In Quantum theory of the fields a non-linear model sigma indicates a theory in which the fundamental fields represent coordinated in a Variété riemannienne called space-target . Together they constitute a Plongement since the space on which they live (for example the Espace of Minkowski) towards the space-target.

Definition

In the simplest case one considers that space on which the fields of the theory live is the space of Minkowki $\backslash \; mathcal\; \{M\}\; \backslash ,$. Its coodonnées are noted by a Greek index $\backslash \; mu=0,1,2,\; n-1\; \backslash ,$ with $n\; \backslash ,$ the Dimension of space (not necessarily equal to four).

If one notes $\backslash \; mathcal\; \{T\}\; \backslash ,$ the space-target and $\backslash \; Sigma:\; \backslash \; mathcal\; \{M\}\; \backslash \; rightarrow\; \backslash \; mathcal\; \{T\}\; \backslash ,$ plunging then the Lagrangien of the theory is written

where $V\; \backslash ,$ is a arbitrary Potentiel.

By choosing coordinates $\backslash \; Sigma^a\; \backslash ,$ on the space-target and $g\_\; \{ab\}\; \backslash ,$ the components of metric then one can rewrite the Lagrangian one like

Let us note that if the space-target is him also the space of Minkowski then this action is that of a simple theory of the fields provided with a potential $V\; \backslash ,$.

Properties

If the dimension of the starting space is higher than two then the model is not renormalisable in general. It is thus not well defined from the quantum point of view and thus has only the statute of effective Théorie of another well defined quantum theory supplementing this one on the scales smaller than the Courbure of the space-target (this discussion does not take account of the properties of the potential which can also break the renormalisability to him).

Applications

In Nuclear physics the chiral Model, which phenomenologic, is described the Méson S without mentioning Quark S (from the point of view of the theory of the strong Interaction that corresponds to take the limit where the masses of the quarks tend towards zero). It is a non-linear model sigma whose target is the Groupe of Dregs $SU\; (NR)\; \backslash ,$ where $N\; \backslash ,$ is the number of savors.

This type of theory is also often used in Physique statistics and Theoretical physics and this particularly in Théorie of the cords which is defined perturbativement like a non-linear model sigma renormalisable in two dimensions.

See too

• Lagrangian