# Gauge Freedom in complex holomorphic systems

## TL;DR

This paper introduces a new gauge symmetry in complex holomorphic systems, linking different real systems via gauge transformations, and extends classical and quantum mechanics frameworks to include these symmetries.

## Contribution

It identifies and analyzes a novel gauge symmetry in complex holomorphic systems, expanding the scope of canonical transformations and their quantum counterparts.

## Key findings

- Gauge transformations relate solutions of different real systems.
- Some gauge transformations are canonical, others alter the symplectic structure.
- Quantum solutions are gauge related via path integrals and Schrödinger equation analysis.

## Abstract

The aim of this paper is to introduce and analyze a new gauge symmetry that appears in complex holomorphic systems. This symmetry allow us to project the system, using different gauge conditions, to several real systems which are connect by gauge transformations in the complex space. We prove that the space of solutions of one system is related to the other by the gauge transformation. The gauge transformations are in some cases canonical transformations. However, in other cases are more general transformations that change the symplectic structure, but there is still a map between the systems. In this way our construction extend the group of canonical transformations in classical mechanics. Also, we show how to extend the analysis to the quantum case using path integrals by means of the Batalin-Fradkin-Vilkovisky theorem and within the canonical formalism, where we show explicitly that solutions of the Schr\"odinger equation are gauge related.