On the precanonical structure of the Schr\"odinger wave functional in curved space-time

TL;DR

This paper derives the functional Schr"odinger equation in curved space-time from a covariant precanonical approach, revealing a new hypercomplex formulation of quantum field theory that generalizes traditional methods.

Contribution

It introduces a precanonical quantization framework that connects to the standard functional Schr"odinger picture in curved space-time, offering a novel geometric perspective.

Findings

Derived the functional Schr"odinger equation from precanonical quantization.

Expressed the wave functional as a trace of a product integral of precanonical wave functions.

Revealed a hypercomplex generalization of quantum field theory.

Abstract

The functional Schr\"odinger equation in curved space-time is derived from the manifestly covariant precanonical Schr\"odinger equation. The Schr\"odinger wave functional is expressed as the trace of the multidimensional product integral of precanonical wave function restricted to a field configuration. The functional Schr\"odinger representation of QFT in curved space-time appears as a singular limiting case of a formulation based on precanonical quantization, which leads to a hypercomplex generalization of quantum formalism in field theory.