Schr\"odinger wave functional in quantum Yang-Mills theory from precanonical quantization

TL;DR

This paper explores the connection between precanonical quantization and the Schr"odinger wave functional in quantum Yang-Mills theory, showing how the latter emerges as a limit of the former with a specific ultraviolet parameter.

Contribution

It demonstrates that the Schr"odinger wave functional can be derived from precanonical quantization by taking the ultraviolet parameter to infinity, linking two different quantization approaches.

Findings

Schr"odinger wave functional expressed as trace of Volterra product integral.

Derivation of Schr"odinger equation and Gauss constraint from precanonical Schr"odinger equation.

Connection established between precanonical quantization and traditional functional Schr"odinger representation.

Abstract

A relation between the precanonical quantization of pure Yang-Mills fields and the functional Schr\"odinger representation in the temporal gauge is discussed. It is shown that the latter can be obtained from the former when the ultraviolet parameter $\varkappa$ introduced in precanonical quantization goes to infinity. In this limiting case, the Schr\"odinger wave functional can be expressed as the trace of the Volterra product integral of Clifford-algebra-valued precanonical wave functions restricted to a certain field configuration, and the canonical functional derivative Schr\"odinger equation together with the quantum Gau\ss\ constraint are derived from the Dirac-like precanonical Schr\"odinger equation.