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Discrete and Continuous Curved Surface Antipodal Paths. Extensions of the Borsuk-Ulam Theorem and Application of the Feynman Path Integral and Wodehouse Contour Integral

Peters, James, Alfano, Roberto, Smith, Peter and Tozzi, Arturo Discrete and Continuous Curved Surface Antipodal Paths. Extensions of the Borsuk-Ulam Theorem and Application of the Feynman Path Integral and Wodehouse Contour Integral.

Item Type: Article

Abstract

Abstract Purpose: This paper introduces discrete and continuous paths over simply-connected surfaces with non-zero curvature as means of comparing and measuring paths between antipodes with either a Feynman path integral
or Wodehouse contour integral, resulting in a number of extensions of the Borsuk Ulam Theorem.
Methods: All paths originate on a Riemannian surface S, which is simply�connected and has non-zero curvature at every point in S. A surface is a simply connected, provided every cross-cut divides the surface into disjoint regions. A cross cut is an arc that runs through the interior of the surface S
without self-intersections and joins one boundary to another.
Results: A fundamental result in this paper is that for any pair of antipodal surface points, a path that begins and ends at the antipodal points can be found. This result is extended to Feynman path integrals on trajectory-of particle curves and to N.M.J. Wodehouse countour integrals for antipodal
vectors on twistor curves. Another fundamental result in this paper is that the Feynman trajectory of a particle is realizable as a Lefschetz arc.

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Depositing User: Michelle Marshall

Identifiers

Item ID: 15671
URI: http://sure.sunderland.ac.uk/id/eprint/15671
Official URL: https://hal.science/hal-03702012/document

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Catalogue record

Date Deposited: 20 Jan 2023 09:17
Last Modified: 20 Jan 2023 09:17

Contributors

Author: James Peters
Author: Roberto Alfano
Author: Peter Smith
Author: Arturo Tozzi

University Divisions

Faculty of Technology

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