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I follow up to getting equation 3.31 but I'm stuck on the comment after : ' But such an algebra of operators can’t be realised on a single vacuum state.'

I have no idea where this comment comes from- (also I then don't understand the smallest representation comment, dimension m, and where the action comes from).

Any links to relevant background knowledge assumed to follow these comments of this section, or any help, would be greatly appreciated. many thanks.