Generalized non-linear σ-models in two dimensions

2021 ◽  
pp. 692-720
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Symmetric Spaces ◽  
Homogeneous Spaces ◽  
Two Dimensions ◽  
Asymptotic Freedom ◽  
Quantum Field Theories ◽  
Field Equations ◽  
Local Conservation ◽  
Non Linear ◽  
Coset Spaces ◽  
Formal Properties

This chapter describes the formal properties, and discusses the renormalization, of quantum field theories (QFT) based on homogeneous spaces: coset spaces of the form G/H, where G is a compact Lie group and H a Lie subgroup. In physics, they appear naturally in the case of spontaneous symmetry breaking, and describe the interaction between Goldstone modes. Homogeneous spaces are associated with non-linear realizations of group representations. There exist natural ways to embed these manifolds in flat Euclidean spaces, spaces in which the symmetry group acts linearly. As in the example of the non-linear σ-model, this embedding is first used, because the renormalization properties are simpler, and the physical interpretation of the more direct correlation functions. Then, in a generic parametrization, the renormalization problem is solved by the introduction of a Becchi–Rouet–Stora–Tyutin (BRST)-like symmetry with anticommuting (Grassmann) parameters, which also plays an essential role in quantized gauge theories. The more specific properties of models corresponding to a special class of homogeneous spaces, symmetric spaces (like the non-linear σ-model), are studied. These models are characterized by the uniqueness of the metric and thus, of the classical action. In two dimensions, from the classical field equations an infinite number of non-local conservation laws can be derived. The field and the unique coupling renormalization group (RG) functions are calculated at one-loop order, in two dimensions, and shown to imply asymptotic freedom.

The non-linear σ-model near two dimensions: Phase structure

2021 ◽  
pp. 458-488
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Field Theory ◽  
Phase Structure ◽  
Symmetric Spaces ◽  
Homogeneous Spaces ◽  
Two Dimensions ◽  
Leading Order ◽  
Non Linear ◽  
Two Parameters ◽  
Infrared Divergences ◽  
Goldstone Modes

This chapter is devoted to the study of the non-linear σ-model, a quantum field theory (QFT) where the (scalar) field is an N-component vector of fixed length, mostly in dimensions close to 2. The model possesses a global, non-linearly realized symmetry, O(N) symmetry: under a group transformation, the transformed field is a non-linear function of the field itself. The non-linear σ-model belongs to a class of models constructed on special homogeneous spaces, symmetric spaces that, as Riemannian manifolds, admit a unique metric. Unlike what happens in a (ϕ2)2 -like field theory with the same symmetry, in the non-linear σ-model, in the tree approximation, the O(N) symmetry is always spontaneously broken: the action describes the interactions of (N−1) massless fields, the Goldstone modes. Since the fields are massless, in two dimensions infrared divergences appear in the perturbative expansion and an infrared regulator is required. To understand the phase structure beyond leading order, a renormalization group (RG) analysis is necessary. This requires understanding how the model renormalizes. Power counting shows that the model is renormalizable in two dimensions. Since the field then is dimensionless, although the degree of divergence of Feynman diagrams is bounded, an infinite number of counterterms is generated, because all correlation functions are divergent. A quadratic master equation satisfied by the generating functional of vertex functions is derived, which makes it possible to prove that the coefficients of all counterterms are related, and that the renormalized theory depends only on two parameters.

A non-linear theory of strong interactions

1958 ◽  
Vol 247 (1249) ◽  
pp. 260-278 ◽  
Keyword(s):  
Linear Theory ◽  
Two Dimensions ◽  
Strong Interactions ◽  
Isotopic Spin ◽  
Meson Field ◽  
Field Equations ◽  
Constant Length ◽  
Cubic Symmetry ◽  
Continuous Rotation ◽  
Non Linear

A non-linear theory of mesons, nucleons and hyperons is proposed. The three independent fields of the usual symmetrical pseudo-scalar pion field are replaced by the three directions of a four-component field vector of constant length, conceived in an Euclidean four-dimensional isotopic spin space. This length provides the universal scaling factor, all other constants being dimensionless; the mass of the meson field is generated by a ϕ 4 term; this destroys the continuous rotation group in the iso-space, leaving a ‘cubic’ symmetry group. Classification of states by this group introduces quantum numbers corresponding to isotopic spin and to ‘strangeness’; one consequence is that, at least in elementary interactions, charge is only conserved modulo 4. Furthermore, particle states have not a well-defined parity, but parity is effectively conserved for meson-nucleon interactions. A simplified model, using only two dimensions of space and iso-space, is considered further; the non-linear meson field has solutions with particle character, and an indication is given of the way in which the particle field variables might be introduced as collective co-ordinates describing the dynamics of these particular solutions of the meson field equations, suggesting a unified theory based on the meson field alone.

scholarly journals Some nozzle flows found by the hodograph method

1959 ◽  
Vol 1 (1) ◽  
pp. 80-94 ◽  
Author(s):  
T. M. Cherry
Keyword(s):  
Exact Solutions ◽  
Two Dimensions ◽  
Nozzle Design ◽  
Perfect Gas ◽  
Field Equations ◽  
Rigid Boundary ◽  
Hodograph Method ◽  
Isentropic Flow ◽  
Fixed Boundaries ◽  
Nozzle Flows

For investigating the steady irrotational isentropic flow of a perfect gas in two dimensions, the hodograph method is to determine in the first instance the position coordinates x, y and the stream function ψ as functions of velocity compoments, conveniently taken as q (the speed) and θ (direction angle). Inversion then gives ψ, q, θ as functions of x, y. The method has the great advantage that its field equations are linear, so that it is practicable to obtain exact solutions, and from any two solutions an infinity of others are obtainable by superposition. For problems of flow past fixed boundaries the linearity of the field equations is usually offset by non-linearity in the boundary conditions, but this objection does not arise in problems of transsonic nozzle design, where the rigid boundary is the end-point of the investigation.

Non-linear dynamics, chaos, complexity and enclaves in granitoid magmas

1996 ◽  
Vol 87 (1-2) ◽  
pp. 217-223 ◽  
Author(s):  
James Flinders ◽  
John D. Clemens
Keyword(s):  
Deterministic Chaos ◽  
Interfacial Area ◽  
Two Dimensions ◽  
Mixing Processes ◽  
Nd Isotope ◽  
Natural Systems ◽  
Non Linear ◽  
The Difference ◽  
Magma System ◽  
Isotope Systematics

ABSTRACT:Most natural systems display non-linear dynamic behaviour. This should be true for magma mingling and mixing processes, which may be chaotic. The equations that most nearly represent how a chaotic natural system behaves are insoluble, so modelling involves linearisation. The difference between the solution of the linearised and ‘true’ equation is assumed to be small because the discarded terms are assumed to be unimportant. This may be very misleading because the importance of such terms is both unknown and unknowable. Linearised equations are generally poor descriptors of nature and are incapable of either predicting or retrodicting the evolution of most natural systems. Viewed in two dimensions, the mixing of two or more visually contrasting fluids produces patterns by folding and stretching. This increases the interfacial area and reduces striation thickness. This provides visual analogues of the deterministic chaos within a dynamic magma system, in which an enclave magma is mingling and mixing with a host magma. Here, two initially adjacent enclave blobs may be driven arbitrarily and exponentially far apart, while undergoing independent (and possibly dissimilar) changes in their composition. Examples are given of the wildly different morphologies, chemical characteristics and Nd isotope systematics of microgranitoid enclaves within individual felsic magmas, and it is concluded that these contrasts represent different stages in the temporal evolution of a complex magma system driven by nonlinear dynamics. If this is true, there are major implications for the interpretation of the parts played by enclaves in the genesis and evolution of granitoid magmas.

The Selberg–Weil–Kobayashi rigidity theorem: The rank one solvable case

2016 ◽  
Vol 27 (10) ◽  
pp. 1650085
Author(s):  
A. Baklouti ◽  
N. Elaloui ◽  
I. Kedim
Keyword(s):  
Homogeneous Space ◽  
Symmetric Spaces ◽  
Homogeneous Spaces ◽  
Rigidity Theorem ◽  
General Context ◽  
Affine Transformations ◽  
Discontinuous Group ◽  
Solvable Case ◽  
Local Rigidity ◽  
Discontinuous Groups

A local rigidity theorem was proved by Selberg and Weil for Riemannian symmetric spaces and generalized by Kobayashi for a non-Riemannian homogeneous space [Formula: see text], determining explicitly which homogeneous spaces [Formula: see text] allow nontrivial continuous deformations of co-compact discontinuous groups. When [Formula: see text] is assumed to be exponential solvable and [Formula: see text] is a maximal subgroup, an analog of such a theorem states that the local rigidity holds if and only if [Formula: see text] is isomorphic to the group Aff([Formula: see text]) of affine transformations of the real line (cf. [L. Abdelmoula, A. Baklouti and I. Kédim, The Selberg–Weil–Kobayashi rigidity theorem for exponential Lie groups, Int. Math. Res. Not. 17 (2012) 4062–4084.]). The present paper deals with the more general context, when [Formula: see text] is a connected solvable Lie group and [Formula: see text] a maximal nonnormal subgroup of [Formula: see text]. We prove that any discontinuous group [Formula: see text] for a homogeneous space [Formula: see text] is abelian and at most of rank 2. Then we discuss an analog of the Selberg–Weil–Kobayashi local rigidity theorem in this solvable setting. In contrast to the semi-simple setting, the [Formula: see text]-action on [Formula: see text] is not always effective, and thus the space of group theoretic deformations (formal deformations) [Formula: see text] could be larger than geometric deformation spaces. We determine [Formula: see text] and also its quotient modulo uneffective parts when the rank [Formula: see text]. Unlike the context of exponential solvable case, we prove the existence of formal colored discontinuous groups. That is, the parameter space admits a mixture of locally rigid and formally nonrigid deformations.

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