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Representation of a group is a homomorphism from the group to the $GL(n,R)$. But I don't quite understand what is the charge of representation. Could someone explain?

Qmechanic
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htr
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  • to the $GL(n,R)$ This is overly restrictive. For example, the representation space doesn’t have to be a vector space over the real numbers. Nor does it have to be finite-dimensional. – Ghoster Dec 06 '22 at 00:14
  • True, but what does charge mean here? – htr Dec 06 '22 at 00:20
  • Where have you read that every group representation has a “charge”? The word “charge” doesn’t even appear in the Wikipedia article on group representations. Are you asking about electric charge in the Standard Model and how it relates to the representation that a particle is in? – Ghoster Dec 06 '22 at 00:21
  • In which context? Which reference? Which page? – Qmechanic Dec 06 '22 at 04:40
  • This Math.SE post seems relevant: https://math.stackexchange.com/q/2714138/11127 – Qmechanic Dec 06 '22 at 04:40

1 Answers1

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I'm not sure how deep your question is. Since you are employing highly ambiguous language, I'll assume the most simple-minded language that physicists talk to each other in.

Typically, a "charge" is a Lie algebra generator, for the Lie group involved, in your case GL(n, R). For instance, if we took the SU(2) subgroup of your GL(2,R), you'd have three charges, the three $\mathfrak{su}(2)$ algebra elements/generators.

In a given representation, e.g. the doublet representation, they'd be, e.g., the three Pauli matrices, suitably normalized. Their eigenvalues on a standardized doublet eigenvector are then called the respective charges of the doublet, e.g. 1 or -1 for $T_3$.

Is this what you are asking about?

Response to comment

You’ll have to choose among these suggestions…

The states of the theory are complex vectors of a given dimension characteristic of the representation; they are operated upon by the representation matrices of the group elements outlined. Such states thus connected are equivalent by the symmetry described by the group discussed… So, if, e.g., the hamiltonian of the system is symmetric under the group, the states thus connected have the same properties conferred by the hamiltonian, such as mass, and they are dubbed "degenerate"... Angular momentum is the archetypal paradigm of such symmetries (rotation group).

Cosmas Zachos
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  • That's, what I was asking. Thank you very much. Could you also briefly describe how Lie group representations are related to the quantum states of a theory? Can you suggest any easy-to-read but rigorous reference to read on such materials? I find these terminologies always confusing (I have more familiarity with math than physics). – htr Dec 08 '22 at 02:41