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arXiv:1908.09194 · Classical and Quantum Gravity

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

Jasel Berra–Montiel

Published Nov 26, 2019• 10 citations
arXivHTMLPDF

Abstract

Abstract In this paper, we analyze 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.

DOI
10.1088/1361-6382/ab5bc7

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Abstract In this paper, we analyze 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.
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 tomographic representation of quantum fields within the deformation quantization formalism is constructed. By employing the Wigner functional we obtain the symplectic tomogram associated with quantum fields. In addition, the tomographic version of the Wigner map allows us to compute the symbols corresponding to field operators. Finally, the functional integral representation of the tomographic star product is determined. Some possible applications of the formalism to loop quantum cosmology and loop quantum gravity are briefly discussed.
"Polymer quantum mechanics" is the name given to a quantization scheme inspired by loop quantum gravity in which the configuration space of the theory is chosen to have a discrete topology. Polymer quantization yields a representation of the canonical commutation relations that is genuinely distinct from the conventional "Schrödinger" representation. In this paper, we summarize the main features of polymer quantum mechanics and investigate in detail the polymer quantization of systems with configuration spaces that are classically compact. We show explicitly how using the standard construction of polymer states leads to a Hilbert space of states defined on a finite graph of points. By way of example, we find the exact energy eigenvalues and eigenfunctions for a particle on a ring and a particle in a box defined on such lattices, and discuss similarities and differences from standard Schrödinger quantum mechanics. We also explore the continuum limit of states in these systems, and demonstrate in detail how the exact eigenfunctions in the position representation approach their continuum counterparts.
Motivated by some well-known results in the phase space description of quantum optics and quantum information theory, we aim to describe the formalism of quantum field theory by explicitly considering the holomorphic representation for a scalar field within the deformation quantization program. Notably, the symbol of a symmetric ordered operator in terms of holomorphic variables may be straightforwardly obtained by the quantum field analogue of the Husimi distribution associated with a normal ordered operator. This relation also allows to establish a [Formula: see text]-equivalence between the Moyal and the normal star-products. In addition, by writing the density operator in terms of coherent states we are able to directly introduce a series representation of the Wigner functional distribution, which may be convenient in order to calculate probability distributions of quantum field observables without performing formal phase space integrals at all.
The area dependence of entanglement entropy of a free scalar field is often understood in terms of coupled harmonic oscillators. In Schrodinger quantization, the Gaussian nature of ground state wave function for these oscillators is sufficient to provide the exact form of the reduced density matrix and its eigenvalues, thus giving the entanglement entropy. However, in polymer quantization the ground state is not Gaussian and the formalism that can provide the exact analytical form of the reduced density matrix is not yet known. In order to address this issue, here we treat the interaction between two coupled harmonic oscillators in the perturbative approach and evaluate the entanglement entropy in Schrodinger and polymer quantization. Contrary to Schrodinger quantization, we show that in the high frequency regime the entanglement entropy decreases for polymer quantization, keeping the ratio of coupling strength to the square of individual oscillator frequency fixed. Furthermore for the free scalar field we validate the area dependence of entanglement entropy in Fock quantization and demonstrate that polymer quantization produces a similar area law.
Abstract We develop a deformation quantization approach to quantum mechanics that exhibits a nonzero minimal uncertainty in position through the modification of commutation relations for the operators of position and momentum. Deformation quantization is a method of quantizing a classical Hamiltonian system by suitably deforming the Poisson algebra of the system. The developed theory of deformation quantization is non-formal. An appropriate integral formula for the star-product is introduced, along with a suitable space of functions on which the star-product is well defined. A C*-algebra of observables and a space of states are constructed. Moreover, an operator representation in momentum space is presented. Finally, an example of states of maximal localization is given in terms of generalized Wigner functions.