An account of transforms on \bar {{\cal A}/{\cal G}}

TL;DR

This paper reviews various transforms in gauge theories of connections, including the loop, inverse loop, coherent state, and Wick rotation transforms, highlighting their roles in quantum gravity and their algebraic properties.

Contribution

It provides a coherent summary of recent transforms in connection theories, emphasizing the Wick rotation transform's role in quantum gravity.

Findings

Unified picture of transforms in connection theories

Wick rotation transform preserves algebraic simplicity

Clarification of reality conditions in quantum gravity

Abstract

In this article we summarize and describe the recently found transforms for theories of connections modulo gauge transformations associated with compact gauge groups. Specifically, we put into a coherent picture the so-called loop transform, the inverse loop transform, the coherent state transform and finally the Wick rotation transform which is the appropriate transform that incorporates the correct reality conditions of quantum gravity when formulated as a dynamical theory of connections while preserving the simple algebraic form of the Hamiltonian constraint.