  Definition of the determinant of Euclidean Dirac operator in the nontrivial
sector of gauge fields suffers from an inherent ambiguity. The popular
Osterwalder-Schrader (OS) recipe for the conjugate Dirac field leads to the
option of a vanishing determinant. We propose a novel representation for the
conjugate field which depends linearly on the Dirac field and yields a
nonvanishing determinant in the nontrivial sector. Physics, it appears, chooses
this second option becuase the novel representation leads to a satisfactory
resolution of two outstanding problems, the strong CP and U(1) problems,
attributed to instanton effects.
