In the number theoretic vision in which WCW is discretized by replacing space-time surfaces with their number theoretical discretizations determined by the points of X4⊂ M8 having the octonionic coordinates of M8 in an extension of rationals and therefore making sense in all p-adic number fields? How could an effective discretization of the real WCW at the geometric H level, making computations easy in contrast to all expectations, take place?
- The key observation is that any functional or path integral with integrand defined as exponent of action, can be formally calculated as an analog of Gaussian integral over the extrema of the action exponential exp(S). The configuration space of fields would be effectively discretized. Unfortunately, this holds true only for the so called integrable quantum field theories and there are very few of them and they have huge symmetries. But could this happen for WCW integration thanks to the maximal symmetries of the WCW metric?
- For the Kähler function K, its maxima (or maybe extrema) would define a natural effective discretization of the sector of WCW corresponding to a given polynomial P defining an extension of rationals.
The discretization of the WCW defined by polynomials P defining the space-time surfaces should be equivalent with the number theoretical discretization induced by the number theoretical discretization of the corresponding space-time surfaces. Various p-adic physics and corresponding discretizations should emerge naturally from the real physics in WCW.
- The physical interpretation is clear. The TGD Universe is analogous to the spin glass phase (see this). The discretized WCW corresponds to the energy landscape of spin glass having an ultrametric topology. Ultrametric topology of WCW means that discretized WCW decomposes to p-adic sectors labelled by polynomials P. The ramified primes of P label various p-adic topologies associated with P.
For a summary of earlier postings see Latest progress in TGD.
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