viii PREFACE Stokes’ property canonical discrete sums are Zd-translation invariant and canon- ical traces vanish on commutators. So all would be well were these canonical in- tegrals, discrete sums, and traces defined on a class of functions and operators appropriate for applications unfortunately most functions and operators arising in most number theory, geometry, or physics do not fall in the class on which the canonical functionals have the desired invariance properties. However, one can ap- proximate any of the functions or operators under consideration by a family of functions or operators in the class on which canonical functionals naturally live this fact is the basic principle which underlies many regularisation procedures. To make this statement more precise, we need to specify the type of functions and operators one comes across. Since we focus on ultraviolet divergences, namely divergences for large values of the momentum, it seems reasonable to pick out a specific class of functions whose controllable behaviour in the large will enable us to integrate and sum them up using appropriate regularisation methods. It turns out that functions of the form σs(ξ) = (1 + |ξ|2)− s 2 which arise in Feynman integrals for s = 2, functions of the form τs(ξ) = |ξ|−s χ(ξ) where χ is a smooth cut-off function that gets rid of infrared divergences, which arise in number theory for negative integer values of s, and operators of the form As = (Δ + 1)− s 2 (whose symbol is σs) for a generalised Laplacian Δ and some integer s, which arise in infinite dimensional geometry and index theory for integer values of s, are all of pseudodifferential nature. Classical and more generally, log-polyhomogeneous pseudodifferential symbols and operators form a natural class to consider in the framework of regularisation. The pseudodifferential symbols and operators that one encounters typically have integer order (−s in the above examples), a feature which is the main source of anomalies in physics and the cause of many a discrepancy. These obstacles disappear when working with noninteger order symbols and operators, for which integrals, sums, and traces are canonically defined. The basic idea behind di- mensional, Riesz, or zeta regularisation is to embed integer order symbols σ or operators A inside holomorphic families of symbols σ(z) or operators A(z) so as to perturb the order of the symbol or the operator away from integers. In the exam- ples mentioned above, natural holomorphic extensions are σs(z) = (1 + |ξ|2)− s+z 2 , τs(z) = |ξ|−(s+z) χ(ξ), and As(z) = (Δ + 1)− s+z 2 , which coincide with the original symbols σs, τs, and operator As at z = 0. Away from integer order valued symbols (resp. operators) ordinary manipula- tions can be carried out on integrals and sums (resp. traces) which legitimise physi- cists’ heuristic computations. Borrowing the physicists’ metaphorical language, this amounts to (holomorphically) embedding the integer11 dimensional world into a complex dimensional one where the canonical functionals mentioned previously have the desired invariance properties, away from an integer dimensional dimen- sional world. Having left integer dimensions using a holomorphic perturbation, the problem remains to get back to integer dimensions or integer orders by means of regularised evaluators at z = 0 which pick up a finite part in a Laurent expansion. The freedom of choice left at this stage is responsible for the one parameter renor- malisation group which plays a central role in quantum field theory. Since we are concerned here with evaluating divergent integrals, discrete sums in one variable, 11 This is 4 for usual space-time.

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