QUANTUM FIELD THEORY in .NET

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QUANTUM FIELD THEORY
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Consider the Lagrange function for a time-independent solution, i.e., the space integral of the lagrangian
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L, - Lz
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d 3 x (Ea Ea - Ba Ba)
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with the notation of Eq. (12-31). The total energy H is the sum L, + Lz; hence its finiteness implies the finiteness of L" L z , and L. A solution if it exists is unstable under the scale transformations
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A~(x) ..... pAA~(.l.x)
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A~(x) ..... AA~(.l.x)
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The Lagrange function is transformed into
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L ..... pZAL, - ALz
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but it should be stationary at p
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1. This implies
Hence
F~v =
and, by a global extension of the local statement of Eq. (12-22), this means that the gauge field is a pure gauge everywhere. This scaling argument (due to Coleman) may be shown to prevent such timeindependent solutions in any space dimension different from four. It has been extended to more general non dissipative configurations. There exist, however, nontrivial four-dimensional euclidean solutions to the classical equations. Before explaining the nature and role of these euclidean solutions, let us analyze further the structure of the ground state in nonabelian gauge theories. To carry out this analysis it is convenient to impose the gauge condition Ao = o. Classically the ground state must correspond to time-independent field configurations of vanishing energy density. We have therefore F"v == 0, which means that the field A is a pure gauge A(x)
5[g(X)Jg-l(X)
(12-33)
Furthermore, we assume that we may restrict ourselves to transformation functions g(x) that have the same limit in all spatial directions. We may take this limit to be the identity in the group (12-34) (there is actually no very convincing argument to justify this restriction). Under these circumstances, all the field configurations of the form (12-33) and (12-34) may be regarded as describing a ground state. We may wonder whether all these copies of the vacuum are equivalent, i.e., whether there exists a continuous gauge transformation vanishing at spatial infinity that connects any two of them. The surprising answer is generally "no." Suppose for definiteness that the gauge group is SU(2). Any matrix of SU(2) may be parametrized in the form
g(x) = Uo
+ iu . t1
(12-35)
in terms of the Pauli matrices, and with u real satisfying u6 + U = 1. Hence, SU(2) is isomorphic to the three-dimensional sphere S3: u6 + uI + u~ + u~ = 1. On the other hand, the whole threedimensional space with all points at infinity identified is also topologically equivalent to S3. Therefore, the gauge transformation g(x) associated with each vacuum is a mapping from S3 onto S3. According to homotopy theory, such mappings fall into equivalence classes. Two mappings x ..... g,(X) and x ..... g2(X) belong to the same class if there exists a continuous deformation from g,(X) to g2(X). In the case at hand, the classes are labeled by a positive or negative integer called the winding number
NONABELIAN GAUGE FIELDS 571
or Pontryagin index of the class. This integer characterizes the number of times S3 is mapped onto itself. It is equal to (12-36) with iA~(x)O"a given by Eq. (12-33). Examples ofrepresentatives of the class n
g,(x) = -exp i n - - = jx2 + 1
1 are
G.X)
x g,(x) = x 2
+ 1 + 2i x 2 + 1
and gn(x) = g,n(x) belongs to the nth class. In the case of other simple groups, the same conclusion holds, namely, that there exists a discrete set of inequivalent vacua In> labeled by an integer As explained in Chap. 11, such a degeneracy of the ground state is intolerable and is actually cured by quantum tunneling. The true vacuum is a linear superposition of the degenerate approximate vacua In>. Since the above gauge transformation shifts the winding number n by one unit, and since the true vacuum must be invariant-up to a phase-under any gauge transformation, it has the form
(12-37) where () is a new arbitrary (and unexpected!) parameter in the theory. We may then try to understand better how tunneling takes place between the degenerate vacua In>. To this end, we recall the Feynman-Kac formula [Eq. (9-198)]
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