Polymer quantum mechanics as a deformation quantization

Abstract

We analyze the polymer representation of quantum mechanics within the deformation quantization formalism. In particular, we construct the Wigner function and the star-product for the polymer representation as a distributional limit of the Schrödinger representation for the Weyl algebra in a Gaussian weighted measure, and we observe that the quasi-probability distribution limit of this Schrödinger representation agrees with the Wigner function for Loop Quantum Cosmology. Further, the introduced polymer star-product fulfills Bohr's correspondence principle even though not all the operators are well defined in the polymer representation. Finally, within our framework, we also derive a generalized uncertainty principle which resembles the one appearing in different scenarios, including theories with a minimal length.

The Polymer representation for the scalar field: a Wigner functional approach

“…In this section we briefly review the polymer representation of quantum mechanics as a limit of the Schrödinger representation for the Weyl algebra in a Gaussian weighted measure. We will closely follow the description of the formalism as described in [16]. For simplicity, we focus on systems with one degree of freedom, nevertheless a generalization to more dimensions follows straightforwardly.…”

“…We turn now to the definition of a quantization prescription on H d . Following [16], there is a linear map Φ from the set of classical observables given by S(R 2 ), the Schwartz space of functions defined on the phase space R 2 whose derivatives are rapidly decreasing, into the linear operator space L(H d ). This map, called the Weyl quantization, is given by the formula…”

“…In addition, the projections on the momentum and position leads to marginal probability densities [16], usually called shadows 1 2π R ρ(ϕ)(p, q)dp = ||ϕ|| 2 H d , and…”

“…respectively. The derived states allow us to construct the corresponding Wigner function associated to polymer representation as a limiting case [16]. Starting with the Wigner function defined in (18) for the vector states ϕ v , the limit 1/d → 0 give us…”

“…According to [16] and [21], in order to construct the Wigner function for the scalar field, we need to provide a quantization map such that it takes the Poisson bracket…”