A First Taste of Quantum Gravity Effects: Deforming Phase Spaces with the Heisenberg Double

quantum gravity;phase space;Heisenberg double;Poisson-Lie group;Applied Mathematics

Maitland, Anson

University of Waterloo

关键词: quantum gravity;    phase space;    Heisenberg double;    Poisson-Lie group;    Applied Mathematics;   

Others  :  https://uwspace.uwaterloo.ca/bitstream/10012/8866/3/Maitland_Anson.pdf

瑞士|英语

来源: UWSPACE Waterloo Institutional Repository

【 摘 要 】

We present a well-defined framework to deform the phase spaces of classical particles. These new phase spaces, called Heisenberg doubles, provide a laboratory to probe theeffects of quantum gravity. In particular, they allow us to equip momentum space witha non-abelian group structure by the introduction of a single deformation parameter. Inorder to connect Heisenberg doubles with classical phase spaces we begin with a review of Hamiltonian systems, symmetries and conservation laws in the classical framework. Next, we provide a comprehensive review of the theory behind Poisson-Lie groups, including Lie bialgebras and the construction of the Drinfeld double. Lastly, we build the Heisenberg double from Poisson-Lie group components. We then identify the Heisenberg double as a deformation of the cotangent bundle of Lie groups and extend many of the notions of classical Hamiltonian systems to this new picture with Poisson-Lie symmetries. As an example, we look at a new presentation of the deformed rotator.