Number theoretical view about unitarity conditions for twistor lift

Twistorialization leads to the proposal that cuts in the scattering amplitudes are replaced with sums over poles, and that also many-particle states have discrete momentum and mass squared spectrum having interpretation in terms of bound states. Gravitation would be the natural physical reason for the discreteness of the mass spectrum and in string models it indeed emerges as "stringy" mass spectrum. The situation is very similar to that in dual resonance models, which were predecessors of super string theories.

Number theoretical discretization based on the hierarchy of extensions of rationals defining extensions of p-adic number fields gives rise to cognitive representatations as discrete sets of space-time surface and discretization of 4-momenta and S-matrix with discrete momentum labels. In number theoretic discretization cuts reduce automatically to sequences of poles. Whether this discretization is an approximation reflecting finite cognitive resolution or whether finite cognitive representation is a property of physical states reflecting itself as a condition that various parameters characterizing them belong to the extension considered, remains an open question.

One can approach the unitarity conditions also number theoretically. In the discretization forced by the extension of rationals the amplitudes are defined between states having a discrete spectrum of 4-momenta. Unitarity condition reduces to a purely algebraic condition involving only sums. In these conditions the Dirac delta functions associated with the mass squared of the resonances are replaced with Kronecker deltas.

1. For given extension of rationals the unitary conditions are purely algebraic equations
   

   i(T_mn+T*_nm)= ∑_r T_mrT*_nr = T_mnT*_nn +T_mmT_mn + ∑_{r≠ m,n} T_mrT*_nr .
   

   where T_mn belongs the extension. Complex imaginary unit i corresponds to that appearing in the extension of octonions in M⁸-H duality (see this).
   
   

2. In the forward direction m=n one obtains
   

   2Im(T_mm)= Re(T_mm)² + Im(T_mm)²+ P_m , P_m= ∑_{r≠ m} T_mrT*_mr .
   

   P_m represents total probability for non-forward scattering.
   
   

3. One can think of solving Im(T_mm) algebraically from this second order polynomial in the lowest order approximation in which T_mn=0 for m ≠ n. This gives
   

   2Im(T_mm)= 1+ (1-P_m-Re(T_mm)²)^1/2 .
   

   Reality requires 1-Re(T_mm)²-P_m≥ 0 giving
   

   Re(T_mm)²+P_m≤ 1 .
   

   This condition is identically true by unitarity since the probability for scattering cannot be larger than 1.
   

   Besides this the real root must belong to the original extension of rationals. For instance, if the extension of rationals is trivial, the quantity 1-P_m-Re(T_mm)² must be a square of rational y giving 1-P_m= y²+Re(T_mm)². In the case of extension y is replaced with a number in the extension. I am not enough of number theorist to guess how powerful this kind of number theoretical conditions might be. In any case, the general ansatz for S is a unitary matrix in extension of rationals and this kind of matrices form a group so that there is no hope about unique solution.
   
   

4. One could think of iterative solution of the conditions by assuming in the zeroth order approximation T_mn=0 for m≠ n giving Re(T_mm)² +Im(T_mm)²= 1 reducing to cos²(θ)+sin²(θ)=1. For trivial extension of rationals θ would correspond to Pythagorean triangle.
   

   For non-diagonal elements of T_mn one would obtain at the next step the conditions
   

   i(T_mn+T*_nm)= T_mnT*_nn + T_mmT*_nm .
   

   This gives a 2 linear equations for T_mn.
   
   

5. These conditions are not enough to give unique solution. Time reversal invariance gives additional conditions and might help in this respect. T invariance is slightly broken but CPT symmetry could replace T symmetry in the general situation.
   

   Time reversal operator T (to be not confused with T_mn above) is anti-unitary operator and one has S^†= T(S). In wave mechanics one can show that T-invariant S-matrix and thus also T-matrix is symmetric. The matrices of this kind do not form a group so that the conditions can be very powerful.
   

   Combined with the above equations symmetry gives
   

   2Im(T_mn)= T_mnT*_nn + T_mmT*_mn .
   

   The two conditions for T_mn in principle fix it completely in this order.
   

   One obtains from the real part of the equation
   

   2Im(T_mn)= Re[T_mnT*_nn + T_mmT*_mn] .
   

   The vanishing of the imaginary part gives
   

   Im[T_mnT*_nn + T_mmT*_mn]=0 .
   

   giving a linear relation between the real and imaginary parts of T_mn. No new number theoretical conditions emerge. This relation requires that real and imaginary parts belong to the extension.
   
   

6. At higher orders one must feed the resulting ansatz to the unitarity conditions for the diagonal elements T_nn. One can hope that the lowest order ansatz leads to rather unique solution by iteration of the unitarity conditions. In higher order conditions the higher order corrections appear linearly so that no new number theoretic conditions emerge at higher orders.
   

   Physical picture suggests that the S-matrices could be obtained by an iterative procedure. Since infinitely long procedure very probably leads out of the extension, one can ask whether the procedure should stop after finite steps. This property would pose an additional conditions to the S-matrix.
   
   

7. Diagonal matrices are solutions to the conditions and for then the diagonal elements are roots of unity in the extension of rationals considered. The automorphisms S_d→ US_dU^-1 produce new S-matrices and if the unitary matrix U is orthogonal real matrix in algebraic extension satisfying therefore UU^T=1, the condition S=S^T is satisfied. There are therefore a large number of solutions.
   

   S-matrices diagonalizable in the extension are not the only solutions. The diagonalization of a unitary matrix S=S^T in general gives a diagonal S-matrix, for which the roots of unity in general do not belong to the extension. Also the diagonalizating matrix fails to be in the extension. This non-diagonalizability might have deep physics content and explain why the physically natural state basis is not the one in which S-matrix is diagonal. In the case of density matrix it would guarantee stability of entanglement.
   
   

To sum up, number theoretic conditions could give rise to highly unique discrete S-matrices, when CPT symmetry can be formulated purely algebraically and be combined with unitarity. CPT symmetry might not however allow formulation in terms of automorphisms of diagonal unitary matrices analogous to orthogonal transformations.

See the article More about the construction of scattering amplitudes in TGD framework or the chapter The Recent View about Twistorialization in TGD Framework.

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

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