# PATH INTEGRALS IN PHASE SPACE

# PATH INTEGRALS IN PHASE SPACE

This chapter contains a few additional results such as a definition of path integrals over phase space trajectories, and the problems generated by the quantization of lagrangians with potentials quadratic in the velocities. The important example of Hamiltonians quadratic in the momentum variables is first examined. In the simplest situations discussed in previous chapters, after an explicit integration over momenta *p(t)* one recovers the usual path integral. More general Hamiltonians are often met, for example, in the quantization of the motion on riemannian manifolds. The analysis is illustrated with the quantization of free motion on a sphere (or hypersphere) *S* _{N-1}. A few relevant elements of classical mechanics are considered first.

*Keywords:*
path integrals, phase space, classical mechanics, quantization, free motion, sphere, lagrangians, Hamiltonians, momentum variables

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