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Quantum "Twin Peaks" or Path Integrals in the Future Light Cone

Vladimir V. Belokurov, Vsevolod V. Chistiakov, Klavdiia A. Lursmanashvili, Evgeniy T. Shavgulidze

Abstract

By analogy with the Wiener measure on the Euclidean plane that is invariant under the group of rotations and quasi-invariant under the group of diffeomorphisms, we construct the path integrals measure that is invariant under the Lorentz group and quasi-invariant under the group of diffeomorphisms. The correspondence between the paths in the future cone of the Minkowskian plane and the paths in the coverings of the Euclidean plane is established.

Quantum "Twin Peaks" or Path Integrals in the Future Light Cone

Abstract

By analogy with the Wiener measure on the Euclidean plane that is invariant under the group of rotations and quasi-invariant under the group of diffeomorphisms, we construct the path integrals measure that is invariant under the Lorentz group and quasi-invariant under the group of diffeomorphisms. The correspondence between the paths in the future cone of the Minkowskian plane and the paths in the coverings of the Euclidean plane is established.
Paper Structure (11 sections, 113 equations, 6 figures)

This paper contains 11 sections, 113 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The Wiener measure on trajectories lying on $\mathbb{R}$
  3. The Wiener measure on trajectories in the Euclidean plane
  4. The measure on trajectories in the future cone
  5. Random process on the infinite-sheeted covering of the plane
  6. Metric on the cone and its properties
  7. Isometric mapping of the cone onto the infinite-sheeted covering of the plane
  8. Geodesics on the cone
  9. Conclusion
  10. Appendix A
  11. Appendix B

Figures (6)

  • Figure 1:
  • Figure 2: Geodesic passing through points $A$ and $B$ in the case where the angular distance between them is less than $\pi$. Figure (a) shows the geodesic on the cone, figure (b) shows its image under the mapping onto the infinite-sheeted covering of the plane.
  • Figure 3: Geodesic passing through points $A$ and $B$ in the case where the angular distance between them is greater than $\pi$ but less than $2\pi$. Figure (a) shows the geodesic on the cone, figure (b) shows its image under the mapping onto the infinite-sheeted covering of the plane. The images of points $A$ and $B$ lie on the same sheet of the covering.
  • Figure 4: Geodesic passing through points $A$ and $B$ in the case where the angular distance between them is greater than $2\pi$. Figure (a) shows the geodesic on the cone, figures (b) and (c) show its image under the mapping onto the infinite-sheeted covering of the plane. The images of points $A$ and $B$ lie on different sheets of the covering.
  • Figure 5: Example of a curve connecting points $A$ and $B$ whose angular distance exceeds $2\pi$. Figure (a) shows the curve on the cone, figures (b) and (c) show its image on the infinite-sheeted covering. The images of points $A$ and $B$ lie on different sheets of the covering.
  • ...and 1 more figures