On Adelic Strings

New approach to p-adic and adelic strings, which takes into account that not only world sheet but also Minkowski space-time and string momenta can be p-adic and adelic, is formulated. p-Adic and adelic string amplitudes are considered within Feynman’s path integral formalism. The adelic Veneziano amplitude is calculated. Some discreteness of string momenta is obtained. Also, adelic coupling constant is equal to unity. 1. Since ancient times, scientists have been interested in unification of human knowledge on Nature. Superstring theory, recently resulted in the M-theory, is the best modern candidate for ”theory of everything”. It unifies all fundamental interactions. Gravity, nonabelian gauge symmetry, supersymmetry and space-time dimension are the most significant predictions of string theory. Superstring theory is a synthesis of deep physical principles and modern mathematical methods. In spite of many significant achievements, there are a number of open questions. One of the main actual problems is the space-time structure at the Planck scale. To find an appropriate answer to this problem it seems unavoidable to extend the Riemannian geometry, which is the Archimedean one and related e-mail: [email protected] 1 to real numbers, with the non-Archimedean geometry described by p-adic numbers. The corresponding mathematical instrument suitable for unification of Archimedean and non-Archimedean geometries, as well as of real and p-adic numbers, is the space of adeles. Recall that the field of rational numbers Q plays an important role not only in mathematics but also in physics. All numerical results of measurements belong to Q. On Q there is the usual (| · |∞) and p-adic (| · |p) absolute value, where p denotes a prime number. Completion of Q with respect to | · |∞ and | · |p yields the field of real (R ≡ Q∞) and p-adic (Qp) numbers, respectively. Many properties of p-adic numbers and their functions [1] are very different in comparison with real numbers and classical analysis. Any x ∈ Qp can be presented in the form x = p(x0 + x1p+ x2p 2 + · · ·), x0 6= 0, ν ∈ Z, (1) where xi ∈ {0, 1, ..., p− 1}. An adele a [2] is an infinite sequence a = (a∞, a2, a3, ..., ap, ...), (2) where a∞ ∈ R and ap ∈ Qp with restriction that ap ∈ Zp = {x ∈ Qp : x = x0 + x1p+ x2p + · · ·} for all but a finite set S of primes p. The set of all adeles A may be given in the form A = ⋃