In mathematics and physics, specifically the study of field theory and partial differential equations, a **Toda field theory**, named after Morikazu Toda, is specified by a choice of Lie algebra and a specific Lagrangian.^{ [1] }

Fixing the Lie algebra to have rank , that is, the Cartan subalgebra of the algebra has dimension , the Lagrangian can be written

The background spacetime is 2-dimensional Minkowski space, with space-like coordinate and timelike coordinate . Greek indices indicate spacetime coordinates.

For some choice of root basis, is the th simple root. This provides a basis for the Cartan subalgebra, allowing it to be identified with .

Then the field content is a collection of scalar fields , which are scalar in the sense that they transform trivially under Lorentz transformations of the underlying spacetime.

The inner product is the restriction of the Killing form to the Cartan subalgebra.

The are integer constants, known as **Kac labels** or **Dynkin labels**.

The physical constants are the mass and the coupling constant .

Toda field theories are classified according to their associated Lie algebra.

Toda field theories usually refer to theories with a finite Lie algebra. If the Lie algebra is an affine Lie algebra, it is called an affine Toda field theory (after the component of φ which decouples is removed). If it is hyperbolic, it is called a hyperbolic Toda field theory.

Toda field theories are integrable models and their solutions describe solitons.

Liouville field theory is associated to the A_{1} Cartan matrix, which corresponds to the Lie algebra in the classification of Lie algebras by Cartan matrices. The algebra has only a single simple root.

The sinh-Gordon model is the affine Toda field theory with the generalized Cartan matrix

and a positive value for β after we project out a component of φ which decouples.

The sine-Gordon model is the model with the same Cartan matrix but an imaginary β. This Cartan matrix corresponds to the Lie algebra . This has a single simple root, and Coxeter label , but the Lagrangian is modified for the affine theory: there is also an *affine root* and Coxeter label . One can expand as , but for the affine root , so the component decouples.

The sum is **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \sum_{i=0}^1 n_i\exp(\beta \alpha_i\phi) = \exp(\beta \phi) + \exp(-\beta\phi).}** Then if is purely imaginary, with real and, without loss of generality, positive, then this is . The Lagrangian is then

which is the sine-Gordon Lagrangian.

In mechanics and geometry, the **3D rotation group**, often denoted **SO(3)**, is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition.

In mathematics, a **root system** is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory. Finally, root systems are important for their own sake, as in spectral graph theory.

In the mathematical field of representation theory, a **weight** of an algebra *A* over a field **F** is an algebra homomorphism from *A* to **F**, or equivalently, a one-dimensional representation of *A* over **F**. It is the algebra analogue of a multiplicative character of a group. The importance of the concept, however, stems from its application to representations of Lie algebras and hence also to representations of algebraic and Lie groups. In this context, a **weight of a representation** is a generalization of the notion of an eigenvalue, and the corresponding eigenspace is called a **weight space**.

In mathematics, in particular the theory of Lie algebras, the **Weyl group** of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection group. In fact it turns out that *most* finite reflection groups are Weyl groups. Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these.

In physics, the ** S-matrix** or

In mathematics, a **Cartan subalgebra**, often abbreviated as **CSA**, is a nilpotent subalgebra of a Lie algebra that is self-normalising. They were introduced by Élie Cartan in his doctoral thesis. It controls the representation theory of a semi-simple Lie algebra over a field of characteristic .

In mathematics, a Lie algebra is **semisimple** if it is a direct sum of simple Lie algebras..

In mathematics, an **affine Lie algebra** is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody algebra, as described below. From a purely mathematical point of view, affine Lie algebras are interesting because their representation theory, like representation theory of finite-dimensional semisimple Lie algebras, is much better understood than that of general Kac–Moody algebras. As observed by Victor Kac, the character formula for representations of affine Lie algebras implies certain combinatorial identities, the Macdonald identities.

In quantum field theory, the **Lehmann–Symanzik–Zimmerman** (**LSZ**) **reduction formula** is a method to calculate *S*-matrix elements from the time-ordered correlation functions of a quantum field theory. It is a step of the path that starts from the Lagrangian of some quantum field theory and leads to prediction of measurable quantities. It is named after the three German physicists Harry Lehmann, Kurt Symanzik and Wolfhart Zimmermann.

In physics, a **sigma model** is a field theory that describes the field as a point particle confined to move on a fixed manifold. This manifold can be taken to be any Riemannian manifold, although it is most commonly taken to be either a Lie group or a symmetric space. The model may or may not be quantized. An example of the non-quantized version is the Skyrme model; it cannot be quantized due to non-linearities of power greater than 4. In general, sigma models admit (classical) topological soliton solutions, for example, the Skyrmion for the Skyrme model. When the sigma field is coupled to a gauge field, the resulting model is described by Ginzburg–Landau theory. 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".

In theoretical physics, a **source field** is a *background* field coupled to the original field as

In a field of mathematics known as differential geometry, a **Courant geometry** was originally introduced by Zhang-Ju Liu, Alan Weinstein and Ping Xu in their investigation of doubles of Lie bialgebroids in 1997. Liu, Weinstein and Xu named it after Courant, who had implicitly devised earlier in 1990 the standard prototype of Courant algebroid through his discovery of a skew symmetric bracket on , called Courant bracket today, which fails to satisfy the Jacobi identity. Both this standard example and the double of a Lie bialgebra are special instances of Courant algebroids.

In mathematics, the **spin representations** are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature. More precisely, they are two equivalent representations of the spin groups, which are double covers of the special orthogonal groups. They are usually studied over the real or complex numbers, but they can be defined over other fields.

In the context of quantum mechanics and quantum information theory, **symmetric, informationally complete, positive operator-valued measures** (SIC-POVMs) are a particular type of generalized measurement (POVM). SIC-POVMs are particularly notable thanks to their defining features of (1) being informationally complete; (2)having the minimal number of outcomes compatible with informational completeness, and (3) being highly symmetric. In this context, informational completeness is the property of a POVM of allowing to fully reconstruct input states from measurement data.

In mathematics, the **structure constants** or **structure coefficients** of an algebra over a field are the coefficients of the basis expansion of the products of basis vectors. Because the product operation in the algebra is bilinear, by linearity knowing the product of basis vectors allows to compute the product of any elements . Therefore, the structure constants can be used to specify the product operation of the algebra. Given the structure constants, the resulting product is obtained by bilinearity and can be uniquely extended to all vectors in the vector space, thus uniquely determining the product for the algebra.

In mathematical physics, **nonlinear realization** of a Lie group *G* possessing a Cartan subgroup *H* is a particular induced representation of *G*. In fact, it is a representation of a Lie algebra of *G* in a neighborhood of its origin. A nonlinear realization, when restricted to the subgroup *H* reduces to a linear representation.

In representation theory, a branch of mathematics, the **theorem of the highest weight** classifies the irreducible representations of a complex semisimple Lie algebra . There is a closely related theorem classifying the irreducible representations of a connected compact Lie group . The theorem states that there is a bijection

This is a glossary for the terminology applied in the mathematical theories of Lie groups and Lie algebras. For the topics in the representation theory of Lie groups and Lie algebras, see Glossary of representation theory. Because of the lack of other options, the glossary also includes some generalizations such as quantum group.

In abstract algebra, specifically the theory of Lie algebras, **Serre's theorem** states: given a root system , there exists a finite-dimensional semisimple Lie algebra whose root system is the given .

**Massless free scalar bosons** are a family of two-dimensional conformal field theories, whose symmetry is described by an abelian affine Lie algebra.

- ↑ Korff, Christian (1 September 2000). "Lie algebraic structures in integrable models, affine Toda field theory". arXiv: hep-th/0008200 .

- Mussardo, Giuseppe (2009),
*Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics*, Oxford University Press, ISBN 978-0-199-54758-6

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