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where H is the hamiltonian of the system and I is the euclidean action between times 0 and T:
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We have extended the definition of the strength tensor in euclidean variables. The functional integral in (12-38) is restricted by the boundary conditions on the winding number n(A):
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n[A(x, r = 0)] = n, n[A(x, r = T)] = n2
(12-40)
We are slightly cheating here, since the measure .01(A) has not yet been properly defined. Its precise definition will be clarified in the quantization procedure of Sec. 12-2. We see now an alternate explanation of the classification of ground states according to homotopy classes of S3 ..... S3. For T ..... 00, we look at those configurations the neighborhood of which yields a finite contribution to the FeynmanKac formula. Since their euclidean action is finite, F., must vanish at infinity in all euclidean directions, which amounts to saying that A" is a pure gauge field and yields a mapping from S3, the surface at infinity in a four-dimensional euclidean space, onto the group, for example, SU(2) ~ S3. Moreover, it may be shown that the winding number n associated with this mapping may be written as the euclidean integral (12-41) where the dual tensor is F~' == -5;e",paFpaa. We observe that (12-42)
QUANTUM FIELD THEORY
For very large T, we expect the integral (12-38) to be dominated by the neighborhood of stationary configurations which are solutions to the euclidean classical equation of motion (12-43) satisfying the boundary condition (12-40) or, equivalently, such that
n == n2 - nl
-1-fd4X F~'a F~'a 32n 2
(12-44)
Such a solution has an action bounded from below in terms of n. This comes from the positivity of (12-45)
whence
(12-46)
The inequality is saturated by the self-dual or anti-self-dual configurations F = F, since the equations of motion are then automatically satisfied as a consequence of the identity (12-21). Explicit solutions of arbitrary winding number have been recently discovered. The solution proposed by Belavin, Polyakov, Schwartz, and Tyupkin for n = 1 reads (12-47)
with Clearly, at infinity A. reduces to a pure gauge, arising from the identity mapping from S3 onto S3. We thus expect and may check by a direct computation that this solution has n = 1. More generally, if we use the parametrization
(SaikOk
+ 0aJJO) In f
i, k
1,2,3
(12-48)
AO a = oalnf
the equation F
F reduces to
(12-49)
The case n = 1 corresponds to number n may be devised:
PI) = 1 + A2/x2
pn)(x) =
but other solutions corresponding to a winding
n+ I
(x -
-'-2
(12-50)
They depend on n + 1 arbitrary scales Ai and positions Xi. Such solutions have been called pseudoparticles (by reference to the imaginary time coordinate) or instantons (due to their local character in time, as compared to the three-dimensional solitons). It is known that further solutions must exist. For SU (2), for instance, the general solution depends on 8n - 3 parameters up to gauge transformation. We will not dwell on this rapidly evolving problem, nor on many other related topics such as the modifications to be brought to this picture in the presence of massless fermions. Although fascinating, these global properties will be omitted in the following. This is legitimate because we shall concentrate on the perturbative expansion, which is insensitive to the choice of the vacuum we expand about.
NONABELIAN GAUGE FIELDS
12-1-4 Gauge Invariance and Constraints
The equations of motion are insufficient to determine the fields AIl(x), given a set of Cauchy conditions at a time to. Two solutions obtained from each other through a gauge transformation g(x) such that g(x) = e for t ::;; to satisfy the same Cauchy conditions but may differ for t> to. Therefore, the gauge arbitrariness has to be restricted through the introduction of auxiliary conditions which do not affect gauge-invariant physical observables. The following discussion, due to Faddeev, will lead us to the functional quantization outlined in Chap. 9. We restrict ourselves to a simple compact Lie group. The action, as in the abelian case, is first rewritten in terms of independent variables F and A:
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