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Register a new userPropagators in Polymer Quantum Mechanics
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Polymer Quantum Mechanics is based on some of the techniques used in the loop quantization of gravity that are adapted to describe systems possessing a finite number of degrees of freedom. It has been used in two ways: on one hand it has been used to represent some aspects of the loop quantization in a simpler context, and, on the other, it has been applied to each of the infinite mechanical modes of other systems. Indeed, this polymer approach was recently implemented for the free scalar field propagator. In this work we compute the polymer propagators of the free particle and a particle in a box; amusingly, just as in the non polymeric case, the one of the particle in a box may be computed also from that of the free particle using the method of images. We verify the propagators hereby obtained satisfy standard properties such as: consistency with initial conditions, composition and Greens function character. Furthermore they are also shown to reduce to the usual Schrodinger propagators in the limit of small parameter $mu_0$, the length scale introduced in the polymer dynamics and which plays a role analog of that of Planck length in Quantum Gravity.
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We study decoherence in a simple quantum mechanical model using two approaches. Firstly, we follow the conventional approach to decoherence where one is interested in solving the reduced density matrix from the perturbative master equation. Secondly, we consider our novel correlator approach to decoherence where entropy is generated by neglecting observationally inaccessible correlators. We show that both methods can accurately predict decoherence time scales. However, the perturbative master equation generically suffers from instabilities which prevents us to reliably calculate the systems total entropy increase. We also discuss the relevance of the results in our quantum mechanical model for interacting field theories.
We characterize point transformations in quantum mechanics from the mathematical viewpoint. To conclude that the canonical variables given by each point transformation in quantum mechanics correctly describe the extended point transformation, we show that they are all selfadjoint operators in $L^2(mathbb{R}^n)$ and that the continuous spectrum of each coincides with $mathbb{R}$. They are also shown to satisfy the canonical commutation relations.
The formulation of Geometric Quantization contains several axioms and assumptions. We show that for real polarizations we can generalize the standard geometric quantization procedure by introducing an arbitrary connection on the polarization bundle. The existence of reducible quantum structures leads to considering the class of Liouville symplectic manifolds. Our main application of this modified geometric quantization scheme is to Quantum Mechanics on Riemannian manifolds. With this method we obtain an energy operator without the scalar curvature term that appears in the standard formulation, thus agreeing with the usual expression found in the Physics literature.
We give a mathematically rigorous derivation of Ehrenfests equations for the evolution of position and momentum expectation values, under general and natural assumptions which include atomic and molecular Hamiltonians with Coulomb interactions.
In physics, experiments ultimately inform us as to what constitutes a good theoretical model of any physical concept: physical space should be no exception. The best picture of physical space in Newtonian physics is given by the configuration space of a free particle (or the center of mass of a closed system of particles). This configuration space (as well as phase space), can be constructed as a representation space for the relativity symmetry. From the corresponding quantum symmetry, we illustrate the construction of a quantum configuration space, similar to that of quantum phase space, and recover the classical picture as an approximation through a contraction of the (relativity) symmetry and its representations. The quantum Hilbert space reduces into a sum of one-dimensional representations for the observable algebra, with the only admissible states given by coherent states and position eigenstates for the phase and configuration space pictures, respectively. This analysis, founded firmly on known physics, provides a quantum picture of physical space beyond that of a finite-dimensional manifold, and provides a crucial first link for any theoretical model of quantum spacetime at levels beyond simple quantum mechanics. It also suggests looking at quantum physics from a different perspective.
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