Comparison of string model and TGD: answer to the question of Harekhrisna Acharaya

Harekhrisna Acharaya asked how TGD compares with string theory. In the following a short answer.

TGD could be also seen as a generalization of string models. Strings replace point-like particles and in TGD 3-surfaces replace them.

A. Symmetries

1. In string models 2-D conformal symmetry and related symmetries are in key roles. They have a generalization to TGD. There is generalization of 2-D conformal invariance to space-time level in terms of Hamilton-Jacobi structure. This also implies a generalization of Kac-Moody type symmetries. There are also supersymplectic symmetries generalizing assignable to δ M⁴₊×CP₂ and reflecting the generalized symplectic structure for the light-cone M⁴₊ and symplectic structure for CP₂. These symmetries act as Noether symmetries in the "world of classical worlds" (WCW).
2. In string models 2-D conformal invariance solves field equations for strings. This generalizes to the TGD framework.
   1. One replaces string world sheets with 4-D surfaces as orbits of 3-D particles replacing strings as particles. The 2-D conformal invariance is replaced with its 4-D generalization. The 2-D complex structure is replaced with its 4-D analog: I call it Hamilton-Jacobi structure. For Minkowski space M⁴ this means composite of 2-D complex structure and 2-D hypercomplex structure. See this .
   2. This allows a general solution to the field equations defining space-time in H=M⁴×CP₂ realizing holography as generalized holomorphy. Space-time surfaces are analogous to Bohr orbits. Path integral is replaced with sum over Bohr orbits assignable to given 3-surface.

      Space-time surfaces are roots of two generalized holomorphic functions of 1 hypercomplex (light-like) coordinate and 3 complex coordinates of H. Space-time surfaces are minimal surfaces irrespective of action as long as it is expressible in terms of the induced geometry.

   3. There are singularities analogous to poles (and cuts) at which generalized holomorphy and minimal surface property fails and they correspond to vertices for particle reactions. There is also a highly suggestive connection with exotic smooth structures possible only in 4-D. This gives rise to a geometric realization of field particle duality. Minimal surface property corresponds to the massless d'Alembert equation and free theory. The singularities correspond to sources and vertices. The minimal surface as the orbit of a 3-surface corresponds to a particle picture.

B. Uniqueness as a TOE

The hope was that string models would give rise to a unique TOE. In string models branes or spontaneous compactification are needed to obtain 4-D or effectively 4-D space-time. This forced to give up hopes for a unique string theory and the outcome was landscape catastrophe.

1. In TGD, space-time surfaces are 4-D and the embedding space is fixed to H=M⁴×CP₂ by standard model symmetries. No compactification is needed and the space-time is 4-D and dynamical at the fundamental level.

   The space-time of general relativity emerges at QFT limit when the sheets of many-sheeted space-time are replaced with single region of M⁴ made slightly curved and carryin gauge potentials sum of those associated with space-time sheets.

2. How to uniquely fix H: this is the basic question. There are many ways to achieve this.
   1. Freed found that loop spaces have a unique geometry from the existence of Riemann connection. The existence of the Kahler geometry of WCW is an equally powerful constraint and also it requires maximal isometries for WCW so that it is analogous to a union of symmetric spaces. The conjecture is that this works only for H=M⁴×CP₂. Physics is unique from its geometric existence.
   2. The existence of the induced twistor structure allows for the twistor lift replacing space-time surfaces with 6-D surfaces as S² bundles as twistor spaces for H=M⁴×CP₂ only. Only the twistor spaces of M⁴ and CP₂ have Kaehler structure and this makes possible the twistor lift of TGD.
   3. Number theoretic vision, something new as compared to string models, leads to M⁸-H duality as an analog of momentum position duality for point-like particles replaced by 3-surface. Also this duality requires H=M⁴×CP₂. M⁸-H duality is strongly reminiscent of Langlands duality.

      Although M⁸-H duality is purely number theoretic and corresponds to momentum position duality and does not make H dynamical, it brings to mind the spontaneous compactification of M¹⁰ to M⁴×S .

   C. Connection with empiria

   String theory was not very successful concerning predictions and the connection with empirical reality. TGD is much more successful: after all it started directly from standard model symmetries.

   1. TGD predicts so called massless extremals as counterparts of classical massless fields. They are analogous to laser light rays. Superposition for massless modes with the same direction of momentum is possible and propagation is dispersion free.
   2. TGD predicts geometric counterparts of elementary particles as wormhole contacts, that is space-time regions of Euclidean signature connecting 2 Minkowskian space-time sheets and having roughly the size of CP₂ and in good approximation having geometry of CP₂. MEs are ideal for precisely targeted communications.
   3. TGD predicts string-like objects (cosmic strings) as 4-D surfaces. They play a key role in TGD in all scales and represent deviation from general relativity in the sense that they do not have 4-D M⁴ projection and are not Einsteinian space-times. The primordial cosmology is cosmic string dominated and the thickening of cosmic strings gives rise to quasars and galaxies. Monopole flux tubes are fundamental also in particle, nuclear and even atomic and molecular physics, biology and astrophysics. See for instance this .
   4. An important deviation of TGD from string models is the notion of field body. The Maxwellian/gauge theoretic view of fields is replaced with the notion of a field body having flux sheets and flux tubes as body parts. Magnetic monopoles flux tubes require no currents to maintain the associated magnetic fields. This explains the existence of magnetic fields in cosmic scales and of huge cosmic structures. Also the stability of the Earth's magnetic field finds an explanation.

      The dark energy of cosmic strings explains the flat velocity spectrum of stars around galacxies and therefore galactic dark matter.

   5. Number theoretic vision predicts a hierarchy of space-time surfaces defined as roots of pairs of polynomials with increasing degree. This gives rise to an evolutionary hierarchy of extensions of rationals and also behind biological evolution.

      The dimensions of extensions correspond to a hierarchy of effective Planck constants h_eff=nh₀ serving as measure of algebraic complexity and giving rise to a hierarchy of increasing quantum coherence scales. Ordinary particles with h_eff>h behave like dark matter. The identification is not as galactic dark matter but as dark phases residing at field bodies and controlling ordinary matter (h_eff serves as a measure for intelligence). They explain the missing baryonic matter.

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

   For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.