Estimation of tunneling times using Nelson's quantum stochastic process

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We have to take into account the half integer spin degree of freedom to the standart Nelson : s quantum stochastic process to estimate the tunneling time measured in the neutron spin echo experiments. In this research, we construct a quantum stochastic process using the Wigner function on a lattice phase space. The lattice composed of even lattice points represents a half integer spin space. The problem has been considered by Cohendet et al. in 1987 for odd lattice. In the first paper, we show that the Wigner functions do exit even on the even lattice contrary to the previous paper by Cohendet et al. There are infinitely many solution satisfying the conditions which reasonable Wigner function should respect and all of them are derived from one Wigner function by a orthogonal transformation. In the second paper, we propose a condition which ensures the correct marginal distributions of the Wigner function along tilted lines on a lattice. Under this condition we get the Wigner function without ambiguity if the lattice is odd. In the third manuscript we construct a quantum Markov process using the Wigner function formulated in the first and second paper. Introducing so-called ghost variables both on even and odd lattices, it is show that we can make the same kind of discussion as Cohendet et al.