# QFT perturbation theory

There is no (general) rigorous non-perturbative definition of a QFT, so there is no rigorous proof of perturbation theory either. Therefore, it makes no sense to claim that the perturbative expansion rests on some analytic assumptions. It rests on no assumptions, because it cannot be derived from anything. There is nothing "more fundamental" that, when expanded in power series, yields a perturbative QFT. That being said, you can proceed as follows:

1. You define a QFT through its perturbative series (say, in the causal approach if you want to be mathematically rigorous). Here, and when regarded as a formal power series, the perturbative expansion is well-defined regardless of any analytic properties of the Hamiltonian, so there are essentially no conditions on the operators.

2. You analyse the problem in standard QM (one-dimensional QFT, if you will: the only spacetime coordinate is time), and assume that the same formalism should hold in QFT, provided we eventually find a good formulation. A canonical reference for rigorous perturbation theory in QM is Kato's Perturbation Theoryfor Linear Operators. It is a tough route, so have fun if you want to go there; no guarantee you will find what you're looking for, but it is hard to imagine you will find anything more explicit than this.

3. Some very specific (lower dimensional) QFTs have been constructed rigorously, from where the perturbative expansion can be derived. The canonical example is Glimm & Jaffe's Quantum physics: A functional Integral point of view. Here the authors deal with two-dimensional (Euclidean) $$\phi^4$$ theory, which has the key property that normal-ordering is all you need to render it finite. Therefore, you cannot really hope to draw general conclusions from this example but, sadly, we don't have many more rigorous (interacting) QFTs that can be analysed explicitly.

Finally, let me mention that a heuristic reason the conditions in the OP are usually assumed is the so-called Gell-Mann and Low theorem, which is sometimes used to justify perturbation theory. This theorem does require the spectrum to be bounded from below, and that interactions are switched on and off adiabatically.

The assumption for a valid perturbation theory in quantum mechanics is that the Hamiltonian of the interacting theory differs from the free theory around which the perturbation is done by a relatively compact term (see the treatise by Reed and Simon), and that the interaction strength (the multiplier of this term) is small enough. [This requires as a necessary condition that the spectrum is bounded from below (since that of the free theory is), and one way of proceeding technically is by switching the interactions on and off adiabatically, thus explaining the comment of your teacher.]

In relativistic quantum field theory, the deviation form a free field is not given by a relatively compact term, making perturbation theory strictly speaking inapplicable. This shows through the fact that all terms in the perturbation series except for those corresponding to the tree diagrams diverge, hence give meaningless results.

What is done instead is to truncate the field theory using some hard or soft cutoff at some energy scale $$\Lambda$$, do perturbation theory on the truncated level (for which the above assumption can be proved in certain cases), and then adjust the interaction constants as a function of $$\Lambda$$ in such a way that order by order, the limit $$\Lambda\to\infty$$ can be taken. This gives a renormalized perturbation series which is then used to extract physcial information, using heuristic resummations and other tricks. Nothing is said about the convergence of the series, which in 4D is unlikely even after Borel summation. The formal renormalized perturbation series exists even when the Hamiltonian is unbounded below, though such a theory (e.g., $$\phi^3$$ in 4 dimensions) has no sensible physical interpretation and is used only for toy calculations to get practice.

In causal perturbation theory one avoids the noncovariant truncations by working with axioms for the S-matrix that are then shown to be satisfiable by formal power series in terms of a mathematically well-defined construction using careful distribution splitting. This gives the same perturbation series, but without the need to take limits (and hence avoiding having to move across Landau poles when resumming the series). Again, nothing is said about the convergence of the series.

In the path integral approach, one starts with a mathematically ill-defined path integral (which simply ignores all possibly needed assumptions) and works with formal rules valid for discretized variants where space-time is replaced by a finite set of points, pretending that they remain valid for a putative functional integral on Euclidean or Minkowski space-time. This explains why one cannot see the assumptions during the derivation of perturbation theory with path integral.