Progress during last year in TGD II: theoretical aspects of TGD
In this posting I will summarize the progress made in TGD itself during this year. One category of problems relates to the interpretation of TGD and its relationship to existing theories, to the understanding of the preferred extremals of Kähler action and solutions of the modified Dirac action, to the construction of the generalized Feynman diagrams and to a more precise view about what particles are in this framework. Zero energy ontology is now a basic pillar of TGD and should be understood better. One should also develop the understanding about the fusion of real physics and various p-adic physics inspired by the success of p-adic mass calculations and required by number theoretical universality, about the effective hierarchy of Planck constants predicting dark matter hierarchy, and about hyperfinite factors allowing to realize the notion of finite measurement/cognitive resolution.
The interpretation of TGD
Last years have meant impressive progress in the interpretation of TGD both at classical and quantum level: here one could speak about refinement of the ontology of TGD. These levels are of course related by quantum classical correspondence and this principle has demonstrated its amazing power. The notion of many-sheeted space-time is central.
Basic argument against TGD
Perhaps the strong objection against TGD is that linear superposition for classical fields is lost. The linear superposition is however central starting point of field theories. Many-sheeted space-time allows to circumvent this argument about which I became conscious of just during this year - about 34 year after the discovery of TGD!
The replacement of linear superposition of fields with the superposition of their effecs meaning that sum is replaced with set theoretic union for space-time sheets. This simple observation has far reaching consequences: it becomes possible to replace the dynamics for a multitude of fields with the dynamics of space-time surfaces with only 4 imbedding space coordinates as primary dynamical variables.
Quantum classical correspondence and quantum ergodicity
Quantum classical correspondence has been one of the guiding principles of TGD. The newest conjecture generated by quantum classical correponds is quantum ergodicity. Quantum ergodicity states that quantal correlation functions for classical field like quantities in zero energy state are identical with the classical correlation functions for a any preferred extremal in their superposition. Single preferred extremal represents entire zero energy state so that all space-time surfaces in the quantum superposition of parallel classical worlds are equivalent observationally.
Although zero energy ontology (S-matrix is relaced with M-matrix definign "square root" of density matrix) and 4-D spin glass degeneracy suggest that this principle is satisfied only by the outcomes of state function reduction, it is extremely powerful if it really works.
The effective hierarchy of Planck constants and dark matter
In TGD framework dark matter could be understand as phases of matter with scaled up value of effective Planck constant. This explanation distinguishes sharply between TGD and competitors and leads to a completely new view about quantum biology based on the notion of magnetic body carrying dark matter. TGD leads also to a view about dark energy as Kähler magnetic energy.
The theoretical understanding of the effective hierarchy of Planck constants in terms of space-time topology has developed during this idea. The newest idea is that the effective n-sheeted cover of imbedding space to which effective value of Planck constant hbareff=nhbar is assigned corresponds to an n-furcation natural in the non-linear dynamics of Kähler action by its enormous vacuum degeneracy. The list about the applications to living matter has been steadily growing.
- Updated view about the hierarchy of Planck constants
- Dark matter hierarchy and fractional quantum Hall effect
- What about the relationship of gravitational Planck constant to ordinary Planck constant
- How the effective hierarchy of Planck constants could reveal itself in condensed matter physics?
Strong form of general coordinate invariance and holography
Strong form of general coordinate invariance (GCI) implies strong form of holography. The outcome is TGD counterpart of AdS/CFT correspondence. Bulk is replaced with space-time surface and strings with 2-D string world sheets carrying fermionic fields (right handed neutrino is exception and is delocalized into entire space-time surface). Partonic 2-surfaces at the boundaries of CDs and the 4-D tangent space data at them would code for the quantum dynamics. Interior dynamics would provide classical correlates for quantum states - say classical correlation functions identical to their quantum counterparts if quantum ergodicity holds true.
One outcome is a new view about black holes replacing the interior of blackhole with a space-time region of Euclidian signature of induced metric and identifiable as analogs of lines of generalized Feynman diagrams. In fact, black hole interiors are only special cases of Eucdlian regions which can be assigned to any physical system. This means that the description of condensed matter as AdS blackholes is replaced in TGD framework with description using Euclidian regions of space-time.
Zero energy ontology
Zero energy ontology (ZEO) has inspired detailed development of the views about the relationship between geometric and subjective time and led to highly non-trivial picture challenging the existing beliefs. No final conclusions are possible yet but I believe that the understanding continues to grow. ZEO has led also to a highly detailed view about generalized Feynman diagrammatics.
- Anatomy of quantum jump in zero energy ontology
- Riemann zeta and quantum theory as square root of thermodynamics
- The mystery of time again
- Could hyperbolic 3-manifolds and hyperbolic lattices be relevant in zero energy ontology?
- Does the square root of p-adic thermodynamics exist?
- Quantum dynamics for the moduli associated with CDs and the arrow of geometric time
p-Adic physics, p-adic mass calculations, p-adic length scale hypothesis and the integration of various p-adic physics and real physics to a bigger whole have continued to be sources of challenges and inspiration. In biological and neuroscience applications number theoretic entropy identifiable as negentropy has led to a vision about living matter as something residing in the intersection of real and p-adic physics.
- p-Adic homology and finite measurement resolution
- Constraints on super-conformal invariance from p-adic mass calculations and ZEO
The mathematics of hyperfinite factors is too difficult for an theoretical physicist like me but still considerable progress has taken place. All new visions generate a great burst of ideas and so did also hyperfinite factors as I realized their importance for about seven years ago. The subsequent progress has been mostly selection of the fittest ones with internal consistency defining the most powerful evolutionary pressure. During this year emerged a Fresh view about hyperfinite factors.
Twistors and TGD
Twistors are the hot topic of recent day theoretical physics. ZEO allows to make conjectures about connections with the twistor approach. I do not have the needed technical competence so that I can only try to see the situation at the level of principles.
The basic physical idea of generalized Feynman diagrammatics is that the fermions propagating along the lines of generalized Feynman diagrams (associated with braids at light-like 3-surfaces at which the signature of the induced metric changes) are massless. The condition that also virtual fermions are massless leads to a non-trivial diagrammatics if the sign of the energy can have both values for the virtual wormhole throats. The condition is extremely powerful and eliminates most diagrams and also both IR and UV divergences and allows to understand how the massivation of external particle can be consistent with conformal invariance.
Masslessness is the basic condition making possible to express kinematics in terms of twistors. What one expects is that after functional integration the resulting amplitudes satisfy the Yangian symmetry characterizing also twistor diagrams and fixing them to a high degree. Also Yangian symmetry has a natural generalization in TGD framework.
- Proposal for a twistorial description of generalized Feynman graphs
- Twistor revolution and TGD
- Still one attempt to understand generalized Feynman diagrams
Preferred extremals and solutions of the modified Dirac equation
The last years have meant rapid progress in the understanding of TGD at the fundamental level. The key concept of TGD is the notion of preferred extremal. It roughly means that one can assign to a collection of 3-surfaces at the ends of a causal diamond (CD) a unique space-time surface. A realization of holography would be in question.
Strong form of GCI leads to the strong form of holography stating that light-like 3-surfaces connecting the ends of CD and space-like 3-surfaces the ends of CD are equivalent choices. Therefore partonic 2-surface defined as their intersections plus 4-D tangent space data at them would fix the quantum physics in ZEO.
One can of course wonder whether the space-like 3-surfaces and light-like 3-surfaces are completely unique from the preferred extremal property or whether they are unique apart from conformal transformations assignable to the light-like coordinate of light-like 3-surface or to the light-like coordinate of δ CD× CP2. This interpretation would conform with the interpretation of these conformal transformations as gauge transformations. These transformations would extend to ordinary conformal transformations at string world sheets (with Minkowskian signature and hypercomplex structure), which are carriers of spinor fields in the proposed general solution of the modified Dirac equation based on the requirement of the conservation of electric charge.
But what are these preferred extremals? This is the key question. The first guess was that they are absolute minima of Kähler action. This option did not resonate with the number theoretic visions for the simple reason that minimization for p-adic valued functions does not make sense. I have gradually gained rather detailed knowledge about propertie of preferred extremals. For instance, the reduction of the Kähler action to Chern-Simons terms coming from Minkowskian and Euclidian regions (and differing by imaginary unit) gives a partial realization of the holography and boils down to the vanishing of jKμAμ, where jK denotes Kähler current.
What I regard as a breakthrough came during this year and reduces the construction of preferred extremals to a generalization of the notion of complex structure. In Minkowskian regions of space-time surface I call this structure Hamilton-Jacobi structure identifiable as a composite of complex structure and hyper-complex structure. In Hamilton-Jacob coordinates the field equations reduce to conditions on 4-metric completely analogous to the condition gzz=0 for Euclidian string world sheets.
The field equations are equivalent to minimal surface equations although action is of course something different from 4-volume which is definitely an unphysical choice. As a consequence, induced gamma matrices satisfy the consistency condition DμΓμ=0 giving additional supercharges besides those associated with modified gamma matrices.
Also Einstein equations with cosmological term follow as a consistency condition requiring that the Maxwell energy momentum tensor has vanishing divergence. Newton's constant and cosmological constant are predictions rather than inputs as in the standard theory. Given space-time sheet has constant value of Ricci scalar which has far reaching consequences.
I already mentioned that also a beautiful solution ansatz for the modified Dirac equations emerges from the conservation of electric charged defined in spinorial sense. All modes of the induced spinor field except right-handed neutrino are localized on 2-D string world sheets. This implies automatically braid picture: the boundaries of the string world sheets at light-like and space-like 3-surfaces are 1-D curves identifiable as light-like and space-like braids strands. This also conforms with the vision that discretization at partonic 2-surfaces identifiable as a space-time correlate of finite measurement resolution follows from the dynamics automatically so that the dynamics is highly self-referential.
- What kind of preferred extremals Maxwell phase could correspond?
- About deformations of known extremals of Kähler action
- Under what conditions electric charge is conserved
- How quaternionicity of space-time could be realized?
- Recent vision about preferred extremals
- The emergence of Yangian symmetry and gauge potentials as duals of Kac-Moody currents
- Quantum criticality and electro-weak gauge symmetries
- The importance of being light-like
- About definition of Hamilton-Jacobi structure
- Gauge symmetries in TGD framework
- M8-H duality, preferred extremals, criticality, and Mandelbrot fractals
- Preferred extremals of Kähler action as constant curvature manifolds whose geometric invariants are topological invariants
- How coupling constant evolution could make itself visible as properties of preferred extremals?
- The counterpart of Ads5 duality in TGD
- Is there a connection between preferred extremals and AdS44/CFT correspondence?
- Electron as a trefoil or something more general?
Mathematical ideas inspired by TGD
I do not regard myself as a mathematician in the technical sense of the word. TGD has however forced to generalize existing mathematical concepts, and to even formulate new mathematical notions besides the use of existing mathematics still relatively new for theoretical physicists.
Classical TGD relies on existing mathematical concepts and the main challenges are the understanding of preferred extremals and solutions of the modified Dirac equation. There are several independent conjectures about preferred extremals which should be proven to be right or wrong.
Quantum TGD provides challenging examples of yet non-existing mathematics such as the generalization of loop space geometry to that of WCW and construction of WCW spinor structure. Infinite-dimensional isometry group fixes these structures more or less uniquely but there is huge amount of work to do. The theory hyper-finite factors seems to require an approach generalizing the existing thermodynamical approach to TGD framework which can be seen as a "square root" of thermodynamics. Number theoretical universality requires the fusion of p-adic and real number fields to a larger structure and there are several challenges involved (say precise definition of the notion of integral in p-adic context) attacked also by the leading mathematicians of our time. Infinite primes is a TGD inspired notion having an interpretation as a repeated second quantization of a super-symmetric arithmetic quantum field theory with states labelled by primes and their generalization to infinite primes. There are strong indications that this hierarchy relates directly to various hierarchies of quantum TGD.
During this year a burst of really crazy ideas was inspired by TGD, and I still feel myself rather uneasy with this stuff. Certainly it takes years to find the surviving ideas if any. The key idea is roughly that the standard arithmetics with its sum and product generalize to an arithmetics of Hilbert spaces with sum replaced with direct sum and product with tensor product. This suggests a calculus of Hilbert spaces: one can define Hilbert spaces with dimension which can be also negative integer, rational, algebraic, or even transcendental, one can define Taylor series of Hilbert spaces. By mapping Hilbert space to a single number defined by its dimension one would obtain the ordinary calculus. Everything that can be done with ordinary numbers could be done with Hilbert spaces. This construction can be repeated just like the construction of infinite primes. One can replace the points of Hilbert space with Hilbert spaces and so on... This of course goes completely over the human head.
The question is whether this crazy construction might have some sensible physical interpretation. What made me to take this crazy idea half-seriously was that for generalized Feynman diagrams there are two basic vertices: the generalization of the 3-vertex of ordinary Feynman diagrams having interpretation as tensor product and the generalization of stringy 3-vertex having very natural interpretation in terms of direct sum (in string models the interpretation is in terms of tensor product).
- Number theoretical universality and quantum arithmetics, renormalization, and relation of TGD to N=4 SYM
- Does 2-adic quantum arithmetics explain p-adic length scale hypothesis?
- Quantum p-adic deformations of space-time surfaces as a representation of finite measurement resolution?
- Quantum mechanics as Quantum mathematics