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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:

  1. 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.

  2. 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).

  3. What is the relation between Weinberg's "specific particle type" and "elementary particle" used in accounts of this correspondence?

  4. 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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1 Answer 1

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

  2. 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 physcists.

  3. In view of Wigner's classification of irreps of Poincare, rederived by Weinberg in Chapter 5: Elementary particles are classified into particle types by their mass and spin. The standard model refines this classification by also specifying the irrep of the gauge group, giving rise to further quantum numbers. (Quantum numbers are nothing else as labels that tell you which irrep is associated with the particle labelled by these numbers.)

  4. A one-particle state is a state in the Hilbert space of an irreducible rep of the Poincare group (extended by the CTP symmetry, for causality reasons). Given the results of 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.

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Thanks for your answer. I am still confused on some points you made. In a logical manner of understanding, as sought by Weinberg in his book, is it a definition of an elementary particle? And I still didn't quite get the correspondence. We have a system of a single particle. Now, all of its states form a Hilbert space. And we can associate a representation of Poincare group with this space. Is this the correspondence? What precisely is the correspondence and why do we associate it to a particle? What about the properties of particles and observables on Hilbert space? Is it bijective or 1 to 1? –  Lakshya Bhardwaj Mar 4 '12 at 7:30
    
And I didn't get your point 4 at all. What do you mean by Hilbert space of an irrep of Poincare group? And how do all particle physicists view this as a correspondence? What motivated us to think of such a correspondence in the first place? –  Lakshya Bhardwaj Mar 4 '12 at 7:33
    
Definition: ''elementary particle = unitary irrep of the symmetry group of the universe''. He ignores internal symmetries, hence his group is Poincare. - 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 $p_0=\sqrt{p^2+m^2}$, with the corresponding irrep of Poincare. - A unitary representation consists of a Hilbert space and operastors on this space generating the group or a homomorphic image of it. For details see chapters B1 and B2 of my FAQ at www.mat.univie.ac.at/~neum/physfaq/physics-faq.html –  Arnold Neumaier Mar 4 '12 at 11:48
    
A single particle can be moved, rotated, and boosted, hence 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 Poincare acts irreducibly on their Hilbert space. In nuclear physics, nuclei have a much more complex Hilbert space and are composite particles. –  Arnold Neumaier Mar 4 '12 at 12:05
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