42
GAUGE FIELDS AND STRINGS
tions of Ay^. Indeed:
Notice, that the operators Vy are generators for the gauge transformaX
Ay ^ = a)y-(Oy^^
(3.35)
=
Therefore:
[ r „ //] = 0
(3.36)
In quantum theory we have to substitute
1 d
i aA„
and to solve the equations:
H il/ = é\l/
Vyip = 0
(3.37)
The strong coupling limit corresponds to neglecting the second, potential term in (3.34). The general solution to (3.37) in this approximation
is:
= exp^i X riy,,Ay,^
(3.38)
with n ^ being integers satisfying the conservation condition
K ":
= 0
> , o t
(3.39)
The vacuum solution corresponds to riy^ = 0. Excited states are
described by a closed loop on the lattice, such that at each site the n are
conserved. We recognize the same set of loops which we had in the
description of the global 0 (2) model, but the interpretation is now
different. Each loop labels the quantum state in the present case. In the
strong coupling limit the energy of this state is given by (3.38) and is
proportional to the total length. As we consider the time propagation of
such loops we obtain the world surfaces of the Euclidean approach.
Physically, the loop is formed of Faraday flux lines (we see from (3.38)
that riy ,^ is an eigenvalue of the electric field E ). The condition (3.39)