Aspects of polymer quantization

D. Demir, Ozan Sargın

Franco Strocchi

Ichiro Ohba

Tommaso F. Demarie, Daniel R. Terno

Polymer quantization is a useful toy model for the mathematical aspects of loop quantum gravity and is interesting in its own right. Analyzing entropies of physically equivalent states in the standard Hilbert space and the polymer Hilbert space we show that they converge in the limit of vanishing polymer scale. We derive a general bound that relates entropies of physically equivalent states in unitarily inequivalent representations.

Subhajit Barman, Gopal Sardar

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.

Maxwell R. Siebersma, Basie Seibert, Samuel Shuman +1 more

"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.