**Sum and product for the space-time surfaces**

In the case of the sum, the basic restriction is the condition that the space-time surfaces appearing as summands allow a common Hamilton-Jacobi structure (see this) in M^{4} degrees of freedom in turn inducing it for the space-time surfaces. The summed space-time surfaces must have a common hypercomplex coordinate with light-like coordinate curves and a common complex coordinate. For the product this is not required.

- One can form analogs of integers as products of polynomials inducing products of space-time surfaces as their roots. The product is defined as the root of (f
_{1},g)*(f_{2},g)=(f_{1}f_{2},g)). The space-time surface defined by the product is the union of the space-time surfaces defined by the factors but an important point is that they have a discrete set of intersection points. In this case there are no restrictions on Hamilton-Jacobi structures.One can argue that the product represents a mere free two-particle state in topological and geometric sense. On the other hand, fermionic n-point functions defining scattering amplitudes are defined in terms of these intersection points and could give a quantum physical realization giving information of the quantum superpositions of space-time surfaces as quantum theorems. This would raise dimensions D=4 and D=8 in a completely unique role.

- Could the sum of space-time surfaces (f
_{1},g)=(0,0) and (f_{2},g)=(0,0) defined as a root of (f_{1},g)+(f_{2},g)=(f_{1}+f_{2},g)) define a topologically and geometrically non-trivial interaction? If the functions f_{1}and f_{2}have interiors of causal diamonds CD_{1}and CD_{2}with different tips as supports (does the complex analyticity allow this?) and CD_{1}and CD_{2}are located within a larger CD then both f_{1}and f_{2}are nonvanishing only in the intersection CD_{1}∩CD_{2}.Generalized complex analyticity requires a Hamilton-Jacobi structure (see this) inside CD. It must have a common hypercomplex coordinate and complex M

^{4}coordinate inside CD and therefore inside CD_{1}∩ CD_{2}and also inside CD_{1}and CD_{2}? Suppose that this condition can be satisfied.Outside CD

_{1}∩ CD_{2}either f_{1}and f_{2}is identically vanishing and one has f_{1}=0 and f_{2}=0 as disjoint roots representing incoming particles in topological sense. In the intersection CD_{1}∩ CD_{2}f_{1}+f_{2}=0 represents a root having interpretation as interaction. f_{i}"interfere" in this region and this interference is consistent with relativistic causality.One could also assign to the sum a tensor product in fermionic degrees of freedom and define n-point functions and restrict their arguments to the self-intersection points of the intersection region CD

_{1}∩ CD_{2}. One could also say that the sum represents z=x+y in such a way that both summands and sum are realized geometrically.

**Could the Hilbert space of pairs (f _{1},f_{2}) have an inner product defined by the intersection of corresponding space-time surfaces?**

The pairs (f_{1},f_{2}) can be formally regarded as elements of a complex Hilbert space. There is however a huge gauge invariance: the multiplication of f_{i} by analytic functions, which are non-vanishing inside the CD, does not affect the space-time surface. The localization of the scalar multiplication means a huge reduction in the number of degrees of freedom. Note that the multiplication with a scalar does not change the spacetime surface but this does not destroy the field property. Since f_{1}= constant=c does not correspond to any space-time surface (this would require c=0) the multiplication with a constant does not correspond to a multiplication with a space-time surface.

The local complex scalings are local variants of complex scalings of Hilbert space vectors which do not affect the state: one cannot however replace Hilbert space by a projective space and the same applies now. Could space-time surfaces define a classical representation for the analogs of local wave functions forming a local counterpart of a Hilbert space?

How could one realize the Hilbert space inner product?

- Could one consider a sensible inner product for the pairs (f
_{1},g) having CD as a dynamic locus (SSFRs) (see this). The only realistic option consistent with the local scaling property seems to be that the locus of the integral defining the inner product must be the intersection of the space-time surfaces defined by (f_{i},g_{i}). By their dimension, the space-time surfaces have in the generic case a discrete set of intersection points so that the inner product is non-trivial. What suggests itself is that the inner product is determined by the intersection form of the space-time surfaces, most naturally its trace. The norm would in turn correspond to self-intersection form. Does this give rise to a positive definite inner product? - The situation would be the same as in the fermionic degrees of freedom where also intersection points would appear as arguments of n-point functions. That 4-D surfaces are in question conforms with the idea of generalized complex and symplectic structures reducing the number of degrees of freedom from 8 to 4.

**Could ordinary arithmetic operations be realized consciously in terms of arithmetic operations for the space-time surfaces?**

Could arithmetic operations be realized at the fundamental level. We have learned in the basic school algorithms for the basic arithmetics as stable associations and the basic arithmetics does not involve conscious thought except in the beginning when we learn the rules by concrete examples. This is very similar to what large language models do.

However, idiot savants (see the books "The man who mistook his wife for this hat" and "Musicophilia" by Oliver Sacks) can decompose numbers into prime factors without any idea about the concept of prime numbers and certainly do not do this consciously by an algorithm or by logical deduction. Could this process occur spontaneously at a fundamental level and for some reason idiot savants could be able to do this consciously, perhaps because they are not able to do this using usual cognitive tools. I have considered the TGD inspired model for this (see this and this). The basic idea of various models is the same. The decomposition of a number to its factors is a spontaneous quantum process observed by the idiot savant.

- The first first thing to notice that division is the time reversal of multiplication: one has co-algebra structure. ZEO (see this and this and this) allows both operations and co-operations and the decomposition of an integer to factors would correspond to a product with a reversed arrow of time. Could pairs of BSFR involving temporary time reversal be involved and be easier for idiot savants than for people with ordinary cognitive abilities? Could the arrow of time in ordinary cognition be highly stable and make these feats impossible? Could the time reversal for the formation of the product of space-time surfaces as generalized numbers make ordinary conscious arithmetics possible?
- M
^{8}-H duality and geometric Langlands correspondence (see this) suggest that the exponent of the Kähler function ex(K) for the region of the space-time surface represented by the polynomial with integer coefficients is some power D^{m}of the discriminant D of a polynomial, which has integer coefficients. D decomposes to a product of powers of ramified primes p_{i}, which are p-adically special. For a product (P_{1},g)*(P_{2},g)= (P_{1}P_{2},g) of space-time surfaces, the exponent of Kähler function is product of those for factors and thus product of powers of D_{m}for f_{1}and f_{2}. A polynomial must be involved and I have considered the possibility that a particular discriminant D could correspond to a partonic 2-surface determining polynomial assignable to the singularity of the space-time surfaces as a minimal surface (see this). - One can say that for polynomials (P
_{1},P_{2}) with integer coefficients, the space-time surface represents an ordinary integer identifiable as D with exp(K) ∝ D^{m}. For a topological single particle state, P is irreducible but can be unstable against a splitting to 2 surfaces unless the D is prime. If exp(K) is conserved in the decay process, the splitting can produce a pair of space-time surfaces such that one has D=D_{1}D_{2}. This would represent physically the factorization of an integer to two factors, co-multiplication as the reversal of the multiplication operation. ZEO allows both.The preservation of the exponeänt of the Kähler function in the splitting reflects quantum criticality meaning that the initial and final states are superpositions of space-time surfaces with the same value of exp(K). The thermodynamic analog is a microcanonical ensemble is a closed system in a thermodynamic equilibrium involving only states of the same energy.

- This consideration generalizes trivially to the case of the sum. The product for the discriminants corresponds to the sum for their logarithms. If the system is able to physically represent the logarithm of the discriminant and also experience this representation consciously, then the product of space-time surfaces corresponds to the product of discriminants and to sum of their logarithms.
The natural base for the logarithm is defined by some ramified prime p appearing in the discriminant. The measurement corresponding to the measurement of the exponent k of p

^{k}would be scaling pd/dp corresponding to the scaling generator of conformal algebra extended to a 4-D algebra in the TGD framework.If discriminant involves only a single ramified prime, the p-adic logarithm is uniquely defined. Just as in the case of co-product, the space-time surface representing integer k=k

_{1}+k_{2}represented by an irreducible polynomial (f,g) splits to two space-time surfaces (f_{1},g) and (f_{2},g) representing representing integers k_{1}and k_{2}.

*viz.*Turing and Gödel or the chapter About Langlands correspondence in the TGD framework.

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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