Limits of regularizations for generalized function solutions to the Schrödinger equation with "square root of delta" initial value

Polymer quantum mechanics as a deformation quantization

“…(or ρ = |ϕ ϕ| in Dirac notation), where φ, ϕ ∈ H d . From (28), we can observe that the operator ρ is an integral operator, then by the Weyl's inversion formula (27) its corresponding phase space representation function is given by…”

“…Then, in order that the Wigner-Weyl map and the Wigner function result well defined under the d → 0 and 1/d → 0 limits, the domain of Φ corresponds to the almost periodic functions on the phase space, that is, the space of finite linear combinations of sine and cosine trigonometric functions closed under the uniform norm [26]. As a consequence of the fact that the integral expression of the Wigner function (55) with respect to the Hilbert space H d contains also a Gaussian measure factor dµ d , it is possible to demonstrate that the limit of this kind of integrals converge to the normalized Haar measure on the Bohr compactification R B with respect to the weak topology [27], [28]. This analysis suggests that, within the deformation quantization approach introduced here, the polymer representation and therefore the Wigner function associated to LQC can be thought as a distributional limit of the standard Schrödinger representation.…”