The author delivers here really interesting lecture series notes, clear, well written, and quite unique in their kind. They reflect many years of work on the subject and a large number of lectures on the same, given to different audiences in different countries. In a way, this can be appreciated in the final result. As the author does explain, regularization techniques implemented in quantum field theory, number theory and geometry may seem very arbitrary and non-canonical at first glance (all practitioners of this subject have indeed suffered from such misconceptions), but they nevertheless conceal rigorous standard concepts, such as canonical integrals, sums and traces. On top of cut-off and dimensional regularization, which are the well-known prototypes of regularization techniques used in quantum field theory, in these notes the author also addresses Riesz and Hadamard finite parts methods used in number theory, zeta regularization used in physics in the form of zeta determinants to compute effective actions, and in geometry, and particularly in the context of infinite dimensional manifolds and index theory as a substitute for the equivalent heat-kernel methods.
Regularized integrals, discrete sums and traces obtained by means of a regularization procedure present many discrepancies responsible for various anomalies but, in contrast, the underlying canonical integrals, discrete sums and traces are well-behaved. All these issues are addressed in this work, which focusses on ultraviolet divergences, namely divergences for large values of the momentum, treated by picking out a specific class of functions whose controllable behavior in the large will enable them to be integrated and summed up using appropriate methods. Classical and log-polyhomogeneous pseudodifferential symbols and operators form a natural class considered in the framework of regularization. The symbols and operators that one encounters typically have integer order, a feature which is the main source of anomalies in physics and the cause of many discrepancies. As the author points out, these obstacles disappear when working with non-integer order symbols and operators, for which integrals, sums and traces are canonically defined.
As well explained in the text, the basic idea behind dimensional, Riesz or zeta regularization is to embed integer order symbols or operators inside holomorphic families of symbols so as to perturb the order of the symbol or the operator away from integers. For order valued symbols (resp. operators) at non-integer values ordinary manipulations can be carried out on the integrals and sums (resp. traces) one encounters, what fully legitimizes the heuristic computations carried out by theoretical physicists. In the language of those, this amounts to holomorphically embedding the integer dimensional world into a complex dimensional one where the objects are canonically defined away from the integer dimensions. The big problem remains then to get back to integer dimensions or integer orders by means of regularized evaluators which pick up a finite part in a Laurent expansion. The freedom of choice left at this stage is responsible for the one-parameter renormalization group which plays a central role in quantum field theory. In the lectures notes, the author helps clarify some crucial aspects of this vast picture in setting some of these heuristic considerations on firm mathematical ground by providing analytic tools to describe regularization techniques in a common framework, whether those used in physics, number theory or geometry. There are not many manuals with purposes similar to those of the present one. The focus is set on the underlying canonical integral, discrete sum and trace which are characterized by natural properties such as Stokes’ and translation invariance or cyclicity. Various anomalies are investigated, which turn out to be local insofar as they can be expressed in terms of the noncommutative residue, which is another central figure in this work.
The author does not claim to present, in these essentially self-contained notes, new breakthrough results but rather to collect and explain, in a unified way, a number of already known ones, which are however very scattered in the physics and mathematics literature, by often giving pedestrian proofs, which the purpose to make the notes accessible to the nonspecialist. The author confesses not to pretend to present an exhaustive work, and mentions a number of important subjects and other approaches which are left aside (as Epstein-Glaser, Pauli-Villars and lattice regularization techniques, and also b-integrals and relative determinants, used to investigate the geometry of certain manifolds with boundaries, and zeta-type regularization procedures in noncommutative geometry).
The lecture notes are organized around nine chapters, the first of which reviews extended homogeneous distributions and serves as a preparation for similar techniques introduced in the subsequent ones. Their titles are as follows:
1. The Gamma function extended to nonpositive integer points 2. The canonical integral and the noncommutative residue on symbols 3. The cut-off regularized integral 4. The noncommutative residue as a complex residue 5. The canonical sum on noninteger order classical symbols 6. Traces on pseudodifferential operators 7. Weighted traces 8. Logarithmic residues 9. Anomalies of regularized determinants
The first chapter addresses the issue of how to extend the Gamma function to nonpositive integer points and serves as a preparation for the more general issue as how to make sense of certain types of divergent integrals. Whereas the divergence is here at zero, later in the notes the divergences take place at infinity, but the way these divergences are cured is similar. The Gamma function offers a good toy model to compare the regularization methods mentioned in the introduction. Extending it to non-positive integers arises as an instance of the more general problem of extending homogeneous distributions to negative integers. The author shows that Riesz and Hadamard’s finite-part regularization methods lead to the same extended homogeneous distributions and hence to the same extended Gamma function, a feature which arises again further in the notes. Discrepancies induced by the regularization procedure are discussed, which provide a first hint to further obstructions to be encountered later while working with regularized integrals.
Chapter 2 focusses on pseudodifferential symbols, and specifically on classical and log-polyhomogeneous symbols with constant coefficients, whose behavior at infinity is polyhomogeneous and log-polyhomogeneous, respectively. Two well-known useful linear forms on certain classes of symbols are introduced: the noncommutative residue on the algebra of classical symbols and the canonical integral on non-integer order log-polyhomogeneous symbols. On the one hand, the canonical integral is characterized as the unique linear form (modulo a multiplicative factor) on certain classes of symbols (including the set of non-integer order symbols) which vanishes on partial derivatives (Stokes’ property). Any other linear extension of the ordinary integral to non-integer order symbols is seen to coincide with the canonical integral. On the other hand, the obstruction that prevents the cut-off regularized integral, obtained by Hadamard finite parts, from obeying Stokes’ property on the whole algebra of classical symbols is measured in terms of a noncommutative residue. The canonical integral is then characterized by means of its translational invariance.
Chap. 3 is a continuation of Chap. 2. There the momentum cut-off regularization procedure (in physical language) which corresponds in mathematics to Hadamard’s finite parts method is explained to provide a realization of the canonical integral on noninteger order symbols. By means of this extended integral a refinement is realized of the first characterization of the noncommutative residue that had been previously obtained via the Stokes property and via its translational invariance. The chapter is mainly dedicated to measuring the obstruction to the usual properties for a given symbol, namely the Stokes property, rescaling and translational invariances, and covariance. A characterization of the canonical integral via its covariance is obtained.
Chap. 4 deals with the evaluation at zero of meromorphic functions presenting a pole at that point, a well-known very common task for theoretical physicists. The minimal subtraction scheme, widely used by them, is one possible, simple way to extract a finite part at zero and which coincides with the ordinary evaluation at zero in the absence of poles. It yields a regularized evaluator at zero which extends ordinary evaluation to that point. In the text regularized evaluators are classified, and the result is then applied to meromorphic extensions of integrals of classical symbols and log-polyhomogeneous symbols of the ordinary integral on \(L^1-\)symbols. The results are then explicitly applied to compare dimensional regularization with cut-off regularization.
In Chap. 5, in the same way the ordinary integral was extended to the algebra of classical symbols, using a cut-off or Hadamard finite part procedure on balls of increasing radius, the ordinary discrete sum is extended to the whole algebra of classical symbols, using the same procedures on polytopes of increasing size. When restricted to the set of non-integer order symbols, cut-off regularized sums obtained in this way are proven to be independent of the chosen polytope and translation invariant, just as cut-off integrals of non-integer order symbols are independent of the rescaling of the radius of the ball they are integrated on. It is also shown in this chapter that translation invariant linear forms on the set of non-integer order classical symbols are proportional to the canonical sum obtained from any of these regularized sums, and the ordinary sum is extended to regularized discrete sums using holomorphic regularizations, for which a regularized Euler-Maclaurin formula is obtained. This formula is then applied to express certain Hurwitz zeta values and certain zeta values associated with quadratic forms.
Chap. 6 discusses the canonical trace and the noncommutative residue on operators. The noncommutative residue on classical symbols and the canonical integral on noninteger order classical symbols give rise respectively to the noncommutative residue on classical pseudodifferential operators and to the canonical trace on those which have noninteger order. In this chapter a preliminary characterization of the noncommutative residue is given. One should note that passing from pseudodifferential symbols to pseudodifferential operators first requires extending the notion of pseudodifferential symbol on the whole d-dimensional space to symbols with varying coefficients on an open subset of this space. When patched up using a partition of unity, these lead to pseudodifferential operators on a closed manifold. Pseudodifferential operators generalize differential operators with smooth coefficients, what is clearly explained in the text, to address then the differences between the symbol of a differential and of a pseudodifferential operator, and finally describe the class of symbols under consideration, the basic properties of pseudodifferential operators, and a first characterization of the noncommutative residue.
Chap. 7 starts with the consideration of the \(L^2-\)trace on trace-class pseudodifferential operators which extends to linear forms called weighted traces and defined on the whole algebra of classical (or log-polyhomogeneous) operators by means of a zeta regularization. This is shown to be an instance of more general holomorphic regularization procedures which amount to embedding an operator in a holomorphic family of them, and picking then the constant term of a meromorphic extension built by means of the canonical trace. Weighted traces obtained in this way using zeta regularization are seen to present discrepancies which can be expressed in terms of the noncommutative residue. The chapter ends with a characterization (uniqueness) of the canonical trace and noncommutative residue which confirms known results relative to the uniqueness of these concepts, but the approach taken in the notes seems to be new and uses weighted traces.
Chap. 8 is devoted to logarithmic residues and it is justified by the fact that logarithms of classical pseudodifferential operators play an important role in describing the local features of regularized traces of many operators. Although weighted traces are expected to be non-local in general, the weighted trace of a differential operator turns to be local insofar as it can be expressed in terms of a noncommutative residue. In particular, the weighted trace and supertrace of the identity map, investigated here in two different theorems, are both local quantities, since they are proportional to the noncommutative residue and to the superresidue of the logarithm of the weight, respectively. In contrast, the weighted trace of a general classical pseudodifferential operator is seen to split into a local part involving a residue density and a global one involving a canonical trace density, what is proven in a theorem. When exponentiated, the logarithmic residue of a classical elliptic pseudodifferential operator gives rise to the residue determinant, which is multiplicative on elliptic operators but actually useless for physicists, since it vanishes on determinant class elliptic operators. No wonder they always use instead the (nonmultiplicative) zeta-determinant, which is considered in the last chapter. In Chap. 9, two types of regularized determinants are investigated, namely weighted determinants and the zeta-determinant, which are related by a local formula. Weighted and zeta-determinants are not multiplicative at all but, as is well-known for zeta determinants the corresponding multiplicative anomaly which measures the obstruction to the multiplicativity is local, in a sense that is made precise in this chapter both for weighted determinants and for zeta determinants, in two corresponding theorems. It is discussed here that the zeta determinant of a conformally covariant operator is again not conformally invariant, either: in a corresponding theorem, the well-known conformal anomaly of conformally covariant Laplacians is computed, after establishing, in another one, a just as well-known formula for the logarithmic variation of the zeta determinant.
The notes, comprising in all exactly 200 pages, end with a five-page quite complete bibliography (I missed there just a few quite influential works, by Voros, Seeley, Dowker, Kirsten,... maybe I am biased towards the physicists’ side here), and a two-page index.