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Abstract. Transport equation of interacting fermions is re-considered in the light of the conservation law. We show that the exact equation for the particle-hole excitation is derived directly from the Ward identity, by simply transforming it. It is almost identical to that derived diagrammatically byÉliashberg, but the collision integral is written in the symmetric form. We further discuss the hidden ultraviolet divergence in the both treatments, claiming that the microscopic justification of the Landau-Silin equation has not yet been completed. The divergence is removed by incorporating the particle-particle and hole-hole pairs of excitations, without sacrificing the rigour. This requires re-identification of the distribution function, but does not alter the basic structure of the fermi liquid theory.

Abstract. Transport process of non-interacting electrons is reformulated, in view of the exact treatment of interacting fermions reported by the present authors in this conference. It makes correspondence to the Boltzmann theory, covering the weakly localized regime, and also eliminates the hidden ultraviolet divergence by incorporating the particle-particle and hole-hole pairs. A mass renormalization of purely two-particle nature appears in the transport equation, giving rise to the conductivity of the Drude form. Taking the maximally crossed diagrams, the 2D conductivity is shown to vanish in the elastic scattering limit, while the distribition function preserves its sharp peak and the transport mass reduces to the free-electron value. The description differs from the self-consistent theory by Vollhardt and Wölfle, in which the Bethe-Salpeter structure of the scattering vertex is abandoned in favour of the time-reversal symmetry so the correspondence to the Boltzmann theory is lost.

Application of orthogonal polynomial expansion to quantum simulations is formulated on a general footing, implementing the regulation technique by Sota and Itoh for treating for the Gibbs oscillation. It is an alternative to the kernel polynomial method using Tchebyshev polynomial, but is simpler to handle and makes it possible to use all the popular orthogonal polynomials, covering finite, semi-infinite and infinite intervals of the eigenvalue spectrum. The accuracy can be made equivalent to direct diagonalization, with the resolution being homogeneous in the whole range of the spectrum. The target quantities can be as diverse as including eigenvectors, as well as all sorts of one-particle properties and correlation functions, involving thermal average and quantum time evolution. It can also be used as a handy tool for solving linear algebraic equations.