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j"(x) = o"</>(x)
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(11-36)
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The vacuum is, of course, not invariant under these transformations. If we substitute </>(x) for A in Eq. (11-30) we find (11-37) Alternatively, the total "charge" lim Qv(t)
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r d x oo</>(x, t)
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(11-38)
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is not well defined. The Goldstone boson in this case is the quantum of the field'</> itself. An interesting point about this example is that we may look at the state obtained from the vacuum through the action of e iAQv . From the Fourier decomposition of the field (11-39) it follows that in the infinite-volume limit the state
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takes the form
v~oo
eiAQv(O) 10)
(11-40)
1.1.) = lim exp
v~oo
2(2n)
d k 3[a(k) e ik ' x
at(k) e-ik'XJ} 10)
= e(J /2)[a(O)-a'(O)]
(11-41)
It appears therefore as a coherent superposition of zero-energy momentum states. To show that <A 1 0) vanishes we have to be more careful in the handling of the infinite-volume limit. It is better
to introduce a smooth spatial cutoff, for instance, (11-42) The reader can easily verify that
<01 A)
= e- n (J.V
)'/64
V~oo
(11-43)
A physically meaningful model exhibiting spontaneous symmetry breaking is based on a set of n coupled scalar fields with a lagrangian
f >(ljJ) = i(8ljJ)2 _ /1 ljJ2 _ ~(ljJ2)2 2 4
(11-44)
QUANTUM FIELD THEORY
<1>2
invariant under an internal O(n) symmetry group. The notation is such that l/J stands for the column vector of these fields and symmetry breaking will occur if the bare-mass parameter /1 2 is negative. At the classical level the ground state is not described by a zero mean field value due to the fact that the potential
/1 A V(l/J) = _l/J2 + _(l/J2 2
(11-45)
has a minimum for a nonvanishing value of the field (Fig. 11-5)
1l/JI=v= ( ---:f
/12)1/2
(11-46)
To insure the stability of the theory we assume, of course, that large values of l/J are damped, which means A > 0. It appears as if the vacuum is degenerate: every state with Il/J 1 = v is a priori a candidate. However, any choice leads to an inequivalent Hilbert space. Note that such a choice of a ground-state value l/Jv will leave an O(n - 1) subgroup unbroken, corresponding to those transformations which act as the identity on l/Jv. To be specific, let us pick a coordinate system in internal space such that l/Jv = <Oll/J 10) has only its nth component nonvanishing, and parametrize the field in terms of a radial displacement p and (n - 1) orthogonal rotations as
l/J = e t {Iv (
(11-47)
SYMMETRIES
The ta with 1 :::; a :::; n - 1 stand for n - 1 generators of the O(n) group acting effectively on the vector lPv. In other words, the Lie algebra of O(n) contains in addition to the O(n - 1) generators the n - 1 ta's. When expressed in terms of the new dynamical variables p and ~ the lagrangian reads
ft' = i(Op)2
+ i(O~2 - 2
+ p - 4 (v + p)4
(11-48)
From this we see that the ~ correspond to (n - 1) decoupled massless fields, as predicted by Goldstone's theorem. In the previous example, to be studied in more detail in Sec. 11-4, the groundstate degeneracy is apparent at the classical level. We have learnt, however, in Chap. 9 that quantum corrections can alter the potential, leading to the possibility of vacuum degeneracy as a result of radiative corrections. This appears to be the case for the following model for scalar electrodynamics, as pointed out by Coleman and Weinberg. A complex massless boson field is coupled minimally to an electromagnetic field. The lagrangian is
ft' = ft'em
+ i(oqh
- eAcP2
+ i(OcP2 + eAcPr)2 -
(cPi + cP~)2
(11-49)
The complex field cP is written in terms of its two real components cPl and cP2. Moreover, the anharmonic coupling constant is denoted A-/4! to compare the oneloop computation of the effective potential with the expression given by Eq. (9-129) for the lagrangian (11-50) which led to the result
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