Tunneling in polymer quantization and quantum zeno effect
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Date
2014
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Izmir Institute of Technology
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Abstract
Polymer quantization is a non-standard and exotic representation of the canonical
commutation relations which is introduced in the context of loop quantum gravity to investigate
the low energy limit of this non-perturbative quantization of gravity. It is one
of the representations of the Weyl-Heisenberg algebra which is inequivalent to the standard
Schr¨odinger representation. Since this representation is inequivalent to Schr¨odinger mechanics,
by Stone-von Neuman uniqueness theorem there should not be a one to one correspondence
between the operators of the two representations. It turns out that, one can not define
the position and momentum operators simultaneously in this construction. In this work, we
use the standard position operator and a second operator which is the analog of ^p. To define
an operator similar to the momentum operator ^p, one needs to use a regularization length
scale which can not be removed and stays as a free parameter in the theory. This free parameter
is interpreted to descend from the fundamental discreteness of space in loop quantum
gravity.
As another application of the polymer quantization scheme, in this work we investigate
the one dimensional quantum mechanical tunneling phenomenon from the perspective
of polymer representation of a non-relativistic point particle, derive the transmission and
reflection coefficients and show that they add up to one which is the requirement of probability
conservation. Since any tunneling phenomenon inevitably evokes a tunneling time we
attempt an analytical calculation of tunneling times by defining an operator well suited in
discrete spatial geometry. We expand our time expression in a Maclaurin series around zero
polymer length scale and arrive at results which hint at appearance of the Quantum Zeno
Effect in polymer framework. Quantum Zeno effect is the inhibition of a quantum system
from making a transition from an initial state to a final state. And in summary, as a result of
our work we can say that discretization of space leads to the Quantum Zeno effect.
Description
Thesis (Master)--Izmir Institute of Technology, Physics, Izmir, 2014
Includes bibliographical references (leaves 47-51)
Text in English; Abstract: Turkish and English
viii, 57 leaves
Includes bibliographical references (leaves 47-51)
Text in English; Abstract: Turkish and English
viii, 57 leaves
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Keywords
Quantum Zeno effect, Polymer quantization