88
GAUGE FIELDS AND STRINGS
¿7 is |a — Z?|. So, it is natural to expect that e% in the above expression
will be replaced by:
e\\a - b\) - 2n/\og(À\a -
(6.12)
obtained from (2.49) for N = 3. The instanton contribution
has to
contain an integral over the parameters a and b. The measure of
integration must have both translational and scale invariance (since a
and b in the classical solution are defined modulo their common scale).
On these grounds we expect:
J \a-b\* b\*
d^a d^b
\a-b\^
exp( -
- b|))
(6.13)
(V is the volume of the system and p = \a — b\). We see that this
integral diverges at large values of |a — hi but we learned in Chapter 2
that for \a — b\>X~^ the formula (6.12 ) is not applicable since
e^(\a — h|) is not small any more. For the same reason we cannot trust
(6.13) in this region, since the WKB approximation used in its derivation is valid only for small couplings.
Nevertheless it makes sense to ask what happens after accounting for
the multi-instanton solution. The tactic for this is to take a classical field
(6.10) and to consider small fluctuations on it as background. After
performing the Gaussian integral, we obtain a determinant of the
corresponding quadratic form which will depend on {Uj} and {bj}. The
logarithm of this determinant can be considered as an interaction
energy between instantons, induced by quantum fluctuations. Finally
we shall have to integrate over the parameters {aj} and {bj}. This
programme can be carried through quite explicitly, the main reason
being that for any self-dual background the kernel of the abovementioned quadratic form simplifies considerably. To show this let us
recall that the quadratic part of the action is described by (2.47). In the
case of 0(3) we can introduce the complex notation:
(P = cp^+ \q>2
S" = ¿ 0 j {|(a, +
= ^ I {l(^^ +
- d,A^)
(6.14)
GAUGE FIELDS AND STRINGS
¿7 is |a — Z?|. So, it is natural to expect that e% in the above expression
will be replaced by:
e\\a - b\) - 2n/\og(À\a -
(6.12)
obtained from (2.49) for N = 3. The instanton contribution
has to
contain an integral over the parameters a and b. The measure of
integration must have both translational and scale invariance (since a
and b in the classical solution are defined modulo their common scale).
On these grounds we expect:
J \a-b\* b\*
d^a d^b
\a-b\^
exp( -
- b|))
(6.13)
(V is the volume of the system and p = \a — b\). We see that this
integral diverges at large values of |a — hi but we learned in Chapter 2
that for \a — b\>X~^ the formula (6.12 ) is not applicable since
e^(\a — h|) is not small any more. For the same reason we cannot trust
(6.13) in this region, since the WKB approximation used in its derivation is valid only for small couplings.
Nevertheless it makes sense to ask what happens after accounting for
the multi-instanton solution. The tactic for this is to take a classical field
(6.10) and to consider small fluctuations on it as background. After
performing the Gaussian integral, we obtain a determinant of the
corresponding quadratic form which will depend on {Uj} and {bj}. The
logarithm of this determinant can be considered as an interaction
energy between instantons, induced by quantum fluctuations. Finally
we shall have to integrate over the parameters {aj} and {bj}. This
programme can be carried through quite explicitly, the main reason
being that for any self-dual background the kernel of the abovementioned quadratic form simplifies considerably. To show this let us
recall that the quadratic part of the action is described by (2.47). In the
case of 0(3) we can introduce the complex notation:
(P = cp^+ \q>2
S" = ¿ 0 j {|(a, +
= ^ I {l(^^ +
- d,A^)
(6.14)