Stokes’ property; canonical discrete sums are
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
which arise in Feynman integrals for s = 2, functions of
the form τs(ξ) =
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 = +
1)− s
(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)−
τs(z) =
χ(ξ), and As(z) = +
, 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
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,
is 4 for usual space-time.
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