Re: please help my confusion about particles and irreps.

Eugene Stefanovich wrote: 

cyberkatru wrote:

I cannot quite get the following two pictures to mesh. Both concern what is an elementary particle in the standard model.

1. One has a principal bundle over a spacetime with group G. Then one considers a FINITE DIMENSIONAL representation of G and obtains an associated bundle. The sections of this bundle are the yet to be quantized particle fields. A connection on such a G-bundle corresponds to the yet to be quantized force particles. Now I get the feeling from somewhere that if the representation was irreducible then we are dealing with an elementary particle. 

No. For an elementary particle in Wigner's sense you need a flat
spacetime, G must be the Lorentz group, and the representation must
be such that the resulting representation of the Poincare group,
with the momentum generating the translations, is irreducible.


On the other hand I have heard that

2. An elementary particle is an INFINTE DIMENSIONAL irrep of SL(2,C) \cross R^{1,3} (semidirect product and cover of the restricted Poincare group). What happened to the internal gauge group G 

If there is a gauge group G, one wants representations of
Poincare x G and not Poincare alone.


and why are we now talking about infinite dimensional irreps?

Quantum mechanics requires unitary representations, and unitary representations of the Poincare groups are necessarily infinite-dimensional.


[...]

So, the *only* purpose for introducing quantum fields is to assist in
construction of interaction terms in H and K. Quantum fields have no
relationship to the description of one-particle states.

As far as I know, there is only one major textbook (S. Weinberg,
"The Quantum Theory of Fields", vol. 1) which offers this correct
perspective on QFT. More details are available in physics/0504062.

This strange view about ''the only purpose for introducing quantum fields'' is Stefanovich's minority view of the ''correct perspective''
on QFT. To claim the authority of Weinberg for it is simply misleading.
He writes in the introduction to his book,
''The traditional approach ... has been to take the existence
of fields for granted.''
''The point of view of this book is that quantun field theory is
the way it is because it is the only way to reconcile the
principles of quantum mechanics with those of special relativity.''
''_any_ relativistic quantum theory will look at sufficiently
low energy like a quantum field theory.''
This is quite different from Stefanovich's claims.


Arnold Neumaier


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Relevant Pages

  • Re: Irreps and gauge fields
    ... Basically im trying to reconcile once and for all, the fiber bundle point of view, with quantum field theory. ... Anyway, particles in Wigners sense are irreducible, unitary ... The Poincare group transformations have very clear physical observable ... they act on multi-component quantum fields. ...
    (sci.physics.research)
  • Re: please help my confusion about particles and irreps.
    ... >>what is an elementary particle in the standard model. ... One has a principal bundle over a spacetime with group G. ... >>somewhere that if the representation was irreducible then we are ... An elementary particle is an INFINTE DIMENSIONAL irrep of SL ...
    (sci.physics.research)
  • Re: please help my confusion about particles and irreps.
    ... > what is an elementary particle in the standard model. ... One has a principal bundle over a spacetime with group G. ... > somewhere that if the representation was irreducible then we are ... An elementary particle is an INFINTE DIMENSIONAL irrep of SL ...
    (sci.physics.research)