36
GAUGE FIELDS AND STRINGS
Diagonalizing Hq + K, we find that in the first order of degenerate
perturbation theory we have the Eigenstates:
y
S(p) = E^(p) - Eo = 2u -I- 2i; ^ cos(pô)
(3.11)
As expected, we obtain point-like elementary excitations with a finite
gap, which are characterized by the quasimomentum p. This conclusion
will be true in all further orders of perturbation theory, which gives an
expansion in v/u. As we have already said, in the Ising model this
expansion will diverge at a certain critical value of v. This value
corresponds to the phase transition point at which the gap
£i(0) - £o = 0
(3.12)
Near this point it is expected (and will be explicity shown for Q ) = 2)
that:
m = £i(0) - Eq Ì
and
< ^ {p ) = (m ^ +p^y , 2^1/2
(3.13)
for \p\
1.
After the phase transition point a condensate of these particles is
formed and the strong coupling expansion is impossible. In this phase
one has to start from the opposite limit, treating the first term in (3.4) as
a perturbation.
In the zero approximation we have a strictly ordered vacuum
\4>y=
n
y - 2 m ô i
(niel)
n
y ô
y + ô
(3.14)
It should be clear from the above that the lines which appear in the
expansion of the Euclidean version of our model are just the world lines
of the particles we treated in the Hamiltonian version. It is an
interesting exercise to establish the correspondence between Shrodinger
perturbation theory for the Hamiltonian and diagrams for the Euclidean approach.