In quantum field theory the path integral is usually formulated in the wave picture, i.e., as a sum over field evolutions. This path integral is difficult to define rigorously because of analytic problems whose resolution may ultimately require knowledge of non-perturbative or even Planck scale physics. Alternatively, QFT can be formulated directly in the particle picture, namely as a sum over all multi-particle paths, i.e., over Feynman graphs. This path integral is well-defined, as a map between rings of formal power series. This suggests a program for determining which structures of QFT are provable for this path integral and thus are combinatorial in nature, and which structures are actually sensitive to analytic issues. For a start, we show that the fact that the Legendre transform of the sum of connected graphs yields the effective action is indeed combinatorial in nature and is thus independent of analytic assumptions. Our proof also leads to new methods for the efficient decomposition of Feynman graphs into $n$-particle irreducible (nPI) subgraphs.
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