  The sign problem is a major obstacle in quantum Monte Carlo simulations for
many-body fermion systems. We examine this problem with a new perspective based
on the Majorana reflection positivity and Majorana Kramers positivity. Two
sufficient conditions are proven for the absence of the fermion sign problem.
Our proof provides a unified description for all the interacting lattice
fermion models previously known to be free of the sign problem based on the
auxiliary field quantum Monte Carlo method. It also allows us to identify a
number of new sign-problem-free interacting fermion models including, but not
limited to, lattice fermion models with repulsive interactions but without
particle-hole symmetry and interacting topological insulators with spin-flip
terms.
