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Physics in Many-Sheeted Space-Time
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Could correlation functions, S-matrix, and coupling constant evolution be coded the statistical properties of preferred extremals?
Quantum classical correspondence states that all aspects of quantum states should have correlates in the geometry of preferred extremals. In particular, various elementary particle propagators should have a representation as properties of preferred extremals. This would allow to realize the old dream about being able to say something interesting about coupling constant evolution although it is not yet possible to calculate the M-matrices and U-matrix. Hitherto everything that has been said about coupling constant evolution has been rather speculative arguments except for the general vision that it reduces to a discrete evolution defined by p-adic length scales. General first principle definitions are much more valuable than ad hoc guesses even if the latter give rise to explicit formulas.
In quantum TGD and also at its QFT limit various correlation functions in given quantum state code for its properties. These correlation functions should have counterparts in the geometry of preferred extremals. Even more: these classical counterparts for a given preferred extremal ought to be identical with the quantum correlation functions for the superposition of preferred extremals.
In zero energy ontology (ZEO) lattices in the 3-D hyperbolic manifold defined by H3 (t2-x2-y2-z2=a2) (and known as hyperbolic space to distinguish it from other hyperbolic manifolds emerge naturally. The interpretation of H3 as a cosmic time=constant slice of space-time of sub-critical Robertson-Walker cosmology (giving future light-cone of M4 at the limit of vanishing mass density) is relevant now. ZEO leads to an argument stating that once the position of the "lower" tip of causal diamond (CD) is fixed and defined as origin, the position of the "upper" tip located at H3 is quantized so that it corresponds to a point of a lattice H3/G, where G is discrete subgroup of SL(2,C) (so called Kleinian group). There is evidence for the quantization of cosmic redshifts: a possible interpretation is in terms of hyperbolic lattice structures assignable to dark matter and energy. Quantum coherence in cosmological scales could be in question. This inspires several questions. How does the crystallography in H3 relate to the standard crystallography in Eucdlidian 3-space E3? Are there general results about tesselations H3? What about hyperbolic counterparts of quasicrystals? In this article standard facts are summarized and some of these questions are briefly discussed.
For details see the article Could hyperbolic 3-manifolds and hyperbolic lattices be relevant in zero energy ontology? or the chapter TGD and Cosmology.
The blackhole information paradox is often believed to have solution in terms of holography stating in the case of blackholes that blackhole horizon can serve as a holographic screen representing the information about the surrounding space as a hologram. The situation is however far from settled. The newest challenge is so called firewall paradox proposed by hrefhttp://http://arxiv.org/abs/1207.3123">Polchinsky et al. Lubos Motl has written several postings about firewall paradox and they inspired me to look the situation in TGD framework.
These paradoxes strengthen the overall impression that the blackhole physics indeed represent the limit at which GRT fails and the outcome is recycling of old arguments leading nowhere. Something very important is lacking. On the other hand, some authors like Susskind claim that the physics of this century more or less reduces to that for blackholes. I however see this endless tinkering with blackholes as a decline of physics. If super string had been a success as a physical theory, we would have got rid of blackholes.
If TGD is to replace GRT, it must also provide new insights to blackholes, blackhole evaporation, information paradox and firewall paradox. This inspired me to look for what blackholes and blackhole evaporation could mean in TGD framework and whether TGD can avoid the paradoxes. This kind of exercises allow also to sharpen the TGD based view about space-time and quantum and build connections to the mainstream views.
For more details see the chapter TGD and Astrophysics or the little article "Do blackholes and Hawking evaporation have TGD counterparts?".
New Scientist tells that spiral galaxies which seem to have tendency to be left handed along two lines which have angle of 85 degrees with respect to each other. Galaxies would be therefore like biomolecules which also have preferred handedness in living matter.
Handedness in geometric sense requires that the mirror image of the galaxy is not identical with galaxy itself. In good approximation galaxies are however rotationally symmetric around the spin axis. In dynamical sense handedness results if the total angular momentum of galaxy is non-vanishing. Spiral galaxies indeed have spin.
What has been observed that along these two lines of sight there are more left- than right handed galaxies. The length for the light of sight was 1.2 billion ly in the survey of Michael Longo and 3.4 billion ly in the survey of Lior Shamir. The scale of our large void is about .1 billion light years so that cosmic length scales are in question. The findings could of course be statistical flukes. Future surveys will resolve this issue.
The existence of a preferred axes of symmetry in cosmic scales does not fit well with isotropy and homogenuity assumptions of the standard cosmology. The TGD based proposal for the formation of galaxies and other astrophysical structures relies on a fractal network of string like objects defined by Kähler magnetic flux tubes. These magnetic flux tubes were present in primordial cosmology and had 1-D M4 projection at that time: they indeed defined string world sheets in M4. During the cosmic expansion the thickness of their M4 projections has increased gradually. These string like objects carry dark energy as magnetic energy and also the magnetic fields have become weaker during expansion. These flux tubes could also correspond to a gigantic value of Planck constant. Various astrophysical structures consisting of ordinary and dark matter would have formed via the decay of the magnetic energy of the flux tubes to ordinary and dark particles. The basic difference with respect to the inflationary scenario is that the energy of inflaton field is replaced with Kähler magnetic field and identified as dark energy.
Galaxies would be like pearls in a necklace. The dark matter and energy along the galactic necklaces causes a logarithmic 2-D gravitational potential producing constant velocity spectrum for distant stars. The basic prediction is that the galaxies can move freely along the flux tubes: this could explain the observed systematic motions in cosmic scale also challenging the basic assumptions of standard cosmology. Galaxies moving along different flux tubes can also collide if the flux tubes go near each other: this could be caused by their gravitational attraction already during the primordial period. One can imagine a cosmic highway network consisting of flux tubes intersecting at nodes and formed during the primordial period. Galaxies not obeying cosmic traffic rules could collide at crossings;-).
Since the necklace would have been much shorter during the primordial period, the proto galaxies possibly existing already at that time would have very near to each other and dynamically strongly coupled. Therefore the correlation of the directions of the angular momenta of proto galaxies - roughly in the direction of the long string like flux tube - could be a remnant from this time. This remnant manifesting itself as a definite handedness would be stabilized by the conservation of angular momentum after the decoupling of the galaxies from each other. The large value of Planck constant could also make possible quantum coherence in astrophysical scales for dark matter and energy and in this manner explain the correlations.
That there are two axes of this kind would suggest that our galaxy resides at junction of cosmic highways as a victim of cosmic traffic accident: that is in the node at which to cosmic necklaces touch. This is what I suggested in the earlier posting inspired by one particular finding challenging the assumption that galactic dark matter forms a spherical halo.
The finding was that near galactic center there is a distribution of satellite galaxies and star clusters, which rotate around Milky Way in a plane orthogonal to the plane of Milky Way. The observation could be interpreted by assuming that two orthogonal magnetic flux tubes (90 degrees is not far form 85 degrees) containing galaxies along them intersect at our galaxy. The newly found distribution of matter would correspond to a matter rotating around the flux tube - call it B - in the same way as the matter of our own galaxy rotates around the second flux tube - call it A. These flux tubes could correspond to the lines of sight found in the two surveys.
For background see the chapter Cosmic Strings.
I have done a considerable amount of speculative guesswork to identify what I have used to call preferred extremals of Kähler action. The problem is that the mathematical problem at hand is extremely non-linear and that there is no existing mathematical literature. One must proceed by trying to guess the general constraints on the preferred extremals which look physically and mathematically plausible. The hope is that this net of constraints could eventually chrystallize to Eureka! Certainly the recent speculative picture involves also wrong guesses. The need to find explicit ansatz for the deformations of known extremals based on some common principles has become pressing. The following considerations represent an attempt to combine the existing information to achieve this.
What might be the common features of the deformations of known extremals?
The dream is to discover the deformations of all known extremals by guessing what is common to all of them. One might hope that the following list summarizes at least some common features.
Effective three-dimensionality at the level of action
Could Einstein's equations emerge dynamically?
For jα satisfying one of the three conditions, the field equations have the same form as the equations for minimal surfaces except that the metric g is replaced with Maxwell energy momentum tensor T.
Are complex structure of CP2 and Hamilton-Jacobi structure of M4 respected by the deformations?
The complex structure of CP2 and Hamilton-Jacobi structure of M4 could be central for the understanding of the preferred extremal property algebraically.
Field equations as purely algebraic conditions
If the proposed picture is correct, field equations would reduce basically to purely algebraically conditions stating that the Maxwellian energy momentum tensor has no common index pairs with the second fundamental form. For the deformations of CP2 type vacuum extremals T is a complex tensor of type (1,1) and second fundamental form Hk a tensor of type (2,0) and (0,2) so that Tr(THk)= is true. This requires that second light-like coordinate of M4 is constant so that the M4 projection is 3-dimensional. For Minkowskian signature of the induced metric Hamilton-Jacobi structure replaces conformal structure. Here the dependence of CP2 coordinates on second light-like coordinate of M2(m) only plays a fundamental role. Note that now Tvv is non-vanishing (and light-like). This picture generalizes to the deformations of cosmic strings and even to the case of vacuum extremals.
For background see the chapter Basic Extremals of Kähler action. See also the article About deformations of known extremals of Kähler action.
I became again interested in finding preferred extremals of Kähler action, which would have 4-D CP2 and perhaps also M4 projections. This would correspond to Maxwell phase that I conjectured long time ago. Deformations of CP2 type vacuum extremals would correspond also to these extremals. The signature of the induced metric might be also Minkowskian. It however turns out that the solution ansatz requires Euclidian signature and that M4 projection is 3-D so that original hope is not realized.
I proceed by the following arguments to the ansatz.
A more realistic guess based on the attempt to construct deformations of CP2 type vacuum extremals is following.
For background see the chapter Basic Extremals of Kähler action.
R. Kiehn has proposed what he calls Topological Thermodynamics (TTD) as a new formulation of thermodynamics. The basic vision is that thermodynamical equations could be translated to differential geometric statements using the notions of differential forms and Pfaffian system. That TTD differs from TGD by a single letter is not enough to ask whether some relationship between them might exist. Quantum TGD can however in a well-defined sense be regarded as a square root of thermodynamics in zero energy ontology (ZEO) and this leads leads to ask seriously whether TTD might help to understand TGD at deeper level. The thermodynamical interpretation of space-time dynamics would obviously generalize black hole thermodynamics to TGD framework and already earlier some concrete proposals have been made in this direction.
One can raise several questions. Could the preferred extremals of Kähler action code for the square root of thermodynamics? Could induced Kähler gauge potential and Kähler form (essentially Maxwell field) have formal thermodynamic interpretation? The vacuum degeneracy of Kähler action implies 4-D spin glass degeneracy and strongly suggests the failure of strict determinism for the dynamics of Kähler action for non-vacuum extremals too. Could thermodynamical irreversibility and preferred arrow of time allow to characterize the notion of preferred extremal more sharply?
It indeed turns out that one can translate Kiehn's notions to TGD framework rather straightforwardly.
For background see the chapter Basic Extermals of K\"ahler action or the article Does thermodynamics have a representation at the level of space-time geometry?.
The standard view about dark matter is in grave difficulties.
The distribution of dark matter would be concentrated around this string rather than forming a spherical halo around galaxy. This would give rise to a gravitational acceleration behaving like 1/ρ, where ρ is transversal distance from the string, explaining constant velocity spectrum for distant stars. The killer prediction is that galaxies could move along the string direction freely. Large scale motions difficult to understand in standard cosmology has been indeed observed. It has been also known for a long time that galaxies tend to concentrate on linear structures.
The third blow against the theory comes from the observation that Milky Way has a distribution of satellite galaxies and star clusters, which rotate around the Milky Way in plane orthogonal to Milky Way's plane. One can visualize the situation in terms of two orthogonal planes such that the second plane contains Milky Way and second one the satellite galaxies and globular clusters. The Milky Way itself has size scale of .1 million light years whereas the newly discovered structure extends from about 33,000 light years to 1 million light years. The study is carried out by astronomers in Bonn University and will be published in journal Monthly Notices of the Royal Astronomical Society. The lead author is Ph. D. student Marcel Pawlowski.
According to the authors, it is not possible to understand the structure in terms of the standard model for dark matter. This model assumes that galactic dark matter forms a spherical halo around galaxy. The problem is the planarity of the newly discovered matter distribution. Not only satellite galaxies and star clusters but also the long streams of material left - stars and also gas - behind them as they orbit around Milky Way move in this plane. Planarity seems to be a basic aspect of the internal dynamics of the system. As a matter fact, quantum view about formation of also galaxies predicts planarity and this allows also to understand approximate planarity of solar system: common quantization axis of angular momentum defined by the direction of string like object in the recent case with a gigantic value of gravitational Planck constant defining the unit of angular momentum would provide a natural explanation for planarity.
The proposal of the researchers is that the situation is an outcome of a collision of two galaxies.
It is encouraging that the TGD based explanation for galactic dark matter survives all these three discoveries meaning grave difficulties for the halo model.
For background see the chapter Cosmic Strings.
Icarus collaboration has replicated the measurement of the neutrino velocity. The abstract summarizes the outcome.
The CERN-SPS accelerator has been briefly operated in a new, lower intensity neutrino mode with about 1012 p.o.t. /pulse and with a beam structure made of four LHC-like extractions, each with a narrow width of about 3 ns, separated by 524 ns. This very tightly bunched beam structure represents a substantial progress with respect to the ordinary operation of the CNGS beam, since it allows a very accurate time-of-flight measurement of neutrinos from CERN to LNGS on an event-to-event basis. The ICARUS T600 detector has collected 7 beam-associated events, consistent with the CNGS delivered neutrino flux of 2.2× 1016 p.o.t. and in agreement with the well known characteristics of neutrino events in the LAr-TPC. The time of flight difference between the speed of light and the arriving neutrino LAr-TPC events has been analyzed. The result is compatible with the simultaneous arrival of all events with equal speed, the one of light. This is in a striking difference with the reported result of OPERA that claimed that high energy neutrinos from CERN should arrive at LNGS about 60 ns earlier than expected from luminal speed.
The TGD based explanation for the anomaly would not have been super-luminality but the dependence of the maximal signal velocity on space-time sheet (see this): the geodesics in induced metric are not geodesics of the 8-D imbedding space. In principle the time taken to move from A (say CERN) to point B (say Gran Sasso) depends on space-time sheets involved. One of these space-time sheets would be that assignable to particle beam - a good guess is "massless extremal": along this the velocity is in in the simplest case (cylindrical "massless extremals") the maximal signal velocity in M4×CP2.
Other space-space-time sheets involved can be assigned to various systems such as Earth, Sun, Galaxy and they contribute to the effect (see this). It is important to understand how the physics of test particle depends on the presence of parallel space-times sheets. Simultaneous topological condensation to all the sheets is extremely probable so that at classical level forces are summed. Same happens at quantum level. The superposition of various fields assignable to parallel space-time sheets is not possible in TGD framework and is replaced with the superposition of their effects. This allows to resolve one of the strongest objections against the notion induced gauge field.
The outcome of ICARUS experiment is not able to kill this prediction since at this moment I am not able to fix the magnitude of the effect. It is really a pity that such a fantastic possibility to wake up the sleeping colleagues is lost. I feel like a parent in a nightmare seeing his child to drown and being unable to do anything.
There are other well-established effects in which the dependence of maximal signal velocity on space-time sheet is visible: one such effect is the observed slow increase of the time spend by light ray to propagate moon and back. The explanation is that the effect is not real but due to the change of the unit for velocity defined by the light-velocity assignable to the distant stars. The maximal signal velocity is for Robertson-Walker cosmology gradually increasing and the anomaly emerges as an apparent anomaly when one assumes that the natural coordinate system assignable to the solar system (Minkowski coordinates) is the natural coordinate system in cosmological scales. The size of the effect is predicted correctly. Since the cosmic signal velocity defining the unit increases, the local maximal signal velocity which is constant seems to be reducing and the measured distance to the Moon seems to be increasing.
For background see the chapter TGD and GRT of "Physics in Many-Sheeted Space-time".
New Scientist reported about the sad fate of the tachyonic explanation of neutrino superluminality. The argument is extremely simple.
What is amusing that this simple kinematic fact was not noticed from beginning. In any case, this finding kills all tachyonic models of neutrino super-luminality assuming energy conservation, and gives additional support for the TGD based explanation in terms of maximal signal velocity, which depends on pair of points of space-time sheet connected by signal and space-time sheet itself characterizing also particular kind of particle.
Even better, one can understand also the jitter in the spectrum of the arrival times which has width of about 50 ns in terms of an effect caused fluctuations in gravitational fields to the maximal signal velocity expressible in terms of the induced metric. The jitter could have interpretation in terms of gravitational waves inducing fluctuation of the maximal signal velocity c#, which in static approximation equals to c#=c(1+Φgr)1/2, where Φgr is gravitational potential.
Suprisingly, effectively super-luminal neutrinos would make possible gravitational wave detector! The gravitational waves would be however gravitational waves in TGD sense having fractal structure since they would correspond to bursts of gravitons resulting from the decays of large hbar gravitons emitted primarily rather than to a continuous flow (see this). The ordinary detection criteria very probably exclude this kind of bursts as noise. The measurements of Witte attempting to detect absolute motion indeed observed this kind of motion identifiable as a motion of Earth with respect to the rest frame of galaxy but superposed with fractal fluctuations proposed to have interpretation in terms of gravitational turbulence - gravitational waves.
For details see the earlier posting, the little article Could the measurements trying to detect absolute motion of Earth allow to test sub-manifold gravity? or the chapter TGD and GRT .
The basic objection against TGD is that induced metrics for space-time surfaces in M4× CP2 form an extremely limited set in the space of all space-time metrics appearing in the path integral formulation of General Relativity. Even special metrics like the metric of a rotating black hole fail to be imbeddable as an induced metric. For instance, one can argue that TGD cannot reproduce the post-Newtonian approximation to General Relativity since it involves linear superposition of gravitational fields of massive objects. As a matter fact, Holger B. Nielsen- one of the very few colleagues who has shown interest in my work - made this objection for at least two decades ago in some conference and I remember vividly the discussion in which I tried to defend TGD with my poor English.
The objection generalizes also to induced gauge fields expressible solely in terms of CP2 coordinates and their gradients. This argument is not so strong as one might think first since in standard model only classical electromagnetic field plays an important role.
The imbeddability constraint has actually highly desirable implications in cosmology. The enormously tight constraints from imbeddability imply that imbeddable Robertson-Walker cosmologies with infinite duration are sub-critical so that the most pressing problem of General Relativity disappears. Critical and over-critical cosmologies are unique apart from a parameter characterizing their duration and critical cosmology replaces both inflationary cosmology and cosmology characterized by accelerating expansion. In inflationary theories the situation is just the opposite of this: one ends up with fine tuning of inflaton potential in order to obtain recent day cosmology.
Despite these and many other nice implications of the induced field concept and of sub-manifold gravity the basic question remains. Is the imbeddability condition too strong physically? What about linear superposition of fields which is exact for Maxwell's electrodynamics in vacuum and a good approximation central also in gauge theories. Can one obtain linear superposition in some sense?
A more detailed formulation goes as follows.
The effective superposition of the CP2 parts of the induced metrics gives rise to an effective metric which is not in general imbeddable to M4× CP2. Therefore many-sheeted space-time makes possible a rather wide repertoire of 4-metrics realized as effective metrics as one might have expected and the basic objection can be circumvented In asymptotic regions where one can expect single sheetedness, only a rather narrow repertoire of "archetypal" field patterns of gauge fields and gravitational fields defined by topological field quanta is possible.
The skeptic can argue that this still need not make possible the imbedding of a rotating black hole metric as induced metric in any physically natural manner. This might be the case but need of course not be a catastrophe. We do not really know whether rotating blackhole metric is realized in Nature. I have indeed proposed that TGD predicts new physics new physics in rotating systems. Unfortunately, gravity probe B could not check whether this new physics is there since it was located at equator where the new effects vanish.
For background and more details see either the article Could the measurements trying to detect absolute motion of Earth allow to test sub-manifold gravity? or the chapter TGD and GRT.
Could the measurements trying to detect absolute motion of Earth allow to test sub-manifold gravity?
The history of the modern measurements of absolute motion has a long - more than century beginning from Michelson-Morley 1887. The reader can find in web a list of important publications giving an overall view about what has happened. The earliest measurements assumed aether hypothesis. Cahill identifies the velocity as a velocity with respect to some preferred rest frame and uses relativistic kinematics although he misleadingly uses the terms absolute velocity and aether. The preferred frame could galaxy, or the system defining rest system in cosmology. It would be easy to dismiss this kind of experiments as attempts to return to the days before Einstein but this is not the case. It might be possible to gain unexpected information by this kind of measurements. Already the analysis of CMB spectrum demonstrated that Earth is not at rest in the Robertson-Walker coordinate system used to analysis CMB data and similar motion with respect to galaxy is quite possible and might serve as a rich source of information also in GRT based theory.
In TGD framework the situation is especially interesting.
Also in Special Relativity the motion relative to the rest system of a larger system is a natural notion. In General Relativistic framework situation should be the same but the mathematical description of the situation is somewhat problematic since Minkowski coordinates are not global due to the loss of Poincare invariance as a global symmetry. In practice one must however introduce linear Minkowski coordinates and this makes sense only if one interprets the general relativistic space-time as a perturbation of Minkowski space. This background dependence is in conflict with general coordinate invariance. For sub-manifold gravity the situation is different.
Could the measurements performed already by Michelson-Morley and followers could provide support for the sub-manifold gravity? This might indeed be the case as the purpose of the following arguments demonstrate. The basic results of this analysis are following.
One must answer several questions before one can make predictions.
For background and more details see either the article Could the measurements trying to detect absolute motion of Earth allow to test sub-manifold gravity? or the chapter TGD and GRT.