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| In the study of [[field theory (physics)|field theory]] and [[partial differential equation]]s, a '''Toda field theory''' (named after [[Morikazu Toda]]) is derived from the following [[Lagrangian]]:
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| :<math>\mathcal{L}=\frac{1}{2}\left[\left({\partial \phi \over \partial t},{\partial \phi \over \partial t}\right)-\left({\partial \phi \over \partial x}, {\partial \phi \over \partial x}\right)\right ]-{m^2 \over \beta^2}\sum_{i=1}^r n_i e^{\beta \alpha_i \cdot \phi}.</math>
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| Here ''x'' and ''t'' are spacetime coordinates, (,) is the [[Killing form]] of a real r-dimensional [[Cartan algebra]] <math>\mathfrak{h}</math> of a [[Kac-Moody algebra]] over <math>\mathfrak{h}</math>, α<sub>i</sub> is the i<sup>th</sup> [[Simple root (root system)|simple root]] in some root basis, n<sub>i</sub> is the [[Coxeter number]], m is the mass (or bare mass in the [[quantum field theory]] version) and β is the [[coupling constant]].
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| Then a '''Toda field theory''' is the study of a function φ mapping 2 dimensional [[Minkowski space]] satisfying the corresponding [[Euler-Lagrange equation]]s.
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| If the [[Kac-Moody algebra]] is finite, it's called a Toda field theory. If it is affine, it is called an affine Toda field theory (after the component of φ which decouples is removed) and if it is [[Hyperbolic algebra|hyperbolic]], it is called a hyperbolic Toda field theory.
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| Toda field theories are [[integrable model]]s and their solutions describe [[soliton]]s.
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| ==Examples==
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| [[Liouville field theory]] is associated to the A<sub>1</sub> [[Cartan matrix]].
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| The [[sinh-Gordon]] model is the affine Toda field theory with the [[generalized Cartan matrix]]
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| :<math>\begin{pmatrix} 2&-2 \\ -2&2 \end{pmatrix}</math> | |
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| and a positive value for β after we project out a component of φ which [[decouple]]s.
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| The [[sine-Gordon]] model is the model with the same Cartan matrix but an imaginary β.
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| ==References== | |
| *{{citation|last=Mussardo|first=Giuseppe|title=Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics|year=2009|publisher=Oxford University Press|isbn=0-199-54758-0}}
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| {{Quantum field theories}}
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| [[Category:Lattice models]]
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| [[Category:Lie algebras]]
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| [[Category:Exactly solvable models]]
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Friends call him Royal Cummins. Camping is some thing that I've carried out for years. Years ago we moved to Arizona but my wife wants us to move. After becoming out of my occupation for many years I grew to become a production and distribution officer but I strategy on altering it.
Here is my weblog - http://Realseo4u.com/