**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= M^{4}× CP_{2} (one of the coordinates is hypercomplex light-like coordinate). Function fields are indeed central in the geometric Langlands correspondence.

- This function field also has a hierarchical structure. There are hierarchies of polymials of various degrees and also rational functions with coefficient fields in different extensions of rationals. Analytical functions for which the Taylor coefficients are in extensions of rationals in the expansion is the next step. At the ultimate limit one has algebraic numbers as coefficients. Also transcendental extensions can be thought of and in this way one eventually ends up with complex numbers.
- For H=M
^{4}× CP_{2}, this would correspond to the lowest level of the hierarchy of infinite primes but the Cartesian powers of H=M^{4}× CP_{2}correspond to the higher levels in the hierarchy of infinite primes. Again, this hierarchy is be analogous to the hierarchy used in the description of condensed matter, 3N-dimensional spaces, N number of particles.

X^{4}_{1} ∪ X^{4}_{2} → X^{4}_{1}*X^{4}_{2} ,

and the tensor product of the fermionic states at the boundaries of CD is formed. This would give

Ψ(X^{4}_{1})⊗ Ψ(X^{4}_{2}) (X^{4}_{1}∪ X^{4}_{2}) → Ψ(X^{4}_{1})⊗ Ψ(X^{4}_{2})(X^{4}_{1}*X^{4}_{2}) .

Here X^{4}_{1}*X^{4}_{2} 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 10^{38}!

**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?

- Let's examine the function pairs (f
_{1},f_{2}) and (g_{1},g_{2}) defined in H=M^{4}× CP_{2}and the corresponding space-time surfaces for which (f_{1},f2)=(0,0) and (g_{1},g_{2})=(0,0) apply. It should be noted that, for example, that the condition f_{1}=0 defines the analog of a 6-D twistor space, and the space-time surface X^{4}is the intersection of the analogs of the twistor bundles of M^{4}and CP_{2}, i.e., its base space. - The product of the function pairs is (f
_{1}g_{1},f_{2}g_{2}). Its components vanish in four cases.- The cases (f
_{1},f_{2})= (0,0) and (g_{1},g_{2})=(0,0) correspond to the union of the incoming surfaces. The corresponding particles are free. - The cases (f
_{1},g_{2})= (0,0) and (f_{2},g_{1})=(0,0) could define space-time regions having an interpretation in terms of the interaction of the particles. Under what conditions could this interpretation makes sense geometrically?Physical intuition suggests that for interacting particles, which do not form a bound state, the product reduces near the passive boundary (initial state) of the CD to the union of the surfaces associated with the free particles. The surfaces (f

_{1},g_{2})= (0,0) and (f_{2},g_{1})=(0,0) would not temporally extend to the passive boundary of the CD. which correspond to the initial state of the particle reaction.This imposes some conditions on the functions involved. f

_{1}=0 and g_{2}=0 (f_{2}=0 and g_{1}=0) are not satisfied near to the passive boundary of the CD simultaneously , so that the intersection of the corresponding 6-D surfaces (analogous to twistori space) is empty near the boundary of the CD.If this condition is not true, the interpretation would be as a bound state. TGD view of nuclei, atoms, and molecules assume that particles forming the bound state are indeed connected by monopole flux tubes (see this).

- The cases (f

**What about the product of spinors fields?**

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 X^{4} 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!

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.

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