the renormalisation group physicists use to make sense of Feynman integrals which
involve multivariables is beyond the scope of this book.
In these lectures, we hope to modestly help clarify a few aspects of this vast pic-
ture in setting some of these heuristic considerations on firm mathematical ground
by providing analytic tools to describe regularisation techniques, whether those
used in physics, number theory, or geometry, in a common framework. The fo-
cus is set on the underlying canonical integral, discrete sum, and trace which are
characterised by natural properties such as Stokes’ property, covariance, transla-
tion invariance, or cyclicity. Various anomalies/discrepancies are investigated, all
of which turn out to be local insofar as they can be expressed in terms of the
noncommutative residue, another central figure in these lectures.
We do not claim to present breakthrough results but rather a unified out-
look with pedestrian proofs on results scattered in the physics and mathematics
literature, which we try to bring to the forefront and to make accessible to the
nonspecialist. Along the way we nevertheless prove yet unpublished original results
such as
a characterisation of the noncommutative residue on classical symbols
(Proposition 2.60 and Theorem 3.39) and of the canonical integral on
noninteger order symbols (Theorem 2.61) in terms of their translation
a characterisation of the noncommutative residue on classical symbols
(Theorem 4.21) and of the canonical integral on noninteger order symbols
(Theorem 3.43) in terms of their covariance;
a characterisation of the noncommutative residue (Proposition 5.40) and
the canonical discrete sum (Theorem 5.41) in terms of their
a regularised Euler-Maclaurin formula on symbols (Theorem 5.29);
Taylor expansions (Theorem 4.16 part (2)) for integrals of holomorphic
families extended to log-polyhomogeneous symbols (this is based on an
unpublished joint work with Simon Scott);
a (local) conformal anomaly formula for the ζ-function at zero of a con-
formally covariant operator in terms of noncommutative residues (Propo-
sition 9.19).
We hope in this way to open new perspectives on and further expand openings
to concepts such as regularised integrals, sums, and traces. Far from being ex-
haustive, these lectures leave out various important regularisation techniques such
as Epstein-Glaser [EG], Pauli-Villars [PV], and lattice regularisation techniques,
as well as other regularisation artefacts such as b-integrals [Mel] and relative de-
terminants [Mu]. Regularisation procedures on manifolds with boundaries or sin-
gularities are further vast topics we do not touch upon in spite of the variety of
applications and extensions they offer. We also leave aside the realm of noncommu-
tative geometry where zeta-type regularisation procedures are extended to abstract
pseudodifferential calculus as well as the ambitious renormalisation issue, which
would be needed to make sense of multiple divergent integrals, such as multiloop
Feynman diagrams in physics, multiple discrete sums, such as multiple zeta val-
ues in number theory, or to count lattice points on convex cones. Here we only
tackle simple integrals, and discrete sums. Also, to keep this presentation down to
a reasonable size, we chose not to report on regularisation methods implemented in
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