Applications of Polymeric Quantisation

Chen Jia Pern Neville,Ching Chee Leong,A Yang,C.H Oh,S.C.C Ng,A.H Chan,W.Y Wang,K.K Phua

doi:10.1051/epjconf/201920609016

Chen Jia Pern Neville, Ching Chee Leong + Show 6 more

Open Access

https://doi.org/10.1051/epjconf/201920609016

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Abstract

Polymer quantisation is a background independent quantisation scheme inspired by loop quantum gravity. Under this quantisation scheme, it predicts that space is discretised and changes in multiples of a fundamental length scale λ. As a result, the momentum operator is not well-defined. However, a new operator can be defined such that a Schrödinger-like equation can be retrieved. The solutions give rise to eigenspectra which are similar to the standard counterparts, with an additional correction term due to λ. We present the basic principles of the polymer representation and apply it to the harmonic oscillator to study the phenomenological implications of such solutions. In addition, we consider an ensemble of such oscillators and calculated the thermodynamical properties for systems that safisty the bosonic and fermionic statistics. The results presented may have physical significance at high energy scales or in exotic matter.

Highlights

Polymer quantisation is a quantisation scheme inspired by the methods of Loop Quantum Gravity (LQG) to quantise gravity in a background independent manner [1,2,3]
We look at two important systems: the free particle and the harmonic oscillator [6]
Using the dimensions of the cavity in [9] and assuming that the harmonic oscillator length scale is on the order of the Bohr radius, the value for an upper bound for λ is estimated to be 10−17m

Summary

Introduction

Polymer quantisation is a quantisation scheme inspired by the methods of Loop Quantum Gravity (LQG) to quantise gravity in a background independent manner [1,2,3]. A crucial result produced by LQG is the introduction of a fundamental length scale λ of which forms the ‘quantum’ of space. Space is discretised into a structure similar to that of points on a lattice.the Hilbert space must be defined differently to accommodate new conditions. This results in an inequivalent representation – the so-called polymer representation. The mathematical structure of the polymer representation is different, which may lead to novel physics. One particular advantage of this method is that the systems that can be solved exactly under the polymer framework can be experimentally studied in laboratories, the bulk properties that arise due to a large collection of the polymer systems can be measured

Polymeric Quantisation

Conclusion