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The aim of this paper is to introduce and analyze a new gauge symmetry thatappears in complex holomorphic systems. This symmetry allow us to project thesystem, using different gauge conditions, to several real systems which areconnect by gauge transformations in the complex space. We prove that the spaceof 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 symplecticstructure, but there is still a map between the systems. In this way ourconstruction extend the group of canonical transformations in classicalmechanics. Also, we show how to extend the analysis to the quantum case usingpath integrals by means of the Batalin-Fradkin-Vilkovisky theorem and withinthe canonical formalism, where we show explicitly that solutions of theSchr\"odinger equation are gauge related.

Gauge Freedom in complex holomorphic systems