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and B(t) is analytic in a plane cut from
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QUANTUM FIELD THEORY
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To obtain a convergent expansion for Z(g), let us map the cut t plane onto a circle, keeping the origin fixed (this is given here by the choice of variable u), and derive the convergent Taylor series for B(u):
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B(t) = B(u) =
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L bk[u(t)Jk
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(9-187)
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from the knowledge of its expansion in t. Then
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Since bk decreases here as k - 3/2 it is easy to see that this new series will converge as e- 3(k/3g 1/2 )2/3. The behavior exhibited in this example bears a close relationship with some divergences encountered in field theory since the expansion of Z(g) in powers of g has coefficients equal to the number of vacuum diagrams in a <p4 model. This follows readily from the fact that Wick's theorem applies to integrals of monomials over a gaussian weight.
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This remark applies to other field theories as well. In electrodynamics, for instance, consider the integral
Z(j, fj,~)
~ fdA dlf dt/l e-A2/2-1f(1-eA)rjJ+If~+~rjJ+jA
f~e-A2/2+~~/(l-eA)+jA
1- eA
(9-189)
where t/I and If are considered as complex conjugate c-number variables. The expansion of In Z in powers of e generates the number of diagrams of the connected functions, except for the cancellations implied by Furry's theorem. These are implemented by symmetrizing the generating function of charged loops with respect to e, In(1 - eA) ->1 In (1 - e2 A 2 ), that is, by replacing Z by
Z(j,fj,~)=f
dA Jl - e 2 A2
e-A2/2+~~/(1-eA)+jA
(9-190)
The integral is meaningful for negative e 2 For the photon and electron propagators G and S, related to the vacuum polarization wand self-energy ~, we find respectively (9-191)
S=(I_~)-l
1_1_) \1-e A2
Z= - 2
where the average is over the measure (dA/Jl - e 2A2) e- A'/2. Surprisingly these expressions coincide:
(9-192)
with Ko(z) the modified Bessel function
Ko(z)
de e-zcosh8
FUNCTIONAL METHODS
Expansion of (9-192) for large z yields
+ 4e4 + 25e6 + 208e 8 + 2146e 10 + 26 368e 12 + ... W = L = e 2 + 3e4 + i8e 6 + 153e 8 + 1638e 10 + 20 898e 12 + ...
G= S
1 + e2
(9-193)
to be compared with the number of diagrams for the vacuum polarization with one charged loop only:
+ 3e4 + 15e 6 + 105e 8 + 945e 10 + 10 395e 12 + '" Similarly, the generating function for vertex diagrams r is equal to
WI =
L (2n -
I)!! e 2n
(9-194)
= 4z(1 -
S)S-2G-
= 1 + e2
+ 7e4 + 72e 6 + 891e 8 + 12 672e 10 + ...
(9-195)
Extremely courageous people are undertaking the computation of the 891 diagrams of the electron anomaly to eighth order, but will anybody ever dream of considering the 12672 ones to tenth order!
9-4-2 Anharmonic Oscillator
Let us apply the previous ideas to the study of a quantum mechanical system. Although the method works for any polynomial potential we shall for definiteness consider the ground-state energy of an anharmonic oscillator with hamiltonian
(9-196)
The configuration variable is denoted <p and its conjugate momentum p to emphasize the formal analogy with higher-dimensional field theories. The problem of the expansion in powers of g may be considered in the framework of the Schrodinger equation. We expect an instability for infinitesimal negative g, which can be investigated by means of the WKB (Wentzel, Kramers, Brillouin) approximation. Since this method will not be available as such in higher dimensions it is instructive to use an alternative approach tailored after the previous example. The matrix elements of the evolution operator e - itH can be expressed as path integrals. So does the trace of e- PH, which may be interpreted as the partition function of a canonical ensemble of oscillators. Here fJ- 1 is equal to the absolute temperature multiplied by Boltzmann's constant and
1 F = - -In (Tr e- PH ) fJ
(9-197)
is the free energy. When the temperature goes to zero, or fJ to infinity, F reduces to the ground-state energy. Thus we may represent the partition function Z(g) as a Feynman-Kac path integral over the exponential of an euclidean action (with a change in the relative sign of kinetic and potential contributions as compared to the usual expression). The configurations <p(t) are periodic in "time": <p(O) = <p(fJ), to comply with the fact that we are computing a trace
(9-198)
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