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要旨:
We continue the semiclassical analysis of the Loop Quantum Gravity (LQG) volume operator that was started in the companion paper [23]. In the first paper we prepared the technical tools, in particular the use of complexifier coherent states that use squares of flux operators as the complexifier. In this paper, the complexifier is chosen for the first time to involve squares of area operators.
Both cases use coherent states that depend on a graph. However, the basic difference between the two choices of complexifier is that in the first case the set of surfaces involved is discrete, while, in the second it is continuous. This raises the important question of whether the second set of states has improved invariance properties with respect to relative orientation of the chosen graph in the set of surfaces on which the complexifier depends. In this paper, we examine this question in detail, including a semiclassical analysis.
The main result is that we obtain the correct semiclassical properties of the volume operator for i) artificial rescaling of the coherent state label; and ii) particular orientations of the 4- and 6-valent graphs that have measure zero in the group SO(3). Since such requirements are not present when analysing dual cell complex states, we conclude that coherent states whose complexifiers are squares of area operators are not an appropriate tool with which to analyse the semiclassical properties of the volume operator. Moreover, if one intends to go further and sample over graphs in order to obtain embedding independence, then the area complexifier coherent states should be ruled out altogether as semiclassical states.
