The Inverse Variational Problem for Autoparallels

TL;DR

This paper investigates the inverse variational problem for autoparallel motion in affine metric spaces with torsion, showing solutions exist only under specific conditions related to the torsion tensor, leading to a dilaton field theory.

Contribution

It establishes conditions under which a scalar field theory can produce autoparallel trajectories in spaces with torsion, revealing solutions only when torsion is a gradient of a scalar.

Findings

No solutions for generic torsion in 4D spacetime.

Solutions exist if torsion is a scalar gradient.

Results connect autoparallel motion to dilaton field theories.

Abstract

We study the problem of the existence of a local quantum scalar field theory in a general affine metric space that in the semiclassical approximation would lead to the autoparallel motion of wave packets, thus providing a deviation of the spinless particle trajectory from the geodesics in the presence of torsion. The problem is shown to be equivalent to the inverse problem of the calculus of variations for the autoparallel motion with additional conditions that the action (if it exists) has to be invariant under time reparametrizations and general coordinate transformations, while depending analytically on the torsion tensor. The problem is proved to have no solution for a generic torsion in four-dimensional spacetime. A solution exists only if the contracted torsion tensor is a gradient of a scalar field. The corresponding field theory describes coupling of matter to the dilaton field.