Group, Symmetries and Representation

[quangtu123]

I'm starting to learn about particle physics but I really want to see the whole picture before going deep. Here is what I know:

- There are symmetries in quantum physics, which are symmetry operators commute with the Hamiltonian (translation operators, rotation operators...) which act on a physics state. (I think these operators form groups)

- The representation theory: each particle corresponds to a representation (in the simplest meaning)

I want to ask is there any relation between the groups of opertors in the first and the groups belonging to representation in the second?

Thank you and I'm sorry if I'm being naive. I'm really not knowing anything about this field. I'm just trying to picturize.

 

[quangtu123]

It seems to me like in the representation theory of particle physics, the groups chosen (to form representation) are groups of symmetry operators.

It starts to make sense. Like the representation theory is the next step of symmetry in physics.

Am I wrong?

 

[Trifis]

The symmetries in nature, either internal or space-like do form groups, which we call symmetry groups. A group, mathematically, is an abstract object. What we do in order to be able to work with them, is to "represent" their elements by linear transformations between vector spaces or simply by matrices. Although the term "representation" refers to the elements of the group, it is usual to call by that name the elements of the underlying vector space, on which the group elements act.
A group can have many representations and each representation belongs to that group (and not the other way around). As it turns out, it is suitable to describe the behavior of different particles under a symmetry transformation by different representations of that symmetry group. For example, if a particle remains fully unaffected by a symmetry transformation, we can represent the whole group with the identity matrix and use this particular representation only for this particle. Therefore, any given particle (or family of particles), as far as a certain symmetry is considered, is associated with a unique representation of the symmetry group and we say the particle "lies in", or "transforms as" the representation.