Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2001-06-15
Physics
High Energy Physics
High Energy Physics - Theory
12 pages, no figures
Scientific paper
We discuss a version of Hamiltonian (2+1)-dimensional dynamics, in which one allows nonvanishing Poisson brackets also between the coordinates, and between the momenta. The resulting equations of motion are not any more derivable from a Lagrangian. However, taking a specific limit, in which the symplectic form becomes singular, one can recover a first-order Lagrangian description. This signals the dimensional reduction of the phase-space to half its initial number of degrees of freedom. We reach the same limit from another point of view, studying a particular form of the Poisson brackets, which is singled out geometrically and easy to handle algebraically. For comparison, a discussion of quantum mechanics with extended Heisenberg algebra is included. The quantum theory constrains the antisymmetric matrix providing the algebra to the above mentioned classically singular limit.
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