Sigma model
Encyclopedia
In physics
, a sigma model is a physical system
that is described by a Lagrangian density of the form:
Depending on the scalars gij it is either a linear sigma model or a non-linear sigma model
. The fields
, in general, provide a map
from a base manifold called the worldsheet
and a target (Riemannian) manifold that is often understood to be the spacetime
.
A basic example is provided by quantum mechanics
which is a quantum field theory
in one dimension. It's a sigma model with a base manifold given by the real line parameterizing the time (or an interval, or the circle, etc.) and a target space that is the real line.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, a sigma model is a physical system
Physical system
In physics, the word system has a technical meaning, namely, it is the portion of the physical universe chosen for analysis. Everything outside the system is known as the environment, which in analysis is ignored except for its effects on the system. The cut between system and the world is a free...
that is described by a Lagrangian density of the form:
Depending on the scalars gij it is either a linear sigma model or a non-linear sigma model
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 fields
, in general, provide a mapMap
A map is a visual representation of an area—a symbolic depiction highlighting relationships between elements of that space such as objects, regions, and themes....
from a base manifold called the worldsheet
Worldsheet
In string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime. The term was coined by Leonard Susskind around 1967 as a direct generalization of the world line concept for a point particle in special and general relativity.The type of string,...
and a target (Riemannian) manifold that is often understood to be the spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
.
A basic example is provided by quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
which is a quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
in one dimension. It's a sigma model with a base manifold given by the real line parameterizing the time (or an interval, or the circle, etc.) and a target space that is the real line.