https://matpitka.blogspot.com/2024/09/

## Tuesday, September 03, 2024

### Extension of Langlands geometric duality to trinity involving also physics-geometry correspondence

The master formula for TGD allowing construction of quantum states using the interpretation of space-time surfaces as numbers realizes the analog of geometric Langlands duality and generalizes it to a trinity. Geometric Langlands correspondence assigns to a pair of elements of a function field, which is a number theoretic object, a geometric object as algebraic surface having interpretation also as a Riemann surface with K\"ahler structure, twistor structure and spinor structure. This extends the number-theory-algebraic geometry duality to trinity and physics becomes the third part of a trinity.
1. The most high level form of number theory corresponds to function fields, which are infinite-D structured. In TGD, the pairs (f1,f2) of two functions of generalized complex coordinates of H=M4×CP2 define a linear space and the functions fi are elements of a function field. This is the number theoretic side of the Langlands geometric duality.
2. A function pair, whose root (f1,f2)=(0,0) defines a space-time surface in H and induces the number field structure of the function field to the space of space-time surfaces, "world of classical worlds" (WCW). Basic arithmetic operations of the number field apply to the component functions fi and induce corresponding operations for space-time surfaces in WCW. The notion of induction, which is the basic principle of TGD, is central also here. It is missing from standard physics and also string models.
3. The root as a space-time surface obeys holography =holomorphy principle and is a minimal surface (as classical representation of generalized massless particle and massless field equations) and represents the geometry side of the geometric Langlands duality. This connection represents geometric Langlands duality in TGD. Riemannian geometries restricted to algebraic geometries is what makes the geometric Langlands duality possible.

It is still unclear whether the choice of the classical action defining space-time surfaces and producing, apart from singularities, a minimal surface as an outcome, is only analogous to a choice of the coordinates and whether the recent choice (volume action + Kaehler action) is only the most convenient choice. If so, the laws of physics boil down to a completely action independent form, that is to the construction of quantum states induced by the products for space-time surfaces regarded as generalized numbers.

4. Space-time surfaces as minimal surfaces with generalized complex structure and are extremals for any variational principle constructible in terms of the induced geometry since extremal property reduces to the generalized complex structure. The action makes itself visible only at the singularities.
5. Langlands geometric duality becomes actually a trinity: number theory<-->geometry<--->physics. The number theory<-->geometry part of this trinity duality corresponds to Langlands geometric duality. The geometry<--->physics part is the TGD counterpart of Einstein's equations identifying geometry and physics.

See the article About Langlands correspondence in the TGD framework or the chapter with the same title.

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.

### Space-time surfaces as numbers and construction of quantum states in terms of products of space-time surfaces

The 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.

1. 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.
2. For H=M4× CP2, this would correspond to the lowest level of the hierarchy of infinite primes but the Cartesian powers of H=M4× CP2 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.
In zero energy ontology (ZEO) (see this), quantum states corresponds to spinor fields of WCW, which consists of space-time surfaces satisfying holography and therefore being analogous to Bohr orbits, and also having interpretation as elements of number field so that one can multiply them (see this and this). WCW spinor fields assign to a given space-time surface a pair of fermionic Fock states at its 3-D ends located at the opposite light-like boundaries of the causal diamond (CD). Could one multiply two WCW spinor fields so that the space-time surfaces appearing as their arguments are multiplied

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?

1. Let's examine the function pairs (f1,f2) and (g1,g2) defined in H=M4× CP2 and the corresponding space-time surfaces for which (f1,f2)=(0,0) and (g1,g2)=(0,0) apply. It should be noted that, for example, that the condition f1=0 defines the analog of a 6-D twistor space, and the space-time surface X4 is the intersection of the analogs of the twistor bundles of M4 and CP2, i.e., its base space.
2. The product of the function pairs is (f1g1,f2g2). Its components vanish in four cases.
1. The cases (f1,f2)= (0,0) and (g1,g2)=(0,0) correspond to the union of the incoming surfaces. The corresponding particles are free.
2. The cases (f1,g2)= (0,0) and (f2,g1)=(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 (f1,g2)= (0,0) and (f2,g1)=(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. f1=0 and g2=0 (f2=0 and g1=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).

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 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!

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.