# PATH INTEGRALS AND QUANTIZATION

# PATH INTEGRALS AND QUANTIZATION

This chapter constructs path integrals for general Hamiltonians with potentials linear in the velocities, like Hamiltonians of particles in a magnetic field. Two examples are considered: a quantum system coupled to a magnetic field, and diffusion as described by the Fokker-Planck equation. In both examples, the Hamiltonian contains products of the position and momentum operators. A quantization problem then arises since these operators do not commute, and the correspondence principle is no longer sufficient to determine the quantum Hamiltonian. The order between quantum operators is determined by additional conditions, such as hermiticity or conservation of probabilities. The calculation of the corresponding path integral then suffers from ambiguities, directly related to this quantization problem. The continuum limit is no longer unique, but depends on the limiting process. This chapter also considers a situation where space has a nontrivial topology, in this case a circle, and shows how this influences the calculation of the path integral.

*Keywords:*
path integrals, quantization, magnetic field, Fokker-Planck equation, Hamiltonians, diffusion, quantum operators, potentials

Oxford Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.

Please, subscribe or login to access full text content.

If you think you should have access to this title, please contact your librarian.

To troubleshoot, please check our FAQs , and if you can't find the answer there, please contact us .