Spatially asymptoticSmatrix from general boundary formulation

Semiclassical regime of Regge calculus and spin foams

“…A standard trick to obtain the vacuum state in conventional quantum mechanics and field theory is to perform a Wick rotation. However the implementation of a meaningful analogue of this procedure (and even the existence of one) in a general-boundary setting is still unclear 17 . A more promising route might be to select the boundary state representing the semiclassical regime in an a posteriori way, by requiring that the results match those of conventional perturbative Regge theory (in which only small fluctuations around a given classical solution for the whole manifold are quantized and summed over; see [60,61,62]).…”

“…Note that this ratio is precisely the parameter that quantifies the kinematical relative dispersion in the boundary Gaussian according to (17). If the action includes a cosmological term or matter sources, then (37) is not generally true, while (36) keeps on holding; in this case, careful study of the classical solution is needed to identify a good expansion parameter, which is no longer a single global scale of the boundary state.…”

“…The starting point for this progress was the suggestion by Rovelli [8,9] of a procedure for computing the graviton propagator from Loop Quantum Gravity (LQG) [1,10,11] with the dynamics implemented covariantly in terms of a spin foam model. On the calculational side, the key ingredients are a boundary semiclassical spin network state peaked on large spins and an analytic expression for the large spin asymptotics of the spin foam vertex amplitude [12]; on the conceptual side, the framework is the boundary state formalism discussed in [1] and in [13,14,15,16,17], which prescribes how to compute observables in the boundary of a spacetime region with a path integral over the interior region only. IfÔ 1 ,Ô 2 are local boundary geometry observables (such as areas, dihedral angles, 3-volumes or lengths [18,19,20,21,22,23]) acting on a space of spin networks s , then the expectation value for their correlation in a boundary geometry q is given in the boundary state formalism by…”

States and amplitudes for finite regions in a two-dimensional Euclidean quantum field theory

“…The plan of this work largely follows that of [13]. We consider in Section 2 the classical field theory in the two settings mentioned above, and we express the classical solutions of the equation of motion in terms of the boundary configurations in the different settings.…”

“…Nevertheless, a consistent probability interpretation was demonstrated. Perturbatively interacting QFT was treated in [12,13] showing that an interacting asymptotic amplitude can be defined from the large radius limit of the hypercylinder. Moreover, it was demonstrated that this amplitude is equivalent to the usual S-matrix when both can be defined.…”