Spectra of Length and Area in 2+1 Lorentzian Loop Quantum Gravity

TL;DR

This work analyzes the spectra of length and area operators in 2+1 Lorentzian loop quantum gravity, showing a continuous spacelike length spectrum and a discrete timelike length spectrum, with the discrete sector tied to SO(2,1) discrete series and the continuous sector to the principal series. The length spectrum is derived from the Casimir eigenvalues of edge representations carried by spin-network edges, while the area spectrum emerges from intertwiners at nodes and is expressible in terms of edge Casimirs, matching a triangle-geometry interpretation. A symmetric quantization variant shifts the spectra such that spacelike lengths are continuous without a gap and timelike lengths become equally spaced, aligning more closely with spin-foam results; the paper also discusses SU(1,1) extensions and potential cosmological-constant deformations. Overall, the results reveal a consistent interplay between canonical loop quantum gravity and covariant spin-foam approaches in 2+1 dimensions, and they raise interesting questions for the interpretation of geometric operators in non-compact gauge groups and higher dimensions. The findings have implications for understanding quantum geometry, the role of matter as a reference frame, and the relationship between discrete and continuous geometric spectra in quantum gravity.

Abstract

We study the spectrum of the length and area operators in Lorentzian loop quantum gravity, in 2+1 spacetime dimensions. We find that the spectrum of spacelike intervals is continuous, whereas the spectrum of timelike intervals is discrete. This result contradicts the expectation that spacelike intervals are always discrete. On the other hand, it is consistent with the results of the spin foam quantization of the same theory.

Figures (2)

• Figure 1: The spectrum of the length operator.
• Figure 2: The spectrum of the symmetric length operator.