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

Abstract

In this paper, we analyze the the polymer representation of the real-valued scalar field theory within the deformation quantization formalism. Specifically, we obtain the polymer Wigner functional by taking the limit of Gaussian measures in the Schrödinger representation. The limiting functional corresponds to the polymer representation derived by using algebraic methods such as the GNS construction, and the Fock quantization procedure.

Star product representation of coherent state path integrals

“…We claim that our developments will be helpful to analyze several inconsistencies related to regularization and continuum limits, encountered in coherent state path integration [14]. From a different perspective, it will also be relevant to implement the techniques described here to establish the star product representation of spin foam models in the loop quantum gravity and loop quantum cosmology framework, since recently, a family of s-parametrized quasi-probability distributions and its relation with the Wigner-Weyl representation has been obtained [15][16][17]. We intend to dedicate a future publication to address this work in progress.…”

“…With the preceding formulas, we now compute the P-representation corresponding to the non-diagonal density operator |α i α f |. By using the explicit expression for P(α) in (16) and the definition of the generalized delta function (25), we obtain…”

“…Determining this star product representation, by means of quasi-probability distributions, may allow us to analyze several aspects related to some inconsistences associated with regularization techniques and the continuum limit, encountered in coherent path integrals [14]. From another perspective, it will also be relevant to implement the formalism described here in order to establish the star product representation of spin foam integrals, in the context of loop quantum gravity and loop quantum cosmology, since recently, a family of quasi-probability distributions has been determined [15][16][17].…”

Quasi-probability distributions in loop quantum cosmology

“…Within the LQC and LQG scenarios, the only quasi-probability distribution that has been recently studied is the Wigner function, specifically in reference [28], where the Wigner function over the Bohr compactification of the real line was defined. Further, its relation to the Schrödinger representation as a limit for systems with finite and infinite degrees of freedom was determined in [29,30]. Being our main aim to extend these results, the purpose of the paper is to obtain a complete family of sparametrized quasi-probability distributions for LQC and their corresponding s-ordered Weyl quantization map.…”

Deformation quantization of constrained systems: a group averaging approach

“…One crucial element within this formulation resides on the definition of the Wigner distribution, which corresponds to a phase space representation of the density matrix that is responsible for all auto-correlation properties and trans ition ampl itudes of a given quantum mechanical system. Despite the deformation quantization formalism have undoubtedly provided important contributions not only in pure mathematics [16,17], but it has also prove to be a reliable technique in the understanding of many physical quantum systems [18,19], including recently certain aspects of the loop representation of quantum cosmology and quantum gravity [20,21]. However, it is important to mention that the application of the deformation quantization formalism to constrained systems has been poorly developed [22][23][24], (see also [25,26] for applications in constrained models associated to field theory and cosmology, respectively).…”