The Reconstruction Problem and Weak Quantum Values

Quantum Mechanical weak values are an interference effect measured by the cross-Wigner transform W({\phi},{\psi}) of the post-and preselected states, leading to a complex quasi-distribution {\rho}_{{\phi},{\psi}}(x,p) on phase space. We show that the knowledge of {\rho}_{{\phi},{\psi}}(z) and of one of the two functions {\phi},{\psi} unambiguously determines the other, thus generalizing a recent reconstruction result of Lundeen and his collaborators.Comment: To appear in J.Phys.: Math. Theo

Weak values of a quantum observable and the cross-Wigner distribution

We study the weak values of a quantum observable from the point of view of the Wigner formalism. The main actor is here the cross-Wigner transform of two functions, which is in disguise the cross-ambiguity function familiar from radar theory and time-frequency analysis. It allows us to express weak values using a complex probability distribution. We suggest that our approach seems to confirm that the weak value of an observable is, as conjectured by several authors, due to the interference of two wavefunctions, one coming from the past, and the other from the future.Comment: Submitted for publicatio

Determinant of Laplacians on Heisenberg Manifolds

We give an integral representaion of the zeta-reguralized determinant of Laplacians on three dimensional Heisenberg manifolds, and study a behaivior of the values when we deform the uniform discrete subgroups. Heiseberg manifolds are the total space of a fiber bundle with a torus as the base space and a circle as a typical fiber, then the deformation of the uniform discrete subgroups means that the "radius" of the fiber goes to zero. We explain the lines of the calculations precisely for three dimensional cases and state the corresponding results for five dimensional Heisenberg manifolds. We see that the values themselves are of the product form with a factor which is that of the flat torus. So in the last half of this paper we derive general formulas of the zeta-regularized determinant for product type manifolds of two Riemannian manifolds, discuss the formulas for flat tori and explain a relation of the formula for the two dimensional flat torus and Kronecker's second limit formula.Comment: 42 pages, no figure