scholarly journals Mathematical Structure of Anomalous Dimensions and QCD Wilson Coefficients in Higher Order

2004 ◽  
Vol 135 ◽  
pp. 225-231 ◽  
Author(s):  
J. Blümlein
Keyword(s):  
Mathematical Structure ◽  
Higher Order ◽  
Anomalous Dimensions

scholarly journals Theoretical framework for higher-order quantum theory

2019 ◽  
Vol 475 (2225) ◽  
pp. 20180706 ◽  
Author(s):  
Alessandro Bisio ◽  
Paolo Perinotti
Keyword(s):  
Quantum Theory ◽  
Convex Sets ◽  
Mathematical Structure ◽  
Higher Order ◽  
Equivalence Relations ◽  
Full Generality ◽  
Special Cases ◽  
Different Types ◽  
Causal Structures

Higher-order quantum theory is an extension of quantum theory where one introduces transformations whose input and output are transformations, thus generalizing the notion of channels and quantum operations. The generalization then goes recursively, with the construction of a full hierarchy of maps of increasingly higher order. The analysis of special cases already showed that higher-order quantum functions exhibit features that cannot be tracked down to the usual circuits, such as indefinite causal structures, providing provable advantages over circuital maps. The present treatment provides a general framework where this kind of analysis can be carried out in full generality. The hierarchy of higher-order quantum maps is introduced axiomatically with a formulation based on the language of types of transformations. Complete positivity of higher-order maps is derived from the general admissibility conditions instead of being postulated as in previous approaches. The recursive characterization of convex sets of maps of a given type is used to prove equivalence relations between different types. The axioms of the framework do not refer to the specific mathematical structure of quantum theory, and can therefore be exported in the context of any operational probabilistic theory.

scholarly journals Conditional Probability, Three-Slit Experiments, and the Jordan Algebra Structure of Quantum Mechanics

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Gerd Niestegge
Keyword(s):  
Quantum Mechanics ◽  
Conditional Probability ◽  
Mathematical Structure ◽  
Higher Order ◽  
Complete Characterization ◽  
Jordan Algebras ◽  
Algebra Structure ◽  
Quantum Logics ◽  
Close Link

Most quantum logics do not allow for a reasonable calculus of conditional probability. However, those ones which do so provide a very general and rich mathematical structure, including classical probabilities, quantum mechanics, and Jordan algebras. This structure exhibits some similarities with Alfsen and Shultz's noncommutative spectral theory, but these two mathematical approaches are not identical. Barnum, Emerson, and Ududec adapted the concept of higher-order interference, introduced by Sorkin in 1994, into a general probabilistic framework. Their adaption is used here to reveal a close link between the existence of the Jordan product and the nonexistence of interference of third or higher order in those quantum logics which entail a reasonable calculus of conditional probability. The complete characterization of the Jordan algebraic structure requires the following three further postulates: a Hahn-Jordan decomposition property for the states, a polynomial functional calculus for the observables, and the positivity of the square of an observable. While classical probabilities are characterized by the absence of any kind of interference, the absence of interference of third (and higher) order thus characterizes a probability calculus which comes close to quantum mechanics but still includes the exceptional Jordan algebras.

scholarly journals Resurgent Analysis of Ward-Schwinger-Dyson Equations

2021 ◽  
Author(s):  
Marc P. Bellon ◽  
◽  
Enrico I. Russo ◽  
◽  
◽  
...  
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Interaction Model ◽  
Higher Order ◽  
Renormalisation Group ◽  
Quantum Field ◽  
Anomalous Dimensions ◽  
Higher Order Corrections ◽  
Resurgent Analysis ◽  
Zumino Model

Building on our recent derivation of the Ward-Schwinger-Dyson equations for the cubic interaction model, we present here the first steps of their resurgent analysis. In our derivation of the WSD equations, we made sure that they had the properties of compatibility with the renormalisation group equations and independence from a regularisation procedure which was known to allow for the comparable studies in the Wess-Zumino model. The interactions between the transseries terms for the anomalous dimensions of the field and the vertex is at the origin of unexpected features, for which the effect of higher order corrections is not precisely known at this stage: we are only at the beginning of the journey to use resurgent methods to decipher non-perturbative effects in quantum field theory.

scholarly journals Comments on contact terms and conformal manifolds in the AdS/CFT correspondence

2020 ◽  
Author(s):  
Tadakatsu Sakai ◽  
Masashi Zenkai
Keyword(s):  
Fixed Point ◽  
Correlation Functions ◽  
Equations Of Motion ◽  
Mathematical Structure ◽  
Regularization Scheme ◽  
Anomalous Dimensions ◽  
Geometrical Data ◽  
Conformal Manifolds ◽  
General Setup ◽  
Marginal Deformation

Abstract We study the contact terms that appear in the correlation functions of exactly marginal operators using the AdS/CFT correspondence. It is known that CFT with an exactly marginal deformation requires the existence of the contact terms is crucial for a consistency of with their coefficients having a geometrical interpretation in the context of conformal manifolds. We show that the AdS/CFT correspondence captures properly the mathematical structure of the correlation functions that is expected from the CFT analysis. For this purpose, we employ holographic RG to formulate a most general setup in the bulk for describing an exactly marginal deformation. The resultant bulk equations of motion are nonlinear and solved perturbatively to obtain the on-shell action. We compute three- and four-point functions of the exactly marginal operators using the GKP-Witten prescription, and show that they match with the expected results precisely. It is pointed out that The cut-off surface prescription in the bulk provides us with a regularization scheme for performing a conformal perturbation. serves as a regularization scheme for conformal perturbation theory in the boundary CFT. around a fixed point is regularized by putting a cut-off surface in the bulk. As an application, we examine a double OPE limit of the four-point functions. The anomalous dimensions of double trace operators are written in terms of the geometrical data of a conformal manifold.

scholarly journals Holographic correlators with multi-particle states

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
Nejc Čeplak ◽  
Stefano Giusto ◽  
Marcel R. R. Hughes ◽  
Rodolfo Russo
Keyword(s):  
Correlation Function ◽  
Single Particle ◽  
Higher Order ◽  
Tree Level ◽  
Linear Combinations ◽  
Anomalous Dimensions ◽  
Formal Limit ◽  
Single Trace

Abstract We derive the connected tree-level part of 4-point holographic correlators in AdS3 × S3 × $$ \mathcal{M} $$ M (where $$ \mathcal{M} $$ M is T4 or K3) involving two multi-trace and two single-trace operators. These connected correlators are obtained by studying a heavy-heavy-light-light correlation function in the formal limit where the heavy operators become light. These results provide a window into higher-point holographic correlators of single-particle operators. We find that the correlators involving multi-trace operators are compactly written in terms of Bloch-Wigner-Ramakrishnan functions — particular linear combinations of higher-order polylogarithm functions. Several consistency checks of the derived expressions are performed in various OPE channels. We also extract the anomalous dimensions and 3-point couplings of the non-BPS double-trace operators of lowest twist at order 1/c and find some positive anomalous dimensions at spin zero and two in the K3 case.

Anomalous dimensions of higher-order operators in the $$\varphi ^4 $$ {-TheoryTheory

1974 ◽  
Vol 9 (12) ◽  
pp. 483-486 ◽  
Author(s):  
E. Brezin ◽  
C. De Dominicis ◽  
J. Zinn-Justin
Keyword(s):  
Higher Order ◽  
Anomalous Dimensions

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

Quantitative EPMA of nitrogen : A tricky element in the electron-probe microanalyzer

1992 ◽  
Vol 50 (2) ◽  
pp. 1622-1623
Author(s):  
G.F. Bastin ◽  
H.J.M. Heijligers
Keyword(s):  
Background Subtraction ◽  
Electron Probe ◽  
Detection System ◽  
Higher Order ◽  
Light Elements ◽  
Strong Suppression ◽  
Electron Probe Microanalyzer ◽  
Quantitative Epma ◽  
Peak Count ◽  
Lead Stearate

Among the ultra-light elements B, C, N, and O nitrogen is the most difficult element to deal with in the electron probe microanalyzer. This is mainly caused by the severe absorption that N-Kα radiation suffers in carbon which is abundantly present in the detection system (lead-stearate crystal, carbonaceous counter window). As a result the peak-to-background ratios for N-Kα measured with a conventional lead-stearate crystal can attain values well below unity in many binary nitrides . An additional complication can be caused by the presence of interfering higher-order reflections from the metal partner in the nitride specimen; notorious examples are elements such as Zr and Nb. In nitrides containing these elements is is virtually impossible to carry out an accurate background subtraction which becomes increasingly important with lower and lower peak-to-background ratios. The use of a synthetic multilayer crystal such as W/Si (2d-spacing 59.8 Å) can bring significant improvements in terms of both higher peak count rates as well as a strong suppression of higher-order reflections.

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