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 p_i. Also quantum superpositions of different Galois singlets with the same incoming momenta p_i 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 p_i, 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: ∑ x_npⁿ→ ∑ x_np^-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.

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 P_i 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 P_i:s. This can be realized if the condition P_i=0 is satisfied and P is a functional composite of polynomials P_i. All permutations π of 1,…,n are allowed: P= P_i1○ P_i2○ ….P_in with (i₁,…i_n)=(π(1),…,π(n)). P possesses the roots of P_i.

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 x_i+1= ∑_k=1ⁱp_k in various orders would corresponds to different orders of functional compositions.

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 P_is 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.