Self Duality and Quantization

TL;DR

This paper develops a novel quantum framework for the Maxwell field using self dual connections, simplifying the representation of quantum states and supporting a non-perturbative approach to quantum gravity.

Contribution

It introduces a holomorphic distribution-based method for quantizing the Maxwell field via self dual variables, extending to linear gravitons and supporting quantum gravity research.

Findings

Quantum states are holomorphic functionals of self dual connections.

The method recovers the full Fock space from self dual variables.

Supports non-perturbative quantization of gravity using self dual variables.

Abstract

Quantum theory of the free Maxwell field in Minkowski space is constructed using a representation in which the self dual connection is diagonal. Quantum states are now holomorphic functionals of self dual connections and a decomposition of fields into positive and negative frequency parts is unnecessary. The construction requires the introduction of new mathematical techniques involving ``holomorphic distributions''. The method extends also to linear gravitons in Minkowski space. The fact that one can recover the entire Fock space --with particles of both helicities-- from self dual connections alone provides independent support for a non-perturbative, canonical quantization program for full general relativity based on self dual variables.