Critical (?4)3,?

Abstract

The Euclidean (φ 4 ) 3, ǫ model in R 3 corresponds to a perturbation by a φ 4 interaction of a Gaussian measure on scalar fields with a covariance depending on a real parameter ε in the range 0 ≤ ε ≤ 1. For ε = 1 one recovers the covariance of a massless scalar field in R 3 . For ε = 0 φ 4 is a marginal interaction. For 0 ≤ ε < 1 the covariance continues to be Osterwalder-Schrader and pointwise positive. After introducing cutoffs we prove that for ε > 0, sufficiently small, there exists a non-gaussian fixed point ( with one unstable direction) of the Renormalization Group iterations. These iterations converge to the fixed point on its stable (critical) manifold which is constructed.

Hints of unitarity at large N in the O(N )3 tensor field theory

“…Generalized free fields are a rather old idea [44], and their interacting counterpart, in the case of a single scalar field, is known as the long-range Ising model [45,46]. The latter has been studied extensively with various methods, including constructive methods [47,48], large-N expansion [49], functional renormalization group [50], and CFT methods [51,52]. Our choice of kinetic term would correspond in the long-range Ising model to the transition point between the mean field theory behavior and the long-range one.…”

The Asymptotic Safety Scenario in Quantum Gravity

“…Identification of the fine-tuned S λ [ χ ] lies at the core of the UV renormalization problem, irrespective of whether Γ λ , k is defined directly via the functional integral or via the FRGE. Beyond perturbation theory the only known techniques to identify the proper S λ start directly from the functional integral and are ‘constructive’ in spirit (see [195, 48]). Unfortunately four-dimensional quantum field theories of interest are still beyond constructive control.…”

Finite range decompositions of positive-definite functions

“…In the case that the covariance of the Gaussian field φ is homogeneous, e.g., |x − y| −α , it is easy to establish existence of these decompositions and we have made this point and used them, for example, in [3]. In [4] we proved that the resolvent of the Laplacian (aI − ) −1 with a 0 admits finite range decompositions and also showed that these decompositions exist when is the finite difference Laplacian on the simple cubic lattice Z d .…”