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The partition function of 3-dimensional quantum gravity has been argued to be 1-loop exact. Here, we verify the vanishing of higher-orders in perturbation theory by explicit computation in the second-order, metric formulation at 3-loops. The number of 1-particle irreducible Feynman diagrams involving both gravitons and ghosts turns out to be 17. Using dimensional regularization, we solve all the diagrams. At 2-loops, we find that all such diagrams vanish separately after regularization. At 3-loops, in contrast, a series of remarkable cancellations between different diagrams takes place, with 9 diagrams beautifully conspiring to yield a vanishing result. Our techniques are suitable to be applied to higher loops as well as to similar computations in higher dimensions., Comment: 6 pages, 8 figures, v2 minor typos corrected

Rindler positivity is a property that holds in any relativistic Quantum Field Theory and implies an infinite set of inequalities involving the exponential of the R��nyi mutual information $I_n(A_i,\bar{A}_j)$ between $A_i$ and $\bar{A}_j$, where $A_i$ is a spacelike region in the right Rindler wedge and $\bar{A}_j$ is the wedge reflection of $A_j$. We explore these inequalities in order to get local inequalities for $I_n(A,\bar{A})$ as a function of the distance between $A$ and its mirror region $\bar{A}$. We show that the assumption, based on the cluster property of the vacuum, that $I_n$ goes to zero when the distance goes to infinity, implies the more stringent and simple condition that $F_n\equiv{e}^{(n-1)I_n}$ should be a completely monotonic function of the distance, meaning that all the even (odd) derivatives are non-negative (non-positive). In the case of a CFT in 1+1 dimensions, we show that conformal invariance implies stronger conditions, including a sort of monotonicity of the R��nyi mutual information for pairs of intervals. An application of these inequalities to obtain constraints for the OPE coefficients of the $4-$point function of certain twist operators is also discussed., 16 pages, 4 figures

During the '70, several relativistic quantum field theory models in $D=1+1$ and also in $D=2+1$ have been constructed in a non-perturbative way. That was done in the so-called {\it constructive quantum field theory} approach, whose main results have been obtained by a clever use of Euclidean functional methods. Although in the construction of a single model there are several technical steps, some of them involving long proofs, the constructive quantum field theory approach contains conceptual insights about relativistic quantum field theory that deserved to be known and which are accessible without entering in technical details. The purpose of this note is to illustrate such insights by providing an oversimplified schematic exposition of the simple case of $����^4$ (with $m>0$) in $D=1+1$. Because of the absence of ultraviolet divergences in its perturbative version, this simple example -although does not capture all the difficulties in the constructive quantum field theory approach- allows to stress those difficulties inherent to the non-perturbative definition. We have made an effort in order to avoid several of the long technical intermediate steps without missing the main ideas and making contact with the usual language of the perturbative approach., 63 pages. Typos corrected

The issue of implementing the principle of general relativity in Einstein equations has been widely discussed, since Kretschmann's well-known criticism stated that general covariance of the Einstein equations is not suffice to express the principle of general relativity (the equivalence of all the coordinate systems). This failure is usually rooted in the fact that metric in Einstein equations is not univocally determined by the matter distribution. We show that the condition of univocal determination of the metric by the matter distribution is stronger than the requirement of equivalence of all coordinate systems. In order to separate the uniqueness problem in Einstein equations from the issue of the principle of general relativity, we define the "equivalence group" instead of the notion of covariance group which is empty of physical content. Moreover, we have complemented in a positive way Kretschmann's objection by supplementing Einstein equations with a sufficient condition for the equivalence of all the coordinate systems., Comment: 10 pages. Minor changes in text. A table was added