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.