Causality, crossing and analyticity in conformal field theories

Analyticity and crossing properties of four-point function are investigated in conformal field theories in the frameworks of Wightman axioms. A Hermitian scalar conformal field, satisfying the Wightman axioms, is considered. The crucial role of microcausality in deriving analyticity domains is discussed and domains of analyticity are presented. A pair of permuted Wightman functions are envisaged. The crossing property is derived by appealing to the technique of analytic completion for the pair of permuted Wightman functions. The operator product expansion of a pair of scalar fields is studied and analyticity property of the matrix elements of composite fields, appearing in the operator product expansion, is investigated. An integral representation is presented for the commutator of composite fields where microcausality is a key ingredient. Three fundamental theorems of axiomatic local field theories; namely, PCT theorem, the theorem proving equivalence between PCT theorem and weak local commutativity and the edge-of-the-wedge theorem are invoked to derive a conformal bootstrap equation rigorously.

Distributions in CFT. Part II. Minkowski space

CFTs in Euclidean signature satisfy well-accepted rules, such as the convergent Euclidean OPE. It is nowadays common to assume that CFT correlators exist and have various properties also in Lorentzian signature. Some of these properties may represent extra assumptions, and it is an open question if they hold for familiar statistical-physics CFTs such as the critical 3d Ising model. Here we consider Wightman 4-point functions of scalar primaries in Lorentzian signature. We derive a minimal set of their properties solely from the Euclidean unitary CFT axioms, without using extra assumptions. We establish all Wightman axioms (temperedness, spectral property, local commutativity, clustering), Lorentzian conformal invariance, and distributional convergence of the s-channel Lorentzian OPE. This is done constructively, by analytically continuing the 4-point functions using the s-channel OPE expansion in the radial cross-ratios ρ, $$ \overline{\rho} $$ ρ ¯ . We prove a key fact that |ρ|, $$ \left|\overline{\rho}\right| $$ ρ ¯ < 1 inside the forward tube, and set bounds on how fast |ρ|, $$ \left|\overline{\rho}\right| $$ ρ ¯ may tend to 1 when approaching the Minkowski space.We also provide a guide to the axiomatic QFT literature for the modern CFT audience. We review the Wightman and Osterwalder-Schrader (OS) axioms for Lorentzian and Euclidean QFTs, and the celebrated OS theorem connecting them. We also review a classic result of Mack about the distributional OPE convergence. Some of the classic arguments turn out useful in our setup. Others fall short of our needs due to Lorentzian assumptions (Mack) or unverifiable Euclidean assumptions (OS theorem).

PCT theorem, Wightman axioms and conformal bootstrap

The axiomatic Wightman formulation for nonderivative conformal field theory is adopted to derive conformal bootstrap equation for the four-point function. The equivalence between PCT theorem and weak local commutativity, due to Jost plays a very crucial role in axiomatic field theory. The theorem is suitably adopted for conformal field theory to derive the desired equations in CFT. We demonstrate that the two Wightman functions are analytic continuation of each other.

Quantization and topology: SL(2,R)-de Sitter invariant fields in two dimensions

We explore the interplay between quantization, local commutativity and the analyticity properties of the two-point functions of a quantum field in a non trivial topological cosmological background in the example of the two-dimensional de Sitter manifold and its double covering. The global topological differences make the many of the well-known features of de Sitter quantum field theory disappear. In particular there is nothing like a Bunch-Davies vacuum and there are no [Formula: see text]-invariant fields whose mass is less than 1/2.

Programming by Composing Filters

We present a formal model for event-processing pipelines. Event-processing pipelines appear in a large number of domains, from control of cyber-physical systems (CPS), to large scale data analysis, to Internet-of-things applications. These applications are characterized by stateful transformations of event streams, for example, for the purposes of sensing, computation, and actuation of inner control loops in CPS applications, and for data cleaning, analysis, training, and querying in data analytics applications.Our formal model provides two abstractions: streams of data, and stateful, probabilistic, filters, which transform input streams to output streams probabilistically. Programs are compositions of filters. The filters are scheduled and run by an explicit, asynchronous, scheduler.We provide a transition system semantics for such programs based on infinite-state Markov decision processes. We characterize when a program is scheduler-independent, that is, provides the same observable behavior under every scheduler, based on local commutativity.

Local extrema of entropy functions under tensor products

We show that under a certain condition of local commutativity the minimum von-Neumann entropy output of a quantum channel is locally additive. We also show that local minima of the 2-norm entropy functions are closed under tensor products if one of the subspaces has dimension 2.