonic 2-surfaces serving as "genes" from which one can construct space-time surfaces as preferred extremals by using infinite number of conditions implying vanishing of classical Noether charges for sub-algebra of super-symplectic algebra.

    For weak form of holography one would have 3-surfaces defined by the light-like orbits or partonic 2-surfaces: at these 3-surfaces the signature of the induced metric changes from Minkowskian to Euclidian and they have partonic 2-surfaces as their ends at the light-like boundaries of causal diamonds (CDs). For SH one has at the boundary of CD fermionic strings and partonic 2-surfaces. Strings serve as geometric correlates for entanglement and SH suggests a map between geometric parameters - say string length - and information theoretic parameters such as entanglement entropy.

  3. The typical size of the partonic 2-surfaces is CP2 scale about 104 Planck lengths for the ordinary value of Planck constant. The naive scaling law for the the area of partonic 2-surfaces would be A ∝ heff2, heff=n×h. An alternative form of the scaling law would be as A ∝heff. CD size scale T would scale as heff and p-adic length scale as its square root ( diffused distance R satisfies R∼ Lp∝ T1/2 in diffusion; p-adic length scale would be analogous to R ).
  4. The most natural identification of entanglement entropy would be as entanglement entropy assignable with the union of partonic 2-surfaces for which the light-like 3-surface representing generalized Feynman diagram is connected. Entanglement would be between ends of strings beginning from different partonic 2-surfaces. There is no bound on the entanglement entropy associated with a given Minkowski 3-volume coming from the area of its outer boundary since interior can contain very large number of partonic 2-surfaces contributing to the area and thus entropy. As a consequence, the correlations between fluctuations are expected to be weak.

  5. Just for fun one can feed numbers into the proposed dimensional estimate, which of course does not make sense now. For R about of order CP2 size it would predict completely negligible effect for ordinary value of Planck constant: this entropy could be interpreted as entropy assignable to single partonic 2-surface. Same is true if R corresponds to Compton scale of elementary particle.
This argument should demonstrate how sensitive the quantitative estimates are for the detailed view about what holography really means. Loose enough definition of holography can produce endless number of non-sense formulas and it is quite possible that AdS/CFT modelled holography in GRT is completely wrong.

The difference between TGD based and GRT inspired holographies is forced by the new view about space-time allowing also Euclidian space-time regions and from new new view about General Coordinate Invariance implying SH. This brings in a natural identification of the 2-surfaces serving as holograms. In GRT framework these surfaces are identified in ad hoc manner as outer surfaces of arbtrarily chosen 3-volume.

For details see the article Quantum fluctuations in geometry as a new kind of noise? or the chapter More about TGD inspired cosmology.



About congruence subgroups

Stephen Crowley made a very interesting observation about Gaussian Mersennes in the comment section of the posting Pion of MG,79 hadron physics at LHC?. I glue the comment below.

Matti, why Low Gaussian primes? Your list of primes is a subset of the factors of the dimension of the friendly giant group.

The monster group was investigated in the 1970s by mathematicians Jean-Pierre Serre, Andrew Ogg and John G. Thompson; they studied the quotient of the hyperbolic plane by subgroups of SL2(R), particularly, the normalizer Γ0(p)+ of Γ0(p) in SL(2,R). They found that the Riemann surface resulting from taking the quotient of the hyperbolic plane by Γ0(p)+ has genus zero if and only if p is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71. When Ogg heard about the monster group later on, and noticed that these were precisely the prime factors of the size of Monster, he published a paper offering a bottle of Jack Daniel's whiskey to anyone who could explain this fact (Ogg (1974)).

I must first try to clarify to myself some definitions so that I have some idea about what I am talking about.

  1. Congruence group Γ0(p) is the kernel of the modulo homomorphism mapping SL(2,Z) to SL(2,Z/pZ) and thus consists of SL(2,Z) matrices which are are unit matrices modulo p. More general congruence subgroups SL(2,Z/nZ) are subgroups of SL(2,Z/pZ) for primes p dividing n. Congruence group can be regarded as subgroup of p-adic variant of SL(2,Z) with elements restricted to be finite as real integers. One can give up the finiteness in real sense by introducing p-adic topology so that one has SL(2,Zp). The points of hyperbolic plane at the orbits of the normalizer of Γ0(p)+ in SL(2,C) are identified.
  2. Normalizer Γ0(p)+ is the subgroup of SL(2,R) commuting with Γ0(p) but not with its individual elements. The quotient of hyperbolic space with the normalizer is sphere for primes k associated with Gaussian Mersennes up to k=47. The normalizer in SL(2,Zp) would also make sense and an interesting question is whether the result can be translated to p-adic context. Also the possible generalization to SL(2,C) is interesting.
First some comments inspired by the observation about Gaussian Mersennes by Stephen.
  1. Gaussian primes are really big but the primes defining them are logarithmically smaller. k=379 defines scale slightly large than that defined by the age of the Universe. Larger ones exist but are not terribly interesting for human physicists for a long time.

    Some primes k define Gaussian Mersenne as MG,k= (1+i)k-1 and the associated real prime defined by its norm is rather large - rather near to 2k and for k= 79 this is already quite big. k=113 characterises muon and nuclear physics, k=151,157,163,167 define a number theoretical miracle in the range cell membrane thickness- size of cell nucleus. Besides this there are astro-physically and cosmoplogically important Gaussian Mersennes (see the earlier posting).

  2. The Gaussian Mersennes below M89 correspond to k=2, 3, 5, 7, 11, 19, 29, 47, 73. Apart from k=73 this list is indeed contained by the list of the lowest monster primes k= 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71. The order d of Monster is product of powers of these primes: d= 246× 320× 59× 76× 112× 133× 17× 19× 23× 29× 31× 41× 47× 59× 71 .

    Amusingly, Monster contains subgroup with order, which is product of exactly those primes k associated with Gaussian Mersennes, which are definitely outside the reach of LHC! Should one call this subgroup Particle Physics Monster? Number theory and particle physics would meet each other! Or actually they would not!

    Speaking seriously, could this mean that the high energy physics above MG,79 energy is somehow different from that below in TGD Universe? Is k=47 somehow special: it correspond to energy scale 17.6× 103 TeV=17.6 PeV (P for Peta). Pessimistic would argue that this scale is the Monster energy scale never reached by human particle physicists.

The continuations of congruence groups and their normalizers to the p-adic variants SL(2,Zp) of SL(2,Z+iZ) (SL(2,C) are very interesting in TGD framework and are expected to appear in the adelization. Now hyperbolic plane is replaced with 3-D hyperbolic space H3 (mass shell for particle physicist and cosmic time constant section for cosmologist).
  1. One can construct hyperbolic manifolds as spaces of the orbits of discrete subgroups in 3-D hyperbolic space H3 if the discrete subgroup defines tesselation/lattice of H3. These lattices are of special interest as the discretizations of the H3 parametrizing the position for the second tip of causal diamond (CD) in zero energy ontology (ZEO), when the second tip is fixed. By number theoretic arguments this moduli space should be indeed discrete.
  2. In TGD inspired cosmology the positions of dark astrophysical objects could tend to be localized in hyperbolic lattice and visible matter could condense around dark matter. There are infinite number of different lattices assignable to the discrete subgroups of SL(2,C). Congruence subgroups and/or their normalizers might define p-adically natural tesselations. In ZEO this kind of lattices could be also associated with the light-like boundaries of CDs obtained as the limit of hyperbolic space defined by cosmic time constant hyperboloid as cosmic time approaches zero (moment of big bang). In biology there is evidence for coordinate grid like structures and I have proposed that they might consist of magnetic flux tubes carrying dark matter.

    Only a finite portion of the light-cone boundary would be included and modulo p arithmetics refined by using congruence subgroups Γ0(p) and their normalizers with the size scale of CD identified as secondary p-adic time scale could allow to describe this limitation mathematically. Γ(n) would correspond to a situation in which the CD has size scale given by n instead of prime: in this case, one would have multi-p p-padicity.

  3. In TGD framework one introduces entire hierarchy of algebraic extensions of rationals. Preferred p-adic primes correspond to so called ramified primes of the extension, and also p-adic length scale hypothesis can be understood and generalized if one accepts Negentropy Maximization Principle (NMP) and the notion of negentropic entanglement. Given extension of rationals induces an extension of p-adic numbers for each p, and one obtains extension of of ordinary adeles. Algebraic extension of rationals leads also an extension of SL(2,Z). Z can be replaced with any extension of rationals and has p-adic counterparts associated with p-adic integers of extensions of p-adic numbers. The notion of primeness generalizes and the congruence subgroups Γ0(p) generalize by replacing p with prime of extension.
Above I have talked only about algebraic extensions of rationals. p-Adic numbers have however also finite-dimensional algebraic extensions, which are not induced by those of rational numbers.
  1. The basic observation is that ep exists as power series p-adically as p-adic integer of norm 1 - ep cannot be regarded as a rational number. One can introduce also roots of ep and define in these manner algebraic extensions of p-adic numbers. For rational numbers the extension would be algebraically infinite-dimensional.

    In real number based Lie group theory e is in special role more or less by convention. In p-adic context the situation changes. p-adic variant of a given Lie group is obtained by exponentiation of elements of Lie algebra which are proportional to p (one obtains hierarchy of sub-Lie groups in powers of p) so that the Taylor series converges p-adically.

    These subgroups and algebraic groups generate more interesting p-adic variants of Lie groups: they would decompose into unions labelled by the elements of algebraic groups, which are multiplied by the p-adic variant of Lie group. The roots of e are mathematically extremely natural serving as hyperbolic counterparts for the roots of unity assignable to ordinary angles necessary if one wants to talk about the notion of angle and perform Fourier analysis in p-adic context: actually one can speak only about trigonometric functions of angles p-adically but not about angles. Same is true in hyperbolic sector.

  2. The extension of p-adics containg roots of e could even have application to cosmology! If the dark astrophysical objects tend to form hyperbolic lattices and visible matter tends to condensed around lattice points, cosmic redshifts tend to have quantized values. This tendency is observed. Also roots of ep could appear. The recently observed evidence for the oscillation of the cosmic scale parameter could be understood if one assumes this kind of dark matter lattice, which can oscillate. Roots of e2 appear in the model! (see the posting Does the rate of cosmic expansion oscillate?). Analogous explanation in terms of dark matter oscillations applies to the recently observed anomalous periodic variations of Newton's constant measured at the surface of Earth and of the length of day (Variation of Newton's constant and of length of day).
  3. Things can get even more complex! eΠ converges Π-adically for any generalized p-adic number field defined by a prime Π of an algebraic extension and one can introduce genuinely p-adic algebraic extensions by introducing roots eΠ/n! This raises interesting questions. How many real transcendentals can be represented in this manner? How well the hierarchy of adeles associated with extensions of rationals allowing also genuinely p-adic finite-dimensionals extensions of p-adics is able to approximate real number system? For instance, can one represent Π in this manner?
For details see the chapter More about TGD inspired cosmology.



Does the rate of cosmic expansion oscillate?

H. I. Ringermacher and L. R. Mead have written a very nice article with title "Observation of discrete oscillations in a model-independent plot of cosmological scale factor versus lookback time and scalar field model". In the article Does the rate of cosmic expansion oscillate? I summarize the contents of the article as I understand it. After that I consider TGD inspired model for the findings based on the assumption that dark matter corresponds to phase with gigantic values of effective Planck constant. Appendix contains summary about Gaussian Mersennes which predict correctly both cosmological, astrophysical, biological, nuclear physics, length scales and predict new important length scales in particle physics.

For details see the chapter More about TGD inspired cosmology or the article Does the rate of cosmic expansion oscillate?.



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.



Planck 2013 bounds for cosmic string tension

Planck 2013 gives bounds on the string tension of cosmic strings too. The bounds depend on the type of string considered: sone can consider Nambu-Goto strings, cosmic strings of gauge theories, string like objects of field theories, etc… The upper bounds for TG are in the range 10-6-10-7 .

One cannot of course directly compare these bounds to cosmic strings in TGD sense (not gauge theory strings but primordial 4-D string like objects). In TGD framework the string tension characterizes the density of Kähler magnetic energy of 4-D string like object with 2-D string world sheet as Minkowski space projection.

Cosmic string tension is inversely proportional to the square of CP2 length scale R and to Kähler coupling strength αK for which the most recent estimate is as equal to fine structure constant: αK≈ 1/137. The value of R is fixed by p-adic mass calculations from the conditions that electron mass comes out correctly. The velocity spectrum of distance stars in galaxy gives the same estimate if the gravitational field created by long cosmic string along which galaxies are located like pearls in string, gives an estimate consistent with this value. The estimate of cosmic string tension is TG= 6.9× 10-7 and is therefore in the interval 10-6-10-7 , where the upper bounds for other string tensions reside.

A comparison with string theory is in order. For Nambu-Goto strings the estimated upper bound for string tension is GT<1.5× 10-7 - not a good news since the Nambu-Goto string tension should satisfy GT=1 in the original approach. The same holds true also for superstrings in the original sense of the word. Therefore the situation is not very promising for superstrings. In fact, it turned out very difficult to find anything concrete about the string tension of superstrings. I however found from web a ten year old estimate estimate TG= 1/3000 for superstring tension involving experimental input. Presumably the Planck 2013 results would lower this estimate by few orders of magnitude.

For background see the chapter Cosmic Strings.



Further progress in the understanding of dark matter and energy in TGD framework

At Thinking Allowed Original (thanks for Ulla!) there was an extremely interesting link to a popular article about a possible explanation of dark matter in terms of vacuum polarization associated with gravitation. The model can make sense only if the sign of the gravitational energy of antimatter is opposite to that of matter and whether this is the case is not known. Since the inertial energies of matter and antimatter are positive, one might expect that this is the case also for gravitational energies by Equivalence Principle but one might also consider alternative and also I have done this in TGD framework.

The popular article lists four observations related to dark matter that neither cold dark matter (CMD) model nor modified gravitation model (MOND) can explain, and the claim is that the vacuum energy model is able to cope with them.

Consider first the TGD based model.

  1. The model assumes that galaxies are like pearls along strings defined by cosmic strings expended to flux tubes during cosmic expansion survives also these tests. This is true also in longer scales due to the fractality if TGD inspired cosmology: for instance, galaxy clusters would be organized in a similar manner.
  2. The dark magnetic energy of the string like object (flux tube) is identifiable as dark energy and the pearls would correspond to dark matter shells with a universal mass density of .8 kg/m2 estimated from Pioneer and Flyby anomalies assuming to be caused by spherical dark matter shells assignable to the orbits of planets. This value follows from the condition that the anomalous acceleration is identical with Hubble acceleration. Even Moon could be accompanied by this kind of shell: if so, the analog of Pioneer anomaly is predicted.
  3. The dark matter shell around galactic core could have decayed to smaller shells by heff reducing phase transition. This phase transition would have created smaller surfaces with smaller values of heff=hgr. One can consider also the possibility that it contains all the galactic matter as dark matter. There would be nothing inside the surface of the gigantic wormhole throat: this would conform with holography oriented thinking.
I checked the four observations listed in the popular article some of which CMD (cold dark matter) scenario and MOND fail to to explain. TGD explains all of them.
  1. It has been found that the effective surface mass density σ = ρ0R0/3 (volume density times volume of ball equals to effective surface density times surface area of the ball for constant volume density) of galactic core region containing possible halo is universal and its value is .9 kg/m2 (see the article). Pioneer and Flyby anomalies fix the surface density to .8 kg/m2. The difference is about 10 per cent! One must of course be cautious here: even the correct order of magnitude would be fine since Hubble acceleration parameter might be different for the cluster than for the solar system now.

    Note that in the article the effective surface density is defined in the article as σ= ρ0r0, where r0 is the radius of the region and ρ0 is the density at its center. The correct definition for a constant 3-D density inside ball is σ= ρ0r0/3 and I use this so that the value given in the article is scaled down by factor 1/3.

  2. The dark matter has been found to be inside core region within few hundred parsecs. This is just what TGD predicts since the velocity spectrum of distant stars is due to the gravitational field created by dark energy identifiable as magnetic energy of cosmic string like object - the thread containing galaxies as pearls.
  3. It has been observed that there is no dark matter halo in the galactic disk. Also this is an obvious prediction of TGD model.
  4. The separation of matter - now plasma clouds between galaxies - and dark matter in the collisions of galaxy clusters (observed for instance for bullet cluster consisting of two colliding clusters) is also explained qualitatively by TGD. The explanation is qualitatively similar to that in the CMD model of the phenomenon. Stars of galaxies are not affected except from gravitational slow-down much but the plasma phase interacts electromagnetically and is slowed down much more in the collision. The dominating dark matter component making itself visible by gravitational lensing separates from the plasma phase and this is indeed observed: the explanation in TGD framework would be that it is macroscopically quantum coherent (heff=hgr) and does not dissipate so that the thermodynamical description does not apply.

    In the case of galaxy clusters also the dark energy of cosmic strings is involved besides the galactic matter and this complicates the situation but the basic point is that dark matter component does not slow down as plasma phase does.

    CMD model has the problem that the velocity of dark matter bullet (smaller cluster of bullet cluster) is higher than predicted by CMD scenario. Attractive fifth force acting between dark matter particles becoming effective at short distances has been proposed as an explanation: intuitively this adds to the potential energy negative component so that kinetic energy is increased. I have proposed that gravitational constant might vary and be roughly twice the standard value: I do not believe this explanation now.

    The most feasible explanation is that the anomaly relates to the presence of thickened cosmic strings carrying dark energy as magnetic energy and dark matter shells instead of 3-D cold dark matter halos. This additional component would contribute to gravitational potential experienced by the smaller cluster and explain the higher velocity.

If I want to believe in something I have two options to choose. TGD indeed allows to understand dark matter and much more or Universe is playing cruel game with me by arranging all these numerical co-incidences.

For details see the chapter TGD and Astrophysics or the article Pioneer and Flyby anomalies for almost ten years later.



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