String Propagator: a Loop Space Representation

TL;DR

This paper introduces a novel loop space representation of string propagation amplitude using a reparametrization invariant string action, providing an exact solution for free strings through a functional Schrödinger equation.

Contribution

It develops a string propagator framework based on Jacobi's variational principle in loop space, avoiding lattice approximations and establishing equivalence with a functional Schrödinger equation.

Findings

Path integral and wave equation are exactly solvable for free strings.

Reparametrization invariance is maintained in the string action.

The approach offers an alternative to traditional quantum field theory methods.

Abstract

The string quantum kernel is normally written as a functional sum over the string coordinates and the world--sheet metrics. As an alternative to this quantum field--inspired approach, we study the closed bosonic string propagation amplitude in the functional space of loop configurations. This functional theory is based entirely on the Jacobi variational formulation of quantum mechanics, {\it without the use of a lattice approximation}. The corresponding Feynman path integral is weighed by a string action which is a {\it reparametrization invariant} version of the Schild action. We show that this path integral formulation is equivalent to a functional ``Schrodinger'' equation defined in loop--space. Finally, for a free string, we show that the path integral and the functional wave equation are {\it exactly } solvable.