On incompleteness of classical field theory

TL;DR

This paper discusses the limitations of classical field theory in handling higher stage Noether identities and gauge symmetries, which impedes quantization of certain reducible degenerate theories.

Contribution

It identifies the incompleteness in the classical formulation related to higher stage symmetries and Noether identities, highlighting a fundamental gap.

Findings

Higher stage Noether identities are not well-defined in generic reducible theories

Gauge symmetries of degenerate Lagrangians face definitional issues

Such limitations prevent the quantization of these field theories

Abstract

Classical field theory is adequately formulated as Lagrangian theory on fibre bundles and graded manifolds. One however observes that non-trivial higher stage Noether identities and gauge symmetries of a generic reducible degenerate Lagrangian field theory fail to be defined. Therefore, such a field theory can not be quantized.