Tuesday, October 23, 2018

Cosmological constant in TGD and in superstring models

Cosmological constant Λ is one of the biggest problems of modern physics.

  1. Einstein proposed non-vanishing value of Λ in Einstein action as a volume term at his time in order to get what could be regarded as a static Universe. It turned out that Universe expanded and Einstein concluded that this proposal was the greatest blunder of his life. For two decades ago it was observed that the expansion of the Universe acclerates and the cosmological constant emerged again. Λ must be extremely small and have correct sign in order to give accelerating rather decelerating expansion in Robertson-Walker cooordinate. Here one must however notice that the time slicing used by Einstein was different and fr this slicing the Universe looked static.

  2. Λ can be however understood in an alternative sense as characterizing the dynamics in the matter sector. Λ could characterize the vacuum energy density of some scalar field, call it quintessense, proportional to 3- volume in quintessence scenario. This Λ would have sign opposite to that in the first scenario since it would appear at opposite side of Einstein's equations.

Remark: This is an updated version of an earlier posting, which represented a slightly wrong view about the interpretation and evolution of cosmological constant in TGD framework. This was due to my laziness to check the details of the earlier version, which is quite near to the version represented here.

Cosmological constant in string models and in TGD

It has turned out that Λ could be the final nail to the coffin of superstring theory.

  1. The most natural prediction of M-theory and superstring models is Λ in Einsteinian sense but with wrong sign and huge value: for instance, in AdS/CFT correspondence this would be the case. There has been however a complex argument suggesting that one could have a cosmological constant with a correct sign and even small enough size.

    This option however predicts landscape and a loss of predictivity, which has led to a total turn of the philosophical coat: the original joy about discovering the unique theory of everything has changed to that for the discovery that there are no laws of physics. Cynic would say that this is a lottery win for theoreticians since theory building reduces to mere artistic activity.

  2. Now however Cumrun Vafa - one of the leading superstring theorists - has proposed that the landscape actually does not exist at all (see this). Λ would have wrong sign in Einsteinian sense but the hope is that quintessence scenario might save the day. Λ should also decrease with time, which as such is not a catastrophe in quintessence scenario.

  3. Theorist D. Wrase et al has in turn published an article (see this) claiming that also the Vafa's quintessential scenario fails. It would not be consistent with Higgs Higgs mechanism. The conclusion suggesting itself is that according to the no-laws-of-physics vision something catastrophic has happened: string theory has made a prediction! Even worse, it is wrong.

    Remark: In TGD framework Higgs is present as a particle but p-adic thermodynamics rather than Higgs mechanism describes at least fermion massivation. The couplings of Higgs to fermions are naturally proportional their masses and fermionic part of Higgs mechanism is seen only as a manner to reproduce the masses at QFT limit.

  4. This has led to a new kind of string war: now inside superstring hegemony and dividing it into two camps. Optimistic outsider dares to hope that this leads to a kind of auto-biopsy and the gloomy period of superstring hegemony in theoretical physics lasted now for 34 years would be finally over.

String era need not be over even now! One could propose that both variants of Λ are present, are large, and compensate each other almost totally! First I took this as a mere nasty joke but I realized that TGD indeed suggests something analogous to this!

The picture in which Λeff parametrizes the total action as dimensionally reduced 6-D twistor lift of Kähler action could be indeed interpreted formally as sum of genuine cosmological term identified as volume action and Kähler action identified as an analog of quintessence. This picture is summarized below.

The picture emerging from the twistor lift of TGD

Consider first the picture emerging from the twistor lift of TGD.

  1. Twistor lift of TGD leads via the analog of dimensional reduction necessary for the induction of 8-D generalization of twistor structure in M4× CP2 to a 4-D action determining space-time surfaces as its preferred extremals. Space-time surface as a preferred extremal defines a unique section of the induced twistor bundle. The dimensionally reduced Kähler action is sum of two terms. Kähler action proportional to the inverse of Kähler coupling strength and volume term proportional to the cosmological constant Λ.

    Remark: The sign of the volume action is negative as the analog of the magnetic part of Maxwell action and opposite to the sign of the area action in string models.

    Kähler and volume actions should have opposite signs. At M4 limit Kähler action is proportional to E2-B2 In Minkowskian regions and to -E2-B2 in Euclidian regions.

  2. Twistor lift forces the introduction of also M4 Kähler form so that the twistor lift of Kähler action contains M4 contribution and gives in dimensional reduction rise to M4 contributions to 4-D Kähler action and volume term.

    It is of crucial importance that the Cartesian decomposition H=M4× CP2 allows the scale of M4 contribution to 6-D Kähler action to be different from CP2 contribution. The size of M4 contribution as compared to CP2 contribution must be very small from the smallness of CP breaking (see this and this.

    For canonically imbedded M4 the action density vanishes. For string like objects the electric part of this action dominates and corresponding contribution to 4-D Kähler action of flux tube extremals is positive unlike the standard contribution so that an almost cancellation of the action is in principle possible.

  3. What about energy? One must consider both Minkowskian and Euclidian space-time regions and be very careful with the signs. Assume that Minkowskian and Eucidian regions have same time orientation.

    1. Since a dimensionally reduced 6-D Kähler action is in question, the sign of energy density is positive Minkowskian space-time regions and of form (E2+B2)/2. Volume energy density proportional to Λ is positive.

    2. In Euclidian regions the sign of g00 is negative and energy density is of form (E2-B2)/2 and is negative when magnetic field dominates. For string like objects the M4 contribution to Kähler action however gives a contribution in which the electric part of Kähler action dominates so that M4 and CP2 contributions to energy have opposite signs. One can even consider the possibility that energies cancel in a good approximation and that the total energy is parameterized by effective cosmological constant Λeff.

The identification of the observed value of cosmological constant is not straightforward and I have considered several options without making explicit their differences even to myself. For Einsteinian option cosmological constant could correspond to the coefficient Λ of the volume term in analogy with Einstein's action. For what I call quintessense option cosmological constant Λeff would approximately parameterize the total action density or energy density.

  1. Cosmological constant - irrespective of whether it is identified as Λ or Λeff - is extremely small in the recent cosmology. The natural looking assumption would be that as a coupling parameter Λ or Λeff depends on p-adic length scale like 1/Lp2 and therefore decreases in average sense as 1/a2, where a is cosmic time identified as light-cone proper time assignable to either tip of CD. This suggests the following rough vision.

    The increase of the thickness of magnetic flux tubes carrying monopole flux liberates energy and this energy can make possible increase of the volume so that one obtains cosmic expansion. As the space-time surface expands, its cosmological constant is eventually reduced in a phase transition changing the p-adic length scale. This phase transition liberates volume energy and leads to an accelerated expansion. The space-time surface would expand by jerks in stepwise manner. This process is analogous to breathing. This process would replace continuous cosmic expansion of GRT. One application is TGD variant of Expanding Earth model explaining Cambrian Explosion, which is really weird event (see this).

    One can however raise a serious objection: since the volume term is part of 6-D Kähler action, the length scale evolution of Λ should be dictated by that for 1/αK and be very slow: therefore cosmological constant identified as Einsteinian Λ seems to be excluded.

  2. This leaves only Λeff option. Λeff would parameterize the value of the total action or energy of the space-time surface. Λeff would be analogous to the sum of Einsteinian and quintessential cosmological constants.

    The gradual reduction of Λeff could be interpreted in terms of the reduction of the total action or energy or both. The reduction of the total action would be by the cancellation of M4 and CP2 parts of Kähler action for string like objects. The reduction of the total energy would be by the cancellation of the contribution of Minkowskian and Euclidian regions. The p-adic length scale evolution of Λ would be slow and induced by that of αK.

This picture still leaves the question whether one should assign Λeff to action or energy or both in which case the assignments should be equivalent and action and energy should be proportional to each other. This identification is however the most detailed one developed hitherto.

Second manner to increase 3-volume

Besides the increase of 3-volume of M4 projection, there is also a second manner to increase volume energy: many-sheetedness. The negative sign of Λ could in fact force many-sheetedness.

  1. Superconductors of type II see this) provide a helpful analogy. In superconductors of type II Meissner effect is not complete unlike for those of type. Below critical value Hc,1 external magnetic field penetrates as flux quanta, which can be cylindrical flux tube and also form complex effectively 2-D thin 3-surfaces maximizing their area near critical field Hc,1 for which magnetic field penetrates the entire superconductor. The reason is that the surface energy for the boundary of non-super-conducting and super-conducting phase is negative for superconductors of type II so that the area in question is maximized. Note that near criticality also volume energy is minimized.

  2. In TGD the negative volume energy associated with Λ is analogous to the surface energy in superconductors of type II. The thin 3-surfaces in superconductors could have similar 3-surface analogs in TGD since their volume is proportional to surface area - note that TGD Universe can be said to be quantum critical.

    This is not the only possibility. The sheets of many-sheeted space-time having overlapping M4 projections provide second mechanism. The emergence of many-sheetedness could also be caused by the increase of n=heff/h0 as a number of sheets of Galois covering.

  3. Could the 3-volume increase during deterministic classical time evolution? If the minimal surface property assumed for the preferred extremals as a realization of quantum criticality is true everywhere, the conservation of volume energy prevents the increase of the volume. Minimal surface property is however assumed to fail at discrete set of points due to the transfer of conserved charged between Kähler and volume degrees of freedom. Could this make possible the increase of volume during classical time evolution so that volume and Kähler energy could increase?

  4. ZEO allows the increase of average 3-volume by quantum jumps. There is no reason why each "big" state function reduction changing the roles of the light-like boundaries of CD could not decrease the average volume energy of space-time surface for the time evolutions in the superposition. This can occur in all scales, and could be achieved also by the increase of heff/h0=n.

  5. The geometry of CD suggests strongly an analogy with Big Bang followed by Big Crunch. The increase of the volume as increase of the volume of M4 projection does not however seem to be consistent with Big Crunch. One must be very cautious here. The point is that the size of CD itself increases during the sequence of small state function reductions leaving the members of state pairs at passive boundary of CD unaffected. The size of 3-surface at the active boundary of CD therefore increases as also its 3-volume.

    The increase of the volume during the Big Crunch period could be also due to the emergence of the many-sheetedness, in particular due to the increase of the value of n for space-time sheets for sub-CDs. In this case, this period could be seen as a transition to quantum criticality accompanied by an emergence of complexity.

  6. In type II superconductivity magnetic energy and negative surface energy for flux quanta compete. Now one has Kähler magnetic energy and negative volume energy. By energy conservation they compete. Could this analog be helpful in TGD? Could the penetration of Kähler magnetic flux tubes to a system give rise to generation of space-time sheets and perhaps increase of n in order to reduce total energy?

See the article TGD View about Coupling Constant Evolution or the chapter with the same title.

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

Articles and other material related to TGD.


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