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Physics in Many-Sheeted Space-Time
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The starting point of TGD was the energy problem of General Relativity (see this). The solution of the problem was proposed in terms of sub-manifold gravity and based on the lifting of the isometries of space-time surface to those of M4× CP2 in which space-times are realized as 4-surfaces so that Poincare transformations act on space-time surface as an 4-D analog of rigid body rather than moving points at space-time surface. It however turned out that the situation is not at all so simple.
There are several conceptual hurdles and I have considered several solutions for them. The basic source of problems has been Equivalence Principle (EP): what does EP mean in TGD framework (see this and this)? A related problem has been the interpretation of gravitational and inertial masses, or more generally the corresponding 4-momenta. In General Relativity based cosmology gravitational mass is not conserved and this seems to be in conflict with the conservation of Noether charges. The resolution is in terms of zero energy ontology (ZEO), which however forces to modify slightly the original view about the action of Poincare transformations.
A further problem has been quantum classical correspondence (QCC): are quantal four-momenta associated with super conformal representations and classical four-momenta associated as Noether charges with Kähler action for preferred extremals identical? Could inertial-gravitational duality - that is EP - be actually equivalent with QCC? Or are EP and QCC independent dualities. A powerful experimental input comes p-adic mass calculations (see this) giving excellent predictions provided the number of tensor factors of super-Virasoro representations is five, and this input together with Occam's razor strongly favors QCC=EP identification.
Twistor Grassmannian approach has meant a technical revolution in quantum field theory (for attempts to understand and generalize the approach in TGD framework (see this and this). This approach seems to be extremely well suited to TGD and I have considered a generalization of this approach from N=4 SUSY to TGD framework by replacing point like particles with string world sheets in TGD sense and super-conformal algebra with its TGD version: the fundamental objects are now massless fermions which can be regarded as on mass shell particles also in internal lines (but with unphysical helicity). The approach solves old problems related to the realization of stringy amplitudes in TGD framework, and avoids some problems of twistorial QFT (IR divergences and the problems due to non-planar diagrams). The Yangian variant of 4-D conformal symmetry is crucial for the approach in N=4 SUSY, and implies the recently introduced notion of amplituhedon (see this). A Yangian generalization of various super-conformal algebras seems more or less a "must" in TGD framework. As a consequence, four-momentum is expected to have characteristic multilocal contributions identifiable as multipart on contributions now and possibly relevant for the understanding of bound states such as hadrons.
1. Scale dependent notion of four-momentum in zero energy ontology
Quite generally, General Relativity does not allow to identify four-momentum as Noether charges but in GRT based cosmology one can speak of non-conserved mass cosmo, which seems to be in conflict with the conservation of four-momentum in TGD framework. The solution of the problem comes in terms of zero energy ontology (ZEO) (see this and this), which transforms four-momentum to a scale dependent notion: to each causal diamond (CD) one can assign four-momentum assigned with say positive energy part of the quantum state defined as a quantum superposition of 4-surfaces inside CD.
ZEO is necessary also for the fusion of real and various p-adic physics to single coherent whole. ZEO also allows maximal "free will" in quantum jump since every zero energy state can be created from vacuum and at the same time allows consistency with the conservation laws. ZEO has rather dramatic implications: in particular the arrow of thermodynamical time is predicted to vary so that second law must be generalized. This has especially important implications in living matter, where this kind of variation is observed.
More precisely, this superposition corresponds to a spinor field in the "world of classical worlds" (WCW) (see this): its components - WCW spinors - correspond to elements of fermionic Fock basis for a given 4-surface - or by holography implied by general coordinate invariance (GCI) - for 3-surface having components at both ends of CD. Strong form of GGI implies strong form of holography (SH) so that partonic 2-surfaces at the ends of space-time surface plus their 4-D tangent space data are enough to fix the quantum state. The classical dynamics in the interior is necessary for the translation of the outcomes of quantum measurements to the language of physics based on classical fields, which in turn is reduced to sub-manifold geometry in the extension of the geometrization program of physics provided by TGD.
Holography is very much reminiscent of QCC suggesting trinity: GCI-holography-QCC. Strong form of holography has strongly stringy flavor: string world sheets connecting the wormhole throats appearing as basic building bricks of particles emerge from the dynamics of induced spinor fields if one requires that the fermionic mode carries well-defined electromagnetic charge (see this).
2. Are the classical and quantal four-momenta identical?
One key question concerns the classical and quantum counterparts of four-momentum. In TGD framework classical theory is an exact part of quantum theory. Classical four-momentum corresponds to Noether charge for preferred extremals of Kähler action. Quantal four-momentum in turn is assigned with the quantum superposition of space-time sheets assigned with CD - actually WCW spinor field analogous to ordinary spinor field carrying fermionic degrees of freedom as analogs of spin. Quantal four-momentum emerges just as it does in super string models - that is as a parameter associated with the representations of super-conformal algebras. The precise action of translations in the representation remains poorly specified. Note that quantal four-momentum does not emerge as Noether charge: at at least it is not at all obvious that this could be the case.
Are these classical and quantal four-momenta identical as QCC would suggest? If so, the Noether four-momentum should be same for all space-time surfaces in the superposition. QCC suggests that also the classical correlation functions for various general coordinate invariant local quantities are same as corresponding quantal correlation functions and thus same for all 4-surfaces in quantum superposition - this at least in the measurement resolution used. This would be an extremely powerful constraint on the quantum states and to a high extend could determined the U-, M-, and S-matrices.
QCC seems to be more or less equivalent with SH stating that in some respects the descriptions based on classical physics defined by Kähler action in the interior of space-time surface and the quantal description in terms of quantum states assignable to the intersections of space-like 3-surfaces at the boundaries of CD and light-like 3-surfaces at which the signature of induced metric changes. SH means effective 2-dimensionality since the four-dimensional tangent space data at partonic 2-surfaces matters. SH could be interpreted as Kac-Mody and symplectic symmetries meaning that apart from central extension they act almost like gauge symmetries in the interiors of space-like 3-surfaces at the ends of CD and in the interiors of light-like 3-surfaces representing orbits of partonic 2-surfaces. Gauge conditions are replaced with Super Virasoro conditions. The word "almost" is of course extremely important.
3. What does Equivalence Principle (EP) mean in TGD?
EP states the equivalence of gravitational and inertial masses in Newtonian theory. A possible generalization would be equivalence of gravitational and inertial four-momenta. In GRT this correspondence cannot be realized in mathematically rigorous manner since these notions are poorly defined and EP reduces to a purely local statement in terms of Einstein's equations. What about TGD? What could EP mean in TGD framework?
Which of the options looks more plausible? The success of p-adic mass calculations (see this) have motivated theuse of them as a guideline in attempts to understand TGD. The basic outcome was that elementary particle spectrum can be understood if Super Virasoro algebra has five tensor factors. Can one decide the fate of the two approaches to EP using this number as an input?
For this option however the GRT inspired interpretation of Equivalence Principle at space-time level remains to be understood. Is it needed at all? The condition that the energy momentum tensor of Kähler action has a vanishing divergence leads in General Relativity to Einstein equations with cosmological term. In TGD framework preferred extremals satisfying the analogs of Einstein's equations with several cosmological constant like parameters can be considered.
Should one give up this idea, which indeed might be wrong? Could the divergence of of energy momentum tensor vanish only asymptotically as was the original proposal? Or should one try to generalize the interpretation? QCC states that quantum physics has classical correlate at space-time level and implies EP. Could also quantum classical correspondence itself have a correlate at space-time level. If so, space-time surface would able to represent abstractions as statements about statements about.... as the many-sheeted structure and the vision about TGD physics as analog of Turing machine able to mimic any other Turing machine suggest.g machine suggests.
4. How translations are represented at the level of WCW?
The four-momentum components appearing in the formulas of super conformal generators correspond to infinitesimal translations. In TGD framework one must be able to identify these infinitesimal translations precisely. As a matter of fact, finite measurement resolution implies that it is probably too much to assume infinitesimal translations. Rather, finite exponentials of translation generators are involved and translations are discretized. This does not have practical signficance since for optimal resolution the discretization step is about CP2 length scale.
Where and how do these translations act at the level of WCW? ZEO provides a possible answer to this question.
1. Discrete Lorentz transformations and time translations act in the space of CDs: inertial four-momentum
Quantum state corresponds also to wave function in moduli space of CDs. The moduli space is obtained from given CD by making all boosts for its non-fixed boundary: boosts correspond to a discrete subgroup of Lorentz group and define a lattice-like structure at the hyperboloid for which proper time distance from the second tip of CD is fixed to Tn=n× T(CP2). The quantization of cosmic redshift for which there is evidence, could relate to this lattice generalizing ordinary 3-D lattices from Euclidian to hyperbolic space by replacing translations with boosts (velocities).
The additional degree of freedom comes from the fact that the integer n>0 obtains all positive values. One has wave functions in the moduli space defined as a pile of these lattices defined at the hyperboloid with constant value of T(CP2): one can say that the points of this pile of lattices correspond to Lorentz boosts and scalings of CDs defining sub-WCW:s.
The interpretation in terms of group which is product of the group of shifts Tn(CP2)→ Tn+m(CP2) and discrete Lorentz boosts is natural. This group has same Cartesian product structure as Galilean group of Newtonian mechanics. This would give a discrete rest energy and by Lorentz boosts discrete set of four-momenta giving a contribution to the four-momentum appearing in the super-conformal representation.
What is important that each state function reduction would mean localisation of either boundary of CD (that is its tip). This localization is analogous to the localization of particle in position measurement in E3 but now discrete Lorentz boosts and discrete translations Tn-->Tn+m replace translations. Since the second end of CD is necessary del-ocalized in moduli space, one has kind of flip-flop: localization at second end implies de-localization at the second end. Could the localization of the second end (tip) of CD in moduli space correspond to our experience that momentum and position can be measured simultaneously? This apparent classicality would be an illusion made possible by ZEO.
The flip-flop character of state function reduction process implies also the alternation of the direction of the thermodynamical time: the asymmetry between the two ends of CDs would induce the quantum arrow of time. This picture also allows to understand what the experience growth of geometric time means in terms of CDs.
2. The action of translations at space-time sheets
The action of imbedding space translations on space-time surfaces possibly becoming trivial at partonic 2-surfaces or reducing to action at δ CD induces action on space-time sheet which becomes ordinary translation far enough from end end of space-time surface. The four-momentum in question is very naturally that associated with Kähler action and would therefore correspond to inertial momentum for PI,class=Pquant,gr option. Indeed, one cannot assign quantal four-momentum to Kähler action as an operator since canonical quantization badly fails. In finite measurement infinitesimal translations are replaced with their exponentials for PI,class=Pquant,gr option.
What looks like a problem is that ordinary translations in the general case lead out from given CD near its boundaries. In the interior one expects that the translation acts like ordinary translation. The Lie-algebra structure of Poincare algebra including sums of translation generators with positive coefficient for time translation is preserved if only timelike superpositions if generators are allowed also the commutators of time-like translation generators with boost generators give time like translations. This defines a Lie-algebraic formulation for the arrow of geometric time. The action of time translation on preferred etxremal would be ordinary translation plus continuation of the translated preferred extremal backwards in time to the boundary of CD. The transversal space-like translations could be made Kac-Moody algebra by multiplying them with functions which vanish at δ CD.
A possible interpretation would be that Pquant,gr corresponds to the momentum assignable to the moduli degrees of freedom and Pcl,I to that assignable to the time like translations. Pquant,gr=Pcl,I would code for QCC. Geometrically quantum classical correspondence would state that timelike translation shift both the interior of space-time surface and second boundary of CD to the geometric future/past while keeping the second boundary of space-time surface and CD fixed.
5. Yangian and four-momentum
Yangian symmetry implies the marvellous results of twistor Grassmannian approach to N=4 SUSY culminating in the notion of amplituhedron which promises to give a nice projective geometry interpretation for the scattering amplitudes (see this). Yangian symmetry is a multilocal generalization of ordinary symmetry based on the notion of co-product and implies that Lie algebra generates receive also multilocal contributions. I have discussed these topics from slightly different point of view (see this).
1. Yangian symmetry
The notion equivalent to that of Yangian was originally introduced by Faddeev and his group in the study of integrable systems. Yangians are Hopf algebras which can be assigned with Lie algebras as the deformations of their universal enveloping algebras. The elegant but rather cryptic looking definition is in terms of the modification of the relations for generating elements (see this). Besides ordinary product in the enveloping algebra there is co-product Δ which maps the elements of the enveloping algebra to its tensor product with itself. One can visualize product and co-product is in terms of particle reactions. Particle annihilation is analogous to annihilation of two particle so single one and co-product is analogous to the decay of particle to two. Δ allows to construct higher generators of the algebra.
Lie-algebra can mean here ordinary finite-dimensional simple Lie algebra, Kac-Moody algebra or Virasoro algebra. In the case of SUSY it means conformal algebra of M4- or rather its super counterpart. Witten, Nappi and Dolan have described the notion of Yangian for super-conformal algebra in very elegant and and concrete manner in the article kenoem Yangian Symmetry in D=4 superconformal Yang-Mills theory (see this). Also Yangians for gauge groups are discussed.
In the general case Yangian resembles Kac-Moody algebra with discrete index n replaced with a continuous one. Discrete index poses conditions on the Lie group and its representation (adjoint representation in the case of N=4 SUSY). One of the conditions conditions is that the tensor product R⊗ R* for representations involved contains adjoint representation only once. This condition is non-trivial. For SU(n) these conditions are satisfied for any representation. In the case of SU(2) the basic branching rule for the tensor product of representations implies that the condition is satisfied for the product of any representations.
Yangian algebra with a discrete basis is in many respects analogous to Kac-Moody algebra. Now however the generators are labelled by non-negative integers labeling the light-like incoming and outgoing momenta of scattering amplitude whereas in in the case of Kac-Moody algebra also negative values are allowed. Note that only the generators with non-negative conformal weight appear in the construction of states of Kac-Moody and Virasoro representations so that the extension to Yangian makes sense.
The generating elements are labelled by the generators of ordinary conformal transformations acting in M4 and their duals acting in momentum space. These two sets of elements can be labelled by conformal weights n=0 and n=1 and and their mutual commutation relations are same as for Kac-Moody algebra. The commutators of n=1 generators with themselves are however something different for a non-vanishing deformation parameter h. Serre's relations characterize the difference and involve the deformation parameter h. Under repeated commutations the generating elements generate infinite-dimensional symmetric algebra, the Yangian. For h=0 one obtains just one half of the Virasoro algebra or Kac-Moody algebra. The generators with n>0 are n+1-local in the sense that they involve n+1-forms of local generators assignable to the ordered set of incoming particles of the scattering amplitude. This non-locality generalizes the notion of local symmetry and is claimed to be powerful enough to fix the scattering amplitudes completely.
2. How to generalize Yangian symmetry in TGD framework?
As far as concrete calculations are considered, it is not much to say. It is however possible to keep discussion at general level and still say something interesting (as I hope!). The key question is whether it could be possible to generalize the proposed Yangian symmetry and geometric picture behind it to TGD framework.
What does this then mean concretely (if this word can be used in this kind of context)?
3.Could Yangian symmetry provide a new view about conserved quantum numbers?
The Yangian algebra has some properties which suggest a new kind of description for bound states. The Cartan algebra generators of n=0 and n=1 levels of Yangian algebra commute. Since the co-product Δ maps n=0 generators to n=1 generators and these in turn to generators with high value of n, it seems that they commute also with n≥ 1 generators. This applies to four-momentum, color isospin and color hyper charge, and also to the Virasoro generator L0 acting on Kac-Moody algebra of isometries and defining mass squared operator.
Could one identify total four momentum and Cartan algebra quantum numbers as sum of contributions from various levels? If so, the four momentum and mass squared would involve besides the local term assignable to wormhole throats also n-local contributions. The interpretation in terms of n-parton bound states would be extremely attractive. n-local contribution would involve interaction energy. For instance, string like object would correspond to n=1 level and give n=2-local contribution to the momentum. For baryonic valence quarks one would have 3-local contribution corresponding to n=2 level. The Yangian view about quantum numbers could give a rigorous formulation for the idea that massive particles are bound states of massless particles.
For details see the article About the Notion of Four-momentum in TGD Framework. For background see the chapter TGD and GRT.
Whether TGD is consistent with Einstein's equations or not has been a continual source of worry and my interpretations have fluctuated. The working with a quite different problem forced however to ask the basic questions again and led to challenge the assumption inspired by the construction of preferred extremals that the local conservation of energy momentum tensor of Kähler action always forces Einstein-Maxwell action with cosmological term. As a matter, fact much milder conditions are enough to achieve this and imply the existence of several cosmological "constants" (not constants actually) multiplying mutually orthogonal projectors, whose sum replaces the ordinary cosmological constant term. This view unifies the two quite different interpretations of cosmological constant.
The first - "Einsteinian" - interpretation is as a fundamental constant not characterizing real energy density. According to the second interpretation cosmological constant characterizes genuine contributions to energy assigned to inflaton type fields. In the recent case these contributions would correspond to topologically condensed matter such as K\"ahler magnetic flux tubes and particles. As a matter fact, this was the original interpretation, which I challenged after the discovery of preferred extremals. I glue below the abstract of the new chapter What are the counterparts of Einstein's equations in TGD?.
The original motivation of this work was related to Platonic solids. The playing with Einstein's equations and the attempts to interpret them physically forced the return to an old interpretational problem of TGD. TGD allows enormous vacuum degeneracy for Kähler action but the vacuum extremals are not gravitational vacua. Could this mean that TGD forces to modify Einstein's equations? Could space-time surfaces carrying energy and momentum in GRT frameword be vacua in TGD context? Of course, also in GRT context cosmological constant means just this and an experimental fact, is that cosmological constant is non-vanishing albeit extremely small.
Trying to understand what is involved led to the realization that the hypothesis that preferred extremals correspond to the solutions of Einstein-Maxwell equations with cosmological constant is too restricted in the case of vacuum extremals and also in the case of standard cosmologies imbedded as vacuum extremals. What one must achieve is the vanishing of the divergence of energy momentum tensor of Kähler action expressing the local conservation of energy momentum currents. The most general analog of Einstein's equations and Equivalence Principle would be just this condition giving in GRT framework rise to the Einstein-Maxwell equations with cosmological constant.
One can however wonder whether it could be possible to find some general ansätze allowing to satisfy this condition. This kind of ansätze can be indeed found and can be written as kG+∑ΛiPi=T, where Λi are cosmological "constants" and Pi are mutually orthogonal projectors such that each projector contribution has a vanishing divergence. One can interpret the projector contribution in terms of topologically condensed matter, whose energy momentum tensor the projectors code in the representation kG=-∑ΛiPi+T. Therefore Einstein's equations with cosmological constant are generalized. This generalization is not possible in General Relativity, where Einstein's equations follow from a variational principle.
The suggested quaternionic preferred extremals and the preferred extremals involving Hamilton-Jacobi structure might allow identification as different families characterized by the little group of particles involved and assignable to time-like/light-like local direction. One should prove that this ansatz works also for all vacuum extremals. If not, the local conservation of K¨hler energy momentum tensor would be the general formulation for the counterpart of Equivalence Principle in TGD framework. This progress - if it really is progress - provides a more refined view about how TGD Universe differs from the Universe according to General Relativity and leads also to a model for how the cosmic honeycomb structure with basic unit cells having size scale 108 ly could be modelled in TGD framework.
For details see the article What are the counterparts of Einstein's equations in TGD? or the new chapter with the same title.
The generation of matter-antimatter asymmetry is still poorly understood. There exists a multitude of models but no convincing one. In TGD framework the generation of matter-antimatter asymmetry can be explained in terms of cosmic strings carrying dark energy identified as Kähler magnetic energy (see this). Their decay to ordinary and dark matter would be the analog for the decay of the inflaton field to matter and the asymmetry would be generated in this process. The details of the process have not been considered hitherto.
The stimulus for constructing a general model for this process came from attempt to understand the notion of sphaleron claimed to allow a non-perturbative description for a separate non-conservation of baryon and lepton numbers in standard model. The separate non-conservation of B and L would make possible models of baryo-genesis and even lepto-genesis assuming that in the primordial situation only right-handed inert neutrinos are present. To my opinion these models however fail mathematically because they equate the non-conservation of axial fermion numbers - which is on a mathematically sound basis - with the non-conservation of fermion numbers. This kind of assumption is unjustified and to my opinion is misuse of the attribute "non-perturbative".
The basic vision about lepto-genesis followed by baryo-genesis is however very attractive. This even more so because right-handed neutrino is in a completely unique role in TGD Universe. The obvious question therefore is whether this vision could make sense also in TGD framework. It would be wonderful if cosmic strings - infinitely thin Kähler magnetic flux tubes carrying magnetic monopole field, which later develop finite sized and expanding M4 projection - carrying only right-handed neutrinos were the fundamental objects from which matter would have emerged in a manner analogous to the decay of vacuum expectations of instanton fields (see this). Even better, Kähler magnetic energy has interpretation as dark energy and magnetic tension gives rise to the negative "pressure" inducing accelerated expansion of the Universe.
The basic question is whether B and L are conserved separately or not. In TGD Universe one can consider two options depending on the answer to this question. For option I - the "official" version of TGD - quarks and leptons correspond to opposite 8-D chiralities of the induced spinor fields and B and L are conserved separately. For option II (see this) only leptonic spinor fields would be fundamental, and the idea is that quarks could be fractionally charged leptons. This option could lead to genuine baryo-genesis, and in the simplest model baryons would be generated from 3-leptons as 3-sheeted structures for which fractionization of color hyper-charge occurs. Leptonic imbedding space spinors moving in triality zero color partial waves would be replaced with triality +/- 1 partial waves assigned with quarks. Whether this replacement is on a mathematically sound basis, is far from obvious since induced spinor fields at space-time level would couple to induced spinor fields with leptonic couplings.
In any case, one can check whether leptogenesis, baryogenesis, and matter antimatter asymmetry are possible for either of both of these options. It turns out that for both option I and II one can construct simple model in terms for the generation of quarks from leptons via emission of lepto-quarks analogous to gauge bosons but differing from their counterparts in GUTs. Option II allows also genuine baryogenesis from leptons. The conclusion is that the "official" version of TGD predicting separate conservation of B and L allows an elegant vision about the generation of matter from cosmic strings containing only right-handed neutrinos in the initial states.
For details see the article Matter-antimatter asymmetry, baryo-genesis, lepto-genesis, and TGD of the chapter TGD and Astrophysics.