# Non-linear sigma model

In quantum field theory, a nonlinear σ model describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T.

The target manifold is equipped with a Riemannian metric g. Σ is a differentiable map from Minkowski space M (or some other space) to T. The Lagrangian density is given by:

$\mathcal{L}={1\over 2}g(\partial^\mu\Sigma_a,\partial_\mu\Sigma_b)-V(\Sigma)$

where here, we have used a + - - - metric signature and the partial derivative $\partial\Sigma$ is given by a section of the jet bundle of T×M and V is the potential.

In the coordinate notation, with the coordinates Σa, a=1,...,n where n is the dimension of T,

$\mathcal{L}={1\over 2}g_{ab}(\Sigma) \partial^\mu \Sigma^{a} \partial_\mu \Sigma^{b} - V(\Sigma)$.

In more than 2 dimensions, nonlinear σ models are nonrenormalizable. This means they can only arise as effective field theories. New physics is needed at around the distance scale where the two point connected correlation function is of the same order as the curvature of the target manifold. This is called the UV completion of the theory.

There is a special class of nonlinear σ models with the internal symmetry group G *. If G is a Lie group and H is a Lie subgroup, then the quotient space G/H is a manifold (subject to certain technical restrictions like H being a closed subset) and is also a homogeneous space of G or in other words, a nonlinear realization of G. In many cases, G/H can be equipped with a Riemannian metric which is G-invariant. This is always the case, for example, if G is compact. A nonlinear σ model with G/H as the target manifold with a G-invariant Riemannian metric and a zero potential is called a quotient space (or coset space) nonlinear σ model.

When computing path integrals, the functional measure needs to be "weighted" by the square root of the determinant of g

$\sqrt{\det g}\mathcal{D}\Sigma.$

This model proved to be relevant in string theory where the two-dimensional manifold is named worldsheet. Proof of renormalizability was given by Daniel Friedan.[1] He showed that the theory admits a renormalization group equation, at the leading order of perturbation theory, in the form

$\lambda\frac{\partial g_{\mu\nu}}{\partial\lambda}=\beta_{\mu\nu}(T^{-1}g)=R_{\mu\nu}+O(T^2).$

being Rμν the Ricci tensor. This represents a Ricci flow having Einstein equations for the target manifold as a fixed point. The existence of such a fixed point is relevant as it grants, at this order of perturbation theory, that conformal invariance is not lost due to quantum corrections and one has a sensible quantum field theory.

## O(3) Non-linear Sigma Model

One of the most famous examples, of interest in for its particularly interesting topological properties, is the O(3) nonlinear sigma model in 1+1 dimension, with the Lagrange density

$\mathcal L= \frac{1}{2}\partial^\mu \hat n \cdot\partial_\mu \hat n$

where $\hat n=(n_1,n_2,n_3)$ with the constraint $\hat n\cdot \hat n=1$ and μ = 1,2. This model allows for topological finite action solutions, as at infinite space-time the Lagrange density must vanish, meaning $\hat n=const.$ at infinity. Therefore in the class of finite-action solutions we may identify the point at the infinity as a single point, i.e. that space-time can be identified with a Riemann Sphere. Since the $\hat n$-field lives on a sphere as well, we have a mapping $S^2\rightarrow S^2$, the solutions of which are classified by the Second Homotopy group of a 2-sphere. These solutions are called the O(3) Instantons.

## Notes

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Sigma model — In physics, a sigma model is a physical system that is described by a Lagrangian density of the form::mathcal{L}(phi 1, phi 2, ldots, phi n) = sum {i=1}^n sum {j=1}^n g {ij} ; mathrm{d}phi i wedge {*mathrm{d}phi j}Depending on the scalars g ij it …   Wikipedia

• Non-linear least squares — is the form of least squares analysis which is used to fit a set of m observations with a model that is non linear in n unknown parameters (m > n). It is used in some forms of non linear regression. The basis of the method is to… …   Wikipedia

• Linear discriminant analysis — (LDA) and the related Fisher s linear discriminant are methods used in statistics, pattern recognition and machine learning to find a linear combination of features which characterize or separate two or more classes of objects or events. The… …   Wikipedia

• Linear least squares — is an important computational problem, that arises primarily in applications when it is desired to fit a linear mathematical model to measurements obtained from experiments. The goals of linear least squares are to extract predictions from the… …   Wikipedia

• Linear least squares/Proposed — Linear least squares is an important computational problem, that arises primarily in applications when it is desired to fit a linear mathematical model to observations obtained from experiments. Mathematically, it can be stated as the problem of… …   Wikipedia

• Linear least squares (mathematics) — This article is about the mathematics that underlie curve fitting using linear least squares. For statistical regression analysis using least squares, see linear regression. For linear regression on a single variable, see simple linear regression …   Wikipedia

• Non-interactive zero-knowledge proof — Non interactive zero knowledge proofs are a variant of zero knowledge proofs. Blum, Feldman, and Micali [1] showed that a common reference string shared between the prover and the verifier is enough to achieve computational zero knowledge without …   Wikipedia

• Linear model — In statistics the linear model is given by:Y = X eta + varepsilonwhere Y is an n times;1 column vector of random variables, X is an n times; p matrix of known (i.e. observable and non random) quantities, whose rows correspond to statistical… …   Wikipedia

• Chiral model — In nuclear physics, the chiral model is a phenomenological model describing mesons in the chiral limit where the masses of the quarks go to zero (without mentioning quarks at all). It s a nonlinear sigma model with the principal homogeneous space …   Wikipedia

• Nambu–Jona-Lasinio model — In quantum field theory, the Nambu–Jona Lasinio model (or more precisely: the Nambu and Jona Lasinio model) is a theory of nucleons and mesons constructed from interacting Dirac fermions with chiral symmetry which parallels the construction of… …   Wikipedia