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 r, that is, the Cartan subalgebra of the algebra has dimension r, the Lagrangian can be written

ℒ=12⟨∂μϕ,∂μϕ⟩−m2β2∑i=1rniexp⁡(β⟨αi,ϕ⟩).

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

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

Then the field content is a collection of r scalar fields ϕi, 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 ni are integer constants, known as Kac labels or Dynkin labels.

The physical constants are the mass m and the coupling constant β.

## 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 𝔰𝔲(2) in the classification of Lie algebras by Cartan matrices. The algebra 𝔰𝔲(2) has only a single simple root.

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

(2−2−22)

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 𝔰𝔲(2). This has a single simple root, α1=1 and Coxeter label n1=1, but the Lagrangian is modified for the affine theory: there is also an affine root α0=−1 and Coxeter label n0=1. One can expand ϕ as ϕ0α0+ϕ1α1, but for the affine root ⟨α0,α0⟩=0, so the ϕ0 component decouples.

The sum is ∑i=01niexp⁡(βαiϕ)=exp⁡(βϕ)+exp⁡(−βϕ). Then if β is purely imaginary, β=ib with b real and, without loss of generality, positive, then this is 2cos⁡(bϕ). The Lagrangian is then ℒ=12∂μϕ∂μϕ+2m2b2cos⁡(bϕ), 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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