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We propose a new treatment of the quantum simplicity constraints appearing inthe general ${SO(D+1)}$ formulation of loop quantum gravity for the${(1+D)}$-dimensional space-time. Instead of strongly imposing the constraints,we construct a specific form of weak solutions by employing the spin net-workstates with specific ${SO(D+1)}$ coherent intertwiners. These states weaklysatisfy the quantum simplicity constraint via the vanishing expectation values,and the quantum Gaussian constraints can be imposed strongly. Remarkably, thosespecific ${SO(D+1)}$ coherent intertwiners used to construct our solutions havenatural interpretations of the $D$-dimensional polytopes, commonly viewed asbasic units of the discrete spatial geometry. Therefore, while the strongimposition of the quantum simplicity constraints leads to an over-constrainedsolution space, our weak solution space for the constraints may contain thecorrect semiclassical degrees of freedom for intrinsic geometry of the spatialhypersurfaces. Moreover, some concrete relations are established between ourconstruction and other existing approaches in solving the simplicityconstraints in all dimensional loop quantum gravity, providing valuableinsights into this unresolved important issue.

Coherent intertwiner solution of simplicity constraint in all dimensional loop quantum gravity