Analytic Continuation of Operators -- operators acting complex s-times -- Applications: from Number Theory and Group Theory to Quantum Field and String Theories

TL;DR

This paper explores the analytic continuation of operators to complex powers, enabling new applications across mathematics and physics, including fractional calculus, number theory, quantum field theory, and string theory.

Contribution

It introduces a framework for analytically continuing operators to complex times, preserving certain properties and enabling novel applications in various scientific fields.

Findings

Extended fractional calculus with preserved commutativity.

Applications to number theory and non-integer power series.

Insights into non-local effects and symmetry deformations in physics.

Abstract

We are used to thinking of an operator acting once, twice, and so on. However, an operator acting integer times can be consistently analytic continued to an operator acting complex times. Applications: (s,r) diagrams and an extension of Fractional Calculus where commutativity of fractional derivatives is preserved, generating integrals and non-standard derivations of theorems in Number Theory, non-integer power series and breaking of Leibniz and Chain rules, pseudo-groups and symmetry deforming models in particle physics and cosmology, non-local effect in analytic continued matrix representations, particle-physics-like scatterings of zeros of analytic continued Bernoulli polynomials (physics/9705021), analytic continuation of operators in QM, QFT and Strings.