Representations of a group

Question

In Griffiths' Introduction to Elementary Particles (2ed), at the end of Sec 4.1, he says that

an ordinary scalar belongs to the one-dimensional representation of the rotation group, $SO(3)$, and a vector belongs to the three-dimensional representation; four-vectors belong to the four-dimensional representation of the Lorentz group;

I don't understand this. To my knowledge, scalars, vectors (four-vectors) are objects on which rotation (lorentz transformation) operations act on. Also, I thought that representations of a group would correspond to square matrices. Please explain.

Answer 1

Griffiths is using common-among-experts but confusing-to-beginners language. When he says, for example, that a four-vector “belongs to” the four-dimensional representation of the Lorentz group, he doesn’t mean that the four-vector is a member of the representation itself; he means that the four-vector is a member of the representation space, the vector space on which the representation acts.

A linear representation maps each group element to a linear transformation on some vector space. Each such transformation can be represented in some basis by a matrix. A four-dimensional representation of the Lorentz group maps Lorentz transformations to $4\times 4$ matrices in the obvious way. These matrices act on four-vectors, transforming them. The set of all possible four-vectors is the four-dimensional representation space.

By the way, there are less obvious representations which map Lorentz transformations onto linear transformations of vector spaces that are not four-dimensional, and thus onto matrices which are not $4\times 4$. For example, traceless symmetric four-tensors with two indices form a 9-dimensional representation space.

Answer 2

Scalar, vectors etc are indeed defined with respect to a group operation (here, $SO(3)$) and the dimensionality of the representation in some cases is enough to identify the representation itself.

It is possible to have representations of $SO(3)$ of dimension $2L+1$. You can just take the states with angular momentum $L$ as basis states for the carrier space. If the dimension is $1$ (i.e. $L=0$) one speaks of a scalar (under this group).

Representation in this context would correspond to a map from abstract operators to $(2L+1)\times (2L+1)$ matrices acting on the carrier space. The basis states transform by or carry a representation, rather than being a representation by themselves.

The obvious example would be the representation by square matrices of size $(2L+1)$ of the angular momentum operators $L_{x,y,z}$, which are usually found as an elementary exercise in basic quantum mechanics.