QUANTUM FIELD THEORY in VS .NET

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QUANTUM FIELD THEORY
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of the question whether or not the series defines a unique mathematical quantity. For certain practical purposes this first step might well be sufficient. For instance, it could happen that quantum electrodynamics, as it now stands, is not completely consistent, but that power series in rx are nevertheless asymptotic within the context of a deeper theory. The second aspect of the problem is indeed to try to reconstruct from these expansions a unique theory according to some definite prescriptions. This is clearly a formidable task which requires some independent information. The latter has to be provided by a different construction. Nevertheless, we might discover some hints in the structure of the series which suggest a sensible way of summing it. The typical nature of the divergence may easily be understood on a simple example. For this purpose we replace path integrals by ordinary ones such as
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f+oo d<p e-(q>2/2 +gq>4)
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(9-177)
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Of course, this case is trivial enough that we might obtain Z(g) in closed form. The point g = 0 is an essential singularity. When Re g becomes negative (and IArg g I<n) we may still rotate the integration contour in (9-177). This is no longer possible when g approaches the negative real axis and the integral blows up for large <po We may think of <p as the value of a field at a point with <p2/2 + g<p4 as a caricature of the action. Negative values of g correspond to an unstable situation where the "potential" is not bounded from below. This is reflected in the perturbative expansion if we write
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Zk = (_l)k
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f+oo d<p <p4k e-q>2/2 = (_W4k r(2k +i)
Jnk!
(9-178)
Using Stirling's formula
k! '"
j'hl eklnk-k
e(k-l/2)lnk-k
for large k, we find that Zk behaves as (9-179) In Nevertheless, the power series in g is asymptotic in the complex g plane cut along the negative real axis since n 4n+1 r(2n+i) Igln+l (9-180) Z(g)- ~Zkr/ < In(n+ 1)! [cos (iArg g>J2n+ 3/2
Zk '" (-16
meaning that for fixed nand g small enough the right-hand side may be made arbitrarily small. The asymptotic behavior of Zk was obtained by applying Stirling's formula
FUNCTIONAL METHODS
to the exact expression. The latter will not be available in more realistic cases. However, this suggests to apply the method of steepest descent for large k to
Zk=(-l)k k! '-l~
Gfoo d<pe- ,z/2+4klnip
(9-181)
The position of the saddle point is
(9-182)
and integration over quadratic deviations from <Pc yields
Z '" (_l)k k k!
12 e2kln4k-2k =
(-16)k e(k-l/2)lnk-k
(9-183)
as before. What can we do from such a divergent series besides using it in an asymptotic sense to evaluate the function for small g In some fortunate circumstances, such as the one discussed here, the power series (9-178) contains in fact enough information to reconstruct the function unambiguously. Of course, it might be argued that we could add any function such as exp (-1/)g) with vanishing derivatives at the origin along the real positive axis. However, within a welldefined class offunctions excluding these pathologies [and to which Z(g) belongs] a Borel transformation enables us to recover Z(g) from its divergent perturbative series. Introduce (9-184) where the choice of r(k + ~) is justified by the growth given in Eq. (9-179) and could be slightly modified in analogous cases. The series on the right-hand side in (9-184) will converge within a circle of finite radius in the complex t plane. If B(t) may be continued along the entire real positive axis and does not grow too fast at infinity, then Z(g) will be given by
Z(g)
too dt t
1 2 /
B(gt)e- t
(9-185)
Knowledge of the perturbative series is insufficient to prove Borel summability.
It is, however, sufficient to disprove it in the case where B(t) would be found
to have a singularity on the positive real axis. This happens, for instance, if asymptotically the Zk have equal phases. For our simple example we find
B(t)=~~ yin
u= - 1 -"-J[=i=+=1=6=-t~+---:-1
J~l~+~16--'--t
(9-186)
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