A study of the action from kinematical integral geometry point of view
Assume a timelike circuit in the \((1+1)\) Galilean (Minkowski) space-time obtained as the union of two future timelike paths from A to B. Then the integral over the set of future timelike straight lines, each of them taken with a weight equal to the oriented number of intersections with the circuit, has a finite value which equals (up to a factor) the difference of the classical (relativistic) action of a free particle along the two trajectories. The result is related to the Gauss-Bonnet theorem. Considering a particle in an external field, one shows that for potentials given by polynomials up to the second order (motion in a uniform field and the harmonic oscillator), the difference of action along two timelike paths with common end points, taking into account the potential term, equals the integral of the number of oriented intersections over the set of real motions. In the last part, these results are extended to the \((1+2)\) and \((1+3)\) cases. According to the authors ``the main moral of the paper is that the quantity with a well defined meaning in the sense of integral geometry is not the action for an open path, but (in a formal way) the action along a closed timelike path, and this is very satisfactory''.
scientific article; zbMATH DE number 4205330
scientific article