Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Proceedings of the Royal Society of London A Mathematical and Physical Sciences

Path integrals in curvilinear coordinates

A. M. Arthurs

Published Sep 22, 1970• 9 citations
Publisher

Abstract

Abstract The problem of summation over Feynman histories in curvilinear coordinates is discussed. Ordinary rules of calculus are used in conjunction with a Hamiltonian approach to the summation over histories. The procedure is illustrated by considering the propagator for dynamical systems in spherical polar coordinates.
DOI
10.1098/rspa.1970.0158

Related papers

Path integrals in polar coordinates

A. M. Arthurs

Abstract The problem of summation over Feynman histories in polar coordinates is discussed. Ordinary rules of calculus are used in conjunction with a Hamiltonian approach to the summation over histories. The procedure is checked by evaluating the propagator for a free particle in two dimensions.

Proceedings of the Royal Society of London A Mathematical and Physical Sciences•Nov 11, 1969•22 citations

Summation over Feynman Histories in Polar Coordinates

David Peak, Akira Inomata

Use of polar coordinates is examined in performing summation over all Feynman histories. Several relationships for the Lagrangian path integral and the Hamiltonian path integral are derived in the central-force problem. Applications are made for a harmonic oscillator, a charged particle in a uniform magnetic field, a particle in an inverse-square potential, and a rigid rotator. Transformations from Cartesian to polar coordinates in path integrals are rather different from those in ordinary calculus and this complicates evaluation of path integrals in polars. However, it is observed that for systems of central symmetry use of polars is often advantageous over Cartesians.

Journal of Mathematical Physics•Aug 1, 1969•225 citations

Hamiltonian approach to the method of summation over Feynman histories

H. Davies

Abstract The Feynman method of quantization by summation over q-histories is extended to a summation over q, p-histories, corresponding to the classical variational method of treating q and p as independent variables. The equivalence of this q-p summation to the usual quantization procedure is shown in the Heisenberg picture by choosing convenient canonical coordinates and in the Schrödinger picture by obtaining the usual representations of the q, p operators.

Mathematical Proceedings of the Cambridge Philosophical Society•Jan 1, 1963•30 citations

Path integral solution of the Schrödinger equation in curvilinear coordinates: A straightforward procedure

J. LaChapelle

A new axiomatic formulation of path integrals is used to construct a path integral solution of the Schrödinger equation in curvilinear coordinates. An important feature of the formalism is that a coordinate transformation in the variables of the wavefunction does not imply a change of variable of integration in the path integral. Consequently, a transformation from Euclidean to curvilinear coordinates is simple to handle; there is no need to introduce ‘‘quantum corrections’’ into the action functional. Furthermore, the paths are differentiable: hence, issues related to stochastic paths do not arise. The procedure for constructing the path integral solution of the Schrödinger equation is straightforward. The case of the Schrödinger equation in spherical coordinates for a free particle is presented in detail.

Journal of Mathematical Physics•Sep 1, 1996•0 citations

New curvilinear integrals along paths of Feynman path integral

Naoto Kumano‐go

We define new curvilinear integrals along paths of Feynman path integral. In order to define curvilinear integrals along paths on a path space, Ito integrals use initial points of line segments of broken line paths, and Stratonovich integrals use middle points of them. However, our new curvilinear integrals use classical curvilinear integrals along broken line paths. Therefore, the classical fundamental theorem of calculus holds in Feynman path integral.

•Jan 1, 2003•0 citations

Transformations of path integrals

Kumar S Narain, Jan Tarski

Several examples of changes of variables in Feynman-type integrals are considered. In particular, path integrals in curvilinear coordinates (of a certain class) are rigorously defined. An appendix is devoted to path integrals for vector potentials.

Journal of Physics A Mathematical and General•Apr 21, 1984•2 citations
Use of polar coordinates is examined in performing summation over all Feynman histories. Several relationships for the Lagrangian path integral and the Hamiltonian path integral are derived in the central-force problem. Applications are made for a harmonic oscillator, a charged particle in a uniform magnetic field, a particle in an inverse-square potential, and a rigid rotator. Transformations from Cartesian to polar coordinates in path integrals are rather different from those in ordinary calculus and this complicates evaluation of path integrals in polars. However, it is observed that for systems of central symmetry use of polars is often advantageous over Cartesians.
A new axiomatic formulation of path integrals is used to construct a path integral solution of the Schrödinger equation in curvilinear coordinates. An important feature of the formalism is that a coordinate transformation in the variables of the wavefunction does not imply a change of variable of integration in the path integral. Consequently, a transformation from Euclidean to curvilinear coordinates is simple to handle; there is no need to introduce ‘‘quantum corrections’’ into the action functional. Furthermore, the paths are differentiable: hence, issues related to stochastic paths do not arise. The procedure for constructing the path integral solution of the Schrödinger equation is straightforward. The case of the Schrödinger equation in spherical coordinates for a free particle is presented in detail.