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What is meant in the following paragraph (this is Yang-Mills original 1954 paper "Conservation of Isotopic Spin and Isotopic Gauge Invariance")

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Where Equation (3) is the transformation of B field

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  1. Why B field is assumed to contain a term that is a linear combination of Pauli matrices?

Is it because equation (3) shows us that B field is a sum of Pauli matrices (the second term) and mixed term (the first term)? but in that case, it is the field in primed coordinate and we cannot really say anything about the un-primed one, right?

  1. What is the parts of B that are scalar or tensor combination of Pauli matrices which are considered irrelevant? for that matter, why are they ignored?

  2. Why equation (6) in the first image is written as the dot product of b field and Pauli matrices?

P.S. As a side note, I am aware that Yang-Mills potential is the pullback of the connection one-form which is Lie algebra valued (that is why we have b field multiplied by two times the T matrices which are the generators of su(2) Lie algebra (I presume T matrices are just Pauli matrices halved, right?)), but I am strictly asking the train of reasoning used in the paragraph of the first image, so I kindly request answering the above question in accord with that reasoning. Thank you.

JON
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    On a side note: It can be considerably harder to learn a topic from the original papers than it is to learn it from a recent textbook. Exposition improves with time. – G. Smith Sep 10 '20 at 17:16

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