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× 2 matrix, which enables us to calculate it for a variety of boundary conditions. PACS: 11.30.Fs; 03.65.-w; 02.70.Hm; 11.30.Pb Fermionic determinant of the operator D is the one we first encounter in the analysis of the quantum physics in the path-integral formalism. It is the exact result of the Grassmann path-integral made from a bilinear Lagrangian ψ̄Dψ over the fermionic degrees of freedom ψ and ψ̄. The determinant carries an important information about the time evolution of the fermions under the influence of bosonic background. Especially when D or its adjoint D† has a zero mode, the zero-frequency eigenmode of the operator, the determinant vanishes and corresponding transition is suppressed. A typical example of such situation happens with D the Dirac operator in SU(2) gauge theory. It possesses zero modes in the instanton background. The consequent suppression of the transition is interpreted to reflect the fermion number violation due to the anomaly on the fermion current [1]. In spite of the importance of the understandings about the role of zero modes in pathintegral calculation, there seems to be a confusion even in the case of fermionic oscillator, the simplest system containing only one fermionic degree of freedom. Its Lagrangian ψ̄Dψ in imaginary time formalism is given by a simple first order differential operator D = d dτ + v(τ) (1)