Quantum gravity, shadow states, and quantum mechanics

Abstract

A program was recently initiated to bridge the gap between the Planck scale physics described by loop quantum gravity and the familiar low energy world. We illustrate the conceptual problems and their solutions through a toy model: quantum mechanics of a point particle. Maxwell fields will be discussed in the second paper of this series which further develops the program and provides details.

35 pages, no figures, RevTeX4 with AMSTeX; v2: three references added. v3: Sections IV and V revised in view of discussions at the recent Schrödinger Institute workshop on quantum field theory in curved space-times. In particular, now this simple quantum mechanical example also serves as a (partial) introduction to the constructions and techniques used in quantum geometry

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