Toda field theory

Short description: Special quantum field theory

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]

## Contents

* 1 Formulation
* 2 Classification of Toda field theories
* 3 Examples
* 4 References

## Formulation

Fixing the Lie algebra to have rank [math]\displaystyle{ r }[/math], that is, the Cartan subalgebra of the algebra has dimension [math]\displaystyle{ r }[/math], the Lagrangian can be written

[math]\displaystyle{ \mathcal{L}=\frac{1}{2}\left\langle \partial_\mu \phi, \partial^\mu \phi \right\rangle -\frac{m^2}{\beta^2}\sum_{i=1}^r n_i \exp(\beta \langle\alpha_i, \phi\rangle). }[/math]

The background spacetime is 2-dimensional Minkowski space, with space-like coordinate [math]\displaystyle{ x }[/math] and timelike coordinate [math]\displaystyle{ t }[/math]. Greek indices indicate spacetime coordinates.

For some choice of root basis, [math]\displaystyle{ \alpha_i }[/math] is the [math]\displaystyle{ i }[/math]th simple root. This provides a basis for the Cartan subalgebra, allowing it to be identified with [math]\displaystyle{ \mathbb{R}^r }[/math].

Then the field content is a collection of [math]\displaystyle{ r }[/math] scalar fields [math]\displaystyle{ \phi_i }[/math], which are scalar in the sense that they transform trivially under Lorentz transformations of the underlying spacetime.

The inner product [math]\displaystyle{ \langle\cdot, \cdot\rangle }[/math] is the restriction of the Killing form to the Cartan subalgebra.

The [math]\displaystyle{ n_i }[/math] are integer constants, known as Kac labels or Dynkin labels.

The physical constants are the mass [math]\displaystyle{ m }[/math] and the coupling constant [math]\displaystyle{ \beta }[/math].

## Classification of Toda field theories

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.

## Examples

Liouville field theory is associated to the A1 Cartan matrix, which corresponds to the Lie algebra [math]\displaystyle{ \mathfrak{su}(2) }[/math] in the classification of Lie algebras by Cartan matrices. The algebra [math]\displaystyle{ \mathfrak{su}(2) }[/math] has only a single simple root.

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

[math]\displaystyle{ \begin{pmatrix} 2&-2 \\\ -2&2 \end{pmatrix} }[/math]

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 [math]\displaystyle{ \mathfrak{su}(2) }[/math]. This has a single simple root, [math]\displaystyle{ \alpha_1 = 1 }[/math] and Coxeter label [math]\displaystyle{ n_1 = 1 }[/math], but the Lagrangian is modified for the affine theory: there is also an affine root [math]\displaystyle{ \alpha_0 = -1 }[/math] and Coxeter label [math]\displaystyle{ n_0 = 1 }[/math]. One can expand [math]\displaystyle{ \phi }[/math] as [math]\displaystyle{ \phi_0 \alpha_0 + \phi_1 \alpha_1 }[/math], but for the affine root [math]\displaystyle{ \langle \alpha_0, \alpha_0 \rangle = 0 }[/math], so the [math]\displaystyle{ \phi_0 }[/math] component decouples.

The sum is [math]\displaystyle{ \sum_{i=0}^1 n_i\exp(\beta \alpha_i\phi) = \exp(\beta \phi) + \exp(-\beta\phi). }[/math] Then if [math]\displaystyle{ \beta }[/math] is purely imaginary, [math]\displaystyle{ \beta = ib }[/math] with [math]\displaystyle{ b }[/math] real and, without loss of generality, positive, then this is [math]\displaystyle{ 2\cos(b\phi) }[/math]. The Lagrangian is then [math]\displaystyle{ \mathcal{L} = \frac{1}{2}\partial_\mu \phi \partial^\mu \phi + \frac{2m^2}{b^2}\cos(b\phi), }[/math] which is the sine-Gordon Lagrangian.

## References

1. ↑ 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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Quantum field theories

Standard|

* Chern–Simons model
* Chiral model
* Conformal field theory
* Ginzburg–Landau theory
* Kondo model
* Local quantum field theory
* Loop quantum gravity
* Loop quantum cosmology
* Noncommutative quantum field theory
* Non-linear sigma model
* Quantum chaos
* Quantum chromodynamics
* Quantum dynamics
* Quantum electrodynamics
* Quantum hadrodynamics
* Quantum hydrodynamics
* Quantum information theory
* Quantum information science
* Quartic interaction
* Quantum logic
* Quantum thermodynamics
* Quantum Yang–Mills theory
* Schwinger model
* sine-Gordon
* Standard Model
* String theory
* Thirring–Wess model
* Toda field theory
* Topological quantum field theory
* Wess–Zumino model
* Wess–Zumino–Witten model
* Yang–Mills theory
* Yang–Mills–Higgs model
* Yukawa model

Four-fermion interactions|

* Thirring model
* Gross–Neveu model
* Nambu–Jona-Lasinio model

Related|

* History of quantum field theory
* Axiomatic quantum field theory
* Quantum field theory in curved spacetime

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