The final piece of work in the Ising model to be mentioned is
what happens to the two point function when a
small magnetic field is put the system for *T*<*T*_{c}. At *H*=0 for
*T*<*T*_{c} the two point function has the important property that it
couples only to states with an even number of particles and thus, in
particular the leading singularity in the Fourier transform is not a
single particle pole but rather a two particle cut.
In 1978 [16], as an
application of our explicit formulas for the *n* spin
correlation functions we did a (singular) perturbation computation
to see what happens when in terms of the scaled variable *h*=*H*/|*T*-*T*_{c}|^{15/8}
a small value of *h* is applied to the system. We found that the two
particle cut breaks up into an infinite number of poles which are
given by the zeroes of Airy functions. These poles are exactly at the
positions of the bound states of a linear potential and are immediately
interpretable as a weak confinement of the particles which are free at
*H*=0. This is perhaps the earliest explicit computation where
confinement is seen. From this result it was natural to conjecture
that as we take the Ising model from *T*>*T*_{c}, *H*=0 to *T*<*T*_{c}, *H*=0 that
as *h* increases from to that bound states
emerge from the two particle cut and that as we further proceed from
*T*=*T*_{c}, *H*>0 down to *T*<*T*_{c}, *H*=0 that bound states continue to emerge
until at *H*=0 an infinite number of bound states have emerged and
formed a two particle cut. What this picture does not indicate is the
remarkable result found 10 years later by A. Zamolodchikov [22]that at *T*=*T*_{c},
*H*=0 the problem can again be studied exactly. This totally unexpected
result will be discussed (if the abstract is correct) be Sasha in the
next presentation.