re: Irreps and gauge fields

Hi, the previous discussion got a little sidetracked and I had a few
things that have been deeply confusing me, similar to the original
posters dilemma. Basically im trying to reconcile once and for all,
the fiber bundle point of view, with quantum field theory (anyone have
good links to where QFT is described in this language, nearly all
treatments are classical that I have read so far, eg no talk of
quantization, and all treatments of QFT tend to lose sight of the
geometry). I mean shouldnt we be able to form a quantized principal
bundle and look at sections thereof, at least in principle?

Anyway, particles in Wigners sense are irreducible, unitary
representations of the universal cover of the Poincare group, good!
Also, classically matter fields arise as sections of the gauge bundle
over spacetime (eg locally you would get a copy of say G * Poincare,
modulo some Z2 factors that are always confusing)

Now, first question. Why is it appropriate for Wigner to just look at
the local picture, for instance he can derive and identify a particle
like an electron with just the irreducible reps from the Poincare grp.
Now, shouldn't it be more accurately the full rep of U(1) * Poincare,
since electrons interact uniquely with the EM field? Obviously U(1) is
a simple abelian group, it seems to me the more complicated topology
for nonabelian groups could completely break the particle picture down
globally, leading to complicated particle representations that are not
directly observed in nature (someone mentioned infraparticles for
instance).

Eg it really kinda bothers me (as much as its been drilled in my head)
that we are locally tensoring internal symmetry groups with the full
spacetime group and identifying particles that way, it seems to me we
should also recover all sorts of other stuff.

Anyone have good reference material to this, already the book Spin
Geometry has helped immensely, where things made a great deal more
sense to me at least.

.

Relevant Pages

  • Re: please help my confusion about particles and irreps.
    ... Single particle states live in Hilbert spaces corresponding to ... two-fold cover of the Poincare group. ... In the Fock space one can define ...
    (sci.physics.research)
  • Re: please help my confusion about particles and irreps.
    ... 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. ... Now I get the feeling from somewhere that if the representation was irreducible then we are dealing with an elementary particle. ... I also thought that the space of L^2 sections for the associate bundle was the single free particle state space and that via the Fock space construction we get to quantization etc. ...
    (sci.physics.research)
  • please help my confusion about particles and irreps.
    ... an elementary particle in the standard model. ... One has a principal bundle over a spacetime with group G. ... An elementary particle is an INFINTE DIMENSIONAL irrep of SL ...
    (sci.physics.research)