The Wigner functions on the one dimensional lattice are studied. Contrary to the previous claim in literature, Wigner functions exist on the lattice with any number of sites, whether it is even or odd. There are infinitely many solutions satisfying the conditions which reasonable Wigner functions should respect. After presenting a heuristic method to obtain Wigner functions, we give the general form of the solutions. Quantum mechanical expectation values in terms of Wigner functions are also discussed.

In order to determine the Wigner function uniquely, we introduce a new condition which ensures that the Wigner function has correct marginal distributions along tilted lines. For a system in N dimensional Hilbert space, whose "phase space" is a lattice with N 2 sites, we get different results depending on whether N is odd or even. Under the new condition, the Wigner function is determined if N is an odd number, but it does not exist if N is even.

We investigate, analytically and numerically, the fermion determinant of a new action on a (1+1)-dimensional Euclidean lattice. In this formulation the discrete chiral symmetry is preserved and the number of fermion components is a half of that of Kogut-Susskind. In particular, we show that our fermion determinant is real and positive for U(1) gauge group under specific conditions, which correspond to gauge conditions on the infinite lattice. It is also shown that the determinant is real and positive for SU(N) gauge group without any condition. PACS number(s): 11.15.Ha

We investigate, analytically and numerically, the fermion determinant of a new action on a (1+1)-dimensional Euclidean lattice. In this formulation the discrete chiral symmetry is preserved and the number of fermion components is a half of that of Kogut-Susskind. In particular, we show that our fermion determinant is real and positive for U(1) gauge group under specific conditions, which correspond to gauge conditions on the infinite lattice. It is also shown that the determinant is real and positive for SU(N) gauge group without any condition. PACS number(s): 11.15.Ha

We investigate the fermion determinant of a new action on a (1ϩD)-dimensional lattice for SU͑2͒ gauge groups. This action possesses the discrete chiral symmetry and provides 2 D -component fermions. We also comment on the numerical results on fermion determinants in the (1ϩD)-dimensional SU͑3͒ gauge fields.