Nonperturbative two-body dynamics in 2+1-dimensional gravity

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Abstract
Two-body dynamics in 2+1-dimensional gravity is analyzed from the point of view of the Poincaré gauge theory without the a priori introduction of a metrical space-time. Making use of the nonuniqueness of the phase space variables, point-like spinless sources are coupled to the Chern–Simons–Witten action in two different ways. In both cases, the exact gravity Hamiltonian is obtained, not by relying on metrical notions, but directly as Wilson loop observables. Then, by viewing the two-body system from a particular frame, and performing a multivalued gauge transformation, it is shown that for both types of matter couplings, the relative coordinate moves on a cone with a deficit angle given by the corresponding classical Hamiltonian. Thus, the metrical space-time emerges as a broken phase of the Poincaré gauge theory. For one type of matter coupling, ’t Hooft’s Hamiltonian is obtained in the low-energy limit. It is also shown that ’t Hooft’s method of obtaining the scattering amplitude can be made applicable to the exact Hamiltonians by an appropriate reinterpretation of parameters.