scholarly journals Some applications of the Jost-Lehmann-Dyson theorem to the study of the global analytic structure of the N point function of quantum field theory

1975 ◽  
pp. 163-184
Author(s):  
R. Stora
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Analytic Structure ◽  
Quantum Field ◽  
Point Function

scholarly journals Exact four-point function and OPE for an interacting quantum field theory with space/time anisotropic scale invariance

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Hidehiko Shimada ◽  
Hirohiko Shimada
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Scale Invariance ◽  
Asymptotic Series ◽  
Space Time ◽  
Quantum Field ◽  
Coupling Constant ◽  
Point Function ◽  
Calogero Model ◽  
Primary Operator

Abstract We identify a nontrivial yet tractable quantum field theory model with space/time anisotropic scale invariance, for which one can exactly compute certain four-point correlation functions and their decompositions via the operator-product expansion(OPE). The model is the Calogero model, non-relativistic particles interacting with a pair potential $$ \frac{g}{{\left|x-y\right|}^2} $$ g x − y 2 in one dimension, considered as a quantum field theory in one space and one time dimension via the second quantisation. This model has the anisotropic scale symmetry with the anisotropy exponent z = 2. The symmetry is also enhanced to the Schrödinger symmetry. The model has one coupling constant g and thus provides an example of a fixed line in the renormalisation group flow of anisotropic theories.We exactly compute a nontrivial four-point function of the fundamental fields of the theory. We decompose the four-point function via OPE in two different ways, thereby explicitly verifying the associativity of OPE for the first time for an interacting quantum field theory with anisotropic scale invariance. From the decompositions, one can read off the OPE coefficients and the scaling dimensions of the operators appearing in the intermediate channels. One of the decompositions is given by a convergent series, and only one primary operator and its descendants appear in the OPE. The scaling dimension of the primary operator we computed depends on the coupling constant. The dimension correctly reproduces the value expected from the well-known spectrum of the Calogero model combined with the so-called state-operator map which is valid for theories with the Schrödinger symmetry. The other decomposition is given by an asymptotic series. The asymptotic series comes with exponentially small correction terms, which also have a natural interpretation in terms of OPE.

scholarly journals Some exact results for the two-point function of an integrable quantum field theory

1981 ◽  
Vol 23 (12) ◽  
pp. 3081-3084 ◽  
Author(s):  
Dennis B. Creamer ◽  
H. B. Thacker ◽  
David Wilkinson
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Exact Results ◽  
Quantum Field ◽  
Point Function ◽  
Integrable Quantum ◽  
Integrable Quantum Field Theory

RENORMALIZATION OF A NONCOMMUTATIVE FIELD THEORY

2012 ◽  
Vol 27 (12) ◽  
pp. 1250067
Author(s):  
HARALD GROSSE ◽  
RAIMAR WULKENHAAR
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Integral Equation ◽  
Beta Function ◽  
Solvable Model ◽  
Linear Integral Equation ◽  
Noncommutative Space ◽  
Quantum Field ◽  
Point Function ◽  
Starting Point

We discuss a special Euclidean [Formula: see text]-quantum field theory over quantized space–time as an example of a renormalizable field theory. Using a Ward identity, it was possible to prove the vanishing of the beta function for the coupling constant to all orders in perturbation theory. We extend this work and obtain from the Schwinger–Dyson equation a nonlinear integral equation for the renormalized two-point function alone. The nontrivial renormalized four-point function fulfills a linear integral equation with the inhomogeneity determined by the two-point function. These integral equations might be the starting point of a nonperturbative construction of a Euclidean quantum field theory on a noncommutative space. We expect to learn about renormalization from this almost solvable model.

RENORMALIZATION OF A NONCOMMUTATIVE FIELD THEORY

2012 ◽  
Vol 13 ◽  
pp. 108-117
Author(s):  
HARALD GROSSE ◽  
RAIMAR WULKENHAAR
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Integral Equation ◽  
Beta Function ◽  
Linear Integral Equation ◽  
Noncommutative Space ◽  
Quantum Field ◽  
Point Function ◽  
Starting Point ◽  
Linear Integral

We discuss a special Euclidean [Formula: see text]-quantum field theory over quantized space-time as an example of a renormalizable field theory. Using a Ward identity, it was possible to prove the vanishing of the beta function for the coupling constant to all orders in perturbation theory. We extend this work and obtain from the Schwinger-Dyson equation a non-linear integral equation for the renormalized two-point function alone. The non-trivial renormalized four-point function fulfils a linear integral equation with the inhomogeneity determined by the two-point function. These integral equations might be the starting point of a nonperturbative construction of a Euclidean quantum field theory on a noncommutative space. We expect to learn about renormalization from this almost solvable model.

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