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Hamiltonian structure of $$2+1$$ dimensional gravity. (English) Zbl 1202.83095
Cianci, R. (ed.) et al., Recent developments in general relativity, Genova 2000. Proceedings of the 14th SIGRAV conference on general relativity and gravitational physics, Genova, September 18–22, 2000. Milano: Springer (ISBN 88-470-0162-5/pbk). 165-177 (2002).
Summary: A summary is given of some results and perspectives of the Hamiltonian ADM approach to $$2+1$$ dimensional gravity. After recalling the classical results for closed universes in absence of matter we go over the the case in which matter is present in the form of point spinless particles. Here the maximally slicing gauge proves most effective by relating $$2+1$$ dimensional gravity to the Riemann–Hilbert problem. It is possible to solve the gravitational field in terms of the particle degrees of freedom thus reaching a reduced dynamics which involves only the particle positions and momenta. Such a dynamics is proven to be Hamiltonian and the Hamiltonian is given by the boundary term in the gravitational action. As an illustration the two body Hamiltonian is used to provide the canonical quantization of the two particle system.
For the entire collection see [Zbl 1063.83501].

##### MSC:
 83C80 Analogues of general relativity in lower dimensions 83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
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