I know that the terms in the Lagrangian needs to be scalars (with respect to Lorentz symmetry etc.). Also I know that [see C. G. Tully (EPP) p. 85]

in general, for $\psi$ in the fundamental representation and $\vec{\lambda}$ the generators of the group, then $$\psi^\dagger\vec{\lambda}\psi$$ will behave like a vector in the adjoint representation and can be dotted into a multiplet of fields in the adjoint representation to form a scalar.

So suppose I want to describe the dynamics or interactions of some particles say $\pi,\Sigma^*, \Lambda$. What would be the first steps here to build a Lagrangian? I want to work with $SU(3)$ here.

The $\Lambda$ and $\Sigma^*$ belong to the octet and decuplet respectively.

So the questions are:

  1. What would be the first steps in writing this Lagrangian?
  2. What shall I take for $\psi$ in this case?

1 Answer 1


The first step is to determine what you want your Lagrangian to describe, such as which particles you have, and which symmetries you have. For example, if you want a nonrelativistic description of some dynamics, you won't need the terms in your Lagrangian to be scalars with respect to Lorentz symmetry. Furthermore, if you have more than one particle, just using "$\psi$" won't suffice, as you will need one field per particle (which you can call $\pi$, $\Sigma^*$ and $\Lambda$ if you want to).

Then you have to figure out what else you know about the theory. If you don't know anything else (yes, it happens), then the next step is to simply write down all possible terms containing the fields you defined that obey the symmetries of the theory. If you have many fields and few symmetries, that may be a whole lot of them $-$ maybe even infinitely many.

If you have more information (such as the values of the masses or which kinds of interactions are actually relevant), you can use that to further constrain your Lagrangian. In most situations I know of, only terms of low degree in the fields and derivatives matter (such that a term $\propto\pi\Lambda^2\partial^2\Sigma^*$ may still be relevant, but one $\propto\pi^{23}\Lambda^{14}{\Sigma^*}^{9}\partial^8\Sigma^*$ will not). But that is not always the case...


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