THE STRONG COUPLING EXPANSION
43
is just the conservation of electric flux (which is true since we consider a
theory without charges). Due to the angular nature of the vector
potential the flux is quantized. In the zero approximation the shape of
the loop remains unchanged by time development. As we include the
second term in (3.34) two different effects arise. First of all, our closed
flux line acquires quasimomentum and begins to move across the
lattice. This is quite analogous to the case of global symmetries where
we had point-like elementary excitations. The second effect is more
tricky. As we perturb the state of a given shape with the cosine term in
(3.34) this shape can change (for instance, the cosine can create a new
small square, formed by the flux lines). Therefore the true quantum
state is a superposition of closed flux lines with different shapes. In later
chapters we shall develop a string theory so as to describe the
continuum limit of such a picture. This continuum limit exists, provided
that no phase transition in el takes place.
Let us clarify the relation of this picture to confinement of charges.
This relation is quite simple. Suppose that we introduce two static
opposite charges into our system. Then, we shall have a flux line which
ends on these charges. The energy of such a state is proportional to the
distance between the charges. If no phase transition takes place, this
picture will remain true even for small coupling. In the next chapter it
will be shown that for ^ = 3 it is indeed so, while for ^ = 4 there is a
phase transition leading to a condensation of strings. After the condensation we obtain the Coulomb law instead of confinement.
Generalization of the above discussion to the Non-Abelian case does
not present any difficulties. The basic variables in this case are matrices
of SU(N) attached to the links of a ^ — 1-dimensional lattice: By^. In
the abelian case we had the electric field operator E, with commutation
relations:
iE„. = dA,
oiAy,, = (e‘^-»)e~
(3.40)
The Non-Abelian generalization of (3.40) is the following. We have two
different electric fields, corresponding to left and right invariant forms
of the matrix
Namely let us introduce:
R
=
^ R
^y,(x ^y,a^y,a.
R
= R ~ ^ I
R
^y,tt ^y,(x^y,(t^y,ei
Jr(LlJ = TriRjJ
(3.41)
43
is just the conservation of electric flux (which is true since we consider a
theory without charges). Due to the angular nature of the vector
potential the flux is quantized. In the zero approximation the shape of
the loop remains unchanged by time development. As we include the
second term in (3.34) two different effects arise. First of all, our closed
flux line acquires quasimomentum and begins to move across the
lattice. This is quite analogous to the case of global symmetries where
we had point-like elementary excitations. The second effect is more
tricky. As we perturb the state of a given shape with the cosine term in
(3.34) this shape can change (for instance, the cosine can create a new
small square, formed by the flux lines). Therefore the true quantum
state is a superposition of closed flux lines with different shapes. In later
chapters we shall develop a string theory so as to describe the
continuum limit of such a picture. This continuum limit exists, provided
that no phase transition in el takes place.
Let us clarify the relation of this picture to confinement of charges.
This relation is quite simple. Suppose that we introduce two static
opposite charges into our system. Then, we shall have a flux line which
ends on these charges. The energy of such a state is proportional to the
distance between the charges. If no phase transition takes place, this
picture will remain true even for small coupling. In the next chapter it
will be shown that for ^ = 3 it is indeed so, while for ^ = 4 there is a
phase transition leading to a condensation of strings. After the condensation we obtain the Coulomb law instead of confinement.
Generalization of the above discussion to the Non-Abelian case does
not present any difficulties. The basic variables in this case are matrices
of SU(N) attached to the links of a ^ — 1-dimensional lattice: By^. In
the abelian case we had the electric field operator E, with commutation
relations:
iE„. = dA,
oiAy,, = (e‘^-»)e~
(3.40)
The Non-Abelian generalization of (3.40) is the following. We have two
different electric fields, corresponding to left and right invariant forms
of the matrix
Namely let us introduce:
R
=
^ R
^y,(x ^y,a^y,a.
R
= R ~ ^ I
R
^y,tt ^y,(x^y,(t^y,ei
Jr(LlJ = TriRjJ
(3.41)