Preface
Regularisation techniques, implemented in quantum field theory, number the-
ory, and geometry to make sense of divergent integrals, discrete sums, or traces,
might seem very arbitrary and uncanonical at first glance. They nevertheless con-
ceal canonical concepts, namely canonical integrals, sums, and traces, which we
want to bring to the forefront in these lectures.
Cut-off and dimensional regularisation are prototypes of regularisation tech-
niques used in quantum field
theory.1
But we also have in mind
Riesz2
and
Hadamard finite
parts3
methods used in number
theory.4
Regularisation techniques
also comprise zeta regularisation used in physics in the form of zeta determinants
to compute effective
actions,5
or in
geometry6
and particularly in the context of
infinite dimensional
manifolds7
and index theory as a substitute for the equivalent
heat-kernel
methods.8
Regularised integrals, discrete sums, and
traces9
obtained by means of a regu-
larisation procedure present many discrepancies responsible for various
anomalies.10
In contrast, the underlying canonical integrals, discrete sums, and traces are well
behaved. Canonical integrals are indeed covariant, translation invariant, and obey
1Just
to quote a few books amongst the vast literature on the subject, see e.g. [Col], [CMa],
[D], [Sm1], [Sm2] as well as more specific references in the context of renormalisation, such as
[Et], [CMa], [He], [HV], [Sp], [Zi].
2Also
called modified dimensional regularisation.
3Which
amounts to cut-off regularisation.
4See
e.g. [Ca] for an introductory presentation.
5Starting with pioneering work by Hawkins [Haw], see other applications in [El], [EORZ],
and further developments in string theory, see e.g. [D] and [AJPS] for a mathematical
presentation.
6With
the work of Ray and Singer [RaSi] on analytic torsion where the zeta determinant
was first introduced in mathematics.
7E.g.
for the geometry of loop groups, see [Fr].
8Starting
with pioneering work by Atiyah and Singer [APS1, APS2, APS3] and later by
Quillen [Q1] and Bismut and Freed [BF], see also more recent work by Scott [Sc1] in the context
of the family index theorem.
9We
use the terminology regularised when a physicist might call this a renormalised in-
tegral, discrete sum, or trace since it is the result of a regularisation procedure combined with
a subtraction scheme used to extract a finite part. We choose not to use of the word “renor-
malisation” because in physics this concept involves much more than merely evaluating divergent
integrals, divergent discrete sums in one variable, or divergent traces that we are concerned with
here.
10For
a treatment of anomalies in physics (see e.g. [D] and [N] for a mathematical presen-
tation) from the point of view of discrepancies also called trace anomalies, see e.g. [CDP] and
[Mi].
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