Topological Geometrodynamics

In this chapter the key ideas behind Topological Geometrodynamics (TGD) are introduced and an overall view about the structure of the book is given.

The observation that profoundly changed my life was that if space-time is microscopically a 4-dimensional surface in certain 8-dimensional imbedding space H, one can solve the "energy" problem of general relativity, which is due to the fact that the notions of energy and momentum become ill-defined in curved space-time since the corresponding symmetries are lost. The simple arguments fixing the choice of the imbedding space to be H = M⁴XCP₂, that is Cartesian product of Minkowski space of special relativity and complex projective space of 2 complex dimensions, are described. Also the basic implications - such as the notion of geometrization of known classical fields in terms of the induction procedure, and the notion of many-sheeted space-time - are described. The basic objections resolved by TGD view about classical fields and their superposition are considered. The relationship of TGD space-time with the space-time of general relativity understood as macroscopic phenomenological concept is briefly depicted.

TGD leads to several generalizations of existing view about the ontology of physics and these modications are described....

This piece of text was written as an attempt to provide a popular summary about TGD. This is of course mission impossible since TGD is something at the top of centuries of evolution which has led from Newton to standard model. This means that there is a background of highly refined conceptual thinking about Universe so that even the best computer graphics and animations fail to help. One can still try to create some inspiring impressions at least. This chapter approaches the challenge by answering the most frequently asked questions. Why TGD? How TGD could help to solve the problems of recent day theoretical physics? What are the basic princples of TGD? What are the basic guidelines in the construction of TGD?

These are examples of this kind of questions which I try to answer in using the only language that I can talk. This language is a dialect of the language used by elementary particle physicists, quantum field theorists, and other people applying modern physics. At the level of practice involves technically heavy mathematics but since it relies on very beautiful and simple basic concepts, one can do with a minimum of formulas, and reader can always to to Wikipedia if it seems that more details are needed. I hope that reader could catch the basic principles and concepts: technical details are not important. And I almost forgot: problems! TGD itself and almost every new idea in the development of TGD has been inspired by a problem.

In this chapter I will discuss three basic visions about quantum Topological Geometrodynamics (TGD). It is somewhat matter of taste which idea one should call a vision and the selection of these three in a special role is what I feel natural just now.

1. The first vision is generalization of Einstein's geometrization program based on the idea that the Kahler geometry of the world of classical worlds (WCW) with physical states identified as classical spinor fields on this space would provide the ultimate formulation of physics.

2. Second vision is number theoretical and involves three threads. The first thread relies on the idea that it should be possible to fuse real number based physics and physics associated with various p-adic number fields to single coherent whole by a proper generalization of number concept. Second thread is based on the hypothesis that classical number fields could allow to understand the fundamental symmetries of physics and and imply quantum TGD from purely number theoretical premises with associativity defining the fundamental dynamical principle both classically and quantum mechanically. The third thread relies on the notion of infinite primes whose construction has amazing structural similarities with second quantization of super-symmetric quantum field theories. In particular, the hierarchy of infinite primes and integers allows to generalize the notion of numbers so that given real number has infinitely rich number theoretic anatomy based on the existence of infinite number of real units.

3. The third vision is based on TGD inspired theory of consciousness, which can be regarded as an extension of quantum measurement theory to a theory of consciousness raising observer from an outsider to a key actor of quantum physics.

In this chapter the classical field equations associated with the Kahler action are studied.

1. Are all extremals actually "preferred"?

The notion of preferred extremal has been central concept in TGD but is there really compelling need to pose any condition to select preferred extremals in zero energy ontology (ZEO) as there would be in positive energy ontology? In ZEO the union of the space-like ends of space-time surfaces at the boundaries of causal diamond (CD) are the first guess for 3-surface. If one includes to this 3-surface also the light-like partonic orbits at which the signature of the induced metric changes to get analog of Wilson loop, one has good reasons to expect that the preferred extremal is highly unique without any additional conditions apart from non-determinism of Kahler action proposed to correspond to sub-algebra of conformal algebra acting on the light-like 3-surface and respecting light-likeness. One expects that there are nite number n of conformal equivalence classes and n corresponds to n in h_eff = nh. These objects would allow also to understand the assignment of discrete physical degrees of freedom to the partonic orbits as required by the assignment of hierarchy of Planck constants to the non-determinism of Kahler action.

2. Preferred extremals and quantum criticality

The identification of preferred extremals of Kahler action defining counterparts of Bohr orbits has been one of the basic challenges of quantum TGD. By quantum classical correspondence the non-deterministic space-time dynamics should mimic the dissipative dynamics of the quantum jump sequence.

The space-time representation for dissipation comes from the interpretation of regions of space-time surface with Euclidian signature of induced metric as generalized Feynman diagrams (or equivalently the light-like 3-surfaces dening boundaries between Euclidian and Minkowskian regions). Dissipation would be represented in terms of Feynman graphs representing irreversible dynamics and expressed in the structure of zero energy state in which positive energy part corresponds to the initial state and negative energy part to the final state. Outside Euclidian regions classical dissipation should be absent and this indeed the case for the known extremals.

The non-determinism should also give rose to space-time correlate for quantum criticality. The study of Kahler-Dirac equations suggests how to dene quantum criticality. Noether currents assignable to the Kahler-Dirac equation are conserved only if the first variation of Kahler-Dirac operator D_K defined by Kahler action vanishes. This is equivalent with the vanishing of the second variation of Kahler action - at least for the variations corresponding to dynamical symmetries having interpretation as dynamical degrees of freedom which are below measurement resolution and therefore efectively gauge symmetries.....

Topological Geometrodynamics is able to make rather precise and often testable predictions. In this and two other articles I want to describe the recent over all view about the aspects of quantum TGD relevant for particle physics.

In the first chapter I concentrate the heuristic picture about TGD with emphasis on particle physics.

1. First I represent briefly the basic ontology: the motivations for TGD and the notion of many-sheeted space-time, the concept of zero energy ontology, the identification of dark matter in terms of hierarchy of Planck constant which now seems to follow as a prediction of quantum TGD, the motivations for p-adic physics and its basic implications, and the identication of space-time surfaces as generalized Feynman diagrams and the basic implications of this identification.

2. Symmetries of quantum TGD are discussed. Besides the basic symmetries of the imbedding space geometry allowing to geometrize standard model quantum numbers and classical fields there are many other symmetries. General Coordinate Invariance is especially powerful in TGD framework allowing to realize quantum classical correspondence and implies effective 2-dimensionality realizing strong form of holography. Super-conformal symmetries of super string models generalize to conformal symmetries of 3-D light-like 3-surfaces.

What GRT limit of TGD and Equivalence Principle mean in TGD framework have are problems which found a solution only quite recently (2014). GRT space-time is obtained by lumping together the sheets of many-sheeted space-time to single piece of M⁴ provided by an effctive metric defined by the sum of Minkowski metric and the deviations of the induced metrics of space-time sheets from Minkowski metric. Same description applies to gauge potentials of gauge theory limit. Equivalence Principle as expressed by Einstein's equations reflects Poincare invariance of TGD. Super-conformal symmetries imply generalization of the space-time supersymmetry in TGD framework consistent with the supersymmetries of minimal supersymmetric variant of the standard model. Twistorial approach to gauge theories has gradually become part of quantum TGD and the natural generalization of the Yangian symmetry identied originally as symmetry of N = 4 SYMs is postulated as basic symmetry of quantum TGD......

Twistors Grassmannian formalism has made a breakthrough in N = 4 supersymmetric gauge theories and the Yangian symmetry suggests that much more than mere technical breakthrough is in question. Twistors seem to be tailor made for TGD but it seems that the generalization of twistor structure to that for 8-D imbedding space H = M4 CP2 is necessary. M4 (and S4 as its Euclidian counterpart) and CP2 are indeed unique in the sense that they are the only 4-D spaces allowing twistor space with Kahler structure.

The Cartesian product of twistor spaces P3 = SU(2; 2)=SU(2; 1)U(1) and F3 denes twistor space for the imbedding space H and one can ask whether this generalized twistor structure could allow to understand both quantum TGD and classical TGD dened by the extremals of Kahler action. In the following I summarize the background and develop a proposal for how to construct extremals of Kahler action in terms of the generalized twistor structure. One ends up with a scenario in which space-time surfaces are lifted to twistor spaces by adding CP1 ber so that the twistor spaces give an alternative representation for generalized Feynman diagrams.

There is also a very closely analogy with superstring models. Twistor spaces replace Calabi-Yau manifolds and the modication recipe for Calabi-Yau manifolds by removal of singularities can be applied to remove self-intersections of twistor spaces and mirror symmetry emerges naturally. The overall important implication is that the methods of algebraic geometry used in super-string theories should apply in TGD framework.

The physical interpretation is totally different in TGD. The landscape is replaced with twistor spaces of space-time surfaces having interpretation as generalized Feynman diagrams and twistor spaces as sub-manifolds of P3 F3 replace Witten's twistor strings.

The classical view about twistorialization of TGD makes possible a more detailed formulation of the previous ideas about the relationship between TGD and Witten's theory and twistor Grassmann approach.

The topics of this chapter are the purely geometric aspects of the vision about physics as an infinite-dimensional Kahler geometry of configuration space or the "world of classical worlds"(WCW), with "classical world" identified either as 3-D surface of the unique Bohr orbit like 4-surface traversing through it. The non-determinism of Kahler action forces to generalize the notion of 3- surfaces so that unions of space-like surfaces with time like separations must be allowed. The considerations are restricted mostly to real context and the problems related to the p-adicization are discussed later.

There are two separate tasks involved.

1. Provide WCW with Kahler geometry which is consistent with 4-dimensional general coordinate invariance so that the metric is Diff4 degenerate. General coordinate invariance implies that the definition of metric must assign to a give 3-surface X³ a 4-surface as a kind of Bohr orbit X⁴(X³).

2. Provide the WCW with a spinor structure. The great idea is to identify WCW gamma matrices in terms of super algebra generators expressible using second quantized fermionic oscillator operators for induced free spinor fields at the space-time surface assignable to a given 3-surface. The isometry generators and contractions of Killing vectors with gamma matrices would thus form a generalization of Super Kac-Moody algebra.....

There are two basic approaches to the construction of quantum TGD. The first approach relies on the vision of quantum physics as infinite-dimensional Kahler geometry for the "world of classical worlds" identified as the space of 3-surfaces in in certain 8-dimensional space. Essentially a generalization of the Einstein's geometrization of physics program is in question.

The second vision identifies physics as a generalized number theory and involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields (in particular, identifying associativity condition as the basic dynamical principle), and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory.

1. p-Adic physics and their fusion with real physics

The basic technical problems of the fusion of real physics and various p-adic physics to single coherent whole relate to the notion of definite integral both at space-time level, imbedding space level and the level of WCW (the "world of classical worlds") . The expressibility of WCW as a union of symmetric spaces leads to a proposal that harmonic analysis of symmetric spaces can be used to define various integrals as sums over Fourier components. This leads to the proposal the p-adic variant of symmetric space is obtained by a algebraic continuation through a common intersection of these spaces, which basically reduces to an algebraic variant of coset space involving algebraic extension of rationals by roots of unity. This brings in the notion of angle measurement resolution coming as = 2=pn for given p-adic prime p. Also a proposal how one can complete the discrete version of symmetric space to a continuous p-adic versions emerges and means that each point is effectively replaced with the p-adic variant of the symmetric space identifiable as a p-adic counterpart of the real discretization volume so that a fractal p-adic variant of symmetric space results.....

The work with TGD inspired model for quantum computation led to the realization that von Neumann algebras, in particular hyper-finite factors, could provide the mathematics needed to develop a more explicit view about the construction of M-matrix generalizing the notion of S-matrix in zero energy ontology (ZEO). In this chapter I will discuss various aspects of hyper-finite factors and their possible physical interpretation in TGD framework.

1. Hyper-finite factors in quantum TGD

The following argument suggests that von Neumann algebras known as hyper-finite factors (HFFs) of type III1 appearing in relativistic quantum field theories provide also the proper mathematical framework for quantum TGD.

1. The Clifford algebra of the infinite-dimensional Hilbert space is a von Neumann algebra known as HFF of type II1. Therefore also the Clifford algebra at a given point (light-like 3-surface) of world of classical worlds (WCW) is HFF of type II1. If the fermionic Fock algebra defined by the fermionic oscillator operators assignable to the induced spinor fields (this is actually not obvious!) is infinite-dimensional it defines a representation for HFF of type II1. Super-conformal symmetry suggests that the extension of the Clifford algebra defining the fermionic part of a super-conformal algebra by adding bosonic super-generators representing symmetries of WCW respects the HFF property. It could however occur that HFF of type II∞ results.

2. WCW is a union of sub-WCWs associated with causal diamonds (CD) defined as intersections of future and past directed light-cones. One can allow also unions of CDs and the proposal is that CDs within CDs are possible. Whether CDs can intersect is not clear......

Quantum criticality is one of the corner stone assumptions of TGD. The value of Kahler coupling strength fixes quantum TGD and is analogous to critical temperature. TGD Universe would be quantum critical. What does this mean is however far from obvious and I have pondered the notion repeatedly both from the point of view of mathematical description and phenomenology. Superfluids exhibit rather mysterious looking effects such as fountain effect and what looks like quantum coherence of superfluid containers which should be classically isolated. These findings serve as a motivation for the proposal that genuine superfluid portion of superfluid corresponds to a large heff phase near criticality at least and that also in other phase transition like phenomena a phase transition to dark phase occurs near the vicinity.

This chapter is devoted to the applications of TGD to astrophysics and cosmology.

1. Many-sheeted cosmology

The many-sheeted space-time concept, the new view about the relationship between inertial and gravitational four-momenta, the basic properties of the paired cosmic strings, the existence of the limiting temperature, the assumption about the existence of the vapor phase dominated by cosmic strings, and quantum criticality imply a rather detailed picture of the cosmic evolution, which differs from that provided by the standard cosmology in several respects but has also strong resemblances with inflationary scenario.

It should be made clear that many-sheeted cosmology involves a vulnerable assumption. It is assumed that single-sheeted space-time surface is enough to model the cosmology. This need not to be the case. GRT limit of TGD is obtained by lumping together the sheets of many-sheeted space-time to a piece of Minkowski space and endowing it with an effective metric, which is sum of Minkowski metric and deviations of the induced metrics of space-time sheets from Minkowski metric. Hence the proposed models make sense only if GRT limits allowing imbedding as a vacuum extremal of Kahler action have special physical role.....

This chapter represents the most recent (2014) view about particle massivation in TGD framework. This topic is necessarily quite extended since many several notions and new mathematics is involved. Therefore the calculation of particle masses involves five chapters. In this chapter my goal is to provide an up-todate summary whereas the chapters are unavoidably a story about evolution of ideas.

The identifcation of the spectrum of light particles reduces to two tasks: the construction of massless states and the identification of the states which remain light in p-adic thermodynamics. The latter task is relatively straightforward. The thorough understanding of the massless spectrum requires however a real understanding of quantum TGD. It would be also highly desirable to understand why p-adic thermodynamics combined with p-adic length scale hypothesis works. A lot of progress has taken place in these respects during last years.

1. Physical states as representations of super-symplectic and Super Kac- Moody algebras.

The basic constraint is that the super-conformal algebra involved must have ve tensor factors. The precise identication of the Kac-Moody type algebra has however turned out to be a dicult task. The recent view is as follows. Electroweak algebra U(2)ew = SU(2)L U(1) and symplectic isometries of light-cone boundary (SU(2)rot SU(3)c) give 2+2 factors and full supersymplectic algebra involving only covariantly constant right-handed neutrino mode would give 1 factor. This algebra could be associated with the 2-D surfaces X2 defined by the intersections of light-like 3-surfaces with δM4± CP2. These 2-surfaces have interpretation as partons.....

TGD predicts a lot of new physics and it is quite possible that this new physics becomes visible at LHC. Although the calculational formalism is still lacking, p-adic length scale hypothesis allows to make precise quantitative predictions for particle masses by using simple scaling arguments.

The basic elements of quantum TGD responsible for new physics are following.

1. The new view about particles relies on their identification as partonic 2-surfaces (plus 4-D tangent space data to be precise). This effective metric 2-dimensionality implies generalizaton of the notion of Feynman diagram and holography in strong sense. One implication is the notion of field identity or field body making sense also for elementary particles and the Lamb shift anomaly of muonic hydrogen could be explained in terms of field bodies of quarks.

4-D tangent space data must relate to the presence of strings connecting partonic 2-surfaces and defining the ends of string world sheets at which the modes of induced spinor fields are localized in the generic case in order to achieve conservation of em charge. The integer characterizing the spinor mode should characterize the tangent space data. Quantum criticality suggests strongly and super-conformal invariance acting as a gauge symmetry at the light-like partonic orbits and leaving the partonic 2-surfaces at their ends invariant. Without the fermionic strings effective 2-dmensionality would degenerate to genuine 2-dimensionality.

2. The topological explanation for family replication phenomenon implies genus generation correspondence and predicts in principle innite number of fermion families.....