Fermion Determinant Calculus

The path-integral of the fermionic oscillator with a time-dependent frequency is analyzed. We give the exact relation between the boundary condition to define the domain in which the path-integral is performed and the transition amplitude that the path-integral calculates. According to this relation, the amplitude suppressed by a zero mode does not indicate any special dynamics, unlike the analogous situation in field theories. It simply says the path-integral picks up a combination of the amplitudes that vanishes. The zero mode that is often neglected in the reason of not being normalizable is necessary to obtain the correct answer for the propagator and to avoid an anomaly on the fermion number. We give a method to obtain the fermionic determinant by the determinant of a simple (2\times 2) matrix, which enables us to calculate it for a variety of boundary conditions.