Extended Body Propagators
Three seemingly distinct extended-body propagators devised by Ansoldi et al., Erbin et al., and Stanford et al.\ respectively arise in completely different settings, though each can be considered a realization of the Polyakov path integral. This paper proves that the Ansoldi propagator corresponds to the characteristic function of the random variable with the probability density function $f(A)=C\,A\,(α^{2}+A^{2})^{-D/2}\mathbf{1}_{A>0}$ in ambient dimension $D$, of scale $α=\sqrt{b/2}$. The exponent is fixed by the dimension and by nothing else in the Ansoldi/geometric comparison; the on-shell two-point amplitude of Erbin et al. and JT gravity agree with its $D=4$ member in the senses made precise below. After an extensive literature review, the author believes that this observation and constructed area measure is novel. The area measure will, however, emerge as the natural candidate for the extension of the earlier non-extended body path space measures to two dimensional simplicial complexes. We further show that the area measure of this paper is exactly the law of the area of a triangle, two of whose sides are drawn from the measure obtained in the companion manuscripts for the one-dimensional Polyakov integral. We close with an argument, rigorous in part, which reduces the measure of tubular worldsheets between two arbitrary boundary curves to that of this triangle.