Master formula for the construction of quantum states using the interpretation of space-time surfaces as numbersThe exact solution of field equations of TGD in terms of holography=holomorphy vision and the recent progress in the understanding of the TGD view of Langlands correspondence allows to propose an explicit recipe, a kind of a master formula, for the construction of states describing the interaction in terms of generalized holomorphic algebraic geometry. Space-time surfaces have the structure of number field As I wrote the most recent article about the recent TGD view of Langlands correspondence (see this), I become convinced that the space-time surfaces indeed have a structure of a number field, induced by the structure of the function field formed by the analytic functions with respect to the four generalized complex coordinates of H= M4× CP2 (one of the coordinates is hypercomplex light-like coordinate). Function fields are indeed central in the geometric Langlands correspondence.
X41 ∪ X42 → X41*X42 , and the tensor product of the fermionic states at the boundaries of CD is formed. This would give Ψ(X41)⊗ Ψ(X42) (X41∪ X42) → Ψ(X41)⊗ Ψ(X42)(X41*X42) . Here X41*X42 would be the product of surfaces induced by the function algebra and the product of fermion states would be tensor product. Could Gods compute using spacetime surfaces as numbers and could our arithmetics be a shadow on the wall of the cave. So: could a believer of TGD dream conclude that these meta-levels and perhaps even mathematical thinking could be described within the framework of the mathematics offered by the infinite dimensional number field formed by the space-time surfaces. This quite a lot more complicated than binary math with a cutoff of the order of 1038! Product of space-time surfaces as geometric counterpart of the tensor product What could the product of space-time surfaces mean concretely? The physical intuition suggest that t corresponds to ae creation of an interacting pair of 3-D particles identified as they 4-D Bohr orbits. The product would be the equivalent of a tensor product, but now with interaction. If so, this product could provide a geometric and algebraic description of the interactions. What would you get?
The WCW spinor field assigns multifermion states to the 3-D ends of a given spacetime surface at the boundaries of the CD. If one can define what happens to the multifermion states associated with the zero energy states in the interaction, then one has a universal construction for the states of WCW as spinor fields of WCW providing a precise description of interactions analogous to an exact solution of an interacting quantum field theory. At the geometric level, the product of the surfaces corresponds to the interaction. At the fermion level, essentially the ordinary tensor product of the multifermion states should correspond to this interaction. Under what conditions does this vision work for fermionic states as WCW spinors, identified in ZEO as pairs of the many-fermion states at the 3-surfaces at the boundaries of the CD? It is obvious that the definition of the fermion state should be universal in the sense that at the fundamental level the fermion state is defined without saying anything about space-time surfaces involved. Induction is a basic principle of TGD and the induction of spinor fields indeed conforms with this idea. The basic building bricks of WCW spinor fields are second quantized spinor fields of H restricted to the 3-surfaces defining the ends of the space-time surfaces at the boundaries of CD. Therefore the multifermion states are restrictions of the multifermion states of H to the spacetime surfaces. The Fourier components (in the general sense) for the second quantized spinor field Ψ of H (not WCW!) and its conjugate Ψ{†} would only be confined to the ends of X4 at the light-like boundaries of CD. The oscillator algebra of H spinor fields makes it possib le to calculate all fermionic propagators and fermionic parts of N-point functions reduce to free fermionic field theory in H but arguments restricted to the space-time surfaces. The dynamics of the formally classical spinor fields of WCW would very concretely be a "shadow" of the dynamics of the second quantized spinor fields of H. One would have a free fermionic field theory in H induced to space-time surfaces! In this way, one could construct multiparticle states containing an arbitrary number of particles. The construction of quantum spaces would reduce to a multiplication in the number field formed by space-time surfaces, accompanied by fermionic tensor product! See the chapter About Langlands correspondence in the TGD framework or the article with the same title.
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