3 research outputs found
Polymer state approximations of Schroedinger wave functions
It is shown how states of a quantum mechanical particle in the Schroedinger
representation can be approximated by states in the so-called polymer
representation. The result may shed some light on the semiclassical limit of
loop quantum gravity.Comment: 11 pages, 1 figure, Conclusions section adde
Hamiltonian and physical Hilbert space in polymer quantum mechanics
In this paper, a version of polymer quantum mechanics, which is inspired by
loop quantum gravity, is considered and shown to be equivalent, in a precise
sense, to the standard, experimentally tested, Schroedinger quantum mechanics.
The kinematical cornerstone of our framework is the so called polymer
representation of the Heisenberg-Weyl (H-W) algebra, which is the starting
point of the construction. The dynamics is constructed as a continuum limit of
effective theories characterized by a scale, and requires a renormalization of
the inner product. The result is a physical Hilbert space in which the
continuum Hamiltonian can be represented and that is unitarily equivalent to
the Schroedinger representation of quantum mechanics. As a concrete
implementation of our formalism, the simple harmonic oscillator is fully
developed.Comment: 19 pages, 2 figures. Comments and references added. Version to be
published in CQ
The LQG -- String: Loop Quantum Gravity Quantization of String Theory I. Flat Target Space
We combine I. background independent Loop Quantum Gravity (LQG) quantization
techniques, II. the mathematically rigorous framework of Algebraic Quantum
Field Theory (AQFT) and III. the theory of integrable systems resulting in the
invariant Pohlmeyer Charges in order to set up the general representation
theory (superselection theory) for the closed bosonic quantum string on flat
target space. While we do not solve the, expectedly, rich representation theory
completely, we present a, to the best of our knowledge new, non -- trivial
solution to the representation problem. This solution exists 1. for any target
space dimension, 2. for Minkowski signature of the target space, 3. without
tachyons, 4. manifestly ghost -- free (no negative norm states), 5. without
fixing a worldsheet or target space gauge, 6. without (Virasoro) anomalies
(zero central charge), 7. while preserving manifest target space Poincar\'e
invariance and 8. without picking up UV divergences. The existence of this
stable solution is exciting because it raises the hope that among all the
solutions to the representation problem (including fermionic degrees of
freedom) we find stable, phenomenologically acceptable ones in lower
dimensional target spaces, possibly without supersymmetry, that are much
simpler than the solutions that arise via compactification of the standard Fock
representation of the string. Moreover, these new representations could solve
some of the major puzzles of string theory such as the cosmological constant
problem. The solution presented in this paper exploits the flatness of the
target space in several important ways. In a companion paper we treat the more
complicated case of curved target spaces.Comment: 46 p., LaTex2e, no figure