104
GAUGE FIELDS AND STRINGS
classically) the interference among these trajectories is highly destructive and the amplitude is exponentially small. The instanton is a
classical solution for imaginary time, which accounts for this smallness
as described in Chapter 4.
Let us explain, after these reminiscences, what kind of states are
connected by the instantons in the nonabelian gauge theories. In order
to do this we can pass to the gauge /Iq = 0 in the instanton solution,
finding that the nonzero part of it, A„(x, has the property:
4„(x, i) -► 0
A„(x,t) -> ■ g \x)d„g(x)
+ 00
(6.75)
Here g{x) is a certain matrix, satisfying the condition g(oo) = /.
Therefore it maps jc-space, which can be considered as due to the last
condition, onto the group G. This is just the same topologically
nontrivial map, which was discussed in Section 6.2, only transcribed to
the ^ 0 = gauge.
Now we can interpret the Yang-Mills instanton as follows. The
configuration space in gauge theory is formed by all possible fields
{v4„(x)}. Let us consider a transition in real time such that it lead from
zero A„ Sit t = — CO to g~^ d„g with topologically nontrivial g{x) at
i = -f 00. Due to this nontrivial g{x), in the process of such a transition
nonzero field strengths necessarily arise. (Otherwise the field g(x, t) in
A„(x, t) = g~^(x, t) d„ g(x, t) would have interpolated between g(x) and
/, which is impossible.)
The Minkowskian interpretation of (6.74) is that those field strengths
are such that they lead to compulsory pair-creation. The word “compulsory” here means that the transition amplitude without pair creation
is exactly zero.
It is interesting to see how these properties are fulfilled in terms of
explicit expressions for the amplitudes.
The vacuum-vacuum amplitude, which as we said must be zero, is
given by (6.61):
Z [4 ] =
exp.{ —
(6.76)
= Det(iy^(a„ + A^))
(We obtain the determinant to the first power and not the inverse
determinant because ip are anticommuting variables, or in other words,
fermionic loops come with a negative sign.)
GAUGE FIELDS AND STRINGS
classically) the interference among these trajectories is highly destructive and the amplitude is exponentially small. The instanton is a
classical solution for imaginary time, which accounts for this smallness
as described in Chapter 4.
Let us explain, after these reminiscences, what kind of states are
connected by the instantons in the nonabelian gauge theories. In order
to do this we can pass to the gauge /Iq = 0 in the instanton solution,
finding that the nonzero part of it, A„(x, has the property:
4„(x, i) -► 0
A„(x,t) -> ■ g \x)d„g(x)
+ 00
(6.75)
Here g{x) is a certain matrix, satisfying the condition g(oo) = /.
Therefore it maps jc-space, which can be considered as due to the last
condition, onto the group G. This is just the same topologically
nontrivial map, which was discussed in Section 6.2, only transcribed to
the ^ 0 = gauge.
Now we can interpret the Yang-Mills instanton as follows. The
configuration space in gauge theory is formed by all possible fields
{v4„(x)}. Let us consider a transition in real time such that it lead from
zero A„ Sit t = — CO to g~^ d„g with topologically nontrivial g{x) at
i = -f 00. Due to this nontrivial g{x), in the process of such a transition
nonzero field strengths necessarily arise. (Otherwise the field g(x, t) in
A„(x, t) = g~^(x, t) d„ g(x, t) would have interpolated between g(x) and
/, which is impossible.)
The Minkowskian interpretation of (6.74) is that those field strengths
are such that they lead to compulsory pair-creation. The word “compulsory” here means that the transition amplitude without pair creation
is exactly zero.
It is interesting to see how these properties are fulfilled in terms of
explicit expressions for the amplitudes.
The vacuum-vacuum amplitude, which as we said must be zero, is
given by (6.61):
Z [4 ] =
exp.{ —
(6.76)
= Det(iy^(a„ + A^))
(We obtain the determinant to the first power and not the inverse
determinant because ip are anticommuting variables, or in other words,
fermionic loops come with a negative sign.)