Gauge-invariant coherent states for Loop Quantum Gravity I: Abelian gauge groups
Benjamin Bahr, Thomas Thiemann
TL;DR
This work develops gauge-invariant coherent states for Loop Quantum Gravity by projecting complexifier coherent states from the kinematical Hilbert space onto the gauge-invariant sector for the abelian group $G=U(1)$. It derives an explicit closed-form for the gauge-invariant states on a fixed graph, revealing dependence on gauge-invariant combinations of edge data and the graph's incidence matrix, and demonstrates strong semiclassical peakedness in gauge-invariant variables as the semiclassicality parameter $t$ tends to zero. The analysis connects the gauge-invariant sector to the graph topology via $H^1(oldsymbol{ au}, G)\, ilde{=} G^{E-V+1}$ and the incidence structure, while showing that the gauge-invariant states are not simply CCS on the reduced configuration space due to nontrivial mode mixing. This first step lays the groundwork for physical coherent states in LQG and sets the stage for extending the construction to $G= ext{SU}(2)$ in a subsequent paper, where the richer non-abelian structure will be addressed.
Abstract
In this paper we investigate the properties of gauge-invariant coherent states for Loop Quantum Gravity, for the gauge group U(1). This is done by projecting the corresponding complexifier coherent states, which have been applied in numerous occasions to investigate the semiclassical limit of the kinematical sector, to the gauge-invariant Hilbert space. This being the first step to construct physical coherent states, we arrive at a set of gauge-invariant states that approximate well the gauge-invariant degrees of freedom of abelian LQG. Furthermore, these states turn out to encode explicit information about the graph topology, and show the same pleasant peakedness properties known from the gauge-variant complexifier coherent states.