In the second chapter of the first volume of his books on QFT, Weinberg writes in the last paragraph of page 63:
In general, it may be possible by using suitable linear combinations of the $\Psi_{p,\sigma}$ to choose the $\sigma$ labels in such a way that the matrix $C_{\sigma'\sigma}(\Lambda,p)$ is block-diagonal; in other words, so that the $\Psi_{p,\sigma}$ with $\sigma$ within any one block by themselves furnish a representation of the inhomogeneous Lorentz group.
He continues:
It is natural to identify the states of a specific particle type with the components of a representation of the inhomogeneous Lorentz group which is irreducible, in the sense that it cannot be further decomposed in this way.
My questions are:
How is the first blockquote true? Why is it possible? Please sketch an outline of proof or refer to some material that might be useful.
What does he even referring to in the second blockquote? I found some material on net and Physics.SE regarding this, but I din't find any treatment upto my satisfaction. Please be precise as to what the correspondence is and whether or not it is bijective (as some accounts seem to indicate).
What is the relation between Weinberg's "specific particle type" and "elementary particle" used in accounts of this correspondence?
What is the definition of "one-particle state"? Is this correspondence a way of defining it? If yes, what is its relation to how we think of such states intuitively? (Of course, the answer of this question depends largely on the answer of 2, but I just asked to emphasize what is my specific query.)
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