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We classify the sign-problem-free relativistic fermion actions on the basisof the Majorana representation. In the Majorana representation, thesign-problem-free condition is given by the semi-positivity of a Pfaffian. Weshow that the known sign-problem-free actions of the Dirac fermions, which areusually understood from the semi-positivity of a determinant, e.g., the actionof quantum chromodynamics with nonzero chiral chemical potential or nonzeroisospin chemical potential, can also be understood from the semi-positivity ofa Pfaffian. We also derive new classes of the sign-problem-free relativisticfermion actions with Majorana-type source terms.

Classification of sign-problem-free relativistic fermions on the basis of the Majorana positivity