How to get from representations to finite or infinitesimal transformations?

[confusio]

Hi all. I have here a reference with a representation of the Lie algebra of my symmetry group in terms the fields in my Lagrangian. In order to calculate Noether currents, I would like to use this representation to derive formulae for the infinitesimal forms of the symmetry transformations described by the elements of the Lie algebra. I know how to get from the finite forms of the transformations to the infinitesimal ones, so those would be great too. Does anybody know how to get from the representation of the algebra to the actual transformations?

Thanks so much for reading.

 

[tom.stoer]

... with a representation of the Lie algebra of my symmetry group in terms the fields in my Lagrangian. ... Does anybody know how to get from the representation of the algebra to the actual transformations?

As far as I understand it you have fields like A^a(x) where a is the algebra-index. Now you want to find the transformation from A(x) to A'(x), correct?

So what you need is the finite transformation. They can be represented via the matrix representation. So you have to find generators of you algebra, i.e. matrices T^a which form a basis of the algebra in some representation (fermions usually in the fundamental rep., bosons like photons, gluons, ... usually in the adjoint rep.). The group-valued transformations are generated as

g[q] = exp iq^aT^a with q^a=q^a(x)

 

[Entropia]

To get from representations to finite or infinitesimal transformations, you can use the following steps:

1. Start by understanding the representation of the Lie algebra in terms of the fields in your Lagrangian. This representation will give you a set of matrices or operators that correspond to the elements of the Lie algebra.

2. Next, you can use these matrices or operators to construct the finite transformations. This can be done by exponentiating the matrices or operators, which will give you a set of finite transformation matrices.

3. To obtain the infinitesimal transformations, you can take the limit as the finite transformation matrices approach the identity matrix. This will give you a set of infinitesimal transformation matrices.

4. Finally, you can use these infinitesimal transformation matrices to derive formulae for the infinitesimal forms of the symmetry transformations. This can be done by taking the derivative of the transformation matrices with respect to a small parameter, such as time or space.

In summary, to get from representations to finite or infinitesimal transformations, you need to use the representation of the Lie algebra to construct finite transformation matrices, take the limit to obtain infinitesimal transformation matrices, and then use these matrices to derive the formulae for the infinitesimal transformations. I hope this helps.