An example of the non-quantized version is the Skyrme model; it cannot be quantized due to non-linearities of power greater than 4.
This article is primarily devoted to the classical field theory of the sigma model; the corresponding quantized theory is presented in the article titled "non-linear sigma model".
The name has roots in particle physics, where a sigma model describes the interactions of pions.
[1] The sigma model was introduced by Gell-Mann & Lévy (1960, section 5); the name σ-model comes from a field in their model corresponding to a spinless meson called σ, a scalar meson introduced earlier by Julian Schwinger.
[2] The model served as the dominant prototype of spontaneous symmetry breaking of O(4) down to O(3): the three axial generators broken are the simplest manifestation of chiral symmetry breaking, the surviving unbroken O(3) representing isospin.
In conventional particle physics settings, the field is generally taken to be SU(N), or the vector subspace of quotient
For the rotation group O(3), the sigma model describes the isotropic ferromagnet; more generally, the O(N) model shows up in the quantum Hall effect, superfluid Helium-3 and spin chains.
This splitting helps propel the dimensional reduction of Kaluza–Klein theories.
The Lagrangian density of the sigma model can be written in a variety of different ways, each suitable to a particular type of application.
must be a differentiable manifold; by convention, it is either Minkowski space in particle physics applications, flat two-dimensional Euclidean space for condensed matter applications, or a Riemann surface, the worldsheet in string theory.
the Kronecker delta, i.e. the scalar dot product in Euclidean space, one gets the
, this is the continuum limit of the isotropic ferromagnet on a lattice, i.e. of the classical Heisenberg model.
The continuum limit is taken by writing as the finite difference on neighboring lattice locations
The sigma model action is then just the conventional inner product on vector-valued k-forms where the
In this way, one may write This makes it explicit and plainly evident that the sigma model is just the kinetic energy of a point particle.
The Hodge star is merely a fancy device for keeping track of the volume form when integrating on curved spacetime.
Classical extrema of the action (the solutions to the Lagrange equations) are then those field configurations that minimize the Dirichlet energy of
Another way to convert this expression into a more easily-recognizable form is to observe that, for a scalar function
More verbosely, The tension between these two inner products can be made even more explicit by noting that is a bilinear form; it is a pullback of the Riemann metric
Several interpretational and foundational remarks can be made about the classical (non-quantized) sigma model.
Assembling all these pieces together, the sigma model action is equivalent to which is just the grand-total kinetic energy of the wave-function
It is very well-known that the geodesic structure of a Riemannian manifold is described by the Hamilton–Jacobi equations.
, define the Hamiltonian function where, as always, one is careful to note that the inverse of the metric is used in this definition:
Caution: the precise definition of a solder form requires it to be an isomorphism; this can only happen if
Note that the Killing form can be written as a trace over two matrices from the corresponding Lie algebra; thus, the Lagrangian can also be written in a form involving the trace.
A common variation of the sigma model is to present it on a symmetric space.
The prototypical example is the chiral model, which takes the product of the "left" and "right" chiral fields, and then constructs the sigma model on the "diagonal" Such a quotient space is a symmetric space, and so one can generically take
In physics, the most common and conventional statement of the sigma model begins with the definition Here, the
Plugging this directly into the above and applying the infinitesimal form of the Baker–Campbell–Hausdorff formula promptly leads to the equivalent expression where
For the sigma model on a symmetric space, as opposed to a Lie group, the