  The predictions of the standard model of particle physics are highly
successful in spite of the fact that several parts of the underlying quantum
field theoretical framework are analytically problematic. Indeed, it has long
been suggested, by Einstein, Schr\"odinger and others, that analytic problems
in the formulation of fundamental laws could be overcome by reformulating these
laws without reliance on analytic methods namely, for example, algebraically.
In this spirit, we focus here on the analytic ill-definedness of the quantum
field theoretic Fourier and Legendre transforms of the generating series of
Feynman graphs, including the path integral. To this end, we develop here
purely algebraic and combinatorial formulations of the Fourier and Legendre
transforms, employing rings of formal power series. These are all-purpose
transform methods and when applied in quantum field theory to the generating
functionals of Feynman graphs, the new transforms are well defined and thereby
help explain the robustness and success of the predictions of perturbative
quantum field theory in spite of analytic difficulties. Technically, we
overcome here the problem of the possible divergence of the various generating
series of Feynman graphs by constructing Fourier and Legendre transforms of
formal power series that operate in a well defined way on the coefficients of
the power series irrespective of whether or not these series converge. Our new
methods could, therefore, provide new algebraic and combinatorial perspectives
on quantum field theoretic structures that are conventionally thought of as
analytic in nature, such as the occurrence of anomalies from the path integral
measure.
