STOCHASTIC QUANTIZATION AND 1/N EXPANSION

1993 ◽  
Vol 08 (02) ◽  
pp. 115-128
Author(s):  
J.C. BRUNELLI ◽  
R.S. MENDES
Keyword(s):  
Perturbation Theory ◽  
Sigma Model ◽  
Functional Approach ◽  
Two Dimensions ◽  
Field Theories ◽  
Nonlinear Sigma Model ◽  
Stochastic Quantization ◽  
Quantization Method ◽  
Systematic Procedure

We study the 1/N expansion of field theories in the stochastic quantization method of Parisi and Wu using the supersymmetric functional approach. This formulation provides a systematic procedure to implement the 1/N expansion which resembles the ones used in the equilibrium. The 1/N perturbation theory for the nonlinear sigma-model in two dimensions is worked out as an example.

NON-RENORMALIZABILITY OF THE MASSIVE N = 2 SUPER-YANG–MILLS THEORY

1991 ◽  
Vol 06 (23) ◽  
pp. 2143-2154 ◽  
Author(s):  
G. A. KHELASHVILI ◽  
V. I. OGIEVETSKY
Keyword(s):  
Perturbation Theory ◽  
Fourth Order ◽  
Sigma Model ◽  
Nonlinear Sigma Model ◽  
Harmonic Superspace ◽  
Yang Mills ◽  
Super Yang Mills Theory

The massive N = 2 supersymmetric Yang–Mills theory is investigated. Its non-renormalizability is revealed starting from the fourth order of the perturbation theory. The N = 2 harmonic superspace approach and the Stueckelberg-like formalism are used. The Stueckelberg fields form some nonlinear sigma model. Non-renormalizability of the latter produces non-renormalizability of the N = 2 supersymmetric Yang–Mills theory.

scholarly journals Statistical inference and string theory

2015 ◽  
Vol 30 (26) ◽  
pp. 1550160 ◽  
Author(s):  
Jonathan J. Heckman
Keyword(s):  
String Theory ◽  
Statistical Inference ◽  
Sigma Model ◽  
Two Dimensions ◽  
Nonlinear Sigma Model ◽  
Dimensional Manifold ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Lorentzian Signature ◽  
Stability Under Perturbations

In this paper, we expose some surprising connections between string theory and statistical inference. We consider a large collective of agents sweeping out a family of nearby statistical models for an [Formula: see text]-dimensional manifold of statistical fitting parameters. When the agents making nearby inferences align along a [Formula: see text]-dimensional grid, we find that the pooled probability that the collective reaches a correct inference is the partition function of a nonlinear sigma model in [Formula: see text] dimensions. Stability under perturbations to the original inference scheme requires the agents of the collective to distribute along two dimensions. Conformal invariance of the sigma model corresponds to the condition of a stable inference scheme, directly leading to the Einstein field equations for classical gravity. By summing over all possible arrangements of the agents in the collective, we reach a string theory. We also use this perspective to quantify how much an observer can hope to learn about the internal geometry of a superstring compactification. Finally, we present some brief speculative remarks on applications to the AdS/CFT correspondence and Lorentzian signature space–times.

THE BOSONIC SIGMA MODEL ON A COMPACT SURFACE

1990 ◽  
Vol 05 (25) ◽  
pp. 2031-2037 ◽  
Author(s):  
M. LEBLANC ◽  
P. MADSEN ◽  
R. B. MANN ◽  
D. G. C. McKEON
Keyword(s):  
Sigma Model ◽  
Stereographic Projection ◽  
Two Dimensions ◽  
Compact Surface ◽  
Three Dimensions ◽  
Dimensional Euclidean Space ◽  
Nonlinear Sigma Model ◽  
Β Function ◽  
The One ◽  
Infrared Divergences

A stereographic projection is used to map the bosonic nonlinear sigma model with torsion from two-dimensional Euclidean space onto a sphere-S2 embedded in three dimensions. The one-loop β-function of the torsionless σ-model is determined using operator regularization to handle ultraviolet divergences. Only by excluding the lowest eigenstate of the rotation operator on the sphere can the usual β-function be recovered; inclusion of this eigenstate leads to severe infrared divergences. Both the ultraviolet and infrared divergences can be regulated by working in n, rather than two, dimensions, in which case the contribution of the lowest mode cancels exactly against the contribution of all other modes, resulting in a vanishing β-function.

scholarly journals On the rotator Hamiltonian for the SU (N) × SU (N) sigma model in the delta regime

2020 ◽  
Vol 2020 (7) ◽  
Author(s):  
J Balog ◽  
F Niedermayer ◽  
P Weisz
Keyword(s):  
Perturbation Theory ◽  
Chiral Perturbation Theory ◽  
Sigma Model ◽  
Two Dimensions ◽  
Leading Order ◽  
Chiral Perturbation ◽  
Casimir Invariant ◽  
The Difference

Abstract We investigate some properties of the standard rotator approximation of the $\mathrm{SU}(N)\times\mathrm{SU}(N)$ sigma-model in the delta regime. In particular, we show that the isospin susceptibility calculated in this framework agrees with that computed by chiral perturbation theory up to next-to-next-to-leading order in the limit $\ell=L_t/L\to\infty$. The difference between the results involves terms vanishing like $1/\ell$, plus terms vanishing exponentially with $\ell$. As we have previously shown for the O($n$) model, this deviation can be described by a correction to the rotator spectrum proportional to the square of the quadratic Casimir invariant. Here we confront this expectation with analytic nonperturbative results on the spectrum in two dimensions for $N=3$.

scholarly journals NEW SPIN-2 GAUGED SIGMA MODELS AND GENERAL CONFORMAL FIELD THEORY

1998 ◽  
Vol 13 (26) ◽  
pp. 4487-4512 ◽  
Author(s):  
J. DE BOER ◽  
M. B. HALPERN
Keyword(s):  
Field Theory ◽  
Sigma Models ◽  
Large Class ◽  
Sigma Model ◽  
Field Theories ◽  
Nonlinear Sigma Model ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Local Gauge Symmetry ◽  
Action Formulation

Recently, we have studied the general Virasoro construction at one loop in the background of the general nonlinear sigma model. Here, we find the action formulation of these new conformal field theories when the background sigma model is itself conformal. In this case, the new conformal field theories are described by a large class of new spin-2 gauged sigma models. As examples of the new actions, we discuss the spin-2 gauged WZW actions, which describe the conformal field theories of the generic affine-Virasoro construction, and the spin-2 gauged g/h coset constructions. We are able to identify the latter as the actions of the local Lie h-invariant conformal field theories, a large class of generically irrational conformal field theories with a local gauge symmetry.

scholarly journals ON FIELD THEORIES OF LOOPS

1995 ◽  
Vol 10 (29) ◽  
pp. 2175-2184 ◽  
Author(s):  
NAOHITO NAKAZAWA
Keyword(s):  
Field Theory ◽  
Stochastic Process ◽  
Loop Space ◽  
Continuum Limit ◽  
Symmetric Matrix ◽  
Field Theories ◽  
Stochastic Quantization ◽  
Quantization Method ◽  
Second Quantization ◽  
The Continuum

We apply stochastic quantization method to real symmetric matrix models for the second quantization of nonorientable loops in both discretized and continuum levels. The stochastic process defined by the Langevin equation in loop space describes the time evolution of the nonorientable loops defined on nonorientable 2-D surfaces. The corresponding Fokker-Planck Hamiltonian deduces a nonorientable string field theory at the continuum limit.

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