80
GAUGE FIELDS AND STRINGS
spectrum in our theory, and it is tempting to find it directly by
considering the correlation functions for operators of higher spin.
Unfortunately in our approximation these correlation functions contain only scalar particle thresholds, and resonances should appear in
higher order approximation, so the resonance problem remains to be
solved even in our model.
An interesting property of the above formulas is that they lead to
confinement of half-integer charges for which the Dirac quantization
condition is true:
e-g ' ■ In
(where g is the minimal charge of a magnetic pole). If we double the
charge of the test particles, then it is easy to check that the contribution
of monopoles to exp(i j
d5^} will be negligible and we have no
confinement of such charges. In the framework of the Georgi-Glashow
model, described in Chapter 4, this result is quite natural since integer
charges can be screened by W~ bosons, while for half-integers this is
impossible. However, in the case of the lattice 0(2) model the above
conclusion is rather surprising. Thus, we have proved the anticipated
result that 0(2) gauge theories for ^ = 3 produce linear confinement of
half-integer charges. For ^ = 4 it is not so for el
since small
instanton loops give very small contributions to the phase factors. This
is easy to verify by calculations analogous to those above.
Let us now describe one more manifestation of confinement, which is
of some interest because of analogies with solid state physics. Namely
let us show that in the confining phase the dielectric constant is zero. In
order to introduce this quantity in abelian systems let us consider an
antisymmetric external field
coupled to our system in the
following way:
ex p (-M /[/J)=
e
i
^ lV f ^ ív \
(5.32)
(where
we take the liberty of using continuum
notation while meaning a lattice field theory). From this definition it is
clear that IF [/] is invariant under gauge transformation of the “third
kind”:
(5.33)
A^
because it can be compensated in (5.32) by a change A^
Therefore, W should depend on the following combinations of
(5.34)
GAUGE FIELDS AND STRINGS
spectrum in our theory, and it is tempting to find it directly by
considering the correlation functions for operators of higher spin.
Unfortunately in our approximation these correlation functions contain only scalar particle thresholds, and resonances should appear in
higher order approximation, so the resonance problem remains to be
solved even in our model.
An interesting property of the above formulas is that they lead to
confinement of half-integer charges for which the Dirac quantization
condition is true:
e-g ' ■ In
(where g is the minimal charge of a magnetic pole). If we double the
charge of the test particles, then it is easy to check that the contribution
of monopoles to exp(i j
d5^} will be negligible and we have no
confinement of such charges. In the framework of the Georgi-Glashow
model, described in Chapter 4, this result is quite natural since integer
charges can be screened by W~ bosons, while for half-integers this is
impossible. However, in the case of the lattice 0(2) model the above
conclusion is rather surprising. Thus, we have proved the anticipated
result that 0(2) gauge theories for ^ = 3 produce linear confinement of
half-integer charges. For ^ = 4 it is not so for el
since small
instanton loops give very small contributions to the phase factors. This
is easy to verify by calculations analogous to those above.
Let us now describe one more manifestation of confinement, which is
of some interest because of analogies with solid state physics. Namely
let us show that in the confining phase the dielectric constant is zero. In
order to introduce this quantity in abelian systems let us consider an
antisymmetric external field
coupled to our system in the
following way:
ex p (-M /[/J)=
e
i
^ lV f ^ ív \
(5.32)
(where
we take the liberty of using continuum
notation while meaning a lattice field theory). From this definition it is
clear that IF [/] is invariant under gauge transformation of the “third
kind”:
(5.33)
A^
because it can be compensated in (5.32) by a change A^
Therefore, W should depend on the following combinations of
(5.34)