scholarly journals A microlocal approach to renormalization in stochastic PDEs

2021 ◽  
pp. 2150075
Author(s):  
Claudio Dappiaggi ◽  
Nicolò Drago ◽  
Paolo Rinaldi ◽  
Lorenzo Zambotti
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Large Class ◽  
Random Fields ◽  
Stochastic Partial Differential Equations ◽  
Algebraic Approach ◽  
Stochastic Pdes ◽  
Quantum Field ◽  
Regularization Scheme ◽  
Text Model

We present a novel framework for the study of a large class of nonlinear stochastic partial differential equations (PDEs), which is inspired by the algebraic approach to quantum field theory. The main merit is that, by realizing random fields within a suitable algebra of functional-valued distributions, we are able to use techniques proper of microlocal analysis which allow us to discuss renormalization and its associated freedom without resorting to any regularization scheme and to the subtraction of infinities. As an example of the effectiveness of the approach we apply it to the perturbative analysis of the stochastic [Formula: see text] model.

scholarly journals Construction of Kink Sectors for Two-Dimensional Quantum Field Theory Models – An Algebraic Approach

1998 ◽  
Vol 10 (06) ◽  
pp. 851-891 ◽  
Author(s):  
Dirk Schlingemann
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Algebraic Approach ◽  
Vacuum State ◽  
Quantum Field ◽  
Two Dimensional ◽  
Algebraic Quantum ◽  
Free Scalar ◽  
Split Property ◽  
Text Model

Several two-dimensional quantum field theory models have more than one vacuum state. Familiar examples are the Sine-Gordon and the [Formula: see text]-model. It is known that in these models there are also states, called kink states, which interpolate different vacua. A general construction scheme for kink states in the framework of algebraic quantum field theory is developed in a previous paper. However, for the application of this method, the crucial condition is the split property for wedge algebras in the vacuum representations of the considered models. It is believed that the vacuum representations of P(ϕ)2-models fulfill this condition, but a rigorous proof is only known for the massive free scalar field. Therefore, we investigate in a construction of kink states which can directly be applied to a large class of quantum field theory models, by making use of the properties of the dynamics of a P(ϕ)2 and Yukawa2 models.

scholarly journals QUANTUM FIELD THEORY ON CURVED BACKGROUNDS — A PRIMER

2013 ◽  
Vol 28 (17) ◽  
pp. 1330023 ◽  
Author(s):  
MARCO BENINI ◽  
CLAUDIO DAPPIAGGI ◽  
THOMAS-PAUL HACK
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hilbert Space ◽  
Degrees Of Freedom ◽  
Algebraic Approach ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
System A ◽  
Step Procedure ◽  
Free Fields

Goal of this paper is to introduce the algebraic approach to quantum field theory on curved backgrounds. Based on a set of axioms, first written down by Haag and Kastler, this method consists of a two-step procedure. In the first one, it is assigned to a physical system a suitable algebra of observables, which is meant to encode all algebraic relations among observables, such as commutation relations. In the second step, one must select an algebraic state in order to recover the standard Hilbert space interpretation of a quantum system. As quantum field theories possess infinitely many degrees of freedom, many unitarily inequivalent Hilbert space representations exist and the power of such approach is the ability to treat them all in a coherent manner. We will discuss in detail the algebraic approach for free fields in order to give the reader all necessary information to deal with the recent literature, which focuses on the applications to specific problems, mostly in cosmology.

HAAG DUALITY IN CONFORMAL QUANTUM FIELD THEORY

1990 ◽  
Vol 02 (01) ◽  
pp. 105-125 ◽  
Author(s):  
DETLEV BUCHHOLZ ◽  
HANNS SCHULZ-MIRBACH
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Central Charge ◽  
Algebraic Approach ◽  
Energy Tensor ◽  
Quantum Field ◽  
Haag Duality ◽  
Conformal Fields ◽  
Stress Energy

Haag duality is established in conformal quantum field theory for observable fields on the compactified light ray S1 and Minkowski space S1×S1, respectively. This result provides the foundation for an algebraic approach to the classification of conformal theories. Haag duality can fail, however, for the restriction of conformal fields to the underlying non-compact spaces ℝ, respectively ℝ×ℝ. A prominent example is the stress energy tensor with central charge c>1.

DIRICHLET FORMS IN TERMS OF WHITE NOISE ANALYSIS I: CONSTRUCTION AND QFT EXAMPLES

1989 ◽  
Vol 01 (02n03) ◽  
pp. 291-312 ◽  
Author(s):  
S. ALBEVERIO ◽  
T. HIDA ◽  
J. POTTHOFF ◽  
M. RÖCKNER ◽  
L. STREIT
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
White Noise ◽  
Random Fields ◽  
Noise Analysis ◽  
Dirichlet Forms ◽  
Quantum Field ◽  
Quantum Fields ◽  
White Noise Analysis ◽  
Nikodym Derivative

Random fields are given in terms of measures which (in general) are singular with respect to that of white noise. However, many such measures can be expressed in terms of white noise through a positive generalized functional acting as a generalized Radon-Nikodym derivative. We give criteria for this to be the case and show that these criteria are fulfilled by Schwinger and Wightman functionals of various nontrivial quantum field theory models. Furthermore a number of closability criteria are given and discussed for the Dirichlet forms associated with positive generalized functionals of white noise. In a second paper we apply these results to the construction of Markov and of quantum fields.

scholarly journals ON THE EXISTENCE OF KINK (SOLITON) STATES

1996 ◽  
Vol 08 (08) ◽  
pp. 1187-1203 ◽  
Author(s):  
DIRK SCHLINGEMANN
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Vacuum State ◽  
Quantum Field ◽  
Algebraic Quantum Field Theory ◽  
Algebraic Quantum ◽  
Vacuum States ◽  
Text Model ◽  
Kink Soliton ◽  
Natural Way

Several two-dimensional quantum field theory models have more than one vacuum state. Familiar examples are the Sine-Gordon and the [Formula: see text]-model. It is known that in these models there are also states, called soliton or kink states, which interpolate different vacua. We investigate the following question: Which are the properties a pair of vacuum states must have, such that an interpolating kink state can be constructed? We discuss the problem in the framework of algebraic quantum field theory which includes, for example, the P(ϕ)2-models. We identify a large class of vacuum states, including the vacua of the P(ϕ)2-models, for which there is a natural way to construct an interpolating kink state.

An Algebraic Approach to Quantum Field Theory

1964 ◽  
Vol 5 (7) ◽  
pp. 848-861 ◽  
Author(s):  
Rudolf Haag ◽  
Daniel Kastler
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Algebraic Approach ◽  
Quantum Field
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