State of a Generator in Representation Theory

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Discussion Overview

The discussion revolves around the concept of the state of a generator in representation theory, particularly in the context of Lie algebras and Lie groups. Participants explore the definitions and implications of generators and states, seeking clarity on their relationship and usage in mathematical notation.

Discussion Character

  • Exploratory
  • Conceptual clarification

Main Points Raised

  • One participant questions the meaning of the state of a generator in representation theory, seeking clarification on how generators relate to states.
  • Another participant suggests that the notation introduced is meant to facilitate discussion about how generators can generate states, implying a need to revisit definitions of "state" and "generator."
  • A third participant elaborates on the role of generators in Lie algebras, noting that they can generate entire Lie groups and expressing confusion about the introduction of states in this context.
  • One participant acknowledges the potential for confusion due to varying terminology and suggests that the notation may refer to a specific state within the group generated by the generator.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the relationship between generators and states, indicating that there is no consensus on the clarity of the concept or the definitions involved.

Contextual Notes

There is an indication that the definitions of "state" and "generator" may depend on the specific context of the material being discussed, which is not fully provided in the thread.

Silviu
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Hello! I am reading something about representation theory (just started) and I encountered this: "We will denote the state in the adjoint representation corresponding to an arbitrary generator ## X_a ## as |## X_a ##>". What is the state of a generator in a certain representation?
Thank you!
 
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They did not talk about the state of a generator.
You have a generator... this can generate a state. They are introducing a notation to help talk about that.
Maybe you need to go back to find the definitions of "state" and "generator", and find out what a generator does?
 
Simon Bridge said:
They did not talk about the state of a generator.
You have a generator... this can generate a state. They are introducing a notation to help talk about that.
Maybe you need to go back to find the definitions of "state" and "generator", and find out what a generator does?
Thank you for your answer. I am still a bit confused. From what I understood generators are part of the Lie Algebra and they can generate the whole Lie Group. For example for SO(2) if S is the generator any rotation by ##\alpha## degrees is written as ##e^{i\alpha S}##. So, I understand a generator generates a whole group which usually acts on a vector space. Please let me know if anything I said is wrong. Now, I just don't understand where the idea of state appears in this as I thought that generators are generating a group not states.
 
I get confusded too and authors can use the same word to refer to slightly different things.
I do not have the context of the passage - but I would see the above as meaning that |A> is a single state belonging to the group that is generated by A.
The passages that follow should include examples that will make the matter clear. If they do not, then get a different book.
 

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