Some method must be found to deal appropriately with the gauge-invariance
problems associated with quantization of gauge fields discussed in section 46.6.
Interacting non-Abelian gauge field theory remains incompletely understood
outside of perturbation theory. The theory of interacting quantum fields shows
that, to get a well-defined theory, one should think of the parametersgjas being
dependent on the distance scale at which the physics is being probed, and one
can calculate the form of this scale-dependence. TheSU(2) andSU(3) gauge
field dynamics is “asymptotically free”, meaning thatg 2 ,g 3 can be defined so as
to go to zero at short-distance scales, with behavior of the theory approaching
that of a free field theory. This indicates that one should be able to consistently
remove short-distance cutoffs necessary to define the theory, at least for those
two terms in the Hamiltonian.
48.2 Fundamental fermions
The Standard Model includes both left-handed and right-handed Weyl spinor
fields, each coming in multiple copies and transforming under theU(1)×SU(2)×
SU(3) group according to a very specific choice of representations. They are
described by the following terms in the Hamiltonian
h=∫
R^3(
∑
a(Ψ†L,aσ·(∇−iAL)ΨL,a)−∑
b(Ψ†R,bσ·(∇−iAR)ΨR,b))d^3 xHere the left-handed fermions take values in three copies (called “genera-
tions”) of the representation
(− 1 ⊗ 2 ⊗ 1 )⊕(
1
3
⊗ 2 ⊗ 3 )
while the right-handed fermions use three copies of
(− 2 ⊗ 1 ⊗ 1 )⊕(−
2
3
⊗ 1 ⊗ 3 )⊕(
4
3
⊗ 1 ⊗ 3 )
with the first term in each tensor product giving the representation ofU(1) (the
fractions indicate a cover is needed), the second the representation ofSU(2), the
third the representation ofSU(3). 1 is the trivial representation, 2 the defining
representation forSU(2), 3 that forSU(3).
TheALandARare vector potential fields acting by the representations
given above, scaled by a constantgjcorresponding to the appropriate term in
the gauge group.
48.3 Spontaneous symmetry breaking
The Higgs field is anR^4 -valued scalar field, with a complex structure chosen so
that it can be taken to beC^2 -valued Φ(x), withU(1) andSU(2) acting by the