Real and complex connections for canonical gravity

TL;DR

The paper analyzes real $SU(2)$ and complex $SL(2,\mathbb{C})$ formulations of canonical gravity, highlighting an arbitrary parameter $\beta$ in the real connection that scales the Poisson structure and the spectra of geometric operators. It shows that area and volume spectra in the loop-quantized theory depend on $\beta$, so physical predictions require fixing this scale, potentially by the Hamiltonian constraint or by a Wick rotation back to the Ashtekar variables. The work develops the $A^{(\beta)}$ connection, discusses the resulting spin-network quantization and its geometric spectra, and argues that without fixing $\beta$ the discreteness cannot yet be meaningfully interpreted. It emphasizes the need to connect kinematic discreteness to dynamics and explores avenues like Thiemann’s Hamiltonian formulation and possible relations to entropy-area relations and the Einstein equations.

Abstract

Both real and complex connections have been used for canonical gravity: the complex connection has SL(2,C) as gauge group, while the real connection has SU(2) as gauge group. We show that there is an arbitrary parameter $β$ which enters in the definition of the real connection, in the Poisson brackets, and therefore in the scale of the discrete spectra one finds for areas and volumes in the corresponding quantum theory. A value for $β$ could be could be singled out in the quantum theory by the Hamiltonian constraint, or by the rotation to the complex Ashtekar connection.