Wednesday, January 12, 2022

Critical questions related to the number theoretical view about fundamental dynamics

One can pose several critical questions helping to further develop the proposed number theoretic picture.

Is mere recombinatorics enough as fundamental dynamics?

Fundamental dynamics as mere re-combination of free quarks to Galois singlets is attractive in its simplicity but might be an over-simplification. Can quarks really continue with the same momenta in each SSFR and even BSFR?

  1. For a given polynomial P, there are several Galois singlets with the same incoming integer-valued total momentum pi. Also quantum superpositions of different Galois singlets with the same incoming momenta pi but fixed quark and antiquark numbers are in principle possible. One must also remember Galois singlet property in spin degrees of freedom.
  2. WCW integration corresponds to a summation over polynomials P with a common ramified prime (RP) defining the p-adic prime. For each P of the Galois singlets have different decomposition to quark momenta. One can even consider the possibility that only the total quark number as the difference of quark and antiquark numbers is fixed so that polynomials P in the superposition could correspond to different numbers of quark-antiquark pairs.
  3. One can also consider a generalization of Galois confinement by replacing classical Galois singlet property with a Galois-singlet wave function in the product of quark momentum spaces allowing classical Galois non-singlets in the superposition.

    Hydrogen atom serves as an illustration: electron at origin would correspond to classical ground state and s-wave correspond to a state invariant under rotations such that the position of electron is not anymore invariant under rotations. The proposal for transition amplitudes remains as such otherwise.

Note however that the basic dynamics at the level of a single polynomial would be recombinatorics for all these options.

General number theoretic picture of scattering

Only the interaction region has been considered hitherto. One must however understand how the interaction region is determined by the 4-surfaces and polynomials associated with incoming Galois singlets. Also the details of the map of p-adic scatting amplitude to a real one must be understood.

The intuitive picture about scattering is as follows.

  1. The incoming particle "i" is characterized by p-adic prime pi, which is RP for the corresponding 4-surface in M8. Also the "adelic" option that all RPs characterize the particle, is considered below.
  2. The interaction region corresponds to a polynomial P. The integration over WCW corresponds to a sum over several P:s with at least one common RP allowing to map the superposition of amplitudes to real amplitude by canonical identification I: ∑ xnpn→ ∑ xnp-n.

    If one gives up the assumption about a shared RP, the real amplitude is obtained by applying I to the amplitudes in the superposition such that RP varies. Mathematically, this is an ugly option.

  3. If there are several shared RPs, in the superposition over polynomials P, one can consider an adelic picture. The amplitude would be mapped by I to a product of the real amplitudes associated with the shared RP:s. This brings in mind the adelic theorem stating that rational number is expressible as a product of the inverses of its p-adic norms. The map I indeed generalizes the p-adic norm as a map of p-adics to reals. Could one say that the real scattering amplitude is a product of canonical images of the p-adic amplitudes for the shared RP:s? Witten has proposed this kind of adelic representation of real string vacuum amplitude.

    Whether the adelization of the scattering amplitudes in this manner makes sense physically is far from clear. In p-adic thermodynamics one must choose a single p-adic prime p as RP. Sum over ramified primes for mass squared values would give CP2 mass scale if there are small p-adic primes present.

The incoming polynomials Pi should determine a unique polynomial P assignable to the interaction regions as CD to which particles arrive. How?
  1. The natural requirement would be that P possess the RPs associated with Pi:s. This can be realized if the condition Pi=0 is satisfied and P is a functional composite of polynomials Pi. All permutations π of 1,...,n are allowed: P= Pi1○ Pi2○ ....Pin with (i1,π(1),...,π(n)). P possesses the roots of Pi.

    Different permutations π could correspond to different permutations of the incoming particles in the proposal for scattering amplitudes so that the formation of area momenta xi+1= ∑k=1ipk in various orders would corresponds to different orders of functional compositions.

  2. Number theoretically, interaction would mean composition of polynomials. I have proposed that so-called cognitive measurements as a model for analysis could be assigned with this kind of interaction (see this and this). The preferred extremal property realized as a simultaneous extremal property for both K\"ahler action and volume action suggests that the classical non-determinism due to singularities as analogs of frames for soap films serves as a classical correlate for quantum non-determinism (see this).
  3. If each incoming state "i" corresponds to a superposition of Pi:s with some common RPs, only the RP:s shared by all compositions P from these would appear in the adelic image. If all polynomials Pi are unique (no integration over WCW for incoming particles), the canonical image of the amplitude could be the product over images associated with common RPs.

    The simplest option is that a complete localization in WCW occurs for each external state, perhaps as a result of cognitive state preparation and reduction, so that P has the RP:s of Pis as RP:s and adelization could be maximal.

See the article About TGD counterparts of twistor amplitudes or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

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