Editing Toda field theory (section)

==Formulation==

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

<math display=block>\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>x</math> and timelike coordinate <math>t</math>. Greek indices indicate spacetime coordinates.

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

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

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

The <math>n_i</math> are integer constants, known as '''Kac labels''' or '''Dynkin labels'''.

The physical constants are the mass <math>m</math> and the [[coupling constant]] <math>\beta</math>.