# Polar decomposition of the Wiener measure: Schwarzian theory versus conformal quantum mechanics

## TL;DR

This paper establishes a mathematical connection between conformal quantum mechanics and Schwarzian theory by deriving the polar decomposition of the Wiener measure, enabling evaluation of complex functional integrals and solving the Schrödinger equation in imaginary time.

## Contribution

It provides an explicit polar decomposition of the Wiener measure and links functional integrals in conformal quantum mechanics to those in Schwarzian theory, offering new computational tools.

## Key findings

- Derived the explicit polar decomposition of the Wiener measure.
- Connected functional integrals in conformal quantum mechanics to Schwarzian theory.
- Obtained the fundamental solution of the Schrödinger equation in imaginary time.

## Abstract

We derive the explicit form of the polar decomposition of the Wiener measure, and obtain the equation connecting functional integrals in conformal quantum mechanics to those in the Schwarzian theory. Using this connection we evaluate some nontrivial functional integrals in the Schwarzian theory and also find the fundamental solution of the Schroedinger equation in imaginary time in the model of conformal quantum mechanics.