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The b function will select a unique saddle point except for a sign ambiguity. Expanding around one of the two solutions (the positive one, for instance) is allowed provided we multiply by a factor two, since from symmetry the two contributions are equal. We set
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cp = CPe
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and keep the dominant terms in the expansion in X. It then follows that
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To leading order we drop
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Sdt ~X and observe that
13k . f dt ljJCPe V4 (f dt (jJ2 )1/2 Jk
while I - Ie to second order in X reads
I - Ie
~ [f dt (i2 + X2 - CO~~22 t) + (f dt ux
j3 -3 u=-cp
With Dirac's bracket notations we find
(9-217) where the Schrodinger operator K is
d2 6 K= - - - - dt 2 cosh 2 t
and Ko corresponds to the free part Ko = - d 2 /dt 2 The instructions are to integrate the normalized version of iP in the subspace orthogonal to ljJ. The spectrum of K + 1 contains a discrete zero mode with eigenfunction given by ljJ. Indeed, by differentiating Eq. (9-204) with respect to time and inserting the solution (9-205), we have d2 6 ).;; -( - dt 2+ 1cosh 2- cp-O t (9-219)
This does not affect the integration due to the b function. Moreover, since ljJ has one node there exists a unique normalized X corresponding to a negative eigenvalue of K + 1. The remaining part of the spectrum is a positive continuum. It may be noted from the definition of u that <ljJ Iu) = O. The operator K corresponds to a new Schrodinger problem with a so-called Bargmann potential which admits explicit solutions. We want to compute the nonvanishing determinant of K + 1 + lu)<ul in the space orthogonal to IljJ). The final result for Zk will then be
f3 (_1)k e- I ,(2k 1 / 2 )N
N- 2
2n det (K + 1 + lu)<ulh det (Ko + 1h
The factor 2n comes from the gaussian integration over the ljJ mode omitted in the denominator when we consider (Ko + 1)1-. This factor would have been modified had we not chosen to normalize ljJ. The extra projector lu)<ul is easily disposed of by noticing that
+ 1)<p =
-2<p 3 = -
As noted above, since lu) belongs to the subspace orthogonal to that Eq. (9-221) may be inverted in this space to yield det (K
l/J, this means
+ 1 + lu)<ulh = det (K + 1)1-(1 + <ul
= det(K + 1)1-(1 =
~ 1 lu )
<<P Iu )
-det (K
+ 1h
The negative sign is welcome because of the remaining unique negative eigenvalue of (K + 1h. In practical terms, to separate the contribution of the transverse modes we may use a limiting procedure by replacing K + 1 by K + z, and extracting the coefficient of (z - 1) as z -+ 1. Thus
N- 2
= -2n det (K + 1)1- = -2n lim _1_ det (K + z)
det (Ko
+ 1h
z - 1 det (Ko
+ z)
Several methods may now be used to obtain the explicit expression of this Fredholm determinant. The most instructive, which may be generalized to an arbitrary one-dimensional potential, would be to relate the evaluation of fluctuations around a saddle point in the path integral and the ordinary WKB approximation for wave functions. We can also note in this special case that the fluctuation problem reduces to a soluble equation. Both methods have been used in the literature to obtain det (K + z) det (Ko + z) from which we derive (9-225) Putting all factors together results in
Zk =
r(1 + Jz)r(Jz) r(3 + Jz)r(Jz - 2)
/3 (_1)k
2k 1 / 2
@ek1n (3k2)-2k
As expected this yields a k! increase as
Zk = /31(k
+ i)( -
3)k n 3
Due to this wild behavior it is easy to translate this result in terms of the expansion for the energy. Indeed, the leading contribution of the coefficient of gk in the series for log (1 + I~ cpg P ) is ck[1 + O(1jk)] when the Ck grow like k!. Thus, applying formula (9-199) we find that the factor /3 drops out and we
are left with
i +Ll
Ek = (-I)k+ 11(k
+ i)3
(:3}/ll + OG)]
a result first derived by Bender and Wu. In the case of the anharmonic oscillator, Graffi., Grecchi, and Simon have been able to prove that the function E(g) is Borel summable, i.e., may be represented in terms of its Borel transform by a relation of the type (9-185). We could then proceed to systematic corrections in powers of l/k using the standard perturbation techniques around the saddle point. The method may also be extended to excited states by keeping terms of order e- f3 in the saddle-point action and expanding e- I , in powers of e- f3 As a final remark we observe that when looking for periodic trajectories of period /3 we have other solutions with fractional periods /3/2, /3/3, ... , which would also lead to saddle points. These have a classical action two, three, ... , times larger than the lowest one; thus they contribute only exponentially small corrections to the results.
The extension of these methods to field theory is in principle straightforward, at least insofar as renormalization is left aside. We look in a similar way to instabilities occurring for small coupling. These are responsible for singularities in the Borel transformed plane. As long as they do not reach the positive real axis (assumed to correspond to the physical situations) the theory is sound. This extra information enables us to use in the most effective way the first few terms of the perturbation series for an accurate determination of physical quantities. This program has encountered a great success in some applications to statistical mechanics of the bosonic <p4 model in three dimensions. It can be generalized to include fermionic fields. Singularities on the positive real axis may, however, be also encountered which originate from actual instabilities even at the semiclassical level (such is the case in the theory of gauge fields presented in Chap. 12). By this we mean that to develop a perturbation series we have to select one among several degenerate minima of the classical energy. The divergence of the series then reflects the quantum tunneling between these ground states. Classical euclidean solutions to the field equations with finite action I interpolate between these states and contribute a transition amplitude e - Ilh. To construct a meaningful theory it is then necessary to remove this degeneracy by introducing additional quantum numbers. It is also possible that extra singularities occur as a result of renormalization. These seem to raise difficult problems about the consistency of renormalizable field theories.
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