# Holomorphic Path Integrals in Tangent Space for Flat Manifolds

## TL;DR

This paper develops a novel approach to quantum evolution on flat manifolds using holomorphic functions on cotangent bundles, employing exponential maps and tangent space integration to construct a holomorphic Feynman integral.

## Contribution

It introduces a new method for quantum evolution in flat manifolds using holomorphic functions and tangent space integration, extending Feynman integral techniques.

## Key findings

- Constructed Hilbert spaces of holomorphic functions with exponential map-based scalar products
- Developed a holomorphic Feynman integral via tangent space integration
- Applied the method to the case of S^1, including relevant paths

## Abstract

In this paper we study the quantum evolution in a flat Riemannian manifold. The holomorphic functions are defined on the cotangent bundle of this manifold. We construct Hilbert spaces of holomorphic functions in which the scalar product is defined using the exponential map. The quantum evolution is proposed by means of an infinitesimal propagator and the holomorphic Feynman integral is developed via the exponential map. The integration corresponding to each step of the Feynman integral is performed in the tangent space. Moreover, in the case of $S^1$, the method proposed in this paper naturally takes into account paths that must be included in the development of the corresponding Feynman integral.