Friday, March 04, 2005

Web of dualities

In TGD there are dual descriptions of the world of classical worlds which I call CH and CHO: number-theoretical compactification states that 3-surfaces can be equivalently regarded as "living" in either H=M^4xCP_2 or HO=M^8 (hyper-octonionic 8-space). There are also other dualities such as HQ-coHQ duality (3-surface can correspond to either hyper-quaternionic or co-hyper-quaternionic space-time surface). The question is how these dualities relate to each other and other dualities, such as electric-magnetic duality conjectured for a long time ago. The understanding of HO-H duality at both configuration space (world of classical worlds) level and concrete hadron physics level has been increasing rapidly and following represents some comments about this.

1. What is Kähler function in the dual of the world of classical worlds?

Space-time surfaces can be either hyper-quaternionic or co-hyper-quaternionic (briefly HQ or coHQ). Vacuum extremals with vanishing Kähler field would be extreme examples of HQ phase whereas CP_2 type vacuum extremal would be an extreme example of coHQ phase. For a generic 3-surface both HQ and coHQ type space-time surface is expected to exist so that the notion of classical space-time depends on phase! The Kähler functions of configuration space for two kinds of surfaces need not be same and the question is how they differ. If HQ-coHQ duality is analogous to a coordinate transformation relating two different coordinate patches, one expects that the two Kähler functions differ only by a real part of holomorphic function of complex configuration space coordinates: this guarantees that Kähler metric of configuration space does not depend on picture used. There are two options concerning the identification of Kähler function on coHQ phase.
  • Kähler calibration is defined by the local minimum of Kähler action density. The dual of Kähler calibration would be naturally defined by its maximum. The dual calibration would maximize the absolute value of contribution to Kähler function in each region where the action density has definite sign. Contrasts between regions of positive and negative action density would be maximized. One could argue that this implies unstable Universe: on the other hand quantum criticality suggests large fluctuations and after 10 years of TGD inspired theory of consciousness I cannot resist the temptation to add that quantum criticality makes Universe optimal perceiving system. One might say that HQ and coHQ pictures correspond to the cautious view of a conservative energy saver and exaggerating view of radical artist about what all this is. The duality of p-adic length scales L_k and L_p, p=about 2^k, suggests that the dual of Kähler coupling strength alpha_K(k) is alpha_K(p) and vice versa.
  • The dual of Kähler function is defined by the dual of Kähler action defined by the projection of CP_2 Kähler form to the normal space of space-time surface rather than to its tangent space. For this option dual action vanishes for CP_2 type extremals. The fact that CP_2 Kähler action appears as a fundamental constant in the basic formulas of TGD (say formula for gravitational constant)disfavors this option.

2. Summary about the web of dualities

It is good idea to try to summarize the recent picture. It seems that HO-H duality involves an entire web of dualities suggested by the general structure of TGD.
  • Electric-magnetic duality conjectured already more than decade ago and corresponding to the possibility for defining flux Hamiltonians coding configuration space geometry by using fluxes associated with space-like partonic 2-surface of timelike string orbits.
  • The purely geometric duality of hyper-quaternionic (HQ) and co-hyper- quaternionic (coHQ) 4-surfaces, which would be naturally equivalent with electric-magnetic duality.
  • Parton-string duality which string theorist would probably call closed-open string duality and equivalent with HQ-coHQ duality.
  • 7-3 duality stating that either space-like 3-surfaces in the intersections of space-time surface with light-like 7-surfaces \delta M^4{\pm}xCP_2 or light-like 3 surfaces X^3_l at which X^4 metric becomes degenerate can be taken as causal determinants.
These dualities relate closely to physical dualities such as
  • ew-color duality described in previous posting,
  • p =about 2^k <--> k duality of long and short p-adic length scales,
  • duality of colored current quarks carrying no spin and ew spin and static quarks carrying no color,
  • duality of hadron and quark level descriptions.

3. Predictions of HO-H duality

HO-H duality leads to concrete predictions. Some of them are already verified (SO(4) chiral symmetry of low energy hadron physics and explanation of proton spin crisis). Some of them might be testable (colored quarks should not contribute to right and left electro-weak isospin of hadrons). The most important prediction is that both standard model of electro-weak interactions and QCD are in a precisely defined sense wrong theories. One might hope that this might allow to develop an experimental arrangements proving or disproving this prediction. Spin-statistics paradox was the crucial observation leading to the introduction of quark color. The 2-dimensionality of partonic and stringy surfaces allowing anyonic statistics, or probably equivalently, the presence of configuration space degrees of freedom, allows to circumvent the spin- statistics paradox otherwise implied by the fact that H spinors do not carry color as spin like quantum number and HO spinors do not carry spin and ew spin. For more details see the chapter TGD as a Generalized Number Theory: Quaternions, Octonions, and their Hyper Counterparts. Matti Pitkänen


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