Perelomov-type coherent states of SO(D+1) in all-dimensional loop quantum gravity

Abstract

We propose a new kind of coherent state for the general SO(D + 1) formulation of loop quantum gravity in the (1 + D)-dimensional space-time. Instead of Thiemann's coherent state for SO(D+1) gauge theory, our coherent spin-network state is given by constructing proper superposition over quantum numbers of the spin-networks with vertices labelled by the coherent intertwiners. Such superposition type coherent states are labelled by the so-called generalized twisted geometric variables which capture the geometric meaning of discretized general relativity. We study the basic properties of this kind of coherent states, i.e., the completeness and peakedness property. Moreover, we show that the superposition type coherent states are consistent with Thiemann's coherent state for SO(D + 1) gauge theory in large η limit.

Polytopes in all dimensional loop quantum gravity

“…It is clear that Eq. ( 42) tends to zero for N → ∞ [33]. The expectation values of the geometric operators composed of the flux operators with respect to the simple coherent intertwiners ||N, V I J have the nice property of semiclassicality.…”

“…is the gauge-fixed simple coherent intertwiner such that the closure constraint and simplicity constraint are weakly satisfied. Here the states |N î , V I J î are SO(D + 1) coherent states of Perelomov type satisfying the peakedness property [33]…”

“…Since the spin network states labelled by coherent intertwiners are well-behaved coherent states for flux operator [14,33], it is natural to discuss the semi-classical property of the usual D-volume operator defined in all dimensional LQG based on flux operators [12]. The usual strategy of constructing D-volume operator in all dimensional LQG is based on the action of the basic flux operator on spin network states.…”

“…is a sub-Hilbert space of the gauge invariant coherent intertwiner space H N . Thus the matrix element of ( 50 A in large N limit [33]. Hence, the integral in (51) only concerns the behaviour of G(A, Z ) on P s.…”

“…has following interesting properties in large N limit. First, using the fact that two coherent intertwiners become orthogonal in large N limit [33], one can show that the expectation value of V for a coherent intertwiner |N, Z reproduces the volume of the classical polytope with shape (A, Z ) in large N limit, i.e.,…”