The Special Theory of Relativity and Quantum Mechanics merge in the key
principle of Quantum Field Theory, the Principle of Locality. We review some
examples of its ``unreasonable effectiveness'' (which shows up best in the
formulation of Quantum Field Theory in terms of operator algebras of local
observables) in digging out the roots of Global Gauge Invariance in the
structure of the local observable quantities alone, at least for purely massive
theories; but to deal with the Principle of Local Gauge Invariance is still a
problem in this frame. This problem emerges also if one attempts to figure out
the fate of the Principle of Locality in theories describing the gravitational
forces between elementary particles as well. Spacetime should then acquire a
quantum structure at the Planck scale, and the Principle of Locality is lost.
It is a crucial open problem to unravel a replacement in such theories which is
equally mathematically sharp and reduces to the Principle of Locality at larger
scales. Besides exploring its fate, many challenges for the Principle of
Locality remain; among them, the analysis of Superselection Structure and
Statistics also in presence of massless particles, and to give a precise
mathematical formulation to the Measurement Process in local and relativistic
terms; for which we outline a qualitative scenario which avoids the EPR
Paradox.Comment: 36 pages. Survey partially based on a talk delivered at the Meeting
"Algebraic Quantum Field Theory: 50 years", Goettingen, July 29-31, 2009, in
honor of Detlev Buchholz. Submitted to Journal of Mathematical Physic
