  It is sometimes speculated that the sign problem that afflicts many quantum
field theories might be reduced or even eliminated by choosing an alternative
domain of integration within a complexified extension of the path integral (in
the spirit of the stationary phase integration method). In this paper we start
to explore this possibility somewhat systematically. A first inspection reveals
the presence of many difficulties but - quite surprisingly - most of them have
an interesting solution. In particular, it is possible to regularize the
lattice theory on a Lefschetz thimble, where the imaginary part of the action
is constant and disappears from all observables. This regularization can be
justified in terms of symmetries and perturbation theory. Moreover, it is
possible to design a Monte Carlo algorithm that samples the configurations in
the thimble. This is done by simulating, effectively, a five dimensional
system. We describe the algorithm in detail and analyze its expected cost and
stability. Unfortunately, the measure term also produces a phase which is not
constant and it is currently very expensive to compute. This residual sign
problem is expected to be much milder, as the dominant part of the integral is
not affected, but we have still no convincing evidence of this. However, the
main goal of this paper is to introduce a new approach to the sign problem,
that seems to offer much room for improvements. An appealing feature of this
approach is its generality. It is illustrated first in the simple case of a
scalar field theory with chemical potential, and then extended to the more
challenging case of QCD at finite baryonic density.
