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- Title
The Riemann hypothesis and tachyonic off-shell string scattering amplitudes.
- Authors
Perelman, Carlos Castro
- Abstract
The study of the 4 -tachyon off-shell string scattering amplitude A 4 (s , t , u) , based on Witten's open string field theory, reveals the existence of poles in the s-channel and associated to a continuum of complex "spins" J. The latter J belong to the Regge trajectories in the t, u channels which are defined by - J (t) = - 1 - 1 2 t = β (t) = 1 2 + i λ ; - J (u) = - 1 - 1 2 u = γ (u) = 1 2 - i λ , with λ = r e a l . These values of β (t) , γ (u) given by 1 2 ± i λ , respectively, coincide precisely with the location of the critical line of nontrivial Riemann zeta zeros ζ (z n = 1 2 ± i λ n) = 0 . It is argued that despite assigning angular momentum (spin) values J to the off-shell mass values of the external off-shell tachyons along their Regge trajectories is not physically meaningful, their net zero-spin value J (k 1) + J (k 2) = J (k 3) + J (k 4) = 0 is physically meaningful because the on-shell tachyon exchanged in the s-channel has a physically well defined zero-spin. We proceed to prove that if there were nontrivial zeta zeros (violating the Riemann Hypothesis) outside the critical line R e a l z = 1 / 2 (but inside the critical strip) these putative zeros d o n ′ t correspond to any poles of the 4 -tachyon off-shell string scattering amplitude A 4 (s , t , u) . We finalize with some concluding remarks on the zeros of sinh(z) given by z = 0 + i 2 π n , continuous spins, non-commutative geometry and other relevant topics.
- Subjects
SCATTERING amplitude (Physics); RIEMANN hypothesis; STRING theory; ANGULAR momentum (Mechanics); TACHYONS
- Publication
European Physical Journal C -- Particles & Fields, 2022, Vol 82, Issue 5, p1
- ISSN
1434-6044
- Publication type
Article
- DOI
10.1140/epjc/s10052-022-10429-3