64
GAUGE FIELDS AND STRINGS
subgroup of some Non-Abelian gauge theory we are necessarily dealing
with the compact version.
After this explanation let us work out the theory of this version
starting from ^ = 3. Following the strategy of the preceeding section
we consider a partition function:
n
Z = X i n
ex p l- ¿ X
(4.59)
{«*.«^1 ^ x,a
f ^^0 x,ciP
J
- n
which takes account of the periodicity of the action. For a given set of
{tix^acfi} we introduce numbers (where z belongs to the centres of the
cubes of our lattice) defined as the flux of n through a given cube :
(4.60)
(the notation in the l.h.s. of (4.60) means the sums of
corresponding to oriented plaquettes forming a cube with centre at z- Its continuum analogue will be an integral of the field
over a closed surface).
Decomposing
as:
+ ^X ,(X T +
' ^X + P.a ^X,p
(4.61)
We obtain from (4.60):
K z'< t> z' = C l,
(4.62)
(Ajj. is the lattice Laplace operator). Substitution of (4.61) and (4.62)
into (4.59) gives:
^ = -^Gauss Z exp(^X
{fl.l V
/
+ (X)
Zoau,.= i n d /4 a .,e x p [ - ' X f L i )
J x,a
\ ^^0 x,a/i
/
(4.63)
We thus obtain the same Coulomb system as (4.30) but in three
dimensions. As we shall see that makes a lot of physical difference.
Before working this out let us explain the meaning of the “charges”
which appeared in (4.63). They can be thought of as instanton solutions