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I'm not asking about the math here, I'm interested in the wording physicists use in QP / QM / QFT. I'm frequently confused, when I'm reading threads here. They often start completely underdetermined and often also just wrong from a mathematical point of view, but seemingly, physicists know what it is about.
- If someone speaks of "representation of G" where G is likewise a simple Lie group as e.g. SU(2) or SU(5), or what I've just read the Poincaré group, or Heisenberg group, do they really mean group representations, or the representations of the corresponding Lie algebras? How is it meant? As said, or as a very questionable use of language here? I often read about group representations, but the equations which sometimes follow are those of Lie algebras.
- Now to the representations of the Lie algebras. Why irreducible? Is it simply because of their minimality, i.e. that all others are sums of them, or is there something with a physical meaning, that irreducible representations have and others don't have? Also, which scalar field can be assumed here? Is there always implicitly meant ##\mathbb{C}## to avoid trouble with weights and eigen values, or at least ##\operatorname{char} \mathbb{F} =0##, because basically nobody feels the need to mention it? I mean at least real or complex should be mentioned, from a mathematical point of view. What is it, that physicists don't feel the need to say what they mean?
- And finally, what about infinite dimensional representations? As soon as someone says quantum, Hilbert spaces come around the corner, so where did they go to, when it comes to representations?