I've been reading Kleinert's book and have been very intrigued by the chapter on variational perturbation theory. Namely, Kleinert derives a very good strong-coupling approximation to the ground state energy of the quantum-mechanical anharmonic oscillator by adding and subtracting a term $\frac{\Omega^2}{2}q^2$ to the Lagrangian and performing an expansion of the path integral different from conventional perturbation theory, then optimizing with respect to $\Omega$. (for details see here). Kleinert notes that his manipulation is mathematically equivalent to re-expanding the traditional perturbation series using a certain substitution for the oscillator frequency to introduce the auxiliary $\Omega$ and then optimizing.

Kleinert also discusses the generalization of this to quantum field theories. His field-theoretic generalization relies on a more obscure re-expansion of the standard perturbation series, but makes no reference to the path integral directly. Can anyone explain to me why adding and subtracting a term like $\frac{M^2\varphi^2}{2}$ to the Lagrangian and expanding the path integral as in the quantum-mechanical case (of course now with renormalization) would fail? Or really what I'm getting at is, why must we be limited to resummation of the weak-coupling series? In the QM case, it was just a trick used for calculational convenience, a shortcut to the underlying path integral expansion, whereas in the QFT case it seems to be fundamental to the method. Kleinert's use of VPT in field theory is with application to critical exponents, and while I understand field theory as relevant to high energy physics and some basic thermal field theory, I'm not well-versed in critical phenomena, so I'm lost. Any help would be appreciated.


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