Identification of the state of particle types with representations of Poincare group 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.)
 A: *

*This just says that you can decompose any unitary representation of the Poincare group (= inhomogeneous Lorentz group) into irreducible representations.


*He suggests to identify the irreducible representations with elementary particles, as suggested by the analogy irreducible = no longer decomposable = elementary. He doesn't really explain why (but only asserts that) it is natural to do that - this is summing up the experience of several generations of particle physicists: A single particle can be moved, rotated, and boosted, hence (in flat space-time) its Hilbert space must carry a unitary rep of the Poincare group. The particle is treated as elementary if this rep is irreducible as it cannot be decomposed. What is considered elementary depends on the resolution: in relativistic chemistry, all nuclei are treated as elementary particles, as the Poincare group acts irreducibly on their Hilbert space. In nuclear physics, nuclei are modelled in more detail as composite particles with a much more complex Hilbert space and a reducible representation of the Poincare group on it. Thus effectively Weinberg defines the notion of an elementary particle (in mathematical models) as being an irreducible representation of the Poincare group.


*In view of Wigner's classification of irreducible unitary representations of Poincare, rederived by Weinberg in Chapter 5, elementary particles are classified into particle types by their mass and spin. The Hilbert space of an elementary particle of mass $m>0$ and spin $s$ is the space of $2s+2$-component wave functions $\psi(p)$ with $p_0=\sqrt{\mathbf{p}^2+(mc)^2}$ (defining the mass shell of mass $m$), with the corresponding irreducible representation of the Poincare group. (For the massless case see point 4 below.) A unitary representation consists of a Hilbert space and operators on this space generating the group (or a homomorphic image of it). For details see chapters B1 and B2 of my FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html
The standard model refines this classification by also specifying the irreducible representation of the gauge group of internal symmetries, giving rise to further quantum numbers. Conserved quantum numbers are nothing else than labels that tell you which irreducible representations is associated with the particle labelled by these numbers.


*A one-particle state is a state in the Hilbert space of an irreducible representation of the Poincare group (extended by the CTP symmetry, for causality reasons). Given the results of Weinberg's Chapter 5, this says that in momentum space, you have a wave function $\psi(p)$ with 4D $p$ on the mass shell with mass $m$, and $2s+2$ components for spin $s$ if $m>0$, but $2$ components (independent of spin) if $m=0$.

I don't think anyone understand Weinberg at first reading; though it is the best QFT book around if you want to understand the deeper reasons for why relativistic QFT is the way it is. So you may need to take some things based on a preliminary understanding, as proper understanding of what it all means requires at least that you covered the first 6 chapters.
