CHAPTER 3
The Strong Coupling Expansion
In the previous chapter we have seen how simple perturbative methods
permit us to examine the short distance behaviour in asymptotically
free theories. The reason for such success was that the effective coupling
for high momentum fluctuations is logarithmically small. In the region
p ^ A however, this coupling becomes of the order of unity (if we take
the formula (2.68), naively it becomes infinite.) In order to examine the
infrared structure of the theory we have to develop some nonperturbative methods. In this chapter we shall describe the simplest (though in
many respects imperfect) method—the strong coupling expansion.
Unfortunately this phrase means expansion not in the physical coupling (which may really be large) but in the bare one, e^. At first sight
the enterprise may seem completely meaningless because as was
explained above, the continuum limit of our lattice models is achieved
when
1
log A ^
0
(3.1)
The reason the large el expansion is interesting is that there are grounds
to believe that in most asymptotically free systems there are no phase
transitions in el. If true, this implies that the qualitative character of the
spectrum and correlation functions is unchanged as we go from small to
large el. For instance, the masses of elementary excitations must have
no singularities in el and hence their X/el expansions can be continued
to rather small values of el. This numerical aspect will not be discussed
here. Instead we shall be concerned with the qualitative picture which
arises from the strong coupling limit. Though our main interest lies in
nonabelian gauge systems, we start from the easier cases because it is
always satisfactory to realize a special theory as belonging to some
larger variety.
33
DOI: 10.1201/9780203755082-3
The Strong Coupling Expansion
In the previous chapter we have seen how simple perturbative methods
permit us to examine the short distance behaviour in asymptotically
free theories. The reason for such success was that the effective coupling
for high momentum fluctuations is logarithmically small. In the region
p ^ A however, this coupling becomes of the order of unity (if we take
the formula (2.68), naively it becomes infinite.) In order to examine the
infrared structure of the theory we have to develop some nonperturbative methods. In this chapter we shall describe the simplest (though in
many respects imperfect) method—the strong coupling expansion.
Unfortunately this phrase means expansion not in the physical coupling (which may really be large) but in the bare one, e^. At first sight
the enterprise may seem completely meaningless because as was
explained above, the continuum limit of our lattice models is achieved
when
1
log A ^
0
(3.1)
The reason the large el expansion is interesting is that there are grounds
to believe that in most asymptotically free systems there are no phase
transitions in el. If true, this implies that the qualitative character of the
spectrum and correlation functions is unchanged as we go from small to
large el. For instance, the masses of elementary excitations must have
no singularities in el and hence their X/el expansions can be continued
to rather small values of el. This numerical aspect will not be discussed
here. Instead we shall be concerned with the qualitative picture which
arises from the strong coupling limit. Though our main interest lies in
nonabelian gauge systems, we start from the easier cases because it is
always satisfactory to realize a special theory as belonging to some
larger variety.
33
DOI: 10.1201/9780203755082-3