The difficulty remains if the field theory is abelian, as in U(1) QED, or if it is nonabelian in the YM sense. This comes from the problem of characterizing the symmetries of the S matrix if there is a mass. Review the Dirac equation and the mass-gap in the two solutions which separate by a gap p ~ m on the momentum light cone.
The S-matrix acts on shift the state or momentum state of a particle. A state with two particle states |p, p'> is acted upon by the S matrix through the T matrix
S = 1 - i(2π)^4 δ^4(p - p')T
So that T|p, p'> != 0. For zero mass plane waves scatter at almost all energy. The Hilbert space is then an infinite product of n-particle subspaces H = (x)_nH^n (here (x) = "otimes" or Cartesian product). As with all Hilbert spaces there exists a unitary operator U, often U = exp(iHt), which transforms the states S acts upon. U transforms n-particle states into n-particle states as tensor products. The unitary operator commutes with the S matrix
SUS^{-1} = [1 - i(2π)^4 δ^4(p - p')T]U[1 + i(2π)^4 δ^4(p - p')T^†]
= U + i(2π)^4 δ^4(p - p')[TU - UT^†] + [(2π)^4 δ^4(p - p')]^2(TUT^†).
By Hermitian properties and unitarity it is not difficult to show the last two terms are zero and that the S-matrix commutes with the unitary matrix. The Lorentz group then defines operator p_μ and L_{μν} for momentum boosts and rotations. The S-matrix defines changes in momentum eigenstates, while the unitary operator is generated by a internal symmetries A_a, where the index a is within some internal space (the circle in the complex plane for example, and we then have with some
[A_a, p_μ] = [A_a, L_{μν}] = 0.
This is a sketch of the infamous "no-go" theorem of Coleman and Mundula. This is what prevents one from being able to place internal and external generators or symmetries on the same footing.
The way around this problem is supesymmetry. The generators of the supergroup, or a graded Lie algebra, have 1/2 commutator group elements [A_a, A_b] = C_{ab}^cA_c (C_{ab}^c = structure constant of some Lie algebra), plus another set of graded operators which obey
{Q_a, Q_b} = γ^μ_{ab}p_μ,
which if one develops the SUSY algebra you find this is a loophole which allows for the intertwining of internal symmetries and spacetime generators. One might think of the above anti-commutator as saying the momentum operator, as a boundary operator p_μ = -iħ∂_μ which has a cohomology, where it results from the application of a Fermi-Dirac operator Q_a. Fermi-Dirac states are such that only one particle can occupy a state, which has the topological content of d^2 = 0. This cohomology is the basis for BRST quantization.
This is why most physicists who work on this stuff take supersymmetry seriously. It is also one reason why many schemes which purport to derive gravitation or unify gravitation with EM in some elementary was can be subject to strong questions. Of course supersymmetry remains a hypothetical, though some signatures of it have been detected. We will have to wait for the LHC to yield such results before anything is conclusive.
Cheers LC
